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+{"text":"\n\\section{Introduction}\n\n\n\nFake audio detection has attracted increasing attention since the organization of a series of automatic speaker verification spoofing and countermeasures challenge (ASVspoof) \\cite{todisco2019asvspoof,kinnunen2017asvspoof,wu2015asvspoof}.\nPlenty of approaches are proposed to alleviate the threat of voice spoofing attacks. The studies on fake audio detection are usually carried out from two aspects. The first is robust acoustic features based on signal processing methods \\cite{patel2015combining, todisco2016new, das2020assessing}. The second is effective classifiers based on neural networks \\cite{lavrentyeva2017audio, Lavrentyeva2019, lei2020siamese, von2020multi}.\nDespite the impressive results that these methods have achieved, we can find that the performance of these models degrades when unseen spoofing attacks occur. For example, in term of the median equal error rate (EER) of the top-10 performing submitted systems in ASVspoof2019 challenge, the median EER under seen attack A16 is $0.02\\%$ while it goes dramatically up to $15.93\\%$ under unseen attack A17 \\cite{todisco2019asvspoof}. This phenomenon reveals the weakness of current countermeasures and urges people to look for ways to improve model's the performance on unseen attacks.\n\n\nSome studies pay attention to fake audio detection on unseen datasets. Monteiro et al. \\cite{monteiro2020ensemble} proposed a model ensemble approach to train three models jointly, which outperforms the model trained with mixed data directly. Wang et al. \\cite{wang2020dual} proposed a dual-adversarial domain adaptation framework to solve out-of-domain dataset problem.\nHowever, these methods require both original and new data. With the growth of voice conversion and speech synthesis technologies as well as the emergence of advanced reply devices, new spoofing attacks keep emerging rapidly. It's storage-consuming to do model ensemble and time-consuming to train model with mixed data involving all the data sources. Besides, for the consideration of privacy, the access to old data may not be allowed in some special cases. Although fine-tuning the trained model is another way to improve the performance on unseen spoofing attack \\cite{wang2020dual}, it will lead to a decrease on the previous spoofing knowledge.\nTherefore, it's necessary to make our detection model have the ability to continually learn new knowledge over time meanwhile maintain the previously learned knowledge.\n\n\\begin{figure}[t]\n\\vspace{-10pt}\n\\centering\n\\includegraphics[width=0.7\\linewidth]{.\/Figure\/ewc.pdf}\n\\vspace{-5pt}\n\\caption{Difference between continual learning methods and fine-tuning.\nBlue and yellow region is the parameter configuration region of source and target models correspondingly. If using the fine-tuning strategy (solid arrow), the results of parameter optimization will easily fall into the region deviation from the source, while continual learning methods (dotted arrow) can learn the parameters in the overlapping region.}\n\\label{ewc}\n\\vspace{-5pt}\n\\end{figure}\n\nContinual learning becomes a hot issue in recent years \\cite{kirkpatrick2017overcoming,rebuffi2017icarl,van2020brain} and has been applied in the field of computer vision \\cite{castro2018end, lee2020residual} and speech recognition \\cite{sadhu2020continual, houston2020continual} already. It aims to overcome the catastrophic forgetting problem existing in fine-tuning, i.e., the model forgets previously learned information after trained on new information. The difference between continual learning and fine-tuning is elaborated in Figure \\ref{ewc}.\n\n\nIn this work, we propose a continual learning method, named detecting fake without forgetting (DFWF), to learn new spoofing attacks incrementally. It is inspired by learning without forgetting (LwF) but takes the characteristics of fake audio detection into account. When model is trained with new unseen data, genuine speech is more consistent in different scenarios while spoofing speech varies widely. Thus, we add an extra positive sample alignment (PSA) constraint on genuine speech.\nDFWF can help model mitigate the forgetting of past knowledge meanwhile focus more on the invariance of genuine speech.\nOur proposed method is investigated on several sequential training tasks across different spoofing types on ASVspoof2019 dataset. The results show that DFWF approach can reduce the forgetting of past knowledge and its performance is much better than fine-tuning.\nTo our best knowledge, this is the first study to apply continual learning to the fake audio detection task to solve the problem that current models have difficulties in distinguishing unseen spoofing audio.\n\n\nThe rest of this paper is organized as follows: Section 2 illustrates our proposed method DFWF from its two components: LwF and PSA. Experiments, results and discussions are reported in Section 3, 4 and 5, respectively. Finally, the paper is concluded in Section 6.\n\n\n\n\n\n\n\n\\section{Proposed methods}\n\\subsection{Proposed detecting fake without forgetting}\n\nUsually, there are three major approaches for continual learning: (1) Regularization approaches \\cite{li2017learning, kirkpatrick2017overcoming} add hand-craft regularization term in loss function to constrain the learning process. (2) Replay approaches \\cite{rebuffi2017icarl,shin2017continual} record some examples in the buffer and replay experience when training new task. (3) Dynamic architecture approaches \\cite{rusu2016progressive, parisi2017lifelong} dynamically change the structure of neural networks according to the new task. We focus on regularization approaches in the study.\n\nSince retraining with mixed data costs a lot of time and fine-tuning leads to catastrophic forgetting, we propose DFWF method to make a trade off between time resource and performance.\nIt doesn't store old data but can ensure the previous information is remembered.\nWhen training, two constrains named learning without forgetting (LwF) and positive sample alignment (PSA) are added to the original loss $\\mathcal{L}_{original}$. Thus, the total loss ${\\mathcal{L}_{total}}$ becomes:\n\\vspace{-2pt}\n\\begin{equation}\n    {\\mathcal{L}_{total}} =  \\mathcal{L}_{original} + \\alpha {\\mathcal{L}_{LwF}} + \\beta {\\mathcal{L}_{PSA}}\n  \\label{totalloss}\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are hyper-parameters to control the importance of respective terms.\nUsually, cross entropy loss is used as $\\mathcal{L}_{original}$ to train the model. The details of LwF loss ${\\mathcal{L}_{LwF}}$ and PSA loss ${\\mathcal{L}_{PSA}}$ are explained in the following subsections.\n\n\\subsection{Learning without forgetting}\n\nLwF\\cite{li2017learning} is introduced to DFWF method. When training new task, LwF uses knowledge distillation loss to make the output probabilities of new data on current network close to the output probabilities of the same data on original network. Thus, it can also be seen as an extension of student-teacher knowledge distillation model \\cite{hinton2015distilling} which is widely adopted in speech recognition \\cite{chebotar2016distilling, shi2019knowledge}. LwF is achieved by including an additional knowledge distillation loss $\\mathcal{L}_{LwF}$:\n\n\\vspace{-5pt}\n\\begin{equation}\n  \\mathcal{L}_{{\\rm{LwF}}} = - \\sum\\limits_{i = 1}^n {{y'}_{old}^{(i)}}\\log {{{y'}_{new}^{(i)}}}\n\\end{equation}\nwhere $n$ is the number of labels. The label includes genuine and fake in this case. $i$ means the i-th class. ${y'}_{old}^{(i)}$ and ${y'}_{new}^{(i)}$ are modified probabilities of original model's output ${y}_{old}^{(i)}$ and current model's output ${y}_{new}^{(i)}$. Specifically, they are calculated as follows:\n\\vspace{-5pt}\n\\begin{equation}\n    {y'}_{old}^{(i)} = \\frac{{{{(y_{old}^{(i)})}^{1\/T}}}}{{\\sum\\nolimits_j {{{(y_{old}^{(j)})}^{1\/T}}} }},\n    \\qquad\n    {y'}_{new}^{(i)} = \\frac{{{{( y_{new}^{(i)})}^{1\/T}}}}{{\\sum\\nolimits_j {{{( y_{new}^{(j)})}^{1\/T}}} }}\n  \\label{LwF2}\n\\end{equation}\nwhere $T$ is a hyper-parameter. Setting $T > 1$ increases the weight of smaller values in logits and encourage the network to produce a relative similar contributions between the old and the new data\\cite{hinton2015distilling}.\n\n\n\\subsection{Positive sample alignment}\nIn the ``genuine or fake\" classification task, the feature distribution of genuine speech is more consistent compared with the feature distribution of various types of fake audio. In a new spoofing scenario, there exists a gap between previous spoofing data and new spoofing data. But the feature of genuine speech in different situations tend to be similar. When meeting new data, the model should focus more on the genuine samples to succeed more knowledge of the original model for genuine samples. Considering this characteristic of fake audio detection, PSA is proposed to constrain the feature distribution of genuine speech among different data resource.\nCosine distance is used to evaluate the similarity of positive (genuine) embeddings between the current model and the original model.\nThe proposed PSA loss ${\\mathcal{L}_{PSA}}$ is denoted as:\n\\vspace{-5pt}\n\\begin{equation}\n    {\\mathcal{L}_{PSA}} = \\frac{1}{{{N_P}}}\\sum\\limits_{k = 1}^{{N_P}} {\\frac{{y_k^{+}\\hat y_k^{+}}}{{\\left\\| {y_k^{+}} \\right\\| \\cdot \\left\\| {\\hat y_k^{+}} \\right\\|}}}\n  \\label{psa1}\n\\end{equation}\nwhere ${N_P}$ is the number of genuine speech. ${y_k^{+}}$ and $\\hat y_k^{+}$ are the embedding vectors (the outputs of the network after removing the last full connection layer) of the $k$-th genuine speech extracted by the original model and the current model, respectively. $\\left\\| \\cdot \\right\\|$ means the norm of the vector. Thus, PSA loss equals to $\\cos {\\theta _{y_k^{+}\\hat y_k^{+}}}$, the cosine distance of genuine speech embeddings between the previous and current model.\n\n\n\\section{Experiments}\n\\subsection{Dataset}\nAll experients are conducted on ASVspoof 2019 dataset \\cite{wang2020asvspoof} which has two subset:  logical access (LA) and physical access (PA). The LA subset includes 19 types spoofing attacks made by voice conversion or speech synthesis. The PA subset contains 27 different acoustic and 9 different replay configurations. To further evaluate the performance of our proposed methods on a series of unseen spoofing attacks, we select four spoofing attacks with big difference in LA and PA subset separately. More specifically, in LA subset, type A13, A17, A10, A19 are chosen. Since only LA evaluation set contains these types, we further redivide them into training, development and evaluation sets. In PA subset, type bbbBB, cccCC, aaaAC, abcAA are chosen. Because all of these PA types occur in the training, development and evaluation sets, their partitioning scheme are according to their divisions in the PA subset. The detailed statistics of the datasets are presented in Table \\ref{tab:dataset}.\n\n\\begin{table}[th]\n\\vspace{-5pt}\n\\setlength\\tabcolsep{3.5pt}\n  \\caption{ Number of utterances in each section of the dataset}\n  \\vspace{-5pt}\n  \\label{tab:dataset}\n  \\centering\n  \\begin{tabular}{ cccc }\n    \\toprule\n    \\multirow{2}*{\\textbf{Dataset}} & \\multicolumn{3}{c}{\\textbf{Number of utterances}} \\\\\n    \\cline{2-4}\n    ~ & \\textbf{Training} & \\textbf{Development}  & \\textbf{Evaluation}\\\\\n    \\midrule\n    ASVspoof2019LA             & 25380 & 24986  & 71933    \\\\\n    ASVspoof2019PA             & 54000  & 33534  & 153522   \\\\\n    LA-A13                        &  2250  & 1250 & 1352     \\\\\n    LA-A17                    & 2250 & 1250  &1321         \\\\\n    LA-A10                 & 3414 & 1500 & 1500         \\\\\n    LA-A19                   & 3414 & 1500 &1500         \\\\\n    PA-bbbBB                    & 356  & 278     & 1078          \\\\\n    PA-cccCC                    & 472  & 336     & 1380          \\\\\n    PA-aaaAC                    & 604  & 402     & 1654          \\\\\n    PA-abcAA                    & 352  & 276     & 916         \\\\\n    \\bottomrule\n  \\end{tabular}\n\\vspace{-10pt}\n\\end{table}\n\n\\subsection{Evaluation metrics}\nThe performance of our proposed methods is evaluated via equal error rate (EER) and average EER (AvgEER).\nThe lower the EER value, the higher the accuracy of the model. AvgEER is introduced when the model is evaluated in sequential trainings. It calculates the arithmetic mean of EER among all the spoofing types.\n\\vspace{-5pt}\n\\begin{equation}\n    AvgEER = \\frac{1}{n}\\sum\\limits_{i = 1}^n {{{EER}_i}}\n  \\label{avgeer}\n\\end{equation}\n\\vspace{-5pt}\n\nwhere $n$ is the number of spoofing types in testing.${EER}_i$ is the EER of trained model tested on the $i$-th spoofing types.\n\n\n\n\n\n\\subsection{Experimental setup}\nWe extract 60-dimensional linear frequency cepstral coefficients (LFCC) \\cite{sahidullah2015comparison} of speech as low-level input feature to the neural network. The extraction way is the same as the ASVspoof2019 offical baseline system \\cite{wang2020asvspoof}. The window length is set to 25 ms. The number of FFT bins is set to 512. The number of filters is set to 20. To avoid the influence of short audio duration, we duplicate the short audio along time-axis up to 320 frames. If the audio is longer than that set length, we just randomly take a segment of the set length.\n\nAs for the network, we adopt light convolutional neural network (LCNN) which is the same architecture as Lavrentyeva et al. \\cite{lavrentyeva2019stc} to generate 80-dimensional high-level embeddings. Then a fully connection layer projects the embeddings to a binary classification.\n\nTo evaluate the performance of DFWF, we compare it with two methods: fine-tuning and multi-condition (retraining) training. Fine-tuning refers to train a new model with new data by taking the weights of previous model as initialization. Multi-condition training refers to retrain the model by incorporating new date with old data.\nIt's worth noting that the EER of multi-condition training is considered to be the lower bound to our continual-learning-based method DFWF \\cite{parisi2019continual}.\nIn addition, we did an ablation study to illustrate the relative contribution of LwF and PSA to the final performance of DFWF. In order to cover comprehensive cases of sequential unseen spoofing attacks, we designed the following 4 types of sequential training settings:\n\n\\begin{enumerate}\n\\item $PA \\rightarrow LA$ : This setting is to evaluate the performance of DFWF method across the spoofing types with obvious gap.\n\\item $LA \\rightarrow PA$: This setting is the contrast to $PA \\rightarrow LA$ to evaluate the influnence of transfer order.\n\\item $A13\\rightarrow A17\\rightarrow A10\\rightarrow A19$: This setting is to evaluate the performance of DFWF method across the spoofing types with relatively subtle difference. These are all LA attacks. In the order $A13\\rightarrow A17\\rightarrow A10\\rightarrow A19$, we train the new model with new spoofing data on the basis of the model trained at the last step.\n\\item $bbbBB\\rightarrow cccCC\\rightarrow aaaAC \\rightarrow abcAA$: This setting is to evaluate the performance of DFWF method across different PA attacks.\n\\end{enumerate}\n\nWe name the model trained at the first step as base model. Then, new model is trained with different method in the following steps. At each step, the model is trained for 100 epochs by Adam optimiser with the learning rate 0.0001. For DFWF method, hyper-parameter $T$ is experimentally set to 2. We set weighting coefficients $\\alpha$ and $\\beta$ ranging from $0.0$ to $2.0$ to explore the optimal parameter for each corresponding task.\n\n\\section{Results and Analysis}\n\n\\subsection{Catastrophic forgetting in fake audio detection}\n\nWe do several experiments to fine-tune the model with unseen spoofing attacks and find that there exists catastrophic forgetting phenomena. Considering the result of ``$LA \\rightarrow PA$\" experiment shown in the first two row of Table \\ref{tab:lapa}, the EER of base model increases significantly when tested on PA. After fine-tuning on PA, the fine-tuned model can perform well on PA, but its EER under LA subset increases sharply from $4.42\\%$ to $19.78\\%$. This is because of the distribution mismatch between LA attacks and PA attacks. When fine-tuned with new data, the model will easily deviate from the previous optimal parameters, resulting in performance decrease on old data.\nThus, it's necessary to apply our continual learning method DFWF to unseen spoofing data scenario. Fine-tuning is used as the baseline of the following experiments.\n\n\\subsection{The effectiveness of our proposed DFWD}\n\nWe compare our proposed DFWF method with fine-tuning and multi-condition training in 4 sequential training experiments.\n\n\\begin{table}[t]\n\\vspace{-5pt}\n\\setlength\\tabcolsep{3pt}\n  \\caption{Performance comparison: results show the of EER\\% on different models for training sequence ``$LA \\rightarrow PA$\" and ``$PA \\rightarrow LA$\". Base model is trained on one dataset with random initialization. Fine-tuning (FT) model and DFWF model are trained on the other dataset on the basis of base model. Multi-condition (MC) training model is trained on the mixture of LA and PA subsets.}\n  \\vspace{-5pt}\n  \\label{tab:lapa}\n  \\centering\n  \\begin{tabular}{ cccccccc }\n    \\toprule\n    \\multirow{2}*{Model} & \\multicolumn{3}{c}{{$LA \\rightarrow PA$}} & \\multicolumn{3}{c}{{$PA \\rightarrow LA$}} \\\\\n    \\cmidrule(r){2-4}  \\cmidrule(r){5-7}\n    ~ & LA & PA & Avg & PA & LA & Avg \\\\\n    \\midrule\n    Base     & 4.42 &  28.02  & 16.22\n                        & 5.47 & 22.31 & 13.89\\\\\n    FT  & 19.78 &  6.74 & 13.26\n                         & 21.22  & 4.61 & 12.92\\\\\n    DFWF (Ours)    & 7.74  &  10.76 & 9.25\n                        & 8.85\t& 12.05 &  10.45 \\\\\n    MC     &  5.92 &  6.29 & 6.11\n                                & 6.29  & 5.92  &6.11\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\\begin{figure}[t]\n\n  \\centering\n  \\subfigure[LA attacks]{\n    \\label{Fig.la4}\n    \\includegraphics[width=0.45\\linewidth]{.\/Figure\/4typesLA.pdf}}\n    \\subfigure[PA attacks]{\n    \\label{Fig.pa4}\n    \\includegraphics[width=0.45\\linewidth]{.\/Figure\/4typesPA.pdf}}\n  \\caption{Performance comparison under incremental increase of spoofing types: results show the AvgEER\\% over the sequential training of LA attacks ($A13\\rightarrow A17\\rightarrow A10\\rightarrow A19$) and PA attacks ($bbbBB\\rightarrow cccCC\\rightarrow aaaAC \\rightarrow abcAA$). FT means fine-tuning model. MC means multi-condition training model. AvgEER when the number of attacks equals to $i$ means the average performance on current and previous data at the $i$-th training step. }\n  \\label{fig:4types}\n\\vspace{-5pt}\n\\end{figure}\n\nTable \\ref{tab:lapa} illustrates the result of sequential training between LA and PA attacks. The AvgEER of DFWF achieves a relative reduction of $30.29\\%$ in ``$LA \\rightarrow PA$\" task, and a relative reduction of $19.08\\%$ in ``$PA \\rightarrow LA$\" task compared to fine-tuning baseline.\nAs this baseline does not consider the previous data in its training procedure, it has the advantage of achieving better performance on current data but damages the performance on old data.\nBesides, the transfer order can influence the performance of DFWF method. The AvgEER of ``$LA \\rightarrow PA$\" task is better than ``$LA \\rightarrow PA$\" task. It can be explained by the distribution of LA attacks and PA attacks provided by the publisher of ASVspoof2019 dataset \\cite{wang2020asvspoof}, different LA attacks can form relatively clear clustering while different PA attacks are completely overlapping. The model trained on LA subset can been seen as a better pre-trained model, providing a set of well-performed model parameters.\n\n\nThe advantage of DFWF is more pronounced in the scenarios where unseen spoofing attacks occur gradually .\nFigure \\ref{fig:4types} shows the results in sequential trainings of LA and PA attacks, respectively. In sequential training``$A13\\rightarrow A17\\rightarrow A10\\rightarrow A19$\", the performance of fine-tuned model becomes worse as the number of spoofing types increases. When the types increase to 4, the AvgEER of fine-tuned model goes dramatically from $0.81\\%$ to $43.72\\%$, while the AvgEER of DFWF model remains to $8.03\\%$. That is to say, DFWF achieves a relative reduction of $81.63\\%$ compared to fine-tuning in this task. In the sequential training ``$bbbBB\\rightarrow cccCC\\rightarrow aaaAC \\rightarrow abcAA$\" , DFWF model achieves a AvgEER of $6.90\\%$ fairly close to the multi-condition training model ($5.67\\%$) at the last step. Besides, we see a obvious fluctuation in the trend of AvgEER with fine-tuning strategy. This can be explained by the existence of similarity among different spoofing types. If the feature new spoofing is similar to some of the previous spoofing types, training with current data can help to maintain some of the previous knowledge.\n\n\n\n\n\\subsection{Ablation Study}\n\\begin{table}[t]\n \\vspace{-10pt}\n\\setlength\\tabcolsep{5.5pt}\n  \\caption{Comparison of AvgEER\\% in different combinations of LwF and PSA in 4 sequential trainings. T1 is the training ``LA $\\rightarrow$ PA\". T2 is the training ``PA $\\rightarrow$ LA\". T3 is the training `` $A13\\rightarrow A17\\rightarrow A10\\rightarrow A19$\". T4 is the training ``$bbbBB\\rightarrow cccCC\\rightarrow aaaAC \\rightarrow abcAA$\". }\n  \\vspace{-5pt}\n  \\label{ablation study}\n  \\centering\n  \\begin{tabular}{ cccccc }\n    \\toprule\n    + LwF & + PSA   & T1 & T2 & T3 & T4 \\\\\n    \\midrule\n   \\XSolidBrush  & \\XSolidBrush   &13.26& 16.01 & 43.72 & 14.38\\\\\n   \\Checkmark  & \\XSolidBrush   & 12.28 & 11.68  &  11.52 &  7.44   \\\\\n     \\XSolidBrush &  \\Checkmark   & 9.64 & 11.79 & 12.28  & 8.48 \\\\\n    \\Checkmark  & \\Checkmark  & 9.25 & 10.45 & 8.03  &  6.90 \\\\\n    \\bottomrule\n  \\end{tabular}\n \\vspace{-5pt}\n\\end{table}\n\nWe carry out an ablation study to evaluate the impact of two components of our method. The first factor LwF is the constraint on the output distribution of model for all input data. The second factor PSA is the constraint on embeddings of genuine speech input. Both of them are used for preserving key parameters consistent with the original model. Table \\ref{ablation study} shows the AvgEER achieved in all combinations of these two components. Training without neither LwF nor PSA equals to fine-tuning, and training with both LwF and PSA equals to DFWF.\nIn training, we adopt the optional hyper-parameters for each setting. The results show that both two components can improve the performance in sequential training. When applied together, DFWF obtains the best AvgEER.\n\n\\section{Discussions}\n\n\\begin{figure}[t]\n\\vspace{-10pt}\n  \\centering\n  \\subfigure[LCNN-BASE]{\n    \\label{Fig.1}\n    \\includegraphics[width=0.45\\linewidth]{.\/Figure\/devLAembedding2.pdf}}\n    \\subfigure[LCNN-FT]{\n    \\label{Fig.2}\n    \\includegraphics[width=0.45\\linewidth]{.\/Figure\/devLAftembedding2.pdf}}\n    \\subfigure[LCNN-DFWF (Ours)]{\n    \\label{Fig.3}\n    \\includegraphics[width=0.45\\linewidth]{.\/Figure\/devCLembedding2.pdf}}\n    \\subfigure[LCNN-MC]{\n    \\label{Fig.4}\n    \\includegraphics[width=0.45\\linewidth]{.\/Figure\/devLPembedding2.pdf}}\n  \\vspace{-10pt}\n  \\caption{The t-SNE visualization of all training data embeddings that are extracted by different models. LCNN-BASE is trained only on LA subset. LCNN-FT is finetuned on PA subset based on LCNN-BASE. LCNN-DFWF is trained on PA by the propoesd DFWF method based on LCNN-BASE. LCNN-MC is train on mixed data of LA and PA subsets. The color red represents genuine speech. The color blue represents spoofing speech of LA attack. The color green represents spoofing speech of PA attack.}\n  \\label{fig:embedding}\n\\vspace{-10pt}\n\\end{figure}\n\nFrom the results, we find that our proposed DFWF doesn't outperform multi-condition training. The reason is that all the data are available to multi-condition model but the previous data is inaccessible to our DFWF systems. In theory, the results of multi-condition model is the upper limit of the performance of continual learning systems\\cite{van2008visualizing}.\n\nBesides, to better understand the mechanism of our proposed DFWF method, we plot the t-SNE figures to visualize the embeddings of genuine and spoofing speech \\cite{van2008visualizing}. Figure \\ref{fig:embedding} shows the embedding distribution of all training data in ``$LA \\rightarrow PA$\" sequential training.\nLCNN-BASE can not distinguish the spoofing speech of PA attack (green dots) since there is an obvious mixture of PA spoofing speech and genuine speech (red dots). After fine-tuned on PA subset, LCNN-FT can distinguish PA attacks but the previous LA attacks (blue dots) are scattered among genuine speech. These illustrate the dilemma of fine-tuning that the fine-tuned model tends to forget the previous knowledegs.\nHowever, after applied DFWF, LCNN-DFWF can distinguish both LA attacks and PA attacks much more better than the LCNN-FT, and the distribution of the data is very similar to that in LCNN-MC.\n\n\n\n\n\n\n\\section{Conclusions}\nIn this paper, we proposed a continual-learning-based method DFWF to solve the unseen spoofing attacks problem in fake audio detection. On the basis of LwF, PSA prevents the embedding of genuine speech from deviating from the original distribution, which further preserves the model's detection capability on previous data.\nThis method can not only save the time and computing resource but also mitigate the catastrophic forgetting problem. In some cases ( i.e., $bbbBB\\rightarrow cccCC\\rightarrow aaaAC \\rightarrow abcAA$), it even performs similarly to multi-condition training but it's faster to train without the need to get access to the previous data.\nExperiments conducted on 4 setting of sequential training show the effectiveness of DFWF method to learn new spoofing attacks incrementally, particularly as spoofing data becomes more and more.\nMore advanced continual learning methods are worth exploring in fake audio detection in the future.\n\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn classical graph problems, we are given a graph and the task is to \\emph{find} a feasible solution. In \\emph{reconfiguration problems}, we are given two feasible solutions -- an input configuration and a target configuration -- and the task is to find a sequence of moves that turns the input configuration into the target configuration.\n\n\\subparagraph{Recoloring problems.}\nPerhaps the most natural example of a reconfiguration problem is \\emph{recoloring}: we are given a graph $G$ and two proper $k$-colorings of $G$, let us call them $s$ and $t$, and the task is to find a way to turn $s$ into $t$ by changing the color of one node at a time, such that each intermediate step is a proper coloring. More formally, the task is to find a sequence of proper $k$-colorings $x_0, x_1, \\dotsc, x_L$ such that $x_0 = s$ and $x_L = t$, and $x_{i-1}$ and $x_i$ differ only at one node. Such problems have been studied extensively from the perspective of graph theory and classical centralized algorithms, but the problems are typically inherently \\emph{global} and solutions are long, i.e., $L$ is large in the worst case.\n\nIn this work we introduce recoloring problems in a \\emph{distributed} setting. We show that there are natural relaxations of the problem that are attractive from the perspective of distributed graph algorithms: they admit solutions that are short and that can be found \\emph{locally} (e.g., in sublinear number of rounds). Distributed recoloring problems are closely related to classical symmetry-breaking problems that have been extensively studied in the area of distributed graph algorithms, but as we will see, they also introduce new kinds of challenges.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[scale=0.8]{3color0.pdf}\n\t\\quad\n\t\\includegraphics[scale=0.8]{3color1.pdf}\n\t\\quad\n\t\\includegraphics[scale=0.8]{3color2.pdf}\n\t\\quad\n\t\\includegraphics[scale=0.8]{3color3.pdf}\n\t\\caption{Distributed recoloring: the input coloring $s$ can be seen on the left and the target coloring $t$ on the very right. The illustration shows one possible way to reach the target coloring in three steps by, in each step, changing the colors of an independent set of nodes.}\n\t\\label{fig:recolor}\n\\end{figure}\n\n\\subparagraph{Distributed recoloring.}\n\nWe will work in the usual LOCAL model of distributed computing: Each node $v \\in V$ of the input graph $G = (V,E)$ is a computer, and each edge $e \\in E$ represents a communication link between two computers. Computation proceeds in synchronous rounds: each node sends a message to each of its neighbors, receives a message from each of its neighbors, and updates its local state. Eventually, all nodes have to announce their local outputs and stop; the running time of the algorithm is the number of communication rounds until all nodes stop. We assume that the algorithm is deterministic, and each node is labeled with a unique identifier.\n\nIn \\emph{distributed recoloring}, each node $v \\in V$ is given two colors, an \\emph{input color} $s(v)$ and a \\emph{target color} $t(v)$. It is guaranteed that both $s$ and $t$ form a proper coloring of $G$, that is, $s(u) \\ne s(v)$ and $t(u) \\ne t(v)$ for all $\\{u,v\\} \\in E$. Each node $v \\in V$ has to output a finite \\emph{recoloring schedule} $x(v) = \\bigl(x_0(v), x_1(v), \\dotsc, x_\\ell(v)\\bigr)$ for some $\\ell = \\ell(v)$. For convenience, we define $x_i(v) = x_\\ell(v)$ for $i > \\ell(v)$. We say that the node \\emph{changes its color at time $i > 0$} if $x_{i-1}(v) \\ne x_i(v)$; let $C_i$ be the set of nodes that change their color at time $i$. Define $L = \\max_v \\ell(v)$; we call $L$ the \\emph{length} of the solution. A solution is feasible if the following holds:\n\\begin{enumerate}\n\\item $x_0 = s$ and $x_L = t$,\n\\item $x_i$ is a proper coloring of $G$ for all $i$,\n\\item $C_i$ is an independent set of $G$ for all $i$.\n\\end{enumerate}\nThe key differences between distributed recoloring and classical recoloring are:\n\\begin{enumerate}\n\\item Input and output are given in a distributed manner: no node knows everything about $G$, $s$, and $t$, and no node needs to know everything about $x_i$ or the length of the solution $L$.\n\\item We do not require that only one node changes its color; it is sufficient that adjacent nodes do not change their colors simultaneously.\n\\end{enumerate}\nSee Figure~\\ref{fig:recolor} for a simple example of distributed recoloring steps.\n\n\n\nNote that a solution to distributed recoloring is locally checkable in the following sense: to check that a solution is feasible, it is enough to check independently for each edge $\\{u,v\\} \\in E$ that the recoloring sequences $x(u)$ and $x(v)$ are compatible with each other, and for each node $v \\in V$ that $x(v)$ agrees with $s(v)$ and $t(v)$. However, distributed recoloring is not necessarily an LCL problem \\cite{naor1995can} in the formal sense, as the length of the output per node is not a priori bounded.\n\nWe emphasize that we keep the following aspects well-separated: what is the complexity of \\emph{finding} the schedule, and how \\emph{long} the schedules are. Hence it makes sense to ask, e.g., if it is possible to find a schedule of length $O(1)$ in $O(\\log n)$ rounds (note that the physical reconfiguration of the color of the node may be much slower than communication and computation).\n\n\\subparagraph{Recoloring with extra colors.}\n\nRecoloring is computationally very hard, as solutions do not always exist, and deciding whether a solution exists is PSPACE-hard. It is in a sense analogous to problems such as finding an \\emph{optimal} node coloring of a given graph; such problems are not particularly interesting in the LOCAL model, as the complexity is trivially global. To make the problem much more interesting we slightly relax it.\n\nWe define a \\emph{$k+c$ recoloring problem} (a.k.a.\\ \\emph{$k$-recoloring with $c$ extra colors}) as follows:\n\\begin{itemize}\n\\item We are given colorings with $s(v), t(v) \\in [k]$.\n\\item All intermediate solutions must satisfy $x_i(v) \\in [k+c]$.\n\\end{itemize}\nHere we use the notation $[n] = \\{1,2,\\dotsc,n\\}$.\n\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics{line.pdf}\n\t\\caption{In the input graph, a bipartition is given. Therefore, the target coloring can be reached by using one extra color in three steps.}\n\t\\label{fig:line}\n\\end{figure}\n\nThe problem of $k+c$ recoloring is meaningful also beyond the specific setting of distributed recoloring. For example, here is an example of a very simple observation:\n\\begin{lem}\\label{lem:bipartite}\nRecoloring with $1$ extra color is always possible in any bipartite graph, with a distributed schedule of length $L = 3$.\n\\end{lem}\n\\begin{proof}\nLet the bipartition be $V = V_1 \\cup V_2$. First each node $v \\in V_1$ switches to $k+1$, then each $v \\in V_2$ switches to color $t(v)$, and finally each $v \\in V_1$ switches to color $t(v)$.\n\\end{proof}\nIncidentally, it is easy to extend this result to show that $k$-recoloring with $c = \\chi-1$ extra colors is always possible with a schedule of length $O(c)$ in a graph with chromatic number $\\chi$, and in particular $k$-recoloring with $c = k-1$ extra colors is trivial.\nFigure~\\ref{fig:line} gives an illustration of recoloring a bipartite graph with one extra color.\n\nAs a corollary, we can solve distributed $k+1$ recoloring in trees in $O(n)$ rounds, with a schedule of length $O(1)$: simply find a bipartition and apply the above lemma. However, is this optimal? Clearly finding a bipartition in a tree requires $\\Omega(n)$ rounds, but can we solve recoloring with $1$ extra color strictly faster?\n\nThese are examples of problems that we study in this work. We initiate the study of distributed complexity of recoloring, with the ultimate objective of finding a complete characterization of graph families and parameters $k$, $c$, and $L$ such that distributed $k+c$ recoloring with schedules of length $L$ can be solved efficiently in a distributed setting.\n\nAs we will see, the problem turns out to be surprisingly rich already in very restricted settings such as grids or $3$-regular trees. Many of the standard lower bound techniques fail; in particular, known results on the hardness of graph coloring do not help here, as we are already given two proper colorings of the input graph.\n\n\\subparagraph{Contributions.}\n\nOur main contribution is a comprehensive study of the complexity of distributed recoloring in various graph families; the results are summarized in Tables \\ref{tab:cycles}--\\ref{tab:3reg}. The highlights of this work are the following results:\n\\begin{enumerate}\n    \\item \\textbf{\\boldmath An algorithm for $3+1$ recoloring on trees.} On trees, $3$-recoloring is inherently global: it is easy to see that the worst-case running time is $\\Theta(n)$ and the worst-case schedule length is $\\Theta(n)$. With one extra color, we can trivially find a schedule of length $O(1)$ in time $O(n)$. However, we show that we can do much better: it is possible to find a schedule of length $O(1)$ in time $O(\\log n)$.\n    \n    Here the key component is a new algorithm that solves the following sub-problem in $O(\\log n)$ rounds: given a tree, find an independent set $I$ such that the removal of $I$ splits the tree in components of size $1$ or $2$. This subroutine may find applications in other contexts as well.\n    \n    These results are presented in Section~\\ref{sec:treepositive}.\n    \\item \\textbf{\\boldmath An algorithm for $3+1$ recoloring for graphs of degree at most $3$.} In general graphs, $3+1$ recoloring is not necessarily possible; we can construct a small $4$-regular graph in which $3+1$ recoloring is not solvable. However, we will show that if the maximum degree of the graph is at most $3$ (i.e., we have a \\emph{subcubic} graph), $3+1$ recoloring is always possible. Moreover, we can find a schedule of length $O(\\log n)$ in time $\\polylog(n)$.\n    \n    This result is presented in Section~\\ref{sec:subcubicpositive}.\n    \n    \\item \\textbf{\\boldmath Complexity of $3+1$ recoloring on toroidal grids.}\n    We also give a complete characterization of $3+1$ recoloring in one particularly interesting family of $4$-regular graphs: $2$-dimensional toroidal grids (a.k.a.\\ torus grid graphs, Cartesian graph products of two cycles). While the case of $1$-dimensional grids (cycles) is easy to characterize completely, the case of $2$-dimensional grids turns out to be much more interesting.\n    \n    Here our main contribution is the following graph-theoretic result: in an $h \\times w$ toroidal grid, $3+1$ recoloring is possible for any input if and only if (i)~both $h$ and $w$ are even, or (ii)~$h = 4$, or (iii)~$w = 4$. In all other cases we can find $3$-colorings $s$ and $t$ such that $t$ is not reachable from $s$ even if we can use $1$ extra color.\n    \n    As a simple corollary, $3+1$ recoloring is inherently global from the perspective of distributed computing, and it takes $\\Theta(n)$ rounds to solve even if we have the promise that e.g.\\ $h$ and $w$ are even (and hence a schedule of length $\\Theta(1)$ trivially exists).\n    \n    This result is presented in Section~\\ref{sec:grids}.\n\\end{enumerate}\nAdditionally, several simple upper and lower bounds and corollaries are given in Sections \\ref{sec:simpleub} and~\\ref{sec:simplecor}.\n\n\\subparagraph{Motivation.}\n\nAs a simple application scenario, consider the task of reconfiguring a system of unmanned aerial vehicles. Here each node is an aircraft, the color corresponds to an altitude range, and an edge corresponds to a pair of aircraft whose paths might cross and hence need to be kept at different cruising altitudes to avoid collisions.\n\nFor each aircraft there are designated areas in which they can safely change their altitude. To reconfigure the entire system, we could take all aircraft to these areas simultaneously. However, this may be a costly maneuver.\n\nAnother possibility is to reserve a longer timespan during which a set $X$ of aircraft may change their altitudes, whenever they happen to be at convenient locations. Now if we let two aircraft $u, v \\in X$ change their altitudes during the same timespan, we need to ensure that any intermediate configuration is safe, regardless of whether $u$ or $v$ happens to change its altitude first. Furthermore, we would like to complete reconfiguration in minimal time (short schedule), and we would like to waste precious airspace as little as possible and hence keep as few altitude levels as possible in reserve for reconfiguration (few extra colors).\n\nThis scenario -- as well as many similar scenarios, such as the task of reconfiguring the frequency bands of radio transmitters in a manner that never causes interference, even if the clocks are not perfectly synchronized -- give rise to the following variant of distributed recoloring that we call \\emph{weak recoloring}: if two adjacent nodes $u$ and $v$ change their color simultaneously at time $i$, then\n$\\bigl\\{ x_{i-1}(u), x_i(u) \\bigr\\} \\cap \\bigl\\{ x_{i-1}(v), x_i(v) \\bigr\\} = \\emptyset$,\nthat is, we have a proper coloring regardless of whether $u$ or $v$ changes its color first.\n\nLet us now contrast weak recoloring with \\emph{strong recoloring}, in which adjacent nodes never change colors simultaneously. Trivially, strong recoloring solves weak recoloring. But the converse is also true up to constant factors: if we have $k$ input colors and a solution to weak recoloring of length $L$, then we can also find a solution to strong recoloring of length $kL$. To see this, we can implement one weak recoloring step in $k$ strong recoloring substeps such that in substep $j$ nodes of input color $j$ change their colors.\n\nAs our focus is on the case of a small number of input colors, we can equally well study strong or weak recoloring here; all of our results hold for either of them. While weak recoloring is closer to applications, we present our results using strong recoloring, as it has a more convenient definition.\n\n\\section{Related work}\n\n\\subparagraph{Reconfiguration and recoloring.}\n\nRecoloring, and more generally combinatorial reconfiguration has received attention over the past few years. Combinatorial reconfiguration problems consist of finding step-by-step transformations between two feasible solutions such that all intermediate results are also feasible. They model dynamic situations where a given solution is in place and has to be modified, but no disruption can be afforded. We refer the reader to the nice survey~\\cite{Jansurvey} for a full overview, and focus here on node coloring as a reference problem. \n\nAs mentioned earlier, we introduce distributed recoloring here, but centralized recoloring has been studied extensively before. Two main models are considered:\n\\begin{enumerate}\n    \\item \\emph{Node recoloring:} at each step, we can recolor a node into a new color that does not appear on its neighborhood\n    \\item \\emph{Kempe recoloring:} at each step, we can switch the colors in a bichromatic component (we operate a Kempe change).\n\\end{enumerate}\n \nThe usual questions are of the form: Given a graph $G$ and an integer $k$, are all its $k$-colorings equivalent (up to node or Kempe recolorings)? What is the complexity of deciding that? What is the maximum number of operations needed to go from to the other? \n \nAll of those questions can also be asked for two specific $k$-colorings $s$ and $t$ of $G$. Are they equivalent (up to node or Kempe recolorings)? What is the complexity of deciding that? What is the maximum number of operations needed to go from $s$ to $t$ in $G$?\n \nWhile the complexity of questions related to Kempe recoloring remains elusive, the problems related to node recoloring are typically PSPACE-hard~\\cite{bonsma2009finding}. The related question of deciding equivalence when a bound on the length of an eligible recoloring sequence is given as part of the input has also been considered \\cite{bonsma2014complexity}. We know that the maximum number of operations needed to go from one $3$-coloring to another in a tree is $\\Theta(n)$~\\cite{cereceda2011finding}. While $(\\Delta+1)$-recoloring a graph with no node of degree more than $\\Delta$ is not always possible, having $\\Delta+2$ colors always suffices \\cite{jerrum1995very}, and there are also meaningful results to obtain for the problem of $(\\Delta+1)$-recoloring~\\cite{feghali2016reconfigurations}. Two other settings have received special attention: characterizing fully when $3$-recoloring is possible \\cite{cereceda2011finding,cereceda2009mixing}, and guaranteeing short reconfiguration sequences in the case of sparse graphs for various notions of sparse \\cite{bonamy2013recoloring,bousquet2016fast}. \n\nKempe changes were introduced in 1879 by Kempe in his attempted proof of the Four Color Theorem~\\cite{kempe79}. Though this proof was fallacious, the Kempe change technique has proved useful in, for example, the proof of the Five Color Theorem and a short proof of Brooks' Theorem. Most works on the topic initially focused on planar graphs, but significant progress was recently obtained in more general settings. We know that all $k$-colorings of a graph with no node of degree more than $k$ are equivalent (w.r.t.\\ Kempe changes), except in the case of one very specific graph: the $3$-prism~\\cite{bonamy2015conjecture,feghali2017kempe,meyniel3}.\n\nNote that some other variants have also been studied, perhaps most notably the question of how many nodes to recolor at once so that the graph can be recolored \\cite{mcdonald2015connectedness}.\n\nWhile we will not discuss Kempe recoloring in our work, we point out that recoloring with extra colors is closely connected to Kempe recoloring: Kempe recolorability implies recolorability with one extra color (while the converse is not true). Hence the negative results related to one extra color also hold for Kempe recoloring.\n\n\n\\subparagraph{Distributed graph coloring.}\n\nPanconesi and Srinivasan~\\cite{panconesi1995local} have used Kempe operations to design efficient distributed algorithms for graph coloring with $\\Delta$ colors. Other than that we are not aware of prior work on distributed recoloring.\nOn the other hand, the literature on the standard distributed coloring is vast. The best overview on the topic is the book by Barenboim and Elkin \\cite{barenboim2013distributed};\nthe most important recent developments include the following results.\nThere is a randomized $O\\bigl(\\log^* n + 2^{\\sqrt{\\log \\log n}}\\bigr)$ -time algorithm for $(\\Delta + 1)$-coloring by Chang et al.~\\cite{Chang2018}.\nIn the case of trees, the number of colors can be reduced to $\\Delta$ with the cost of increasing the runtime to $O(\\log_{\\Delta} \\log n)$~\\cite{Chang2016}.\nOn the deterministic side, the best known $(\\Delta + 1)$-coloring algorithm requires $O(\\Delta^{3\/4} \\log \\Delta + \\log^* n)$ communication rounds~\\cite{Barenboim2015}.\nIn the case of trees, the \\emph{rake-and-compress} -method by Miller and Reif gives a $3$-coloring in time $O(\\log n)$~\\cite{MillerR89}.\n\nHowever, there seems to be surprisingly little technology that one can directly transfer between the coloring domain and recoloring domain. Toroidal grids are a good example: by prior work \\cite{brandt2017lcl}, $3$-coloring is an inherently global problem, and by the present work, $3+1$ recoloring is an inherently global problem, but the arguments that are used in these proofs are very different (despite the fact that both of them are related to the idea that a ``parity'' is preserved).\n\n\n\n\n\\section{Preliminaries}\nIn this article, each graph $G=(V,E)$ is a simple undirected graph where $V$ represents its node set and $E$ its edge set. For a subset of nodes $S \\subseteq V$, we denote by $G[S]$ the subgraph induced by $S$. For a node $u \\in V$, we denote by $N(u)$ the \\emph{open neighborhood} of $u$ that is the set of all the neighbors of $u$ and by $N[u]$ its \\emph{closed neighborhood} i.e.\\ the set $N(u) \\cup \\{u\\}$. For a subset $S \\subseteq V$, its closed neighborhood corresponds to the set $\\bigcup_{u \\in S} N[u]$.\n\nThe \\emph{degree} of a node is the number of neighbors. A \\emph{$k$-regular graph} is a graph in which all nodes have degree $k$, a \\emph{cubic graph} is the same thing as a $3$-regular graph, and a \\emph{subcubic graph} is a graph in which all nodes have degree at most $3$. A \\emph{tree} is a connected acyclic graph, and a \\emph{$k$-regular tree} is a tree in which each node has degree $1$ or $k$.\n\nA \\emph{maximal independent set} (MIS) $S \\subseteq V$ is an independent set (i.e.\\ a set of  pairwise non-adjacent nodes) such that for each non-MIS node $u \\notin S, N(u) \\cap S \\neq \\emptyset$. \n\nGiven a graph $G=(V,E)$, a \\emph{list-assignment} is a function which assigns to each node $v \\in V$ a list of colors $L(v)$. An \\emph{$L$-coloring} of $G$ is a function $c$ that assigns to each node $v \\in V$ a color $c(v) \\in L(v)$ such that for any two adjacent nodes $u, v \\in V$, we have $c(u) \\neq c(v)$. A graph $G$ is \\emph{$k$-list-colorable} if it admits an \\emph{$L$-coloring} for every list-assignment where the list of each node is of size at least $k$. Therefore, list-coloring generalizes node-coloring if we consider the special case where each node receives the same input list. The notion of \\emph{$L$-recoloring} is the natural generalization of $k$-recoloring: the same elementary steps are considered, and every intermediate coloring must be an $L$-coloring. \n\nIn order to output a recoloring schedule, it is convenient to consider the question of recoloring a graph $G$ from a coloring $s$ to a coloring $t$, rather than the more symmetric question of whether the two colorings are equivalent in the given setting. We take this opportunity to note that we can reverse time and hence recoloring schedule from $s$ to $t$ also yields a recoloring schedule from $t$ to $s$. In the rest of the paper, we therefore address the two questions as one.\n\n\n\\section{Warmup -- simple results}\\label{sec:simpleub}\n\nWe will start by presenting a number of simpler upper and lower bounds that also serve as an introduction to the topic of distributed recoloring.\n\n\n\\subsection{Upper bounds}\n\n\\begin{lem}\\label{lem:minusone}\nIn any graph, $k+c$ recoloring for $c = k-1$ is possible in $0$ communication rounds, with a schedule of length $O(k)$.\n\\end{lem}\n\\begin{proof}\nGeneralize the idea of Lemma~\\ref{lem:bipartite}; note that the schedule of node $v$ depends only on $s(v)$ and $t(v)$, and not on the colors of any other node around it.\n\\end{proof}\n\n\\begin{lem}\\label{lem:3paths}\nIn paths and trees, $3$-recoloring is possible in $O(n)$ rounds, with a schedule of length $O(n)$.\n\\end{lem}\n\\begin{proof}\nEvery node has full knowledge of the graph. The statement can be intuited by induction on the size of the tree, but we delay a formal proof to Section \\ref{sec:treepositive} and more precisely Lemma \\ref{lem:recoltree}.\n\\end{proof}\n\n\\begin{lem}\\label{lem:3plus1paths}\nIn cycles and paths, $3+1$ recoloring is possible in $O(1)$ rounds, with a schedule of length $O(1)$.\n\\end{lem}\n\\begin{proof}\nUse the input coloring to find a maximal independent set $I$. Nodes of $I$ switch to color $4$. Nodes of $V\\setminus I$ induce paths of length $O(1)$, apply Lemma~\\ref{lem:3paths} there to recolor each of the paths by brute force, without using the extra color $4$. Finally, nodes of $I$ switch to their target colors.\n\\end{proof}\n\n\\begin{lem}\\label{lem:beyonddelta}\nLet $G$ be a graph of maximum degree at most $\\Delta$, and let $k \\ge \\Delta+2$. Then $k$-recoloring with $c$ extra colors is at least as easy as $(k-1)$-recoloring with $c+1$ extra colors.\n\\end{lem}\n\\begin{proof}\nGiven a $k$-coloring $x$, we can construct a $(k-1)$-coloring $x'$ as follows: all nodes of color $k$ pick a new color from $\\{1,2,\\dotsc,k-1\\}$ that is not used by any of their neighbors. Note that $x \\to x'$ is a valid step in distributed recoloring (nodes of color $k$ form an independent set), and by reversing the time, also $x' \\to x$ is a valid step.\n\nHence to recolor $s \\to t$ with $c$ extra colors, it is sufficient to recolor $s' \\to t'$ with $c+1$ extra colors (color $k$ no longer appears in the input and target colorings and can be used as an auxiliary color during recoloring). Then we can put everything together to form a recoloring schedule $s \\to s' \\to t' \\to t$, with only constant overhead in the running time and schedule length.\n\\end{proof}\n\n\\begin{lem}\\label{lem:4plus1subcubic}\nIn subcubic graphs, $4+1$ recoloring is possible in $O(1)$ rounds, with a schedule of length $O(1)$.\n\\end{lem}\n\\begin{proof}\nUse the input coloring to find a maximal independent set $I$ in constant time. Nodes of $I$ switch to color $5$. Delete $I$; we are left with a graph $G'$ that consists of paths and isolated nodes. Apply Lemmas \\ref{lem:beyonddelta} and \\ref{lem:3plus1paths} to solve $4+0$ recoloring in each connected component of $G'$. Finally nodes of $I$ can switch to their target colors.\n\\end{proof}\n\n\\begin{lem}\\label{lem:4plus2grids}\nIn toroidal grids, $4+2$ recoloring is possible in $O(1)$ rounds, with a schedule of length $O(1)$.\n\\end{lem}\n\\begin{proof}\nPick a maximal independent set $I$, color it with color $6$, and delete; we have a graph of degree at most $3$ and $1$ extra color. Apply Lemma~\\ref{lem:4plus1subcubic} to recolor it, and finally nodes of $I$ can switch to their target colors.\n\\end{proof}\n\n\\begin{lem}\\label{lem:5plus1grids}\nIn toroidal grids, $5+1$ recoloring is possible in $O(1)$ rounds, with a schedule of length $O(1)$.\n\\end{lem}\n\\begin{proof}\nPick a maximal independent set $I$, color it with color $6$, and delete; we have a graph of degree at most $3$ and $5+0$ colors remaining. Apply Lemma~\\ref{lem:beyonddelta} to reduce to the case of $4+1$ colors, and then use Lemma~\\ref{lem:4plus1subcubic}.\n\\end{proof}\n\n\\begin{lem}\\label{lem:MISplusforest}\nFor any graph $G$ on $n$ nodes, for any two $k$-colorings $\\alpha, \\beta$ of $G$, if we can compute in $O(f(n))$ rounds an MIS $S$ such that $V\\setminus S$ induces a forest of trees of depth at most $O(d(n))$,\nwe can compute in $O(f(n)+d(n))$ rounds  how to $(k+1)$-recolor $G$ from $\\alpha$ to $\\beta$ with schedule of length $O(d(n))$.\n\\end{lem}\n\\begin{proof}\nThe idea is quite simple: each node in $S$ goes into color $k+1$. We then use the algorithm described in the proof of Lemma \\ref{lem:recoltree} to find a recoloring with schedule of\nlength $O(d(n))$ for each connected component after the removal of $S$. After that, each node of $S$ can go to their final color.\n\\end{proof}\n\n\n\\subsection{Lower bounds}\n\n\\begin{lem}\\label{lem:needsextra}\nRecoloring without any extra colors is not possible in the following settings for some pairs of input and target colorings:\n\\begin{enumerate}[label=(\\alph*),noitemsep]\n\\item $2$-recoloring paths or trees.\n\\item $2$-recoloring cycles.\n\\item $3$-recoloring cycles.\n\\item $2$-recoloring toroidal grids.\n\\item $3$-recoloring toroidal grids.\n\\item $4$-recoloring toroidal grids.\n\\item $5$-recoloring toroidal grids.\n\\item $2$-recoloring cubic graphs.\n\\item $3$-recoloring cubic graphs.\n\\item $4$-recoloring cubic graphs.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nWe can construct a source coloring in which no node can make a move, and a target coloring different from the input coloring. Here we show examples of the source coloring $s$; the target coloring can be constructed by $t(v) \\equiv s(v) + 1 \\bmod k$:\n\\begin{enumerate}[label=(\\alph*)]\n\\item A path with $2$ nodes, $s = \\begin{bmatrix}1 & 2\\end{bmatrix}$.\n\\item A $4$-cycle, $s = \\begin{bmatrix}1 & 2 & 1 & 2\\end{bmatrix}$.\n\\item A $4$-cycle, $s = \\begin{bmatrix}1 & 2 & 3\\end{bmatrix}$.\n\\item A $4 \\times 4$ grid,\n$\n  s = \\begin{bsmallmatrix}\n  1 & 2 & 1 & 2 \\\\\n  2 & 1 & 2 & 1 \\\\\n  1 & 2 & 1 & 2 \\\\\n  2 & 1 & 2 & 1\n  \\end{bsmallmatrix}\n$.\n\\item A $3 \\times 3$ grid,\n$\n  s = \\begin{bsmallmatrix}\n  1 & 2 & 3 \\\\\n  2 & 3 & 1 \\\\\n  3 & 1 & 2\n  \\end{bsmallmatrix}\n$.\n\\item A $4 \\times 4$ grid,\n$\n  s = \\begin{bsmallmatrix}\n  1 & 2 & 3 & 4 \\\\\n  3 & 4 & 1 & 2 \\\\\n  1 & 2 & 3 & 4 \\\\\n  3 & 4 & 1 & 2\n  \\end{bsmallmatrix}\n$.\n\\item A $5 \\times 5$ grid,\n$\n  s = \\begin{bsmallmatrix}\n  1 & 2 & 3 & 4 & 5 \\\\\n  3 & 4 & 5 & 1 & 2 \\\\\n  5 & 1 & 2 & 3 & 4 \\\\\n  2 & 3 & 4 & 5 & 1 \\\\\n  4 & 5 & 1 & 2 & 3\n  \\end{bsmallmatrix}\n$.\n\\item Complete bipartite graph $K_{3,3}$, with $s$ constructed from the bipartition.\n\\item Prism graph: connect the nodes of a $3$-cycle colored with $\\begin{bmatrix}1 & 2 & 3\\end{bmatrix}$ to another $3$-cycle colored with $\\begin{bmatrix}2 & 3 & 1\\end{bmatrix}$, in this order.\n\\item Complete graph $K_4$. \\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{lem}\\label{lem:3pathslb}\nIn paths and trees, $3$-recoloring without extra colors requires $\\Omega(n)$ rounds and produces schedules of length $\\Omega(n)$ in the worst case. This holds also in the case of $3$-regular trees.\n\\end{lem}\n\\begin{proof}\nConsider a long path with the input coloring $1,2,3,1,2,3,\\dotsc,1,2,3$ and observe that a node of degree $2$ can change its color only after at least one neighbor has changed colors. We can embed such a path also in a $3$-regular tree.\n\\end{proof}\n\n\\begin{lem}\\label{lem:4treelb}\nIn trees, $4$-recoloring without extra colors requires $\\Omega(\\log n)$ time and produces schedules of length $\\Omega(\\log n)$ in the worst case.\n\\end{lem}\n\\begin{proof}\nIt is sufficient to construct a $3$-regular tree in which each node is surrounded by nodes of all other colors: color the root with color $1$ and its neighbors with colors $2$, $3$, and $4$. Then recursively for each leaf node of color $x$ that is already adjacent to a node of color $y$, add two new neighbors with the colors $\\{1,2,3,4\\} \\setminus \\{x,y\\}$, etc., and continue in a balanced manner such that the distance between the root and the nearest leaf is logarithmic. Now a non-leaf node can change its color only once its neighbor has changed its color.\n\\end{proof}\n\n\n\n\n\n\n\\section{Recoloring algorithm for trees}\\label{sec:treepositive}\n\nIn this section, we provide two efficient algorithms for recoloring and list-recoloring trees. Note that Theorem~\\ref{thm:tree4} is tight; see the full version for more details.\n\n\\begin{theorem}\\label{thm:tree3-1}\nFor any $k\\in \\mathbb{N}$, for every tree $T$ on $n$ nodes, for any two $k$-colorings $\\alpha, \\beta$ of $T$, we can compute in $O(\\log n)$ rounds how to recolor $T$ from $\\alpha$ to $\\beta$ with $1$ extra color and a schedule of length $O(1)$.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:tree4}\nFor every tree $T$ on $n$ nodes and any list assignment $L$ of at least $4$ colors to every node of $T$, for any two $L$-colorings $\\alpha, \\beta$ of $T$, we can compute in $O(\\log n)$ rounds how to $L$-recolor $T$ from $\\alpha$ to $\\beta$ with schedule of length $O(\\log n)$.\n\\end{theorem}\n\n\nWe first discuss how to compute efficiently an independent set with some desirable properties. For this, we use a simple modification of the \\emph{rake and compress} method by Reif and Miller~\\cite{MillerR89}. More precisely, we iterate rake and compress operations, and label nodes based on the step at which they are reached. We then use the labels to compute an independent set satisfying given properties. We finally explain how to make use of the special independent set to obtain an efficient recoloring algorithm, in each case.\n\n\\begin{defn}\\label{def:hlabel}\nA \\emph{light $h$-labeling} is a labeling $V \\to [h]$ such that for any $i \\in [h]$:\n\\begin{enumerate}\n    \\item Any node labeled $i$ has at most two neighbors with label $\\geq i$, at most one of which with label $\\geq i+1$.\n    \\item No two adjacent nodes labeled $i$ both have a neighbor with label $\\geq i+1$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{lem}\\label{lem:treelabeling}\nThere is an $O(\\log n)$-round algorithm that finds a light $(2 \\log n)$-labeling of a tree.\n\\end{lem}\n\n\\begin{proof}\nAs discussed above, we merely use a small variant of the \\emph{rake and compress} method. At step $i$, we remove all nodes of degree $1$ and all nodes of degree $2$ that belong to a chain of at least three nodes of degree $2$, and assign them label $i$. \n\nOne can check that this yields a light labeling. It remains to discuss how many different labels are used, i.e. how many steps it takes to delete the whole tree. Let us argue that no node remains after $2 \\log n$ rounds. Let $T$ be a tree, let $V_1$ (resp.\\ $V_2$, $V_3$) be the number of nodes of degree $1$ (resp.\\ $2$, $\\geq 3$) in the tree, and let $T'$ be the tree obtained from $T$ by replacing any maximal path of nodes of degree $2$ with an edge. Note that $|V(T')|=|V_1|+|V_3|$. Let $W$ be the set of nodes in $T$ that have degree $2$ with both neighbors of degree $2$. Note that $|V_2 \\setminus W|\\leq 2 |E(T')| =2(|V_1|+|V_3|-1)$. Note also that $|V_1|\\geq |V_3|$, simply by the fact that there are fewer edges than nodes in a tree. It follows that $|W|\\geq |V_2|- 2(|V_1|+|V_3|-1)= |V(T)|-|V_1|-|V_3|-2(|V_1|+|V_3|-1) \\geq |V(T)|-6|V_1|$. Consequently, we obtain $|W|+|V_1|\\geq \\frac{|V|}6$. In other words, at every step, we remove in particular $W \\cup V_1$, hence at least a sixth of the nodes. It follows that at after $k$ steps, the number of remaining nodes is at most $n \\cdot \\bigl(\\frac56\\bigr)^k$. Note that this is less than $1$ once $k \\geq 2 \\log n$.\n\\end{proof}\n\nWe now discuss how to make use of light $h$-labelings.\n\n\\begin{lem}\\label{lem:uselightlabels}\nFor any graph $T$, any $3$-coloring $\\alpha$ of $T$, and any integer $h$, let $L$ be a light $h$-labeling of $T$. There is an $O(h)$-round algorithm that finds a maximal independent set $S$ such that $T \\setminus S$ only has connected components on $1$ or $2$ nodes.\n\\end{lem}\n\n\\begin{proof}\nIn brief, we proceed as follows: at step $i = h, h-1, \\dotsc, 1$, we first add all nodes of label $i$ which have a neighbor of label $\\geq i+1$ that is not in $S$ (they form an independent set by definition of a light label), then use the $3$-coloring to obtain a fast greedy algorithm to make $S$ maximal on the nodes of label $\\geq i$. The detailed algorithm can be found in the full version.\n\n\\begin{algorithm}[t]\n\\caption{\\label{algo:stabletreedecomposition}\\textsc{Decomposing into an independent set and components of size $\\le2$}}\n\\begin{algorithmic}[1] \n\\REQUIRE A tree $T$, a 3-coloring $\\alpha$ and a light $h$-label of $T$.\n\\ENSURE A set $S$ of $V(T)$ such that $G[S]$ is an independent set and every connected component of $G[V\\setminus S]$ has size at most $2$.\n\\FOR{$i$ from $h$ down to 1}\n\\FOR{$u$ with label $i$ (in parallel)}\n\\STATE If $u$ has a neighbor of higher label that is not in $S$, add $u$ to $S$\n\\ENDFOR\n\\FOR{$j$ from $1$ to 3}\n\\FOR{$u$ with label $i$ and color $j$ (in parallel)}\n\\STATE If $N(u)\\cap S=\\emptyset$, add $u$ to $S$\n\\ENDFOR\n\\ENDFOR\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\nThe fact that the output $S$ is an independent set follows directly from the construction, as does the fact that the running time in $O(h)$ rounds. We note that no connected component of $T \\setminus S$ contains nodes of different labels, due to the first operation at step $i$. \n\nIt remains to argue that for any $i$, the nodes of label $i$ that do not belong to $S$ only form connected components of size $1$ or $2$. Assume for a contradiction that there is a node $u$ of label $i$ which has two neighbors $v$ and $w$, also of label $i$, such that none of $\\{u,v,w\\}$ belongs to $S$. By definition of a light label, the node $u$ has no other neighbor of label $\\geq i$, a contradiction to the fact that we build $S$ to be an MIS among the nodes of label $\\geq i$.\n\\end{proof}\n\nCombining Lemmas~\\ref{lem:treelabeling} and~\\ref{lem:uselightlabels}, and observing that a $3$-coloring of a tree can be obtained in $O(\\log n)$ rounds, we immediately obtain the following.\n\n\\begin{lem}\\label{lem:treeMIS}\nThere is an $O(\\log n)$-round algorithm that finds an MIS in a tree, such that every component induced by non-MIS nodes is of size one or two.\n\\end{lem}\n\nWe are now ready to prove Theorem~\\ref{thm:tree3-1}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:tree3-1}]\nFirst, we use Lemma~\\ref{lem:treeMIS} to obtain in $O(\\log n)$ rounds an MIS $S$ such that $T \\setminus S$ only has connected components of size $1$ or $2$. We recolor each node in $S$ with the extra color. Remove $S$, and recolor each component from $\\alpha$ to $\\beta$ without using any extra colors; this can be done in $O(1)$ recoloring rounds. Each node in $S$ can then go directly to its color in~$\\beta$.\n\\end{proof}\n\nMoving on to the list setting, we have to use a more convoluted approach since there is no global extra color that we can use. Before discussing $4$-list-recoloring, we discuss $3$-list-recoloring. For the sake of intuition, we start by presenting an algorithm for $3$-recoloring trees, and explain afterwards how to adapt it for the list setting.\n\n\\begin{lem}\\label{lem:recoltree}\nFor every tree $T$ with radius at most $p$ and for any two 3-colorings $\\alpha, \\beta$ of $T$, we can compute in $O(p)$ rounds how to $3$-recolor $T$ from $\\alpha$ to $\\beta$ with a schedule of length $O(p)$.\n\\end{lem}\n\n\\begin{proof}\nLet $c\\colon V \\to [3]$ be a $3$-coloring of $T$. We introduce an identification operation: Given a leaf $u$ and a node $v$ such that $u$ and $v$ have a common neighbor $w$, we recolor $u$ with $c(v)$, and from then on we pretend that $u$ and $v$ are a single node. In other words, we delete $u$ from the tree we are considering, and reflect any recoloring of $v$ to the node $u$. Note that these operations can stack up: the recoloring of a single node might be reflected on an arbitrarily large independent set in the initial tree. \n\nWe now briefly describe an algorithm to recolor a $3$-coloring into a $2$-coloring $c'$ in $O(p)$ rounds, with schedule $O(p)$. First, root $T$ on a node $r$ which is at distance at most $p$ of any node of $T$. Any node of $T$ which is not adjacent to the root has a \\emph{grandparent}, which is defined as its parent's parent.\n\nThen, at each step, we consider the set $A$ of leaves of $T$ which have a grandparent, if any. We identify each leaf in $A$ with its grandparent (note that the notion of grandparent guarantees that this operation is well-defined, and that the operation results in $A$ being deleted).\n\nThis process stops when $T$ consists only of the root $r$ and its children. We select one of the children arbitrarily and identify the others with it. This results in $T$ being a single edge. Note that the color partition of $c'$ is compatible with the identification operations, as we only ever identify nodes at even distance of each other.\n\nWe then recolor $T$ into $c'$: this is straightforward in the realm of $3$-recoloring.\n\nWe can now choose a $2$-coloring of $T$ (this can be done in $O(p)$ rounds), and apply the above algorithm to $3$-recolor both $\\alpha$ and $\\beta$ to that $2$-coloring. This results in a $3$-recoloring between $\\alpha$ and $\\beta$ with schedule $O(p)$.\n\\end{proof}\n\nThe same idea can be adapted to list coloring:\n\n\\begin{lem}\\label{lem:recoltreelist}\nFor every tree $T$ with radius at most $p$, for any list assignment $L$ of at least $3$ colors to each node, for any two $L$-colorings $\\alpha, \\beta$ of $T$, we can compute in $O(p)$ rounds how to $L$-recolor $T$ from $\\alpha$ to $\\beta$ with schedule $O(p)$.\n\\end{lem}\n\n\\begin{proof}\nWe adapt the identification operation introduced in the proof of  Lemma~\\ref{lem:recoltree}, merely by adapting the notion of having the same color. Let $u$ and $v$ be two nodes with a common neighbor $w$. We say $u$ has the same color as $v$ \\emph{with respect to $w$} in the following cases:\n\\begin{itemize}\n    \\item If $L(u)\\neq L(w)$, then $u$ is colored with the smallest element of $L(u) \\setminus L(w)$\n    \\item If $L(u)=L(w)$ and the color of $v$ belongs to $L(u)$, then $u$ is colored the same as $v$\n    \\item If $L(u)=L(w)$ and the color of $v$ does not belong to $L(u)$, then $u$ is colored with the smallest element of $L(w)$ that differs from the color of $w$\n\\end{itemize}\n\nTherefore, when we identify a leaf $u$ with a node $v$ that has a common neighbor $w$ with $u$, we first assign to $u$ the same color as $v$ with respect to $w$, and from then on we pretend that $u$ and $v$ are a single node. In other words, any recoloring of $v$ is mirrored on $u$ so that at each step, the node $u$ has the same color as $v$ with respect to $w$. Note that in some cases it may be that the color of $u$ does not actually change when the color of $v$ does.\n\nWhen the operations stack up, i.e. a node $u$ is identified with a node $v$ which is identified with a node $x$, we do not claim transitivity of the relation. In particular, $u$ and $x$ have no common neighbor, hence them having the same color is not well-defined. We merely enforce that $u$ has the same color as $v$ with respect to their common neighbor, and that $v$ has the same color as $w$ with respect to their common neighbor.\n\nWe insist on the fact that the definition of having the same color only depends on the list assignment. In particular, let us consider the situation once no more identification operation can be operated, i.e. the tree has been identified into an edge (see the proof of Lemma~\\ref{lem:recoltree}). The coloring of the edge characterizes entirely the coloring of the whole tree, regardless of the initial coloring. Therefore, we can pick an arbitrary $L$-coloring of the edge, and recolor both $\\alpha$ and $\\beta$ into the corresponding $L$-coloring of the tree in $O(p)$ rounds with schedule $O(p)$.\n\nThis results in computing in $O(p)$ rounds an $L$-recoloring between $\\alpha$ and $\\beta$ with a schedule of length~$O(p)$.\n\\end{proof}\n\nTo prove Theorem~\\ref{thm:tree4}, we first split the tree in small components. We slightly adapt the proof of Lemma~\\ref{lem:uselightlabels}:\n\n\\begin{lem}\\label{lem:uselightlabelsforlists}\nFor any tree $T$, any $3$-coloring $\\alpha$ of $T$, and any integer $h$, let $L$ be a light $h$-label of $T$. There is a $O(h)$-round algorithm that finds a maximal independent set $S$ such that no node has two neighbors in $S$ and $T \\setminus S$ only has connected components of radius $O(h)$.\n\\end{lem}\n\n\\begin{proof}\nThe algorithm is far simpler than Algorithm~\\ref{algo:stabletreedecomposition}. We compute the set $R$ of nodes with no neighbor of higher label. We note that $G[R]$ is a collection of paths and cycles. We compute an independent set $S \\subseteq R$ that is maximal subject to the property that no node in $R$ has two neighbors in $S$. Note that by definition of light label, no node outside of $R$ may have two neighbors in $R$ (hence in $S$). It remains to argue that $T \\setminus S$ only has connected components of radius $O(h)$. We point out that every connected component of $T[R]$ contains an element of $S$. Therefore, any connected subset of nodes of $T[R]$ has at most one neighbor of higher label, since $T$ is a tree. Together with the fact that any connected component of $T[R \\setminus S]$ has at most $2$ nodes, we derive the conclusion.\n\\end{proof}\n\nNow we are ready to prove Theorem~\\ref{thm:tree4}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:tree4}]\nCompute (in $O(\\log n)$ rounds) an independent set $S$ such any two elements of $S$ are at distance at least $2$ of each other and every connected component of $T \\setminus S$ has radius $O(\\log n)$. By Lemmas~\\ref{lem:treelabeling} and~\\ref{lem:uselightlabelsforlists} and the fact that a $3$-coloring of a tree can be computed in $O(\\log n)$ rounds, we compute (in $O(\\log n)$ rounds) an $L$-coloring $\\gamma$ of $T \\setminus S$ such that every node adjacent to an element $u \\in S$ has a color different from $\\alpha(u)$ and $\\beta(u)$. Note that this coloring exists since any tree is $2$-list-colorable. Use Lemma~\\ref{lem:recoltreelist} to recolor each connected component of $T \\setminus S$ from $\\alpha$ to $\\gamma$. Recolor every element of $S$ with its color in $\\beta$. Use Lemma~\\ref{lem:recoltreelist} to recolor each connected component $T \\setminus S$ from $\\gamma$ to $\\beta$. Note that this yields an $L$-recoloring of $T$ from $\\alpha$ to $\\beta$ with schedule $O(\\log n)$.\n\\end{proof}\n\nNote that a direct corollary of Theorem \\ref{thm:tree4} is that for any $k-$coloring $\\alpha$, $\\beta$ of a trees with $k\\ge4$, a schedule of length  $\\Theta(\\log n)$  can be found in $\\Theta(\\log n)$ rounds.\n\n\\section{Recoloring algorithm for subcubic graphs}\\label{sec:subcubicpositive}\n\nIn this section we study recoloring in subcubic graphs (graphs of maximum degree at most $3$); our main result is summarized in the following theorem:\n\n\\begin{theorem}\\label{thm:cubic3-1}\nFor every subcubic graph $G$ on $n$ nodes, for any two $3$-colorings $\\alpha, \\beta$ of $G$, we can compute in $O(\\log^2 n)$ rounds how to recolor $G$ from $\\alpha$ to $\\beta$ with $1$ extra color and a schedule of length $O(\\log n)$.\n\\end{theorem}\n\n\nA \\emph{theta} is formed of three node-disjoint paths between two nodes. Note that in particular if a graph contains two cycles sharing at least one edge, then it contains a theta. We note $B^{k}(u)$ the set of nodes at distance at most $k$ to $u$.\n\nWe show here, roughly, that there is around every node a nice structure that we can use to design a valid greedy algorithm for the whole graph. This proof is loosely inspired by one in~\\cite{ABBE}.\n\n\\begin{lem}\\label{lem:thereisthetaor2}\nFor every subcubic graph $G$ on $n$ nodes, for every node $u \\in V(G)$, there is a node $v$ with degree at most $2$ or a theta that is contained in $B^{2 \\log n}(u)$.\n\\end{lem}\n\n\\begin{proof}\nAssume for a contradiction that there is a subcubic graph $G$ on $n$ nodes with a node $u$ such that $B^{2 \\log n}(u)$ contains no node of degree $2$ nor any theta. Let $B$ be the set of nodes at distance at most $2 \\log n$ from $u$, and $B^-$ the set of nodes at distance at most $2 \\log n-1$ from $u$. Let $\\mathcal{C}$ be the set of cycles of $G$ contained in $B$. Note that cycles in $\\mathcal{C}$ are edge-disjoint by assumption on $u$ and thus node-disjoint since $G$ is cubic. We select a set $\\mathcal{E}$ by picking for every $C \\in \\mathcal{C}$ an arbitrary edge in $E(C)$ among those with both endpoints farthest from $u$. Note that $|\\mathcal{E}|=|\\mathcal{C}|$, and that by choice of $\\mathcal{E}$, every edge in $B$ with both endpoints at the same distance of $u$ is selected in~$\\mathcal{E}$. Therefore, the distance to $u$ yields a natural orientation of the edges in $B \\setminus \\mathcal{E}$, orientation from closer node to $u$ toward further node. We also note that by choice of $\\mathcal{E}$, for any edge $wx$ in $\\mathcal{E}$ such that $x$ is farther away from $u$ than $w$, the node $x$ has another neighbor $y$ at the same distance of $u$ as $w$. In that case, note that the edge $xy$ does not belong to $\\mathcal{E}$.\nWe claim as a consequence that the distance from $u$ is the same in $B$ as in $B \\setminus \\mathcal{E}$.\n\nFor any node $w \\in B$, we say an outgoing edge is \\emph{useful} if it does not belong to $\\mathcal{E}$.\nIn addition to the above remarks, we make two observations:\n\\begin{enumerate}\n    \\item Every node in $B^-$ has at least one useful edge.\n    \\item If a node $w$ in $B^-$ has only one useful edge $wx$, then $x$ has two outgoing useful edges.\n   \n\\end{enumerate}\n\nLet us consider the graph $H$ obtained from $G[B]$ by removing all edges in $\\mathcal{E}$. We claim that every node in $B$ has degree at least $2$ in $H$, and that no two adjacent nodes in $H$ have degree $2$: this is immediate from the observations and remarks above. We also observe that $H$ is a tree. Let $H'$ be the graph obtained from $H$ by replacing every node of degree two with an edge. We note that $H'$ is a $3$-regular tree of root $u$ and with no leaf at distance less than $\\log n$ of $u$. It follows that $H'$ contains at least $1+3 \\cdot 2^{\\log n} > n$ nodes, a contradiction.\n\\end{proof}\n\n\\begin{lem}\\label{lem:decompositionextension}\nLet $G$ be a subcubic graph, let $p$ be an integer, and let $\\mathcal{A}$ be a collection of thetas and nodes of degree $\\leq 2$ in $G$ each at distance at least $2$ of each other. Let $r \\geq 1$ be such that no element of $\\mathcal{A}$ has diameter more than $\\frac{r}2$. If the nodes of $G \\setminus (\\bigcup_{A \\in \\mathcal{A}} A)$ can be partitioned into $S$ and $F$ such that $G[S]$ is an independent set and $G[F]$ is a forest of radius at most $p$, then there is a partition $(S',F')$ of $\\bigcup_{A \\in \\mathcal{A}} A$ such that $G[S \\cup S']$ is an independent set and $G[F \\cup F']$ is a forest of radius at most $p+r$.\n\\end{lem}\n\n\\begin{proof}\nOur construction ensures that any pair of nodes that are not connected in $G[F]$ are not connected in $G[F \\cup F']$ neither. Hence, it suffices to prove that the statement holds for a single element of $\\mathcal{A}$, since the elements of $\\mathcal{A}$ are by hypothesis non-adjacent. Let $A$ be an element of  $\\mathcal{A}$. We consider two cases depending on whether $A$ is a node of degree at most $2$ or is a theta.\n\n\\begin{itemize}\n    \\item If $A$ consists of a node $v$ of degree $1$, or $2$, we set $v$ to be in $F'$ if it has a neighbor in $S$, in $S'$ otherwise. Note that since $v$ has at most one neighbor in $F$, the radius of $F \\cup F'$ is at most one more than that of $F$.\n    \\item If $A$ consists of a theta with endpoints $u$ and $v$ and three node-disjoint paths $P_1, P_2, P_3$, we prove independently that each $P_i$ admits a partition that is compatible with $u$ being set to $S'$ and $v$ to $F'$, in such a way that the connected component of $F \\cup F'$ that contains $v$ is contained in $F'$. We do this by induction on the number of nodes in $P_i$. If $P_i$ has no internal node, the conclusion immediately follows. If all the neighbors of $P_i$ at distance $\\geq 3$ of $u$ through $P_i$ are in $S$, we set all of $P_i$ to $F'$. Otherwise, let $w$ be the neighbor of $P_i$ in $F$ that with smallest distance ($\\geq 3)$ to $u$ through $P_i$. Let $x$ be the neighbor of $w$ in $P_i$. We apply induction on $P_i \\setminus \\{\\textrm{nodes closer to $u$ than $x$}\\}$, with $x$ in the role of $u$. Note that $x$ is distinct from $v$ and not adjacent to $u$, by construction. The nodes between $u$ and $x$ on $P_i$ are added to $F'$. Note that these nodes are connected to at most one component of $G[F]$, on the first node of $P_i$.\n    We extend the resulting decomposition to the rest of $P_i$ by setting all corresponding nodes to $F'$. \\qedhere\n\\end{itemize}\n\\end{proof}\n\n\\begin{lem}\\label{lem:algostableforestdecompositionrunningtime}\nLet $G$ be a subcubic graph on $n$ nodes. We can compute in $O(\\log^2 n)$ rounds a partition $(S,F)$ of the nodes of $G$ that $G[S]$ is an independent set and $G[F]$ is a forest of radius $O(\\log n)$.\n\\end{lem}\n\n\\begin{proof}\nTo that purpose, we combine the previous lemmas in Algorithm~\\ref{algo:stableforestdecomposition}.\nThe algorithm computes a decomposition as desired and runs in $O(\\log n)+RS(n)$ rounds, where $RS(n)$ is the number of rounds necessary to compute a $(4 \\log n, 8 \\log n)$-ruling set in a subcubic graph. We derive from~\\cite{panconesi1992improved} that $RS(n) = O(\\log^2 (n))$, hence the conclusion.\n\\end{proof}\n\n\\begin{algorithm}\n\\caption{\\label{algo:stableforestdecomposition}\\textsc{Decomposing into a small forest and an independent set}}\n\\begin{algorithmic}[1] \n\\REQUIRE A subcubic graph $G$.\n\\ENSURE A decomposition $(F,S)$ of $V(G)$ such that $G[S]$ is an independent set and every connected component of $G[F]$ has radius at most $\\log n$.\n\\FOR{$u$ in $V(G)$ (in parallel)}\n\\STATE Acquire knowledge on $B^{2 \\log n}(u)$\n\\STATE Select in the node set of $B^{2 \\log n}(u)$ a configuration $C(u)$ that is a minimal theta or a node of degree $1$ or $2$\n\\ENDFOR\n\\STATE Compute a $(4 \\log n, 8 \\log n)$-ruling set $X$ in $G$\n\\STATE Define $\\mathcal{A}=\\cup_{u \\in X}\\{C(u)\\}$\n\\STATE Compute the distance of every node in $G$ to an element of $\\mathcal{A}$\n\\STATE Let $F=S=\\emptyset$\n\\FOR{$i=8 \\log n$ downto $1$}\n\\STATE Extend the partition $(F,S)$ to the nodes at distance $i$ from $\\mathcal{A}$, more precisely:\n\\STATE Each connected component is a path or cycle where no internal node has an already assigned neighbor, let $U_i$ be the set of the internal nodes\n\\STATE Assuming a pre-computed MIS on each layer for the sets $U_i$, assign that MIS to $S$\n\\STATE Extend greedily on the remaining nodes (which form bounded-size components), assigning nodes to $S$ when possible, to $F$ when not\n\\ENDFOR\n\\STATE Extend the partition $(F,S)$ to the nodes belonging to an element of $\\mathcal{A}$ using Lemma~\\ref{lem:decompositionextension}\n\\end{algorithmic}\n\\end{algorithm}\n\nWe are now ready to prove Theorem \\ref{thm:cubic3-1}, which we do in a similar fashion as Theorem \\ref{thm:tree3-1}.\n\n\\begin{proof}\nUse Lemma \\ref{lem:algostableforestdecompositionrunningtime}, and obtain a decomposition $(S,F)$ as stated. Recolor all of $S$ to the extra color, then use Lemma \\ref{lem:recoltreelist} on each connected component of $G[F]$ so that all nodes of $F$ reach their target color (remember that each connected component of $G[F]$ has radius $O(\\log n)$). Finally recolor each node of $S$ with its target color.\n\\end{proof}\n\n\n\\section{Recoloring in toroidal grids}\\label{sec:grids}\n\n\nIn this section we study toroidal grids (torus grid graphs). Throughout this section, an $h \\times w$ toroidal grid is the Cartesian graph product of cycles of lengths $h$ and $w$; we assume $h \\ge 3$ and $w \\ge 3$. A toroidal grid can be constructed from an $h \\times w$ grid by wrapping both boundaries around into a torus. In the full version, we show that e.g.\\ $2+0$, $3+0$, and $4+0$ recoloring is not always possible, and by Lemma~\\ref{lem:minusone} e.g.\\ $2+1$, $3+2$, and $4+3$ recoloring is trivial. The first nontrivial case is $3+1$ recoloring; in this section we give a complete characterization of $3+1$ recolorability in toroidal grids:\n\n\\begin{theorem}\\label{thm:grids}\nLet $G$ be the $h \\times w$ toroidal grid graph. Then $3+1$ recoloring is possible for any source and target coloring in the following cases:\n(i)~both $h$ and $w$ are even, or\n(ii)~$h = 4$, or\n(iii)~$w = 4$.\nFor all other cases it is possible to construct $3$-colorings $s$ and $t$ such that $t$ is not reachable from $s$ by valid recoloring operations using $1$ extra color.\n\\end{theorem}\n\nThis also shows that $3+1$ recoloring is an inherently global problem in toroidal grids, even if we have a promise that recoloring is possible. For example, if there was a sublinear-time distributed recoloring algorithm $A$ for $6 \\times w$ grids for an even $w$, we could apply the same algorithm in a $6 \\times w$ grid with an odd $w$ (the algorithm cannot tell the difference between these two cases in time $o(w)$), and hence we could solve recoloring in $6 \\times w$ grids for all $w$, which contradicts Theorem~\\ref{thm:grids}. By a similar argument, distributed recoloring in non-toroidal grids is also an inherently global problem.\n\n\\subparagraph{Existence.}\n\nTo prove Theorem~\\ref{thm:grids}, let us start with the positive results. If $h$ and $w$ are even, the graph is bipartite and recoloring is always possible by Lemma~\\ref{lem:bipartite}. The remaining cases are covered by the following lemma.\n\\begin{lem}\\label{lem:grids4xw}\n    Let $G$ be a $4 \\times w$ toroidal grid for any $w \\ge 3$, and let $s$ and $t$ be any $3$-colorings. Then there exists a recoloring from $s$ to $t$ with one extra color.\n\\end{lem}\n\\begin{proof}\nWe first take an MIS $S$ over pairs of consecutive columns, i.e.\\ a set of indices of the form $(i,i+1)$ such that every column $j \\notin S$ is such that at least one of $j-1$ and $j+2$ belongs to $S$, every column $i \\in S$ is such that precisely one of $i-1$ and $i+1$ is in $S$. Note that indices are taken modulo $w$. \nFor every pair in $S$, we select a maximal independent set of the corresponding columns. The resulting union yields an independent set $R$. We then greedily make $R$ maximal columnwise away from $S$. We recolor $R$ with the extra color. It remains to argue that $G\\setminus R$ can reach its targeted coloring. We note that since leaves are not problematic, removing $R$ essentially boils down to removing the columns with index in $S$. Note that the remaining connected components are cycles of length 4. Cycles of length 4 can be always $3$-recolored.\n\nNote that the above proof yields in fact an $O(\\log n)$ rounds algorithm that outputs an $O(1)$ schedule. We can improve it into an $O(1)$-round algorithm, simply by pointing out that there is only a finite number of possible colorings for a column, and two adjacent columns cannot have the same coloring. This allows us to compute $S$ in constant time. \n\\end{proof}\n\n \\subparagraph{Non-existence.}\n\nLet us now prove the negative result. Our high-level plan is as follows. Let $G$ be an $h \\times w$ toroidal grid. We will look at all \\emph{tiles} of size $2 \\times 2$. If $G$ is properly colored with $k$ colors, so is each tile. The following two tiles are of special importance to us; we call these tiles of \\emph{type A}:\n\\[\n\\begin{bmatrix}\n{\\color{blue} 2} & {\\color{red} 3} \\\\\n{\\color{red} 3} & {\\color{black} 1}\n\\end{bmatrix},\n\\quad\n\\begin{bmatrix}\n{\\color{black} 1} & {\\color{red} 3} \\\\\n{\\color{red} 3} & {\\color{blue} 2}\n\\end{bmatrix}.\n\\]\nWe are interested in the \\emph{number} of type-A tiles. For example, consider the following colorings of the $3 \\times 3$ toroidal grid:\n\\[\ns = \\begin{bmatrix}\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2}\n\\end{bmatrix},\n\\quad\nt = \\begin{bmatrix}\n{\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3}\n\\end{bmatrix}.\n\\]\nHere $s$ contains $3$ tiles of type A (recall that we wrap around at the boundaries), while $t$ does not have any tiles of type A. In particular, $s$ has an odd number of type-A tiles and $t$ has an even number of type-A tiles. In brief, we say that the \\emph{A-parity} of $s$ is odd and the A-parity of $t$ is even. It turns out that this is sufficient to show that recoloring from $s$ to $t$ with one extra color is not possible (see the full version of the article for the proof of this lemma):\n\n\\begin{lem}\\label{lem:gridsAparity}\n    Let $G$ be a toroidal grid, and let $s$ and $t$ be two $3$-colorings. If $s$ and $t$ have different A-parities, then it is not possible to recolor $G$ from $s$ to $t$ with $1$ extra color.\n\\end{lem}\n\n\\begin{proof}\nLet us now return to the last missing ingredient: the proof of Lemma~\\ref{lem:gridsAparity}, which says that $3+1$ recoloring from $s$ to $t$ is not possible if we have different A-parities.\n\nIntuitively, we would like to prove that recoloring operations preserve the A-parity. Unfortunately, this is not the case, as we can use the extra color. For example, if you take a proper $3$-coloring $s$ with an odd A-parity and replace all nodes of color $3$ with color $4$, you will have a proper $4$-coloring $t$ with an even A-parity.\n\nIt turns out that recoloring operations do preserve a certain kind of parity of $2 \\times 2$ tiles, but it is much more involved than merely preserving type-A parity. We introduce a new set of $2 \\times 2$ tiles that we call \\emph{type-B} tiles; the parity of the number of type-B tiles is called {B-parity}:\n\\input{typeb.tex}\n\nThe collection of type-B tiles looks indeed somewhat arbitrary, but the following property is easy to verify: there is exactly one node of color $4$ in each type-A tile. Therefore if $x$ is a proper $3$-coloring, then B-parity of $x$ is even. In particular, in $3+1$ recoloring, the initial coloring $s$ and the target coloring $t$ both have even B-parities.\n\nThe magic behind the choice of type-B tiles is that it happens to satisfy the following property.\n\n\\begin{lem}\\label{lem:ABparity}\nIf you change the color of one node in a properly $4$-colored grid so that the result is also a proper $4$-coloring, A-parity changes if and only if B-parity changes.\n\\end{lem}\n\n\\begin{proof}\nEnumerate all $3\\times 3$ neighborhoods and all possible ways to change the middle node, and check that the claim holds. We have made a computer program that verifies the claim and a human-readable list of all cases available online.\\footnote{\\url{https:\/\/github.com\/suomela\/recoloring}}\n\\end{proof}\n\nNow any $3+1$ recoloring from $s$ to $t$ can be serialized so that we change the color of one node at a time. We start with an even B-parity (no type-B tiles in $s$), apply Lemma~\\ref{lem:ABparity} at each step, and eventually we arrive at even B-parity (no type-B tiles in $t$). As B-parity did not change between $s$ and $t$, also A-parity cannot change between $s$ and $t$. This concludes the proof of Lemma~\\ref{lem:gridsAparity}.\n\\end{proof}\n\nHence the A-parity of a coloring partitions the space of colorings in two components that are not connected by $3+1$ recoloring operations. To complete the proof of Theorem~\\ref{thm:grids}, it now suffices to construct a pair of $3$-colorings with different A-parities for each relevant combination of $h$ and $w$.\n\n\n\\subparagraph{\\boldmath Odd $h$, odd $w$.} First assume that both $h$ and $w$ are odd. The simplest case is $h = w$. In that case we can simply have $3$s on the anti-diagonal and color all remaining areas with colors $1$ and $2$; this gives a coloring $s$ with $h$ type-A tiles (odd type-A parity). If we put $3$s on the diagonal, we can construct a coloring $t$ with $0$ type-A tiles (even type-A parity). Here are examples for $h = w = 5$:\n\\[\ns = \\begin{bmatrix}\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} \\\\\n{\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} \\\\\n{\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \n\\end{bmatrix},\n\\quad\nt = \\begin{bmatrix}\n{\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} \n\\end{bmatrix}.\n\\]\nIf $h \\ne w$, assume w.l.o.g.\\ that $h < w$, and in particular, $w = h + 2\\ell$ for some $\\ell$. Then we can take the diagonal construction for $h \\times h$ and add $\\ell$ copies of the two rightmost columns. For example, for $h = 5$ and $w = 9$ (and hence $\\ell = 2$) we get\n\\[\ns = \\begin{bmatrix}\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} \\\\\n{\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} \\\\\n{\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \n\\end{bmatrix},\n\\quad\nt = \\begin{bmatrix}\n{\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} \n\\end{bmatrix}.\n\\]\nNote that each additional pair of columns results in one new tile of type A in both $s$ and $t$; hence overall we will have $h + \\ell$ type-A tiles in $s$ and $\\ell$ type-A tiles in $t$, and as $h$ was odd, the parity differs.\n\n\\subparagraph{\\boldmath Odd $h$, even $w \\ne 4$.} The case that remains to be considered is that exactly one of $h$ and $w$ is odd; w.l.o.g., assume that $h$ is odd and $w$ is even. Also recall that $w \\ne 4$, and hence we can focus on the case $h \\ge 3$ and $w \\ge 6$.\n\nFor the base case of $h = 3$ and $w = 6$ we can use the following configuration; here in $s$ there is a sequence of $3$s that \\emph{wraps around vertically twice}, while in $t$ there is a sequence of $3$s that does not wrap around vertically. Here $s$ has got $6$ type-A tiles (even), while $t$ has got $3$ type-A tiles (odd):\n\\[\ns = \\begin{bmatrix}\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} \n\\end{bmatrix},\n\\quad\nt = \\begin{bmatrix}\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} \n\\end{bmatrix}.\n\\]\n\nIf $h = 3$ and $w = 6 + 2\\ell$, we can take the above construction and pad it by duplicating the two leftmost columns $\\ell$ times. Each such duplication results in one new type-A tile in both configurations, maintaining the difference in parities. For example, for $h = 3$ and $w = 8$ we get\n\\[\ns = \\begin{bmatrix}\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} \n\\end{bmatrix},\n\\quad\nt = \\begin{bmatrix}\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} \n\\end{bmatrix}.\n\\]\nFinally, if $h = 3 + 2\\ell$, we can take the above construction for $h = 3$ and take $2\\ell$ copies of the top row, shifting it back and forth to preserve the coloring. For example, for $h = 7$ and $w = 8$ we get\n\\[\ns = \\begin{bmatrix}\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} \\\\\n{\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} \\\\\n{\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{black} 1} & {\\color{blue} 2} \n\\end{bmatrix},\n\\quad\nt = \\begin{bmatrix}\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} \\\\\n{\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} & {\\color{black} 1} & {\\color{blue} 2} \\\\\n{\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} & {\\color{red} 3} & {\\color{black} 1} \\\\\n{\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} & {\\color{blue} 2} & {\\color{red} 3} \n\\end{bmatrix}.\n\\]\nThis way we preserve the basic topological structure, with a sequence of $3$s wrapping around vertically twice in $s$ and zero times in $t$. Note that in $s$ we will get $2\\ell$ new type-A tiles (as there are $2$ nodes of color $3$ in the top row) and in $t$ we will get $0$ new type-A tiles, again preserving the parity difference. This concludes the proof of Theorem~\\ref{thm:grids}.\n\n\n\n\n\\section{Simple corollaries}\\label{sec:simplecor}\n\n\\begin{lem}\\label{lem:MISplusforest}\nAssume that we are given a graph $G$ and input and target colorings with $k \\ge 3$ colors. Assume that in $O(f(n))$ rounds we can find an independent set $I$ of $G$ such that $V \\setminus I$ induces a forest of trees of depth at most $O(d(n))$. Then in $O(f(n)+d(n))$ rounds we can solve $k+1$ recoloring, with a schedule of length $O(d(n))$.\n\\end{lem}\n\\begin{proof}\nEach node in $I$ switches to color $k+1$. We then use the algorithm described in the proof of Lemma \\ref{lem:recoltree} to find a recoloring with schedule of length $O(d(n))$ for each connected component after the removal of $I$. After that, each node of $I$ can switch to its final color.\n\\end{proof}\n\\subparagraph{Acknowledgments.}\n\nThe authors would like to thank Nicolas Bousquet for helpful discussions regarding the proof of Lemma~\\ref{lem:thereisthetaor2}.\n\n\\input{results.tex}\n\n\\clearpage\n\n\\bibliographystyle{plainurl}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this note, we consider two problems related to directed polymers with a localization property.\nDirected $d+1$-dimensional polymers are modeled by random walks in a random potential \n$\\phi:\\mathbb{Z}^d\\times\\mathbb{Z}\\times \\Omega\\to\\mathbb{R}$\ndefined for all times $n\\in\\mathbb{Z}$ on the lattice $\\mathbb{Z}^d$ via\n\\begin{equation}\n\\phi_n(x)=\\phi_n(x,\\omega)=V(x)B_n(\\omega).\n\\label{eq:definition_of_potential}\n\\end{equation}\nHere $V:\\mathbb{Z}^d\\to\\mathbb{R}$ is a deterministic bounded function, and $B_n(\\omega)$ is a sequence of\ni.i.d. random variables defined on a probability space $(\\Omega,\\mathcal{F},\\mathsf{P})$ and \ntaking values $\\pm1$ with probabilities $1\/2$. It is convenient\nto suppose that $\\Omega=\\{-1,1\\}^\\mathbb{Z}$, $\\mathcal{F}$ is the cylindric $\\sigma$-algebra, and $\\mathsf{P}$ is the\n$1\/2$-Bernoulli product measure on $\\Omega$. The random variables $B_n$ coincide with coordinate\nmaps: $B_n(\\omega)=\\omega_n$ for $\\omega\\in\\Omega, n\\in\\mathbb{Z}$. \n\nFor any integers $n_1\\le n_2$ the set of {\\it paths} that we consider consists of all possible trajectories\nof the so called ``lazy random walk''. Namely, we deal only with paths\n $\\gamma:[n_1,n_2]\\cap\\mathbb{Z}\\to \\mathbb{Z}^d$ such that\nfor any $n$, $\\gamma_{n+1}$ equals either~$\\gamma_n$ or one of the~$2d$ neighbors of $\\gamma_n$ in $\\mathbb{Z}^d$, i.e., $|\\gamma_{n+1}-\\gamma_n|\\le 1$. \n\n\nWe\nstudy two main objects based on paths in the random potential $\\phi$. First, we\nuse the random potential~$\\phi$ and averaging over paths to define a parabolic model via a random Feynman--Kac evolution operator. We show that \nfor the resulting cocycle, there is a unique positive cocycle eigenfunction serving as a forward and pullback attractor. \nSecondly, we use~$\\phi$ to define a Gibbs specification on paths for any bounded time interval in the\nusual way and study the thermodynamic limit and a.s.-existence and uniqueness of an infinite volume Gibbs measure.  \nBoth main results claim that the local structure of interaction leads to a unique macroscopic object for almost every\nrealization of the random potential.\n\nThe\nanalogues of our results for a setting with a compact spin space replacing $\\mathbb{Z}^d$ are based entirely on elementary\ncontraction arguments. The non-compactness of our spin space $\\mathbb{Z}^d$ is the crucial feature that makes the problem\nmore interesting and difficult. We have to combine the contraction arguments with those based on localization. It is the\nlocalization that makes the evolution of the system essentially compact.\nThe localization arguments we use are similar to those used by Sinai in his paper~\\cite{SinaiMR1317991}\non Nechaev's model. However, the mechanism of localization in Nechaev's model is different from ours.\nThe result on existence and uniqueness of a Gibbs measure is also related to\nthe characterization of stationary Markov chains with finitely many states as Gibbs distributions with zero-range potential, see e.g.~\\cite[Chapter 3]{GeorgiiMR956646}.\n\nLocalization for random walks in a random potential has been studied intensively in the past 10 years, see \n\\cite{Carmona-Hu:MR1939654}, \\cite{Comets-et-al:MR1996276}, \\cite{Comets-et-al:MR2073332}, \\cite{Alexander-Sidoravicius:MR2244428},\n\\cite{Giacomin:MR2380992},\nand references therein. In our setting, the localization properties are based on a specific structure of the spatial\npotential $V$. We assume that it has a large maximum (or minimum) located (without loss of generality) at the origin and has the\nfollowing form: $V(x)=V_0(x)+\\Lambda \\delta_0(x)$. Here $V_0$ is an arbitrary bounded potential and the delta-potential\n$\\delta_0$ takes value 1 at the origin and 0 everywhere else. All the results in this paper are proven under the assumption that\n$\\Lambda$ is large, the exact condition being expressed in terms of $\\max|V_0(x)|$ and the dimension $d$. \nA special case when $V_0(x)\\equiv 0$ was considered in \\cite{Alexander-Sidoravicius:MR2244428},\n\\cite{Giacomin:MR2380992}.\nIt turn out that in dimension one and two localization occurs\nfor an arbitrary $\\Lambda>0$.\n\nIn the theory of directed polymers it is often assumed that the dependence of the random potential on the space variable $x$\nis non-deterministic. Of course, we can also think that $V_0$ is a realization of a bounded stationary potential $V_0^\\omega$. However,\nsince our results hold for all such realizations we fix a particular realization and assume that $V$ is deterministic. \n\n\nIn the physics literature, mostly the model with independent noise values~$B_{x,n}$ for all space-time points $(x,n)$ has been considered. \nIn the model that we suggest it is essential that one realization of the noise serves all lattice points $x$ at once, thus\nmodeling a spatially disordered structure represented by $V$ and embedded into an external potential $B_n$ fluctuating in time. \nThe conditions we impose on the random potential~$\\phi$ can be significantly relaxed though, and the results we prove hold true for a wider class of potentials with localization properties. \n\nFor example, one can consider the following random potential:\n\\[\n \\phi_n(x,\\omega)=\\sum_{i=1}^{m} V_i(x)B_n^{(i)}(\\omega).\n\\]\nHere $(V_i)_{1\\le i\\le m}$ is a collection of bounded potentials and $(B^{(i)})_{1\\le i\\le m}$ is a family of $m$ independent copies of the ``random sign'' process $B_n(\\omega)$. To guarantee the localization it is sufficient to assume that one of the potentials~$V_i$ has a large\nmaximum or minimum at the origin. \n\nIt is also possible to study the continuous time case where $B_n(\\omega)$ is replaced by the white noise\n$\\dot W(t)$. However, this case is technically more complicated and it will be considered in a forthcoming publication. \n\nThe paper is organized as follows. In this section, we describe the setting in detail and state our two main results. \nThe crucial localization lemma is proven in Section~\\ref{sec:main_loc_lemma}, and it is used to prove the main results in\nSections~\\ref{sec:cocycle} (cocycle eigenfunctions) and~\\ref{sec:Gibbs} (infinite volume Gibbs distributions) respectively.\nSection~\\ref{sec:auxi_lemmas} is devoted to auxiliary technical lemmas. \n\n\\subsection{The parabolic model}\n\nThe Feynman--Kac operator associated with $\\phi$ is defined for two integers $n_1<n_2$ and all\nbounded functions $u$ as\n\\begin{align*}\nT^{n_1,n_2}u(x)=(T^{n_1,n_2}(\\omega) u)(x)=\\mathop{\\mathsf E} e^{\\Phi_{n_1,n_2}(\\gamma)} u(\\gamma(n_1)),\n\\end{align*}\nwhere \n\\begin{equation*}\n\\Phi_{n_1,n_2}(\\gamma)=\\Phi_{n_1,n_2}(\\gamma,\\omega)=\\sum_{n=n_1+1}^{n_2} \\phi_n(\\gamma(n),\\omega)\n=\\sum_{n=n_1+1}^{n_2} V(\\gamma(n))\\omega_n,\n\\end{equation*} \nand the expectation $\\mathop{\\mathsf E}$ is taken with respect to the uniform distribution on the space of\ndirected polymer realizations $\\Gamma_{n_1,n_2}(x)$ which is \nthe set of admissible paths $\\gamma:[n_1,n_2]\\to\\mathbb{Z}^d$ with $\\gamma(n_2)=x$, i.e.\n\\begin{equation}\nT^{n_1,n_2}u(x)=\n\\frac{1}{(2d+1)^{n_2-n_1}}\\sum_{\\gamma\\in\\Gamma_{n_1,n_2}(x)} e^{\\Phi_{n_1,n_2}(\\gamma)} u(\\gamma(n_1)). \n\\label{eq:T}\n\\end{equation}\nWe recall that throughout the paper, by {\\it admissible paths} or, simply, {\\it paths} we mean lazy random walk trajectories (see the\nIntroduction), and by \n$[n_1,n_2]$ we always mean a discrete set $\\{n_1,\\ldots,n_2\\}$.\n\n\n\n\n\nIt is easy to see that $T^{n_1,n_2}$ can be considered as a discrete time analogue of the Feynman-Kac kernel for\nthe linear heat equation.\nThe family of operators $T^{n_1,n_2}$ defines a random dynamical system (cocycle) on $L^\\infty=L^\\infty(\\mathbb{Z}^d)$\n(we shall denote the $L^\\infty$ norm by $\\|\\cdot\\|$):\nfor every $\\omega\\in\\Omega$\n\\begin{equation*}\nT^{n_2,n_3}(\\omega)T^{n_1,n_2}(\\omega)=T^{n_1,n_3}(\\omega),\n\\end{equation*} \nor, equivalently,\n\\[\nT^{n_2}(\\theta^{n_1}\\omega)T^{n_1}(\\omega)=T^{n_1+n_2}(\\omega), \n\\]\nwhere $T^n(\\omega)=T^{0,n}(\\omega)$ and the time shift $\\theta^k:\\Omega\\to\\Omega$ is defined via\n\\begin{equation*}\n\\theta^k \\omega(n)=\\omega(k+n),\\quad k,n\\in\\mathbb{Z}.\n\\end{equation*}\n\n\n\nOur goal is to prove a Perron--Frobenius theorem for the cocycle $T$ defined above. The classical notion\nof an eigenfunction has to be modified for the cocycle setting.\n\nWe say that an $L^\\infty$-valued random variable $u^\\omega$ is an eigenfunction for the cocycle $T$\nif with probability 1 and for all $n\\in{\\mathbb N}$, $T^n(\\omega)u^\\omega$ is equal to~$u^{\\theta^n\\omega}$ up to\na random scalar factor. It is sufficient to impose this requirement for $n=1$. Obviously, $u^\\omega$ is defined up to a multiplication by an arbitrary $c(\\omega)\\ne0$. Thus, it\nis convenient to introduce normalized eigenfunctions assuming $\\|u^\\omega\\|=1$. Then\n\\[\n T^n(\\omega)u^{\\omega}=\\kappa_n(\\omega)u^{\\theta^n\\omega},\\quad n\\in{\\mathbb N}.\n\\]\nfor some process $(\\kappa_n(\\omega))_{n\\in{\\mathbb N}}$. When working with normalized functions, it is useful to consider\na normalized version of the cocycle $T^n$ defined by\n\\[\n \\overline T^n(\\omega) \\phi =\\frac{T^n(\\omega) \\phi}{\\|T^n(\\omega) \\phi\\|}.\n\\]\n\n \n\n\n\n\nFor a positive $\\lambda$, an eigenfunction $u^\\omega(x)$ is called $\\lambda$-localized if there is a random variable $c=c(\\omega)>0$ such that with probability 1,\nfor all $x\\in\\mathbb{Z}^d$,\n\\[\n |u^\\omega(x)|\\le c(\\omega)e^{-\\lambda|x|},\n\\]\nwhere $|x|=\\max\\{|x_1|,\\ldots,|x_d|\\}$.\n \n \n\n\n\n\n\n\n\n\n\n\nThroughout the paper we shall require the following conditions on the potential function $V$:\nthere are constants $M_0,M_1$ and $\\lambda_0$\nsuch that \n\\begin{align}\n\\label{eq:conditions_on_V_1}\n&V(0)=M_0,\n\\\\\n&|V(x)|\\le M_1,\\quad x\\ne 0,\n\\label{eq:conditions_on_V_2}\\\\\n&\\lambda_0{=}\\frac12 (M_0-3M_1)-\\ln(2d+1)>0,\n\\label{eq:conditions_on_V_3}\n\\end{align}\nFor our first main theorem we shall need one more condition:\n\\begin{equation}\n\\lambda_1 \\stackrel{def}{=} 2M_1+\\ln(2d+1)<\\lambda_\n\\label{eq:conditions_on_V_4}\n\\end{equation}\n\\begin{theorem}\\label{thm:perron-frobenius-cocycle} \nSuppose conditions \\eqref{eq:conditions_on_V_1}---\\eqref{eq:conditions_on_V_4} on $V$ are satisfied.\nThen there is a unique normalized,  positive  eigenfunction~$u^\\omega$ for the cocycle $T$ satisfying\n\\begin{equation}\n\\label{eq:uniqueness_condition}\n\\limsup_{n\\to-\\infty}u^{\\theta^n\\omega}(0)>0.\n\\end{equation}\nThis eigenfunction is $\\lambda$-localized for any $\\lambda\\in(0,\\lambda_0)$.\n\nMoreover, almost surely,\nfor any nonnegative function $v\\in L^\\infty$ not identically equal to zero,\n\\begin{equation*}\n\\lim_{n\\to\\infty}\\left\\|{u^{\\theta^n\\omega}}-\\overline T^{n}v\\right\\|= 0\\quad\\text{\\rm (forward attraction),}\n\\end{equation*}\nand\n\\begin{equation*}\n\\lim_{n\\to\\infty}\\overline T^{-n,0}v= u^\\omega\\quad\\text{\\rm (pullback attraction)}.\n\\end{equation*}\n\\end{theorem}\nA proof of this result is given in Section~\\ref{sec:cocycle}. \n\n\n\n\n\n\n\n\n\n\\begin{remark} It follows straightforwardly from the ergodicity of the shift operator $\\theta$ that there is a non-random Lyapunov\nexponent $\\lambda_L$ so that with probability 1,\n\\[ \n \\lim_{n\\to\\infty} \\frac{\\ln \\kappa_n(\\omega)}{n}=\\lambda_L.\n\\]\n\\end{remark}\n\n\n\\begin{remark} Assumption \\eqref{eq:conditions_on_V_1} can be replaced by $|V(0)|=M_0$. \nWe require $V(0)$ to be positive\nwithout loss of generality and only to simplify the notations in the proofs. \n\\end{remark}\n\n\\begin{remark} Condition~\\eqref{eq:conditions_on_V_4} may be relaxed if the forward and pullback attraction\nhold for functions $v$ that satisfy certain localization conditions (belong to $L^1$, $L^2$, or have compact support).\n\\end{remark}\n\n\n\n\\subsection{Localized Gibbs distributions}\nLet $\\Gamma_{n_1,n_2}(x_1,x_2)$ be the set of all admissible paths on $[n_1,n_2]$ with fixed endpoints\n$\\gamma(n_1)=x_1,\\ \\gamma(n_2)=x_2$.\n\nWe say that a measure $\\mu$ on $(\\mathbb{Z}^d)^\\mathbb{Z}$ (i.e. on paths $\\alpha:\\mathbb{Z}\\to\\mathbb{Z}^d$) is a Gibbs measure corresponding to\na realization of the potential $\\phi:\\mathbb{Z}\\times\\mathbb{Z}^d\\to\\mathbb{R}$ if it satisfies the DLR condition with\nGibbs specification given by\n\\[\n\\mu_{n_1,n_2}(\\gamma|\\ x_1,x_2)=\\frac{1}{Z_{n_1,n_2}(x_1,x_2)}e^{\\Phi_{n_1,n_2}(\\gamma)},\\quad \\gamma\\in\\Gamma_{n_1,n_2}(x_1,x_2), \n\\]\nwith\n\\begin{equation*}\nZ_{n_1,n_2}(x_1,x_2)=\\sum_{\\gamma\\in\\Gamma_{n_1,n_2}(x_1,x_2)}e^{\\Phi_{n_1,n_2}(\\gamma)}.\n\\end{equation*}\nNamely, for any times $n_1<n_2$ and any points $x_1,x_2\\in\\mathbb{Z}$ \nthe conditional distribution on paths in $\\Gamma_{n_1,n_2}(x_1,x_2)$\ndefined by $\\mu$ \nagrees with this specification: \nfor any path $\\gamma\\in\\Gamma_{n_1,n_2}(x_1,x_2)$, \n\\begin{equation*}\n\\mu\\left\\{\\alpha_{[n_1,n_2]}=\\gamma \\ |\\ \\alpha_{[n_1,n_2]}\\in\\Gamma_{n_1,n_2}(x_1,x_2)\\right\\}=\n\\mu_{n_1,n_2}(\\gamma|\\ x_1,x_2),\n\\end{equation*}\nwhere $\\alpha_{[n_1,n_2]}$ is the restriction of $\\alpha$ on $[n_1,n_2]$.\n\nNotice that $Z_{n_1,n_2}(x_1,x_2)$ can be expressed via the cocycle $T$:\n\\begin{equation}\nZ_{n_1,n_2}(x_1,x_2)=(2d+1)^{n_2-n_1}T^{n_1,n_2}\\delta_{x_1}(x_2),\n\\end{equation}\nwhere\n\\begin{equation*}\n\\delta_{x_1}(x)=\\begin{cases}1,&x=x_1,\\\\ 0,&x=0.\\end{cases}\n\\end{equation*}\n\n\n\n\\begin{theorem}\\label{th:gibbs-localiz} There is a set $\\Omega'\\subset \\Omega$ of probability 1 with the following properties: \n\\begin{enumerate}\n\\item  For every $\\omega\\in\\Omega'$, there is a Gibbs measure $\\mu=\\mu^\\omega$ corresponding to the realization\nof the potential $\\phi$. \n\n\\item For every $\\omega\\in\\Omega'$, the measure $\\mu$ is a unique Gibbs measure with the following property: for every $\\varepsilon>0$\nthere is a number $r_\\varepsilon$ such that\n\\begin{equation}\n\\label{eq:weak_localization_condition}\n \\liminf_{n\\to\\pm\\infty} \\mu\\{\\alpha:\\ |\\alpha_n|>r_\\varepsilon\\}<\\varepsilon.\n\\end{equation}\n\\end{enumerate}\nMoreover, for any $\\lambda\\in(0,\\lambda_0)$, the measure $\\mu$ is $2\\lambda$-localized: there is a random variable $c=c(\\omega)>0$\nsuch that for every $\\omega\\in\\Omega'$, every $n\\in\\mathbb{Z}$,\n\\begin{equation}\n\\label{eq:loc_for_gibbs}\n \\mu\\{\\alpha: \\alpha_n=x\\}< c(\\theta^n \\omega) e^{-2\\lambda |x|},\\quad x\\in\\mathbb{Z}^d.\n\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\n Notice \nthat the exponent in the r.h.s.\\ of~\\eqref{eq:loc_for_gibbs} is $2\\lambda$ rather than~$\\lambda$ that\nappears in the statement of Theorem~\\ref{thm:perron-frobenius-cocycle}. This is due to two-sided estimates that we can use\nin the proof of Theorem~\\ref{th:gibbs-localiz}.\n\\end{remark}\n\n\n\n\n\n\n\n\n\nA proof of Theorem~\\ref{th:gibbs-localiz} is given in Section~\\ref{sec:Gibbs}.\n\n\n\n\n\\section{Main localization lemma: the optimal path vs.\\ entropy}\\label{sec:main_loc_lemma}\n\nThis section is devoted to the estimate that plays the central role in the proofs given in forthcoming sections.\nWe shall need the following ``optimal'' path $\\gamma^*$:\n\\begin{equation}\n\\label{eq:optimal_gamma}\n\\gamma^*_n(\\omega)=\n\\begin{cases}0,& \\omega_n=1\\\\\n             e_1, &\\omega_n=-1.\n\\end{cases}\n\\end{equation}\nwhere $e_1=(1,0,\\ldots,0)\\in\\mathbb{Z}^d$.\nClearly,\n\\[\n \\Phi_{n_1,n_2}(\\gamma^*)\\ge \\sum_{n_1+1}^{n_2}\\xi_m,\n\\]\nwhere the random variables $(\\xi_m)_{m\\in\\mathbb{Z}}$ are defined by\n\\[\n \\xi_m(\\omega)=M_0\\mathbf{1}_{\\{\\omega_m=1\\}}-M_1\\mathbf{1}_{\\{\\omega_m=-1\\}}.\n\\]\n\n\n\n\\begin{lemma}\\label{lm:main_localization_lemma}  Let $\\lambda\\in(0,\\lambda_0)$. Then\nthere are random variables $\\nu^-(\\omega)=\\nu^-(\\ldots,\\omega_{-1},\\omega_0)$ and $\\nu^+(\\omega)=\\nu^+(\\omega_1,\\omega_2,\\ldots)$  such that with probability 1, for any $k\\ge \\nu^-(\\omega)$,\n\\[\ne^{\\Phi_{-k,0}(\\gamma^*)}\\ge e^{\\sum_{m=-k+1}^0\\xi_m} > (2d+1)^k e^{M_1k} e^{\\lambda k},\n\\]\nand for any $k\\ge \\nu^+(\\omega)$,\n\\[\ne^{\\Phi_{0,k}(\\gamma^*)}\\ge e^{\\sum_{m=1}^{k}\\xi_m} > (2d+1)^k  e^{M_1k} e^{\\lambda k}.\n\\]\nIn addition, $\\mathsf{P}\\{\\nu^-(\\omega)=1\\}>0$, and $\\mathsf{P}\\{\\nu^+(\\omega)=1\\}>0$.  \n\\end{lemma}\n\\bpf Both  $\\sum_{m=1}^{k}\\xi_m$ and $\\sum_{m=-k+1}^0\\xi_m$ are\nsums of i.i.d.\\ random variables with mean equal to $(M_0-M_1)\/2$. \nLet\n\\begin{equation}\n\\label{eq:epsilon}\n\\varepsilon=\\frac{M_0-M_1}{2}-M_1-\\ln(2d+1)-\\lambda.\n\\end{equation}\nNotice that $\\varepsilon>0$ due to~\\eqref{eq:conditions_on_V_3}. Now the strong law of large numbers implies the existence of random variables $\\nu^\\pm$\nsuch that for almost every $\\omega$ if $k\\ge\\nu^-(\\omega)$ then \n\\[\n\\sum_{m=-k+1}^0\\xi_m >((M_0-M_1)\/2-\\varepsilon)k.\n\\]\nand if $k\\ge\\nu^+(\\omega)$ then \n\\[\n\\sum_{m=1}^{k}\\xi_m >((M_0-M_1)\/2-\\varepsilon)k.\n\\]\nTherefore, for these values of $k$, respectively,\n\\begin{align*}\n\\frac{1}{(2d+1)^k} \\cdot e^{\\sum_{m=-k+1}^0\\xi_m}>e^{((M_0-M_1)\/2-\\varepsilon - \\ln(2d+1))k}= e^{M_1k} e^{\\lambda k},\\\\\n\\frac{1}{(2d+1)^k} \\cdot e^{\\sum_{m=1}^{k}\\xi_m}>e^{((M_0-M_1)\/2-\\varepsilon - \\ln(2d+1))k}= e^{M_1k} e^{\\lambda k},\n\\end{align*}\nand the proof is complete.\n{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\n\\section{Proof of Theorem~\\ref{thm:perron-frobenius-cocycle}}\\label{sec:cocycle}\n\nOur plan is  to find, for any large value of $r\\in{\\mathbb N}$, a sequence of numbers $(n_i)$  decreasing to $-\\infty$ and such that, $T^{n_{i+1},n_i}$ is essentially a contraction in Hilbert projective metric on functions restricted to $B_r=[-r,r]^d$,\nthe ball of radius $r$ in the  sup-norm $|\\cdot|$ on $\\mathbb{Z}^d$.\n\n\nThe proof relies on several lemmas. The first part of the proof is devoted to finding an invariant and almost compact set \nessentially supporting the dynamics.\n\nWe begin with the main localization result. For any $c\\ge 1$, we denote\n\\[\nF(c)=\\{\\phi:\\mathbb{Z}^d\\to\\mathbb{R}_+:\\ \\|\\phi\\|\\le c\\phi(0)\\}. \n\\]\n\n\\begin{lemma}\\label{lm:radius_loc} Suppose $\\lambda\\in(0,\\lambda_0)$. There is a number $K_1(\\lambda)\\ge 1$ such that if\n$c\\ge1$, $\\phi\\in F(c)$, $|y|\\ge \\nu^-(\\omega)$, and $n\\ge\\nu^-(\\omega)$, then \n\\[\n T^{-n,0}\\phi(y)\\le K_1(\\lambda) (e^{-\\lambda|y|}+ce^{-\\lambda n})T^{-n,0}\\phi(0).\n\\]\n\\end{lemma}\n \nThe proof of this Lemma is given in Section~\\ref{sec:auxi_lemmas}.\n\nFor $\\lambda\\in(0,\\lambda_0)$, $r\\in{\\mathbb N}$, and $c\\ge 1$, we define\n\\[\n n_0(\\lambda,r,c)=\\frac{\\ln c}{\\lambda}+r+1.\n\\]\n\n\\begin{lemma}\\label{lm:initialization_in_the_past} Suppose $\\lambda\\in(0,\\lambda_0)$, $r\\in{\\mathbb N}$, and $c\\ge 1$.\nIf $n>n_0(\\lambda,r,c)$,\n$r\\ge \\nu^-(\\omega)$, and $\\phi\\in F(c)$, then\n\\[\n \\|(T^{-n,0}\\phi)\\mathbf{1}_{B_r^c}\\|\\le 2K_1(\\lambda)e^{-\\lambda (r+1)} T^{-n,0}\\phi(0).\n\\]\n\\end{lemma}\n\nThis lemma is a direct implication of Lemma~\\ref{lm:radius_loc} since its conditions automatically imply $n>\\nu^-(\\omega)$.\n\nFor the rest of the proof we fix $\\lambda\\in(0,\\lambda_0)$. Lemma~\\ref{lm:main_localization_lemma} implies that for any $r\\in{\\mathbb N}$, there is an event $A$ with $\\mathsf{P}(A)>0$\nsuch that for every $\\omega\\in A$ the following conditions hold true:\n\n\\begin{enumerate}\n\\item $\\omega_{1}=\\ldots=\\omega_{r}=1$; \\label{cnd:first_pluses-1.5}\n\\item $\\nu_+(\\theta^{r}\\omega)=1$; \\label{cnd:nu_plus_immediately-1.5}\n\\item $\\omega_{-r+1}=\\omega_{-r+2}=\\ldots=\\omega_{0}=1$; \\label{cnd:terminal_pluses-1.5}\n\\item $\\nu_-(\\theta^{-r}\\omega)=1$. \\label{cnd:nu_minus_immediately-1.5}\n\\end{enumerate}\n\n\n\nTherefore, with probability 1, we can choose a sequence $(n_i)_{i\\in{\\mathbb N}}$ (depending on $\\omega$) decreasing to $-\\infty$ and\nsuch that the following Conditions 1--5 are satisfied for each $i\\in{\\mathbb N}$:\n\\begin{enumerate}\n\\item $\\omega_{n_i+1}=\\ldots=\\omega_{n_i+r}=1$; \\label{cnd:first_pluses-2}\n\\item $\\nu_+(\\theta^{n_i+r}\\omega)=1$; \\label{cnd:nu_plus_immediately-2}\n\\item $\\omega_{n_i-r+1}=\\omega_{n_i-r_0+2}=\\ldots=\\omega_{n_i}=1$; \\label{cnd:terminal_pluses-2}\n\\item $\\nu_-(\\theta^{n_i-r}\\omega)=1$. \\label{cnd:nu_minus_immediately-2}\n\\item $n_{i-1}-n_{i}>2 n_0(\\lambda,r,2K_1(\\lambda))$.\\label{cnd:spacing_in_sequence}\n\\end{enumerate}\nThe sequence can be chosen in a measurable way. Notice that if it satisfies conditions 1--5 for some $r$ it also satisfies the same conditions with $r$ replaced by\nany nonnegative $r'<r$. \n \nFor $r\\in{\\mathbb N}$, we define\n\\[\nG(\\lambda,r)=\\{\\phi\\in F(2K_1(\\lambda)):\\ \\|\\phi\\mathbf{1}_{B_r^c}\\|\\le \\|\\phi \\mathbf{1}_{B_r}\\|\\}.\n\\]\n\n\\begin{lemma}\\label{lm:G_preserved} There is a nonrandom $r_0\\in{\\mathbb N}$ such that  \n\\begin{enumerate}\n \\item \nIf $c\\ge 1$,  $r>r_0$, and $-n<n_i-n_0(\\lambda,r,c)$, then\n\\[\nT^{-n,n_i} F(c)\\subset G(\\lambda,r).\n\\] \n\\item If $r>r_0$, then for any $i\\in{\\mathbb N}$,\n\\[\nT^{n_{i},n_{i-1}} G(\\lambda,r)\\subset G(\\lambda,r). \n\\]\n\\end{enumerate}\n\\end{lemma}\n\n\\bpf The first part of the lemma follows from Lemma~\\ref{lm:initialization_in_the_past}.\nThe second part is a consequence of the first one and Condition~\\ref{cnd:spacing_in_sequence}. {{\\hspace{4 ex} $\\Box$ \\smallskip}} \n\n\n\\begin{lemma}\\label{lm:boundedness_in_Hilbert} For $r\\in{\\mathbb N}$, there is a number $K_2(\\lambda, r)\\ge 1$ such that if\n$\\|\\phi\\mathbf{1}_{B_r^c}\\|\\le \\|\\phi\\mathbf{1}_{B_r}\\|$ then for any $y_1$ and $y_2$ with $|y_1|,|y_2|\\le r$,\nand any $i\\in{\\mathbb N}$,\n\\[\n\\frac{1}{K_2(\\lambda, r)}\\le\\frac{T^{n_i,n_{i-1}}\\phi(y_1)}{T^{n_i,n_{i-1}}\\phi(y_2)}\\le K_2(\\lambda, r).\n \\]\n\\end{lemma}\n\nThe proof is given in Section~\\ref{sec:auxi_lemmas}.\n\n\nWe introduce now\n\\[\n H(\\lambda,r)=\\left\\{\\phi\\in G(\\lambda,r):\\ \\|\\phi\\|=1\\ \\text{and}\\ \\phi(y)\\ge \\frac{1}{K_2(\\lambda,r)}\\ \\text{for}\\ |y|\\le r\\right\\}.\n\\]\n\n\n\\begin{lemma} \\label{lm:image_F(c)subsetH}\n\\begin{enumerate}\\item\nIf $r>r_0$, then for any $i\\in{\\mathbb N}$,\n \\begin{align*} \n\\overline T^{n_i,n_{i-1}} H(\\lambda,r)\\subset H(\\lambda,r).\n\\end{align*}\n\\item\\label{it:image_F(c)subsetH} If $c>0$,  $r>r_0$, $-n<n_i-n_0(\\lambda,r,c)$, then\n\\[\n \\overline T^{-n,n_{i-1}}F(c)\\subset H(\\lambda,r).\n\\]\n\\end{enumerate}\n\\end{lemma}\n\\bpf Lemmas~\\ref{lm:G_preserved} and~\\ref{lm:boundedness_in_Hilbert} imply \n\\[\n\\overline T^{n_i,n_{i-1}} G(\\lambda,r)\\subset H(\\lambda,r),\n\\]\nand the first part of the lemma follows since $H(\\lambda,r)\\subset G(\\lambda,r)$. The second part of the lemma follows from the first one and Lemma~\\ref{lm:G_preserved}.{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\n\n\n\n\n\n\n\n\n\nThe Hilbert projective metric $\\rho_r$ between \ntwo functions $\\phi,\\psi:B_r\\to\\mathbb{R}_+$ is defined by\n\\begin{equation*}\n\\rho_r(\\phi,\\psi)=\\ln\\left(\\max_{|x|\\le r} \\frac{\\phi(x)}{\\psi(x)}\\cdot \\max_{|x|\\le r}\\frac{\\psi(x)}{\\phi(x)}\\right).\n\\end{equation*}\nThe Hilbert metric $\\rho_r$ is not a true metric since it does not distinguish functions\nproportional to each other. However it does define a metric on normalized positive functions defined on $B_r$.\n\nFor a function $\\phi:\\mathbb{Z}^d\\to\\mathbb{Z}_+$, we denote its restriction onto $B_r$ by $\\pi_r\\phi$. \n\nWe notice that $\\pi_r H(\\lambda,r)$ is closed in $\\rho_r$, and ${\\rm diam}_r( H(\\lambda,r))<\\infty$, where for a set $A$ we denote\n\\[{\\rm diam}_r (A)=\\sup\\{\\rho_r(\\pi_r\\phi,\\pi_r\\psi):\\ \\phi,\\psi\\in A\\}.\\]\n\n\n\nThe following is the main contraction estimate:\n\\begin{lemma}\n\\label{lm:contraction_and_correction}  There are numbers $K_3(\\lambda), K_4(\\lambda)$ such that\nfor any $r$, any set $A\\subset H(\\lambda,r)$, and any $i\\ge 2$,\n\\[\n {\\rm diam}_r (T^{n_i,n_{i-1}} A)\\le (1-K_3(\\lambda)e^{-2\\lambda_1 r}){\\rm diam}_r(A)+K_4(\\lambda)e^{-2\\lambda_0r}.\n\\]\n\\end{lemma}\n\nThe proof is given in Section~\\ref{sec:auxi_lemmas}. \n\nWe can apply this estimate recursively along the sequence $(n_i)$. We need the following lemma to make connection to time 0 which is not included in~$(n_i)$:\n\n\\begin{lemma}\\label{lm:terminal} There is a number $K_5(\\lambda)$ such that  for any $r\\in{\\mathbb N}$ and\nsufficiently large $i$, if $\\phi,\\psi\\in F(2K_1(\\lambda))$, \nthen\n\\[\n \\rho_r(\\pi_rT^{n_i,0}\\phi, \\pi_r T^{n_i,0}\\psi) \\le \\rho_r(\\pi_r\\phi,\\pi_r\\psi)+K_5(\\lambda)e^{-\\lambda_2 r},\n\\]\nwhere\n\\[\n\\lambda_2=2(M_0-M_1-\\ln(2d+1))>0.\n\\]\n\\end{lemma}\n\nThe proof is given in Section~\\ref{sec:auxi_lemmas}.\n\n\n\n\n\n\\bpf[Proof of Theorem~\\ref{thm:perron-frobenius-cocycle}] \nLemma~\\ref{lm:image_F(c)subsetH} implies that for sufficiently large $n$,\n\\begin{equation}\n\\label{eq:inclusionTH1}\n T^{-n,0} F(c)\\subset T^{n_{i-1},0}H(\\lambda,r),\n\\end{equation}\nand if $i_1<i_2$,\n\\begin{equation}\n\\label{eq:inclusionTH2}\n T^{n_{i_2},0}H(\\lambda,r)\\subset T^{n_{i_1},0}H(\\lambda,r).\n\\end{equation}\n\nTake any sequence $\\phi=(\\phi_n)_{n\\in{\\mathbb N}}$ in $F(c)$ and consider the sequence $( T^{-n,0}\\phi_n)$. \nRelations~\\eqref{eq:inclusionTH1} and~\\eqref{eq:inclusionTH2} imply that for each $r$, $\\pi_r T^{-n,0}\\phi_n$ are uniformly bounded\nin $n$ and $\\phi$. \nWe consider the all pointwise limit points of $T^{-n,0}\\phi_n$. Since pointwise limit points are not necessarily normalized in the uniform norm,\nwe normalize them and denote the resulting set by $S_\\phi$. We denote by $S$ the union of $S_\\phi$ over all possible\nsequences $\\phi$.   \n\nIt follows from the classical diagonal method that $S$ is not empty. Let us show that $S$, in fact, consists of a single point.\n\nSuppose that on the contrary\n$S$ contains at least two different points $\\psi$ and $\\psi'$. Then there exists $\\bar \\rho>0$ such that\n$\\rho_r(\\pi_r\\psi,\\pi_r\\psi')\\geq \\bar\\rho$ for all large enough $r$. \nSince $\\psi$ and $\\psi'$ are\nlimiting points, there exist two sequences of functions $(\\psi_{k})$ and $(\\psi'_k)$ and two sequences of times\n$(m_k)$ and $(m'_k)$ decaying to $-\\infty$ such that for all large enough $k$,\n\\begin{equation}\n\\rho_r(\\pi_rT^{-m_k,0}\\psi_k,\\pi_rT^{-m'_k,0}\\psi'_{k})\\geq \\bar\\rho\/2. \n\\label{eq:barrho}\n\\end{equation}\nOn the other hand, for any $i$, if $k$ is large enough, then both $T^{-m_k,n_i}\\psi_k$ and $T^{-m'_{k},n_i}\\psi'_k$\nbelong to $F(2K_1(\\lambda))$. Lemmas~\\ref{lm:contraction_and_correction} and~\\ref{lm:terminal} imply that \n\\begin{equation*}\n\\limsup_{i\\to\\infty} {\\rm diam}_r(T^{-n_i,0}F(2K_1(\\lambda)))\\leq d(\\lambda,r)=\\frac{K_4(\\lambda)e^{-2\\lambda_0r}}{K_3(\\lambda)e^{-2\\lambda_1 r}}+K_5(\\lambda)e^{-\\lambda_2 r}.\n\\end{equation*} \nTaking $r$ large enough so that\n$d(\\lambda,r)<\\bar\\rho\/2$, we obtain a contradiction with~\\eqref{eq:barrho}. We conclude that $S_\\infty$ cannot contain two distinct elements. Therefore, $S_\\infty=\\{\\psi_\\infty\\}$ for some $\\phi_\\infty$, and it is easy to see\nthat $\\psi_\\infty$ does not depend on $c$.\n\nWe now set $u^\\omega=\\psi_\\infty$, where $\\psi_\\infty$ is the unique element of $S$. The uniqueness\nabove ensures that $u$ is a positive cocycle eigenfunction satisfying~\\eqref{eq:uniqueness_condition}. It is also obviously unique. The desired localization property follows from Lemma~\\ref{lm:radius_loc}.\nThe pullback attraction follows since for any $\\varepsilon$, we can find $r_0$ and $n_0$ such that\n\\[\n \\|u\\mathbf{1}_{B_{r_0}^c}\\|< \\varepsilon,\n\\]\n\\[\n \\|(\\overline T^{-n,0}v)\\mathbf{1}_{B_{r_0}^c}\\|< \\varepsilon,\\quad n>n_0,\n\\]\nand\n\\[\n \\rho_{r_0}(\\overline T^{-n,0} v,u)<\\varepsilon, \\quad n>n_0.\n\\]\nThe forward attraction is proven similarly.{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem~\\ref{th:gibbs-localiz}}\\label{sec:Gibbs}\nFor a fixed $\\omega$, we say that a measure $\\mu_{n_1,n_2}$ on $\\Gamma_{n_1,n_2}$ is a finite volume Gibbs distribution on $[n_1,n_2]$ for the realization of the potential $\\phi$ if\nfor any points $x_1,x_2$ and any path $\\gamma\\in\\Gamma_{n_1,n_2}(x_1,x_2)$,\n\\[\n \\mu_{n_1,n_2}\\bigl(\\{\\gamma\\}\\ |\\ \\Gamma_{n_1,n_2}(x_1,x_2)\\bigr)=\\frac{e^{\\Phi_{n_1,n_2}(\\gamma)}}{Z_{n_1,n_2}(x_1,x_2)}.\n\\]\n\nLet us introduce $\\nu=\\max\\{\\nu^+,\\nu^-\\}$, where $\\nu^\\pm$ are introduced in Section~\\ref{sec:main_loc_lemma}.\n\nThe proof of the following localization lemma for finite volume Gibbs distributions is given in Section~\\ref{sec:auxi_lemmas}:\n\\begin{lemma}\\label{lm:concentration_for_Gibbs_marginals} For any $\\lambda\\in(0,\\lambda_0)$, there is a constant $C>0$ and a function $N_1:{\\mathbb N}\\to{\\mathbb N}$ with the following property.\nSuppose $\\mu_{n_1,n_2}$ is a finite volume Gibbs measure on an interval $[n_1,n_2]$ for a realization of the potential $\\phi$. If \n$r\\in{\\mathbb N}$ and $n\\in[n_1,n_2]$ satisfy \\[\\mu_{n_1,n_2}\\{\\alpha:\\ \\alpha_{n_1},\\alpha_{n_2}\\in B_r\\}=1,\\] \n\\[\n\\nu(\\theta^n\\omega)<r,\n\\]\n\\[\nn_2-n>N_1(r),\\]\\[n-n_1>N_1(r),\n\\]\nthen\n\\[\n\\mu_{n_1,n_2}\\{\\alpha:\\ |\\alpha_n|>r\\}< C e^{-2 \\lambda r}.\n\\]\n\\end{lemma}\n\n\\medskip\n\nLet us prove the existence first. \nWe fix an $\\omega\\in\\Omega$ and for each $m\\in{\\mathbb N}$ consider a unique measure $\\mu^m$ on $(\\mathbb{Z}^d)^\\mathbb{Z}$ such that \n\\begin{enumerate}\n\\item $\\mu^m\\{\\alpha:\\ \\alpha_k=0\\}=1$ for all $k\\le -m$ and all $k\\ge m$ \n\\item The projection of $\\mu^m$ on $[-m,m]$ is a finite volume Gibbs measure.\n\\end{enumerate}\n\nThe following statement is a direct consequence of Lemma~\\ref{lm:concentration_for_Gibbs_marginals}.\n\n\\begin{lemma}\\label{lm:basic_localization_lemma2} For any $\\lambda\\in(0,\\lambda_0)$, there is a random variable $c(\\omega)$ \nsuch that for almost every $\\omega\\in\\Omega$ and every $r$,\n\\begin{equation*}\n\\mu^m\\{\\alpha: |\\alpha_n|>r\\}\\le c(\\theta^n \\omega)e^{-2\\lambda r}.\n\\end{equation*} \n\\end{lemma}\n\nApplying this result, we conclude that with probability 1, the sequence of measures $(\\mu^m)$ is tight in $(\\mathbb{Z}^d)^{\\mathbb{Z}}$, and, due to the Prokhorov criterion, it contains a weakly convergent subsequence. Denoting the limit of this subsequence by $\\mu$, one can easily verify that\n$\\mu$ is a Gibbs measure satisfying~\\eqref{eq:loc_for_gibbs}.\n\n\n\\medskip\n\nTo prove the uniqueness, we must show that any two Gibbs measures $\\mu$ and $\\mu'$ satisfying~\\eqref{eq:weak_localization_condition} coincide. The plan is as follows. We shall consider a sequence of restrictions of $\\mu$ and $\\mu'$ on intervals $[n_j^-,n_j^+]$, with $n_j^+\\to\\infty$\nand\n$n_j^-\\to -\\infty$ as $j\\to\\infty$. We shall iteratively estimate the proximity of these restrictions to each other in total variation, by showing\n that the restrictions on  $[n_{j-1}^-,n_{j-1}^+]$  are  (up to a small correction) closer to each other than the respective restrictions on $[n_{j}^-,n_{j}^+]$,\nby a multiplicative factor that is less than~1. The multiplier and the correction can be controlled by the\nchoice of the sequences $(n_j^+)$ and $(n_j^-)$.\n\n\n\\begin{lemma}\\label{lm:coupling_minorant} There is a constant $c>0$ and a function $N_2:{\\mathbb N}\\to{\\mathbb N}$ with the following property. Suppose $r\\in{\\mathbb N}$, , and\n\\[\n\\omega_{-N_2(r)+1}=\\omega_{-N_2(r)+2}=\\ldots=\\omega_{N_2(r)}=1.\n\\] If $n>N_2(r)$ and $\\mu$ a finite volume Gibbs measure on $[-n,n]$ such that\n\\[\n\\mu\\{\\gamma:\\ \\gamma_{-N_2(r)},\\gamma_{N_2(r)}\\in B_r\\}=1, \n\\]\n then\n\\[\n \\mu\\{\\gamma:\\ \\gamma_0=0\\}>c \\mu\\{\\gamma:\\ \\gamma_0=x\\},\\quad x\\ne 0. \n\\]\n\\end{lemma}\nThe proof is analogous to that of Lemma~\\ref{lm:radius_loc}.\n\\begin{lemma}\\label{lm:coupling_times} Let $\\lambda\\in(0,\\lambda_0)$. Then for almost every $\\omega\\in\\Omega$ and for every $r>0$,\nthere a doubly infinite sequence $(n_i)_{i\\in\\mathbb{Z}}$ such that \n\\[\\lim_{i\\to+\\infty} n_i=\\infty,\\quad \\lim_{i\\to-\\infty} n_i=-\\infty,\\]  \nand for any $i$, \n\\[\n\\nu^-(\\theta^{-N_2(r)+n_i}\\omega)=1,\n\\]\n\\[\n\\omega_{-N_2(r)+1+n_i}=\\omega_{-N_2(r)+2+n_i}=\\ldots=\\omega_{N_2(r)+n_i}=1,\n\\]\n\\[\n \\nu^+(\\theta^{N_2(r)+n_i}\\omega)=1,\n\\]\nand for all $i$, \n\\[n_{i+1}-n_i>2N_2(r).\\]\n\\end{lemma}\n\\bpf Given $r>0$.\n\\begin{align*}\n \\mathsf{P}\\bigl\\{\\nu^-(\\theta^{-N_2(r)}\\omega)=1;\\ \\omega_{-N_2(r)+1}=\\ldots=\\omega_{N_2(r)}=1;\\nu^+(\\theta^{N_2(r)}\\omega)=1\\bigr\\}\n\\\\=\\mathsf{P}\\{\\nu^-(\\theta^{-N_2(r)}\\omega)=1\\}\\mathsf{P}\\{\\omega_{-N_2(r)+1}=\\omega_{-N_2(r)+2}=\\ldots=\\omega_{N_2(r)}=1\\}\\\\\n\\times \\mathsf{P}\\{\\nu^+(\\theta^{N_2(r)}\\omega)=1\\}>0,\n\\end{align*}\nso that the lemma follows from the Bernoulli property.\n{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\nNow we return to the proof of the uniqueness in Theorem~\\ref{th:gibbs-localiz}.\nFor a measure $\\mu$ on $(\\mathbb{Z}^d)^\\mathbb{Z}$ and any set $S\\subset\\mathbb{Z}$ we denote by $\\mu_S$  the measure induced by $\\mu$ on paths restricted to $S$.\n\nLet now $\\mu$ and $\\mu'$ be two Gibbs measures satisfying~\\eqref{eq:weak_localization_condition} for a given $\\omega$.\n\n\nWe have to show that for any $l>0$, the distributions induced by $\\mu$ and $\\mu'$ on trajectories defined on $[-l,l]$ coincide: $\\mu_{[-l,l]}=\\mu'_{[-l,l]}$. Since $\\mu$ and $\\mu'$ are Gibbs measures with nearest neighbor interaction,  it is sufficient to\ncheck that the two-dimensional boundary distributions coincide: $\\mu_{\\{-l,l\\}}=\\mu'_{\\{-l,l\\}}$.\n\n\nWe fix an arbitrary $\\varepsilon>0$ and use~\\eqref{eq:weak_localization_condition} to find $r_\\varepsilon>0$ and sequences $(m_k)_{k\\in\\mathbb{Z}}$, $(m'_k)_{k\\in\\mathbb{Z}}$ \nsuch that\n\\[\n \\mu\\bigl\\{\\alpha:\\ |\\alpha_{m_k}|>r_\\varepsilon\\bigr\\}<\\varepsilon,\\quad k\\in\\mathbb{Z},\n\\]\nand\n\\[\n \\mu'\\bigl\\{\\alpha:\\ |\\alpha_{m_k}|>r_\\varepsilon\\bigr\\}<\\varepsilon,\\quad k\\in\\mathbb{Z}.\n\\]\nFor $r\\ge r_\\varepsilon$ and $k\\in{\\mathbb N}$ consider measures $\\mu^{k,r}$ and $\\mu'^{k,r}$ obtained from\n$\\mu$ and $\\mu'$ by conditioning, respectively, on $\\{\\alpha_{m^\\pm_k}\\le r\\}$ and $\\{\\alpha_{m'^\\pm_k}\\le r\\}$. \nDue to the arbitrary choice of $\\varepsilon$, it is sufficient to show that the total variation distance\nbetween $\\mu^{k,r}_{\\{-l,l\\}}$ and $\\mu'^{k,r}_{\\{-l,l\\}}$ can be made arbitrarily small by choosing sufficiently large $r$ and $k$.\n\nLet us fix $r$ and find the sequence $(n_i)_{i\\in\\mathbb{Z}}$ provided by Lemma~\\ref{lm:coupling_times}. For any given $i_0\\in{\\mathbb N}$, one can find\n$k$ such that \\[n_{i_0}+N(r)<\\min\\{m_k,m'_k\\}\\] and \\[n_{-i_0}-N(r)>\\max\\{m_{-k},m'_{-k}\\}.\\]   \nLemma~\\ref{lm:concentration_for_Gibbs_marginals} implies that if $|i|<i_0$, then\n\\begin{align}\n\\mu^{k,r}\\{\\alpha:\\ |\\alpha_{n_i}|>r\\}< C e^{-2 \\lambda r},\\label{eq:loc_gibbs1}\\\\\n\\mu^{k,r}\\{\\alpha:\\ |\\alpha_{n_i+N_2(r)}|>r\\}< C e^{-2 \\lambda r},\\label{eq:loc_gibbs2}\\\\\n\\mu^{k,r}\\{\\alpha:\\ |\\alpha_{n_i-N_2(r)}|>r\\}< C e^{-2 \\lambda r},\\label{eq:loc_gibbs3}\n\\end{align}\nand same estimates hold for $\\mu'^{k,r}$.\nFor $0<i<i_0$, let us denote\n\\[\n \\mu_i=\\mu_{\\{n_{-i},n_{i}\\}}^{k,r},\\quad \\mu'_i=\\mu'^{k,r}_{\\{n_{-i},n_{i}\\}}.\n\\]\nWe are going to estimate the total variation distance $d_{TV}(\\mu_{i-1},\\mu'_{i-1})$ via $d_{TV}(\\mu_{i},\\mu'_{i})$.\n\n\nFor any $i$ we introduce $\\sigma_i$ to be the maximal measure minorizing both $\\mu_i$ and $\\mu'_i$ and concentrated on $B_r\\times B_r$: \n\\begin{align*}\n\\mu_i&=\\sigma_i+\\rho_i+\\delta_i\\\\\n\\mu'_i&=\\sigma_i+\\rho'_i+\\delta'_i,\n\\end{align*}\nHere $\\rho_i$ and $\\rho'_i$ are mutually singular measures on $B_r\\times B_r$, and measures $\\delta_i,\\delta'_i$ are supported on\n$\\mathbb{Z}^d\\times \\mathbb{Z}^d\\setminus B_r\\times B_r$.\nWe have\n\\[\nd_{TV}(\\mu_{i},\\mu'_{i})\\ge\\rho_i(B_r),\n\\]\nand, due to~\\eqref{eq:loc_gibbs2},\\eqref{eq:loc_gibbs3},\n\\begin{align*}\n \\delta_i(\\mathbb{Z}^d\\times\\mathbb{Z}^d)\\le 2Ce^{-2\\lambda r},\\\\\n\\delta'_i(\\mathbb{Z}^d\\times\\mathbb{Z}^d)\\le 2Ce^{-2\\lambda r}.\n\\end{align*}\n\nCombining a basic coupling estimate based on Lemma~\\ref{lm:coupling_minorant}, the fact that there are $(2r+1)^{2}$ points in $B_r$,\nand the estimate~\\eqref{eq:loc_gibbs1},\nwe obtain:\n\\[\nd_{TV}(\\mu_{i-1},\\mu'_{i-1})\\le d_{TV}(\\mu_{i},\\mu'_{i})\\left(1-\\left(\\frac{c}{(2r+1)^d-1+c}\\right)^2\\right)+2Ce^{-2\\lambda r}.\n\\]\n\nSince the total variation distance is always bounded by 1,  applying this estimate iteratively, we obtain\n\\[\nd_{TV}(\\mu_i,\\mu'_i)< 2\\frac{C}{c^2}e^{-2\\lambda r}((2r+1)^d-1+c)^2,\\quad i\\in{\\mathbb N}.\n\\]\nSince the choice of $r$ is arbitrary and the r.h.s.\\ converges to 0 as $r\\to\\infty$, the proof is completed.{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proofs of auxiliary lemmas}\\label{sec:auxi_lemmas}\n\\bpf[Proof of Lemma \\ref{lm:radius_loc}]\nLet us write\n\\begin{align}\\notag\n(2d+1)^{n}T^{-n,0}\\phi(y)&\\le \\sum_{k=|y|}^{n}\\sum_x \\phi(x) Z_{-n,-k}(x,0) \\tilde Z_{-k,0}(0,y)+\\bar Z_{-n,0}(y)\\|\\phi\\|\\\\\n&\\le \\sum_{k=|y|}^{n} I_k +\\bar I.\\label{eq:decomposition_in_I_k}\n\\end{align}\nHere $\\tilde Z_{-k,0}(0,y)$ is the partition function over all paths on $[-k,0]$ connecting $0$ to $y$ and avoiding $0$ after time $-k$;\n$\\bar Z_{-n,0}(y)$ is the partition function over paths on $[-n,0]$ ending up at $y$ and never visiting $0$.\n \nConsidering all paths that coincide with $\\gamma^*$ on $[-k+1,0]$, we can write \n\\[\n(2d+1)^{n}T^{-n,0}\\phi(0)\\ge  \\sum_{x}\\phi(x)Z_{-n,-k}(x,0)e^{\\sum_{m=-k+1}^{0}\\xi_m}e^{-(M_0+M_1)},\n\\]\nwhere the factor of $e^{-(M_0+M_1)}$ appears since it is possible that  $\\gamma^*(0)\\ne 0$. \nWe also notice that\n\\[\n\\tilde Z_{-k,0}(0,y)\\le (2d+1)^{k}e^{M_1k} \n\\]\nsince there are at most $(2d+1)^{k}$ paths contributing to $\\tilde Z_{-k,0}(0,y)$ and each path contributes at most $e^{M_1k}$. \nCombining these two estimates we get\n\\begin{align}\\notag\n I_k&\\le e^{M_0+M_1}\\frac{(2d+1)^{k}e^{M_1k}}{e^{\\sum_{m=-k+1}^{0}\\xi_m}}(2d+1)^nT^{-n,0}\\phi(0)\\\\\n&\\le e^{M_0+M_1} e^{-\\lambda k}(2d+1)^nT^{-n,0}\\phi(0),\n\\label{eq:I_k-improved}\n\\end{align}\nwhere the second inequality follows from Lemma~\\ref{lm:main_localization_lemma} and condition $k\\ge|y|\\ge \\nu^-(\\omega)$.\nSimilarly, for $\\bar I$ we get:\n\\begin{align*}\n\\bar I=\\bar Z_{-n,0}(y)\\|\\phi\\|\\le (2d+1)^n e^{M_1n}\\|\\phi\\|\\le (2d+1)^n e^{M_1n}c\\phi(0). \n\\end{align*}\nOn the other hand,\n\\begin{align*}\n(2d+1)^n T^{-n,0}\\phi(0)&\\ge \\phi(0) e^{\\sum_{m=-n+1}^{0}\\xi_m}e^{-(M_0+M_1)}\n\\\\ &\\ge \\phi(0) (2d+1)^{n}e^{M_1n}e^{\\lambda n}e^{-(M_0+M_1)},\n\\end{align*}\nwhere the second inequality follows from Lemma~\\ref{lm:main_localization_lemma} and condition $n\\ge \\nu^-(\\omega)$.\nCombining these estimates, we get\n\\begin{align}\\notag\n \\bar I\n&\\le (2d+1)^n e^{M_1n}c \\frac{(2d+1)^n T^{-n,0}\\phi(0)}{(2d+1)^{n}e^{M_1n}e^{\\lambda n}e^{-(M_0+M_1)}}\\\\\n&\\le c e^{M_0+M_1} e^{-\\lambda n}(2d+1)^n T^{-n,0}\\phi(0).\\label{eq:barI}\n\\end{align}\nThe lemma now follows from~\\eqref{eq:decomposition_in_I_k},\\eqref{eq:I_k-improved},and \\eqref{eq:barI}.\n{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\\bpf[Proof of Lemma~\\ref{lm:boundedness_in_Hilbert}] \nIt is sufficient to prove the upper bound, so we write\n\\begin{align}\\notag\n(2d+1)^{n_{i-1}-n_i} &T^{n_i,n_{i-1}}\\phi(y_1)\\\\ \\le &\\left(\\sum_{n_i+r+1\\le k\\le l\\le n_{i-1}-r} Z_{n_i,n_{i-1}}^{k,l}(y_1) + \\bar Z_{n_i,n_{i-1}}(y_1)\\right)\\|\\phi\\|,\n\\label{eq:decomp_Z_again-for-boundedness-in-hilbert-norm}\n\\end{align}\nwhere $Z_{n_i,n_{i-1}}^{k,l}(y_1)$ is the partition function over all paths $\\gamma$ defined on interval~$[n_i,n_{i-1}]$ that terminate at~$y_1$ and satisfy\n\\[\\min\\{m> n_i+r: \\gamma(m)=0\\}=k,\\]\n\\[\\max\\{m\\le n_{i-1}-r: \\gamma(m)=0\\}=l,\\]\nand $\\bar Z_{n_i,n_{i-1}}(y_1)$ is the partition function over all paths that terminate at~$y_1$ and do not visit 0 between $r$\nand $n-r$.\nWe have\n\\begin{align*}\n\\notag\nZ_{n_i,n_{i-1}}^{k,l}(y_1)\\le& (2d+1)^{r}e^{M_0r}(2d+1)^{k-(n_i+r)} e^{M_1(k-(n_i+r)-1)}e^{M_0}\\\\&\\times Z_{k,l}(0,0)(2d+1)^{n_{i-1}-r-l}e^{M_1(n_{i-1}-r-l)} (2d+1)^{r}e^{M_0r},\n\\end{align*}\nand, considering a point $x^*$ such that $|x^*|\\le r$ and $\\phi(x^*)=\\|\\phi\\|$,\n\\begin{align}\n\\notag\n(2d&+1)^{n_{i-1}-n_i} T^{n_i,n_{i-1}}\\phi(y_2)\\\\ &\\ge \\notag e^{-M_0r}e^{\\sum_{m=n_i+r+1}^{k-1}\\xi_m}e^{-M_0}Z_{k,l}(0,0)e^{\\sum_{m=l+1}^{n_{i-1}-r}\\xi_m}e^{-M_0r}\\phi(x^*),\n\\end{align}\nso that\n\\begin{multline\n\\frac{Z_{n_i,n_{i-1}}^{k,l}(y)\\|\\phi\\|}{(2d+1)^{n_i-n_{i-1}} T^{n_i,n_{i-1}}\\phi(y_2)}\\\\ \\le (2d+1)^{2r+2}e^{M_0(4r+2)}e^{-\\lambda(k-(n_i+r)-1)} e^{-\\lambda(n-r-l)}. \\label{eq:Z_four_indices_lower_bound}\n\\end{multline}\nSince\n\\[\n \\sum_{n_i+r+1\\le k\\le l\\le n_{i-1}-r}e^{-\\lambda(k-(n_i+r)-1)} e^{-\\lambda(n-r-l)}<\\infty,\n\\]\nthe lemma follows from inequality~\\eqref{eq:Z_four_indices_lower_bound} combined with an analogous estimate for $\\bar Z_{n_i,n_{i-1}}(y_1)$ \nand~\\eqref{eq:decomp_Z_again-for-boundedness-in-hilbert-norm}.\n{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\nNow we begin preparations for the proof of the main contraction estimate, Lemma~\\ref{lm:contraction_and_correction}.\nFor any $i$ and $r$, we define truncated operators\n\\begin{align*}\nT_r^{n_i,n_{i-1}} u(y)&=\\frac{1}{(2d+1)^{n_{i-1}-n_i}}\\sum_{|x|\\le r}Z_{n_i,n_{i-1}}(x,y)u(x)=T^{n_i,n_{i-1}}(u\\mathbf{1}_{B_r})(y),\\\\\n\\hat T^{n_i,n_{i-1}} u(y)&=\\frac{1}{(2d+1)^{n_{i-1}-n_i}}\\sum_{|x|> r}Z_{n_i,n_{i-1}}(x,y)u(x)=T^{n_i,n_{i-1}}(u\\mathbf{1}_{B_r^c})(y).\n\\end{align*}\nThe restriction of $T^{n_i,n_{i-1}}$ on $B_r$ can be viewed as a linear finite-dimensional operator in $\\mathbb{R}^{B_r}$\ngiven by a matrix   $(2d+1)^{n_{i}-n_{i-1}}Z_{n_i,n_{i-1}}(x,y)$, $x,y\\in B_r$, with positive entries.\nTherefore, we can apply a classical estimate on contraction in Hilbert metric, see e.g.,\nTheorem 3.12 in \\cite{SenetaMR719544}: for any functions $\\phi,\\psi:\\mathbb{Z}^d\\to\\mathbb{R}_+$,\n\\begin{equation}\n\\rho_r(T_r^{n_i,n_{i-1}}\\phi,T_r^{n_i,n_{i-1}}\\psi) \\le \\frac{1-\\sqrt{L^{n_i,n_{i-1}}_r}}{1+\\sqrt{L^{n_i,n_{i-1}}_r}}\\rho_{r}(\\phi,\\psi),\n\\label{eq:contraction_via_L}\n\\end{equation}\nwhere\n\\begin{equation*}\nL^{n_i,n_{i-1}}_r=\\min_{|x_1|,|x_2|,|y_1|,|y_2|\\le r}\\left(\\frac{Z_{n_i,n_{i-1}}(x_1,y_1)}{Z_{n_i,n_{i-1}}(x_2,y_1)}\\cdot \n\\frac{Z_{n_i,n_{i-1}}(x_2,y_2)}{Z_{n_i,n_{i-1}}(x_1,y_2)}\\right).\n\\end{equation*}\n\n\\begin{lemma}\\label{lm:contraction_coef} There is a positive constant $K_6(\\lambda)$ such that\nfor any $i$ and $r$,\n\\[\nL^{n_i,n_{i-1}}_r\\ge K_6(\\lambda) e^{-4\\lambda_1 r}\n\\] \n\\end{lemma}\n\n \n\nSince the dynamics of the actual system is not restricted to $B_r$ we have to estimate the influence of $B_r^c$:\n\\begin{lemma}\\label{lm:influx} There is a positive number $K_7(\\lambda)$ such that if \n$|y|\\le r$, then\n\\[\n \\sum_{x:|x|>r} Z_{n_i,n_{i-1}}(x,y)\\le K_7(\\lambda)e^{-2\\lambda_0 r} Z_{n_i,n_{i-1}}(0,y).\n\\]\n\\end{lemma}\n\nWe postpone the proof of  Lemmas~\\ref{lm:contraction_coef} and~\\ref{lm:influx}  till the end of this section.\n\n\\bpf[Proof of Lemma~\\ref{lm:contraction_and_correction}] Lemma~\\ref{lm:contraction_coef} and estimate \\eqref{eq:contraction_via_L} imply\nthat for any positive functions $\\phi$ and $\\psi$,\n\\begin{equation}\n\\label{eq:contraction_constants}\n\\rho_r(T_r^{n_i,n_{i-1}}\\phi,T_r^{n_i,n_{i-1}}\\psi)<\\left(1-\\sqrt{K_6(\\lambda)}e^{-2\\lambda_1}\\right) \\rho_r(\\phi,\\psi).\n\\end{equation}\nTo estimate the full untruncated operators, we write:\n\\begin{align}\\label{eq:hilbert-metric}\n\\rho_r(T^{n_i,n_{i-1}}\\phi,T^{n_i,n_{i-1}}\\psi)=&\n\\ln\\left(\\max_{|y|\\le r} \\frac{T^{n_i,n_{i-1}}\\phi(y)}{T^{n_i,n_{i-1}}\\psi(y)}\\cdot \n\\max_{|y|\\le r}\\frac{T^{n_i,n_{i-1}}\\psi(y)}{T^{n_i,n_{i-1}}\\phi(y)}\\right)\n\\\\\\notag \\le& \n\\ln\\left(\\max_{|y|\\le r} \\frac{T_r^{n_i,n_{i-1}}\\phi(y)+\\hat T_r^{n_i,n_{i-1}}\\phi(y)}{T_r^{n_i,n_{i-1}}\\psi(y)}\\right)\n\\\\ \\notag &+\\ln\\left(\n\\max_{|y|\\le r}\\frac{T_r^{n_i,n_{i-1}}\\psi(y)+\\hat T_r^{n_i,n_{i-1}}\\psi(y)}{T_r^{n_i,n_{i-1}}\\phi(y)}\\right)\n\\end{align}\nLemma \\ref{lm:influx} implies\n\\begin{align*}\n\\hat T_r^{n_i,n_{i-1}}\\phi(y)&\\le \\frac{1}{(2d+1)^{n_{i-1}-n_i}}K_7(\\lambda)e^{-2\\lambda_0 r} Z_{n_i,n_{i-1}}(0,y)\\|\\phi\\mathbf{1}_{B_r}\\|\\\\\n&\\le  K_7(\\lambda)e^{-2\\lambda_0 r} \\frac{1}{(2d+1)^{n_{i-1}-n_i}} Z_{n_i,n_{i-1}}(0,y)\\cdot 2K_1(\\lambda)\\phi(0)\\\\\n&\\le 2K_1(\\lambda) K_7(\\lambda)e^{-2\\lambda_0 r} T_r^{n_i,n_{i-1}}\\phi(y).\n\\end{align*}\nAnalogously,\n\\begin{align*}\n\\hat T_r^{n_i,n_{i-1}}\\psi(y)&\\le 2K_1(\\lambda) K_7(\\lambda)e^{-2\\lambda_0 r} T_r^{n_i,n_{i-1}}\\psi(y).\n\\end{align*}\nPlugging the last two inequalities into~\\eqref{eq:hilbert-metric}, we get \n\\begin{align*}\n\\rho_r(T_r^{n_i,n_{i-1}}\\phi,T_r^{n_i,n_{i-1}}\\psi) &\\le \n\\ln\\left(\\max_{|y|\\le r} \\frac{T_r^{n_i,n_{i-1}}\\phi(y)(1+2K_1(\\lambda)K_7(\\lambda)e^{-2\\lambda_0 r})}{T_r^{n_i,n_{i-1}}\\psi(y)}\\right) \n\\\\&+\\ln\\left(\\max_{|y|\\le r}\\frac{T_r^{n_i,n_{i-1}}\\psi(y)(1+2K_1(\\lambda)K_7(\\lambda)e^{-2\\lambda_0 r})}{T_r^{n_i,n_{i-1}}\\phi(y)}\\right)\\\\ \n&\\le \\rho_r(T_r^{n_i,n_{i-1}}\\phi,T_r^{n_i,n_{i-1}}\\psi)+ 4K_1(\\lambda)K_7(\\lambda)e^{-2\\lambda_0 r}\n\\end{align*}\nThis estimate along with~\\eqref{eq:contraction_constants} implies the lemma with $K_3(\\lambda)=\\sqrt{K_6(\\lambda)}$ and $K_4(\\lambda)=4K_1(\\lambda)K_7(\\lambda)$.{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\n\n\n\n\n\n\n\n\n\\bpf[Proof of Lemma~\\ref{lm:contraction_coef}]\nLet us estimate the ratios in the r.h.s.\\ of the definition of $L^{n_i,n_{i-1}}_r$.\nFor $n_i+r< k\\le l\\le n-r$, we define $Z_{n_i,n_{i-1}}^{k,l}(x,y)$ as the partition function over all paths $\\gamma$ on $[n_i,n_{i-1}]$ connecting $x$ to $y$\nsatisfying \n\\[\\min\\{m> n_i+r:\\ \\gamma(m)=0\\}=k,\\]\n\\[\\max\\{m\\le n_{i-1}-r:\\ \\gamma(m)=0\\}=l,\\]\nwe also define $\\bar Z_{n_i,n_{i-1}}(x,y)$ to be the partition function over all paths on $[n_i,n_{i-1}]$ connecting $x$ to $y$ that do not visit~0 between $n_{i}+r+1$ and $n_{i-1}-r$, so that\n\\begin{equation}\\label{eq:decompositionwrttoching_zero}\nZ_{n_i,n_{i-1}}(x,y)=\\sum_{n_i+r < k \\le l \\le n_{i-1}-r} Z_{n_i,n_{i-1}}^{k,l}(x,y) + \\bar Z_{n_i,n_{i-1}}(x,y).\n\\end{equation}\nFor $n_i+r< k\\le l\\le n_{i-1}-r$ and any points $x,y$ with $|x|,|y|\\le r$,\n\\begin{align}\\notag\nZ_{n_i,n_{i-1}}^{k,l}(x,y)\\le &(2d+1)^{|x|-1}e^{M_1(|x|-1)}(2d+1)^{r-|x|+1}e^{M_0(r-|x|+1)}\\\\ \\notag\n&\\times (2d+1)^{k-1-(n_i+r)}e^{M_1(k-1-(n_i+r))}e^{M_0}Z_{k,l}(0,0)\\\\\\notag&\\times (2d+1)^{n-r-l}e^{M_1(n-r-l)}\\\\\n&\\times (2d+1)^{r-|y|}e^{M_0(r-|y|)}(2d+1)^{|y|}e^{M_1|y|}.\\label{eq:sum_for_touching_0}\n\\end{align}\n\nOn the other hand, considering all paths that start at $x$, go straight to~$0$ (which takes $|x|$ steps),\nstay at $0$ until time $n_i+r$ accumulating $M_0$ at each time step, follow the optimal path $\\gamma^*$ from\n$n_i+r+1$ to $k-1$ (if $k=n_i+r+1$ this part is empty), at $k$ visit $0$, return to~$0$ at time $l$, follow $\\gamma^*$ up to $n_{i-1}-r$, stay at $0$ up to\n$n_{i-1}-|y|$, go straight to $y$ where they terminate at time $n_{i-1}$, \n\\begin{align}\n\\notag Z_{n_i,n_{i-1}}(x,y)\\ge &e^{-M_1(|x|-1)}e^{M_0(r-|x|+1)}e^{\\sum_{m=n_i+r+1}^{k-1} \\xi_m} e^{-M_0}\\\\ &\\times Z_{k,l}(0,0)\ne^{\\sum_{m=l+1}^{n_{i-1}-r}\\xi_m} e^{M_0(r-|y|)}e^{-M_1|y|}.\\label{eq:partition_along_0}\n\\end{align}\nThe definitions of $\\nu^+$ and $\\nu^-$ from Lemma~\\ref{lm:main_localization_lemma} imply that,\ndue to our assumptions on $n_i$ and $n_{i-1}$,\n\\begin{equation}\n\\label{eq:lln-consequence-1}\n e^{\\sum_{m=n_i+r+1}^{k-1} \\xi_m}\\ge (2d+1)^{k-1-(n_i+r)}e^{M_1(k-1-(n_i+r))} e^{\\lambda (k-1-(n_i+r))},\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:lln-consequence-2}\n e^{\\sum_{m=l+1}^{n_{i-1}-r} \\xi_m}\\ge (2d+1)^{n_{i-1}-r-l}e^{M_1(n_{i-1}-r-l)}e^{\\lambda(n_{i-1}-r-l)},\n\\end{equation}\nso that\n\\begin{align}\\notag\n\\frac{Z_{n_i,n_{i-1}}^{k,l}(x_2,y_1)}{Z_{0,n}(x_1,y_1)}\n\\le& (2d+1)^{2r+1}e^{M_1(|x_1|+|x_2|)}e^{M_0(|x_1|-|x_2|)}e^{M_0+M_1}\n\\\\&\\times e^{2 M_1|y_1|}e^{-\\lambda((k-r)+(n-r-l)-2)}\\notag\n\\\\\n\\le& (2d+1)^{2r+1}e^{4M_1r}e^{M_0(|x_1|-|x_2|+2)} \\notag\\\\  &\\times e^{-\\lambda((k-1-(n_i-r))+(n_{i-1}-r-l))}. \\label{eq:generic_term_fraction}\n\\end{align}\nFor $\\bar Z_{n_i,n_{i-1}}(x,y)$ we have\n\\begin{align*}\n\\bar Z_{n_i,n_{i-1}}(x,y)\\le &(2d+1)^{|x|-1}e^{M_1(|x|-1)}(2d+1)^{r-|x|+1}e^{M_0(r-|x|+1)}\\\\\n&\\times (2d+1)^{n_{i-1}-n_{i}-2r}e^{M_1(n_{i-1}-n_{i}-2r)} \\\\\n&\\times (2d+1)^{r-|y|}e^{M_0(r-|y|)}(2d+1)^{|y|}e^{M_1|y|},\n\\end{align*}\nand\n\\begin{align*}\nZ_{n_i,n_{i-1}}(x,y)\\ge &e^{-M_1(|x|-1)}e^{M_0(r-|x|+1)}e^{\\sum_{m=n_i+r+1}^{n_{i-1}-r} \\xi_m} e^{M_0(r-|y|)}e^{-M_1|y|},\n\\end{align*}\nso that\n\\begin{align}\n\\frac{\\bar Z_{n_i,n_{i-1}}(x_2,y_1)}{Z_{n_i,n_{i-1}}(x_1,y_1)}\\le &(2d+1)^{2r+1}e^{4M_1r}e^{M_0(|x_1|-|x_2|)}.\\label{eq:terminal_term_fraction}\n\\end{align}\nPlugging \\eqref{eq:generic_term_fraction} and \\eqref{eq:terminal_term_fraction} into~\\eqref{eq:decompositionwrttoching_zero}, we see that\n\\begin{align*}\n\\frac{Z_{n_i,n_{i-1}}(x_2,y_1)}{Z_{n_i,n_{i-1}}(x_1,y_1)}&\\le (2d+1)^{2r+1}e^{4M_1r}e^{M_0(|x_1|-|x_2|)}K_8(\\lambda),\n\\end{align*}\nfor some $K_8(\\lambda)>0$. This inequality and its counterpart with $x_1,x_2$, and, respectively, $y_1$ replaced by $x_2,x_1$, and, respectively, $y_2$, immediately\nimplies:\n\\[\nL^{n_i,n_{i-1}}_r\\ge (2d+1)^{-4r-2}e^{-8M_1r} K^{-2}_8(\\lambda),\n\\]\nand the lemma holds true with $K_6(\\lambda)=(2d+1)^{-2}K^{-2}_8(\\lambda)$.\n{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\\bpf[Proof of Lemma~\\ref{lm:influx}] The following decomposition is analogous to\n~\\eqref{eq:decompositionwrttoching_zero}:\n\\begin{equation}\n\\label{eq:decompositionwrttoching zero2}\n\\sum_{x:|x|>r} Z_{n_i,n_{i-1}}(x,y)\\le \\sum_{n_i+r<k\\le l\\le n_{i-1}-r} Z_{n_i,n_{i-1}}^{k,l}(y)+ \\bar Z_{n_i,n_{i-1}}(y),  \n\\end{equation}\nwhere for $n_i+r<k\\le l\\le n_{i-1}-r$,  $Z_{n_i,n_{i-1}}^{k,l}(y)$ is the partition function over all paths $\\gamma$ on $[n_i,n_{i-1}]$ terminating at $y$ and satisfying \n\\[\\min\\{m:\\ \\gamma(m)=0\\}=k,\\]\n\\[\\max\\{m\\le n_{i-1}-r:\\ \\gamma(m)=0\\}=l,\\]\nand $\\bar Z_{n_i,n_{i-1}}(y)$ is the partition function over all paths terminating at $y$ that do not visit 0 between $n_i$ and $n_{i-1}-r$.\nAnalogously  to \\eqref{eq:sum_for_touching_0},\n\\begin{align}\nZ_{n_i,n_{i-1}}^{k,l}(y)\\le &(2d+1)^{r}e^{M_1r} (2d+1)^{k-1-(n_i+r)}e^{M_1(k-1-(n_i+r))}e^{M_0}\\notag\n\\\\&\\times Z_{k,l}(0,0) (2d+1)^{n_{i-1}-r-l}e^{M_1(n_{i-1}-r-l)}\\notag\\\\\n&\\times (2d+1)^{r-|y|}e^{M_0(r-|y|)}(2d+1)^{|y|}e^{M_1|y|}.\\label{eq:sum_for_touching_0-2}\n\\end{align}\nAnalogously  to \\eqref{eq:partition_along_0},\n\\begin{align}\n\\notag Z_{n_i,n_{i-1}}(0,y)\\ge & e^{M_0 r}e^{\\sum_{m=n_i+r+1}^{k-1} \\xi_m} e^{-M_0}\\\\ &\\times Z_{k,l}(0,0)\ne^{\\sum_{m=l+1}^{n_{i-1}-r}\\xi_m} e^{M_0(r-|y|)}e^{-M_1|y|}.\\label{eq:partition_along_0-2}\n\\end{align}\nDividing \\eqref{eq:sum_for_touching_0-2} by \\eqref{eq:partition_along_0-2} and taking into account \\eqref{eq:lln-consequence-1}, \\eqref{eq:lln-consequence-2}, we obtain:\n\\begin{align}\n \\frac{Z_{n_i,n_{i-1}}^{k,l}(y)}{Z_{n_i,n_{i-1}}(0,y)}&\\le e^{(3M_1-M_0)r}(2d+1)^{2r}e^{2M_0}e^{-\\lambda((k-1-(n_i+r))+(n_{i-1}-r-l))}\\notag\\\\\n&\\le e^{-2\\lambda_0 r}e^{2M_0}e^{-(\\lambda(k-1-(n_i+r))+(n_{i-1}-r-l))}\\label{eq:generic_frac}.\n\\end{align}\nFor $\\bar Z_{n_i,n_{i-1}}(y)$ we have\n\\begin{align*}\n\\bar Z_{n_i,n_{i-1}}(y)\\le &(2d+1)^{r}e^{M_1r}(2d+1)^{n_{i-1}-n_{i}-2r}e^{M_1(n_{i-1}-n_{i}-2r)}(2d+1)^{r}e^{M_1r}\n\\end{align*}\nand\n\\begin{align*}\nZ_{n_i,n_{i-1}}(y)\\ge &e^{M_0 r}e^{\\sum_{m=n_i+r+1}^{n_{i-1}-r} \\xi_m} e^{M_0(r-|y|)}e^{-M_1|y|},\n\\end{align*}\nso that\n\\begin{equation}\n\\frac{\\bar Z_{n_i,n_{i-1}}(y)}{Z_{n_i,n_{i-1}}(0,y)}\\le (2d+1)^{2r}e^{(3M_1-M_0) r} e^{-\\lambda(n_{i-1}-n_{i}-2r)}\\le e^{-2\\lambda_0 r}.\n\\label{eq:special-frac}\n\\end{equation}\nNow the lemma follows from \\eqref{eq:decompositionwrttoching zero2} and \\eqref{eq:generic_frac},\\eqref{eq:special-frac} \n{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\\bpf[Proof of Lemma~\\ref{lm:terminal}]\nProceeding as above, one can use the fact that all paths contributing to $\\hat T_r^{n_i,0}\\phi(y)$ do not visit $0$\nat first $r$ steps, and \nshow that for some $K_9(\\lambda)$ and sufficiently large $i$,\n\\[\n\\hat T_r^{n_i,0}\\phi(y)\\le  K_9(\\lambda)(2d+1)^re^{(M_1-M_0)r} T_r^{0,n}\\phi(y), \n\\]\nwhenever $|y|\\le r$. \nSince \\eqref{eq:hilbert-metric} implies:\n\\begin{align*}\n&\\rho_r(T^{n_i,0}\\phi, T^{n_i,0}\\phi)\\\\ \n\\le& \\ln\\left(\\max_{|y|\\le r} \n\\frac{T_r^{n_i,0}\\phi(y)\\left(1+\\frac{\\hat T_r^{n_i,0}\\phi(y)}{T_r^{n_i,0}\\phi(y)}\\right)}{T_r^{n_i,0}\\psi(y)}\\cdot \n\\max_{|y|\\le r}\\frac{T_r^{n_i,0}\\psi(y)\\left(1+\\frac{\\hat T_r^{n_i,0}\\psi(y)}{T_r^{n_i,0}\\psi(y)}\\right)}{T_r^{n_i,0}\\phi(y)}\\right)\n\\\\\\le& \\rho_r(\\phi,\\psi)+\\max_{|y|\\le r} \\frac{\\hat T_r^{n_i,0}\\psi(y)}{T_r^{n_i,0}\\psi(y)}\\cdot \\max_{|y|\\le r} \\frac{\\hat T_r^{n_i,0}\\phi(y)}{T_r^{n_i,0}\\phi(y)},\n\\end{align*}\nthe lemma follows with $K_5(\\lambda)=K_9(\\lambda)^2$.\n{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\n\n\n\\bpf[Proof of Lemma~\\ref{lm:concentration_for_Gibbs_marginals}]\nWe can write\n\\[\n \\mu_{n_1,n_2}\\{|\\alpha_{n}|>r\\}=\\sum_{x_1,x_2\\in B_r}\\mu_{n_1,n_2}\\{\\alpha_{n_1}=x_1,\\alpha_{n_2}=x_2\\}\\frac{\\tilde Z_{n_1,n_2}(x_1,x_2)}{Z_{n_1,n_2}(x_1,x_2)},\n\\]\nwhere $\\tilde Z_{n_1,n_2}(x_1,x_2)$ denotes the partition function over all paths $\\gamma$ defined on $[n_1,n_2]$ with $\\gamma(n_1)=x_1$,\n$\\gamma(n_2)=x_2$, and such that $|\\gamma(n)|>r$.\nFor any $x_1,x_2\\in B_r$,\n\\[\n\\frac{\\tilde Z_{n_1,n_2}(x_1,x_2)}{Z_{n_1,n_2}(x_1,x_2)}\\le \\sum_{\\substack{k_1,k_2:\\\\|x_1|\\le n_1+k_1 \\le n-r\\\\ n+r\\le n_1+k_2 \\le n_2-|x_2|}}\n\\frac{\\tilde Z^{k_1,k_2}_{n_1,n_2}(x_1,x_2)}{Z^{k_1,k_2}_{n_1,n_2}(x_1,x_2)}+ \\frac{\\bar Z_{n_1,n_2}(x_1,x_2)}{Z_{n_1,n_2}(x_1,x_2)},\n\\]\nwhere $\\tilde Z^{k_1,k_2}_{n_1,n_2}(x_1,x_2)$ and $Z^{k_1,k_2}_{n_1,n_2}(x_1,x_2)$ are partition functions taken over paths $\\gamma$\ncontributing to $\\tilde Z_{n_1,n_2}(x_1,x_2)$ and $Z_{n_1,n_2}(x_1,x_2)$ respectively, with the following restriction:\n\\begin{align*}\n \\sup\\{k\\le n:\\ \\gamma_k=0\\}=k_1,\\\\\n \\inf\\{k\\ge n:\\ \\gamma_k=0\\}=k_2,\n\\end{align*}\nand $\\bar Z_{n_1,n_2}(x_1,x_2)$ is defined as the partition function over paths contributing to $\\tilde Z_{n_1,n_2}(x_1,x_2)$ and never visiting the\norigin between $n_1$ and $n_2$. Using Lemma~\\ref{lm:main_localization_lemma} to estimate the contribution of  the optimal path $\\gamma^*$ \nto the denominator,\nwe can write:\n\\begin{align*}\n\\frac{\\tilde Z^{k_1,k_2}_{n_1,n_2}(x_1,x_2)}{Z^{k_1,k_2}_{n_1,n_2}(x_1,x_2)}\n&\\le \\frac{e^{M_1(k_2-k_1)}(2d+1)^{k_2-k_1}}{(2d+1)^{k_2-k_1}\ne^{M_1(k_2-k_1)}e^{\\lambda(k_2-k_1)}e^{-(M_0+M_1)}}\n\\\\&\\le e^{-(M_0+M_1)} e^{-\\lambda(k_2-k_1)}. \n\\end{align*}\nFor the last term, we get:\n\\begin{align*}\n \\frac{\\bar Z_{n_1,n_2}(x_1,x_2)}{Z_{n_1,n_2}(x_1,x_2)}\\le&\\frac{e^{M_1(n_2-n_1)}(2d+1)^{n_2-n_1}}{e^{-M_1|x_1|}e^{-(M_0+M_1)}\n(2d+1)^{n_2-n_1-|x_1|-|x_2|}}\n\\\\ &\\times \\frac{1}{e^{M_1(n_2-n_1-|x_1|-|x_2|)}e^{\\lambda(n_2-n_1-|x_1|-|x_2|)}e^{-M_1|x_2|}}\\\\\n\\le& (2d+1)^{2r} e^{4M_1r}e^{M_0+M_1}e^{-\\lambda(n_2-n_1-2r)}.\n\\end{align*}\nand the lemma follows by combining the estimates above.{{\\hspace{4 ex} $\\Box$ \\smallskip}}\n\n\\section{Acknowledgements} \nThe authors would like to thank Leonid Koralov for reading the manuscript and suggesting several useful corrections.\nThe research of Yuri Bakhtin is partially supported by NSF through CAREER grant DMS-0742424. The research of Konstantin Khanin is partially\nsupported by NSERC.\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe briefly describe the few\nbasic notions of Nielsen fixed point theory(see \\cite{f1}).\nWe assume  $X$ to be a connected, compact\npolyhedron and $f:X\\rightarrow X$ to be a continuous map.\nLet $p:\\tilde{X}\\rightarrow X$ be the universal cover of $X$\nand $\\tilde{f}:\\tilde{X}\\rightarrow \\tilde{X}$ a lifting\nof $f$, i.e. $p\\circ\\tilde{f}=f\\circ p$.\nTwo liftings $\\tilde{f}$ and $\\tilde{f}^\\prime$ are called\n{\\sl conjugate} if there is a $\\gamma\\in\\Gamma\\cong\\pi_1(X)$\nsuch that $\\tilde{f}^\\prime = \\gamma\\circ\\tilde{f}\\circ\\gamma^{-1}$.\nThe subset $p(Fix(\\tilde{f}))\\subset Fix(f)$ is called\n{\\sl the fixed point class of $f$ determined by the lifting class $[\\tilde{f}]$}.Two fixed points $x_0$ and $x_1$ of $f$ belong to the same fixed point class iff  there is a path $c$ from $x_0$ to $x_1$ such that $c \\cong f\\circ c $ (homotopy relative endpoints). This fact can be considered as an equivalent definition of a non-empty fixed point class.\n Every map $f$  has only finitely many non-empty fixed point classes, each a compact  subset of $X$.\nA fixed point class is called {\\sl essential} if its index is nonzero.\nThe number of essential fixed point classes is called the {\\sl Nielsen number}\nof $f$, denoted by $N(f)$.The Nielsen number is always finite.  $N(f)$ is a  homotopy invariant.\nIn the category of compact, connected polyhedra, the Nielsen number\nof a map is, apart from certain exceptional cases,\n equal to the least number of fixed points\n of maps with the same homotopy type as $f$.\\\\\nLet us consider a braid representative of a knot and induced map of\ncorresponding representation variety(see section 2).We prove in section 3\nthat  the Nielsen number of induced map is a invariant under Markov moves and\nso is a knot invariant.\\\\\nThe author came to the  idea that Nielsen number is a  knot  invariant  at the summer  2003, after conversations with Jochen Kroll and Uwe Kaiser.\nThe author would like to thank the Max-Planck-Institute f\\\"ur Mathematik, Bonn\nfor kind hospitality and support.\n\n\n\\section{ Casson-Lin invariant}\n\n\nWe recall firstly the  Lin's construction in \\cite{lin}\nfor the\nintersection number of the representation spaces corresponding to a braid\nrepresentative of a knot $K$ in $S^3$. \nLet $(S^3, D^3_+, D^3_-, S^2)$ be a Heegaard decomposition of $S^3$ with genus\n$0$, where\n\\[ S^3 = D^3_+ \\cup_{S^2} D^3_- , \\ \\ \\ \\partial D^3_+ = \\partial D^3_- =\nD^3_+ \\cap D^3_- = S^2 .\\]\nSuppose that a knot  $K \\subset S^3$ is\nin general position with respect to this Heegaard decomposition. So\n$K \\cap S^2 = \\{x_1, \\cdots, x_n, y_1, \\cdots, y_n\\}$, $K \\cap D^3_{\\pm }$\nis a collection of unknotted, unlinked arcs\n$\\{ \\gamma_1^{\\pm }, \\cdots, \\gamma_n^{\\pm } \\} \\subset D^3_{\\pm }$,\nwhere $\\partial \\gamma_i^- = \\{x_i, y_i\\}$ and $\\{\\gamma_1^+, \\cdots, \\gamma_n^+\\} = K \\cap\nD^3_+$ becomes a braid of $n$ strands inside $D^3_+$. Denote by  $\\beta $ a corresponding word  in the\nbraid group $B_n$. For the top end points  $x_i$ of $\\gamma_i^+$, the bottom\nend points of $\\{\\gamma_1^+, \\cdots, \\gamma_n^+\\}$ give  a permutation of\n$\\{y_1, \\cdots , y_n\\}$ which generates a map\n\\[ \\pi : B_n \\to S_n ,\\]\nwhere $\\pi (\\beta )$ is the permutation of $\\{y_1, \\cdots , y_n\\}$ in the symmetric\ngroup of $n$ letters. Let $K = \\overline{\\beta }$ be the closure of $\\beta$.\nIt is well-known that there is a correspondence between a knot and a braid $\\beta$\nwith $\\pi (\\beta )$ is a complete cycle of the $n$ letters (see \\cite{bi}).\n\nThere is a corresponding {Heegaard decomposition} for the complement of a  $K$,\n\\[ S^3 \\setminus K = (D^3_+ \\setminus K) \\cup_{(S^2 \\setminus K)} (D^3_-\n\\setminus K ) , \\]\n\\[  D^3_{\\pm } \\setminus K = D^3_{\\pm } \\setminus (D^3_{\\pm }\n\\cap K), \\ \\ \\ S^2 \\setminus K = S^2 \\setminus (S^2 \\cap K) . \\]\nThus by  Seifert-van Kampen theorem we have following diagramm\n\\[ \\begin{array} {ccc}\n\\pi_1 (S^2 \\setminus K) & \\to & \\pi_1 (D^3_+ \\setminus K)\\\\\n\\downarrow & & \\downarrow \\\\\n\\pi_1 (D^3_- \\setminus K ) & \\to & \\pi_1 (S^3 \\setminus K) ,\n\\end{array} \\]\nand a corresponding diagramm  of {representation} spaces\n\\begin{equation} \\label{pullback}\n \\begin{array} {ccc}\n{\\mathcal R}(S^2 \\setminus K)& \\leftarrow & {\\mathcal R}(D^3_+ \\setminus K)\\\\\n\\uparrow & & \\uparrow \\\\\n{\\mathcal R}(D^3_- \\setminus K ) & \\leftarrow & {\\mathcal R} (S^3 \\setminus K) ,\n\\end{array} \\end{equation}\nwhere ${\\mathcal R }(X) = Hom (\\pi_1 (X), SU(2))\/SU(2)$ for $X =\nS^2 \\setminus K, D^3_{\\pm } \\setminus K, S^3 \\setminus K$.\n\nIn \\cite{ma}, Magnus used the trace free matrices to represent the\ngenerators of a free group to show that the faithfulness of a\n{representation} of braid groups in the automorphism groups of the rings generated\nby the character functions on free groups. This is original idea\nto have {representation}s with trace free along all meridians which Lin worked in\n\\cite{lin} to define the knot invariant. It has been carried out by\nM. Heusener and J. Kroll  in \\cite{hk} for the {representation} of knot groups with\nthe trace of the meridian fixed (not necessary zero).\nLet ${\\mathcal R}(S^2 \\setminus K)^{[i]}$ be the\nspace of $SU(2)$ {representation}s $\\rho : \\pi_1 (S^2 \\setminus K) \\to SU(2)$\nsuch that\n\\begin{equation} \\label{trace}\n \\rho ([m_{x_i}]) \\sim \\left( \\begin{array}{cc}\ni & 0 \\\\ 0 & -i \\end{array} \\right) , \\ \\ \\\n\\rho ([m_{y_i}]) \\sim \\left( \\begin{array}{cc}\ni & 0 \\\\ 0 & -i \\end{array} \\right) , \\end{equation}\nwhere $m_{x_i}, m_{y_i}, i =1, 2, \\cdots, n$ are the meridian circles around\n$x_i, y_i$ respectively. Note that $\\pi_1(S^2 \\setminus K)$ is generated by\n$m_{x_i}, m_{y_i}, i =1, 2, \\cdots, n$\nand one relation $\\prod^n_{i=1} m_{x_i} = \\prod^n_{i=1} m_{y_i}$.\nCorresponding to (\\ref{pullback}), we have\n\\begin{equation} \\label{irre}\n \\begin{array} {ccc}\n{\\mathcal R}(S^2 \\setminus K)^{[i]} & \\leftarrow & {\\mathcal R}(D^3_+ \\setminus K)^{[i]}\\\\\n\\uparrow & & \\uparrow \\\\\n{\\mathcal R}(D^3_- \\setminus K )^{[i]} & \\leftarrow & {\\mathcal R} (S^3 \\setminus K)^{[i]} .\n\\end{array}\n\\end{equation}\nThe conjugacy class in $SU(2)$ is completely determined by its trace.\nSo the condition (\\ref{trace}) can be reformulated for $\\rho \\in {\\mathcal R}(X)^{[i]}$,\n\\begin{equation} \\label{22}\ntrace \\rho ([m_{x_i}]) = trace \\rho ([m_{y_i}]) = 0 . \\end{equation}\nThe space ${\\mathcal R}(S^2 \\setminus K)^{[i]}$ can be identified with the space\nof $2n$ matrices $X_1 \\cdots, X_n, Y_1, \\cdots, Y_n$ in $SU(2)$ satisfying\n\\begin{equation} \\label{zero}\ntrace (X_i) = trace (Y_i) = 0, \\ \\ \\ \\ \\mbox{for $i =1, \\cdots, n$},\n\\end{equation}\n\\begin{equation} \\label{product}\nX_1 \\cdot X_2 \\cdots X_n = Y_1 \\cdot Y_2 \\cdots Y_n .\n\\end{equation}\nLet $Q_n$ be the space $\\{(X_1, \\cdots , X_n) \\in SU(2)^n | \\ \\\ntrace (X_i) = 0, i =1, \\cdots, n \\}$.\nLet  ${\\mathcal R}^*(S^2 \\setminus K)^{[i]}$ be the subset of  ${\\mathcal R}(S^2 \\setminus K)^{[i]}$ consisting of irreducible representations.\nNote that ${\\mathcal R}^*(S^2 \\setminus K)^{[i]} = (H_n \\setminus S_n)\/SU(2)$\nin Lin's notation \\cite{lin}, where\n\\[H_n = \\{(X_1, \\cdots, X_n, Y_1, \\cdots, Y_n) \\in Q_n \\times Q_n | \\ \\\nX_1 \\cdots X_n = Y_1 \\cdots Y_n \\},\\]\n$S_n$ is the subspace of $H_n$ consisting of all the reducible points.\nHere $H_n \\setminus S_n$ is the total space of a $SU(2)$-fiber bundle over\n${\\mathcal R}^*(S^2 \\setminus K)^{[i]}$.\n\nGiven $\\beta \\in B_n$, we denote by ${\\Gamma}_{\\beta}$ the graph of $\\beta $ in $Q_n \\times Q_n$,\ni.e.\n\\[{\\Gamma}_{\\beta} = \\{(X_1, \\cdots, X_n, \\beta (X_1) , \\cdots, \\beta (X_n)) \\in Q_n \\times Q_n\\}.\n\\]\nAs an automorphism of the free group $Z[m_{x_1}]*Z[m_{x_2}]* \\cdots *Z[m_{x_n}]$, this\nelement $\\beta \\in B_n$ preserves the word $[m_{x_1}] \\cdots [m_{x_n}]$. Thus we have\n$ X_1 \\cdots X_n = \\beta (X_1) \\cdots \\beta (X_n), $\nor in other words ${\\Gamma}_{\\beta}$ is a subspace of $H_n$. In fact, for $\\overline{\\beta}\n= K$, this subspace ${\\Gamma}_{\\beta}$ coincides with  the subspace of {representation}s\n$\\rho : \\pi_1(S^2 \\setminus K) \\to SU(2)$ in $H_n$ which can be extended\nto $\\pi_1(D^3_+ \\setminus K)$,\n${\\Gamma}_{\\beta} = Hom (\\pi_1(D^3_+ \\setminus K), SU(2))^{[i]}.$\nHence the space ${\\mathcal R}^*(D^3_+ \\setminus K)^{[i]} = {\\Gamma}_{\\beta, irre}\/SU(2)$\nis the irreducible $SU(2)$ {representation}s with traceless condition over\n$D^3_+ \\setminus K$.\n\nIn the special case $\\beta = id$, then ${\\Gamma}_{id}$ represents the diagonal in\n$Q_n \\times Q_n$,\n\\[{\\Gamma}_{id} = \\{ (X_1, \\cdots, X_n, X_1, \\cdots, X_n) \\in Q_n \\times Q_n\\} . \\]\nSince $K \\cap D^3_-$ represents the trivial braid, this space ${\\Gamma}_{id}\n\\subset H_n$ can be identified with the subspace of {representation}s\nin $Hom(\\pi_1(S^2 \\setminus K), SU(2))^{[i]}$ which  can be extended to\n$\\pi_1(D^3_- \\setminus K)$, i.e.\n$ {\\Gamma}_{id} = Hom (\\pi_1(D^3_- \\setminus K), SU(2))^{[i]} . $\nBy Seifert, Van-Kampen Theorem, the intersection\n${\\Gamma}_{\\beta} \\cap {\\Gamma}_{id}$ is the same as the space of {representation}s of\n$\\pi_1(S^3 \\setminus K)$ satisfying the monodromy condition $[i]$\n(see (\\ref{pullback})),\n\\[{\\Gamma}_{\\beta} \\cap {\\Gamma}_{id} = Hom (\\pi_1(S^3 \\setminus K), SU(2))^{[i]} . \\]\n\nGiven $\\beta \\in B_n$ with $\\overline{\\beta } = K$, there is an induced diffeomorphism\n(still denoted by $\\beta $) from $Q_n$ to itself. Such a diffeomorphism also\ninduces a diffeomorphism $f_{\\beta }:  {\\mathcal R}^*(S^2 \\setminus K)^{[i]} \\to\n{\\mathcal R}^*(S^2 \\setminus K)^{[i]}$of the representation variety.\n\nNote that\n$\\overline{{\\Gamma}}_{\\beta} = ({{\\Gamma}}_{\\beta} \\setminus ({{\\Gamma}}_{\\beta} \\cap S_n))\/SU(2)$\nis the image  of the  ``diagonal'' $\\overline{{\\Gamma}}_{id}$ under diffeomorphism  $f_{\\beta }$.\nBy Seifert- van Kampen theorem (\\ref{irre}), it is clear that the fixed point\nset of $f_{\\beta }$ is\n\\[\\ Fix (f_{\\beta }|_{{\\mathcal R}^*(S^2 \\setminus K)^{[i]}}) =\n\\overline{{\\Gamma}}_{\\beta} \\cap \\overline{{\\Gamma}}_{id} =\n{\\mathcal R}^*(S^3 \\setminus K)^{[i]} .\\]\n \nThe oriented submanifolds\n$\\overline{{\\Gamma}}_{\\beta}={\\mathcal R}^*(D^3_+ \\setminus K)^{[i]},  \\overline{{\\Gamma}}_{id}={\\mathcal R}^*(D^3_-\\setminus K)^{[i]}$\nintersects each other in a compact subspace of\n${\\mathcal R}^*(S^2 \\setminus K)^{[i]}$ from Lemma 1.6 in \\cite{lin}.\nHence we can perturb $f_{\\beta }$, or\nin another words perturb ${\\mathcal R}^*(D^3_+ \\setminus K)^{[i]}$ to\n$\\hat{{\\mathcal R}}^*(D^3_+ \\setminus K)^{[i]}$ by a compactly support isotopy so that\n$\\hat{{\\mathcal R}}^*(D^3_+ \\setminus K)^{[i]}$ intersects\n${\\mathcal R}^*(D^3_-\\setminus K)^{[i]}$ transversally at a finite number of\nintersection points. Denote the perturbed  diffeomorphism\nby $\\hat{f }_{\\beta }$. So its fixed points are all nondegenerated.\n\nThe Casson-Lin invariant of a knot $K = \\overline{\\beta}$ is given by\ncounting the algebraic intersection number of\n$\\hat{{\\mathcal R}}^*(D^3_+ \\setminus K)^{[i]}$ and\n${\\mathcal R}^*(D^3_-\\setminus K)^{[i]}$, or the algebraic number of\n$\\operatorname{Fix}(\\hat{f }_{\\beta })$,\n\\[\\lambda_{CL} (K) =  \\lambda_{CL}(\\beta ) = Algebraic( \\# \\operatorname{Fix}(\\hat{f }_{\\beta })) =\nAlgebraic( \\# (\n\\hat{{\\mathcal R}}^*(D^3_+ \\setminus K)^{[i]} \\cap {\\mathcal R}^*(D^3_-\\setminus K)^{[i]})) .\\]\n\nThe results proved by Lin in \\cite{lin} show that the Casson-Lin invariant \n$\\lambda_{CL} (K) = \\lambda_{CL}(\\beta )$\nis independent of its braid representatives, i.e. $\\lambda_{CL}(\\beta )$ is invariant\nunder the Markov moves of type I and type II on $\\beta$ and is one half of\nthe classical signature of the knot $K$.\n\n\\section{Nielsen number is a knot  invariant}\n\n\nIn this article we propose to count fixed points of  ${f }_{\\beta }$ in a Nielsen way  - using the classical  Nielsen numbers of ${f }_{\\beta }$.\n Nielsen counting of fixed points is a counting in the presence of the fundamental group.\nIn order to get an invariant of knots from braids, we have to verify that\nNielsen number\n$ N(f_{\\beta})$  is invariant\nunder Markov moves. A Markov move of type I changes $\\sigma \\in B_n$ to\n$\\xi^{-1} \\sigma \\xi \\in B_n$ for any $\\xi \\in B_n$, and the Markov move of\ntype II changes $\\sigma \\in B_n$ to $\\sigma_n^{\\pm } \\sigma \\in B_{n+1}$, or the inverses\nof these operations. It is well-known that two braids $\\beta_1$ and $\\beta_2$ has\nisotopic closure if and only if $\\beta_1$ can be changed to $\\beta_2$ by a sequence\nof finitely many Markov moves \\cite{bi}.\n\\begin{theorem} \\label{invar}\nIf  $\\overline{\\beta_1} = \\overline{\\beta_2} = K$ as a knot, $\\beta_1 \\in B_n, \\beta_2 \\in\nB_m$, then\n\\[ N(f_{\\beta_1})=N(f_{\\beta_2}).\\]\nSo the Nielsen number \n$ N(f_{\\beta})$ is a knot invariant.\n\\end{theorem}\nProof: We only need to show that for $\\beta \\in B_n$ with $\\overline{\\beta}$ being\na knot $K$, the Markov moves of type I and type II on $\\beta$ provide either a cojugacy or a\nisotopy of $f_{\\beta}$. Hence from the commutativity and  the invariance property under isotopy of the Nielsen numbers, we get that $ N(f_{\\beta})$\nis an invariant of knot $K = \\overline{\\beta}$.\n\nSuppose we have the Markov move of type I: change $\\beta$ to $\\xi^{-1} \\beta \\xi$\nfor some $\\xi \\in B_n$. The element $\\xi$ in $B_n$ induces a diffeomorphism\n$\\xi: Q_n \\to Q_n$ is orientation preserving as observed by Lin in \\cite{lin}.\nNote that $B_n$ is generated by $\\sigma_1, \\cdots, \\sigma_{n-1}$. For any $\\sigma_i^{\\pm }$,\nthe induced diffeomorphism $\\sigma_i^{\\pm } \\times \\sigma_i^{\\pm }:\nQ_n \\times Q_n \\to Q_n \\times  Q_n$ is an orientation preserving  diffeomorphism. So $\\xi $ is also a  orientation preserving\ndiffeomorphism since orientation  preserving\nproperties are invariant under the composition operation. Hence there is a\nhomeomorphism\n\\[\\xi \\times \\xi : Q_n \\times Q_n \\to Q_n \\times Q_n , \\]\nwhich commutes with the $SU(2)$-action and\n\\[ \\xi \\times \\xi ({\\mathcal R}^*(S^2 \\setminus K)^{[i]}) = {\\mathcal R}^*(S^2 \\setminus K)^{[i]}\n \\ \\ \\ (\\mbox{changing variables by $\\xi \\times \\xi$}),\\]\n\\[\\xi \\times \\xi ({\\mathcal R}^*(D_-^3 \\setminus K)^{[i]}) =\n{\\mathcal R}^*(D_-^3 \\setminus K)^{[i]} \\ \\ \\\n(\\mbox{in new coordinate $\\xi(X_1), \\cdots, \\xi (X_n)$}),\\]\n\\[ \\xi \\times \\xi ({\\mathcal R}^*(D_+^3 \\setminus K)^{[i]}) = \n{\\mathcal R}^*(D_+^3 \\setminus K)^{[i]}\\ \\ \\\n(\\mbox{in new coordinate $\\xi(X_1), \\cdots, \\xi (X_n)$}),\\]\nas oriented manifolds. Let $g_{\\xi} : {\\mathcal R}^*(S^2 \\setminus K)^{[i]} \\to\n{\\mathcal R}^*(S^2 \\setminus K)^{[i]}$ be the induced  homeomorphism, induced   from\n$\\xi \\times \\xi$ as coordinate changes . Hence we get a conjugacy relation\n\\[g_{\\xi}^{-1}\\circ f _{\\beta}\\circ g_{\\xi}  =  f_{\\xi^{-1} \\beta \\xi} , \\]\nfrom changing variables via $g_{\\xi}$. Note that $Fix(f_{\\xi^{-1} \\beta \\xi})$\nis identified with $Fix(f_{\\beta})$ under $g_{\\xi}$. Thus the Markov move of type I preserves the conjugacy class of $ f_{\\beta}$\n Therefore by commutativity of the Nielsen number (see \\cite{jb}) we have,\n\\begin{equation} \\label{type1}\nN(f_{\\xi^{-1} \\beta \\xi}) =\nN(g_{\\xi}^{-1}\\circ f _{\\beta}\\circ g_{\\xi})\n= N(f_{\\beta}) .\n\\end{equation}\nIt is clear that the argument goes through for the inverse operation of Markov\nmove of type I.\n\nSuppose we have the Markov move of type II: change $\\beta$ to \n$\\sigma_n \\beta \\in B_{n+1}$. Recall that $\\sigma_n (x_i) = x_i, 1 \\leq i \\leq n-1,\n\\sigma_n (x_n) = x_n x_{n+1} x_n^{-1}$ and $\\sigma_n (x_{n+1}) = x_n$. We need to \nidentify the Nielsen number  from the construction in\n$\\hat{H}_n$ into the one from $\\hat{H}_{n+1}$. Following Lin \\cite{lin}, there\nis an imbedding $q: Q_n \\times Q_n \\to Q_{n+1} \\times Q_{n+1}$ given by\n\\[q(X_1, \\cdots, X_n, Y_1, \\cdots, Y_n) = (X_1, \\cdots, X_n,Y_n,\nY_1, \\cdots, Y_n, Y_n) .\\]\nSuch an imbedding commutes with the $SU(2)$-action and $q(H_n) \\subset H_{n+1}$,\nand induces an imbedding\n\\[ \\hat{q}: \\hat{H}_n (={\\mathcal R}^*(S^2 \\setminus \\overline{\\beta})^{[i]}) \\to\n\\hat{H}_{n+1} (= {\\mathcal R}^*(S^2 \\setminus \\overline{\\sigma_n \\beta})^{[i]}) .\\]\nNote that the symplectic structure of $\\hat{H}_{n+1}$ restricted on\n$\\hat{q}(\\hat{H}_n)$ is the symplectic structure on $\\hat{H}_n$.\nUnder this imbedding, we have\n$\\hat{q}(f_{\\beta}): \\hat{H}_{n+1} \\to \\hat{H}_{n+1}$ is given by\n\\begin{equation} \\label{pb}\n(X_1, \\cdots, X_n, X_1, \\cdots, X_n) \\mapsto (X_1, \\cdots, X_n, \\beta(X_n),\n\\beta(X_1), \\cdots, \\beta(X_n), \\beta(X_n)).\n\\end{equation}\nThe image of $\\hat{q}(f_{\\beta})$ is invariant under the operation of $\\sigma_n$.\nAlso the corresponding  diffeomorphism $f_{\\sigma_n \\beta}$ is given by\n\\[f_{\\sigma_n \\beta}(X_1, \\cdots, X_n, X_{n+1}, X_1, \\cdots, X_n, X_{n+1}) \\]\n\\begin{equation} \\label{pnb}\n = (X_1, \\cdots, X_{n+1}, \\beta(X_1), \\cdots, \\beta(X_{n-1}), \\beta(X_n) X_{n+1}\n\\beta(X_n)^{-1}, \\beta(X_n) ) .\n\\end{equation}\nThus we have\n\\[ \\hat{q}({\\mathcal R}^*(D_-^3  \\setminus \\overline{\\beta})^{[i]}) \\subset\n{\\mathcal R}^*(D_-^3  \\setminus \\overline{\\sigma_n \\beta})^{[i]}, \\ \\\n\\hat{q}({\\mathcal R}^*(D_+^3  \\setminus \\overline{\\beta})^{[i]}) \\subset\n{\\mathcal R}^*(D_+^3  \\setminus \\overline{\\sigma_n \\beta})^{[i]} . \\]\nThe fixed points of $f_{\\sigma_n \\beta}$ are elements\n\\[ \\beta(X_i) = X_i, 1 \\leq i \\leq n_1; \\ \\ \\\n \\beta(X_n) X_{n+1} \\beta(X_n)^{-1} = X_n, \\ \\ \\\n\\beta(X_n) = X_{n+1} , \\]\nwhich is equivalent to $\\beta(X_i) = X_i, 1\\leq i \\leq n$, i.e.\n\\[ \\operatorname{Fix}(f_{\\sigma_n \\beta}) =\\ Fix (\\hat{q}(f_{\\beta})) = \\operatorname{Fix} (f_{\\beta}) . \\]\n\nThen there is  a (Hamiltonian) isotopy\n$\\psi_t : \\hat{H}_{n+1} (= (H_{n+1} \\setminus S_{n+1})\/SU(2)) \\to \\hat{H}_{n+1}$\nbetween $\\psi_{t_0} = \\hat{q}(f_{\\beta})$ by (\\ref{pb}) and\n$\\psi_{t_1} = f_{\\sigma_n \\beta}$ (\\ref{pnb})(see \\cite{li,lin} for the explicit\nconstructions).  So we have\n\\begin{equation} \\label{type2}\nN(f_{\\sigma_n \\beta}) = N(\\hat{g}(f_{\\beta})) =\nN(f_{\\beta}) .\n\\end{equation}\nThe first equality  is from the invariance property of Nielsen numbers under the  isotopy $\\psi_t$ and the second\nfrom the natural identification. We can similarly prove that\n\\[N(f_{\\sigma_n^{-1}\\beta}) =\nN(f_{\\beta}) .\\]\n\n\\begin{rk}\nIt is known for a long time, that the problem of  computation of  Nielsen numbers is a  very difficult problem. By this reason, we strongly believe that the  Nielsen number $N(f_{\\beta})$ is a new knots invariant, which cannot be reduced to the  known knots invariants,  as it  happened   in the  case of the Casson-Lin invariant of knots.\n\n\\end{rk}\n\n\\begin{ex}\n``Pillowcase''. For $n=2$ the irreducible variety  ${\\mathcal R}^*(S^2 \\setminus K)^{[i]}$ is a 2-sphere with four cone points deleted(see \\cite{lin}). So, in this case the space\n${\\mathcal R}^*(S^2 \\setminus K)^{[i]}$ is non-simply-connected  and the  Nielsen number $N(f_{\\beta})$ is   not trivial  for  general $\\beta$.\n\n\\end{ex}\n\\begin{que}\nDoes  the space ${\\mathcal R}^*(S^2 \\setminus K)^{[i]}$ is simply-connected\n if  $n>2$?\n\\end{que}\nThe author was informed by Hans Boden  that the theorem 8.3 in \\cite{furste} about  a moduli space of stable parabolic bundles over 2- sphere\n with marked points with given degree  and weights may be very usefull for the full   answer\non  this question.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nPure exploration~\\cite{even2006action,best_arm_COLT2010,kalyanakrishnan2012pac,bubeck2013multiple,kaufmann2016complexity,towards_COLT2017} is a fundamental online learning problem in multi-armed bandits, where an agent sequentially chooses options (often called arms) and observes random feedback, with the objective of identifying the best option (arm). This problem finds various applications such as recommendation systems~\\cite{distributed_recommendation_system}, online advertising~\\cite{online_advertising} and neural architecture search~\\cite{neural_architecture_search}.\nHowever, the traditional single-agent pure exploration problem~\\cite{even2006action,best_arm_COLT2010,kalyanakrishnan2012pac,bubeck2013multiple,kaufmann2016complexity,towards_COLT2017} cannot be directly applied to many real-world distributed online learning platforms, which often face a large volume of user requests and need to coordinate multiple \\emph{distributed} computing devices to process the requests, e.g., geographically distributed data centers~\\cite{zhenhua2011} and Web servers~\\cite{web_servers2003}.\nThese computing devices communicate with each other to exchange information in order to attain globally optimal performance.\n\n\nTo handle such distributed pure exploration problem, prior works \\cite{distributed2013,tao2019collaborative,top_collaborative2020} have developed the Collaborative Pure Exploration (CoPE) model, where there are multiple agents that communicate and cooperate in order to identify the best arm with learning speedup. Yet, existing results only investigate the classic multi-armed bandit (MAB)  setting~\\cite{even2006action,best_arm_COLT2010,kalyanakrishnan2012pac}, %\nand focus only on the \\emph{fully-collaborative}  setting, i.e., the agents aim to solve a common task.  \nHowever, in many real-world applications such as recommendation  systems~\\cite{distributed_recommendation_system}, it is often the case that different computing devices  face \\emph{different but correlated} recommendation tasks.  Moreover, there usually exists some \\emph{structured dependency} of user utilities on the recommended items.\nIn such applications, \nit is important to develop a more general CoPE model that allows heterogeneous tasks and complex reward structures, and quantitatively investigate how task similarities impact learning acceleration. \n\n\nMotivated by the above facts,   we propose a novel \\emph{Collaborative Pure Exploration in Kernel Bandit (CoPE-KB)} problem, which generalizes traditional single-task CoPE problems~\\cite{distributed2013,tao2019collaborative,top_collaborative2020} to the multi-task setting. It also generalizes the classic MAB model to allow  general  (linear or nonlinear) reward structures via the powerful kernel representation. \nSpecifically, each agent is given a set of arms, \nand the expected reward of each arm is generated by a task-dependent reward function with a low norm in a high-dimensional (possibly infinite-dimensional) Reproducing Kernel Hilbert Space (RKHS)~\\cite{RKHS1990,learning_with_kernel2002}, by which we can represent real-world nonlinear reward dependency as some linear function in a high-dimensional space, and can go beyond linear rewards as commonly done in the literature, e.g., \\cite{GP_UCB_Srinivas2010,kernelUCB_valko2013,IGP_UCB2017,multi_task_2017,high_dimensional_ICML2021}. \nEach agent sequentially chooses arms to sample and observes noisy outcomes. The agents can broadcast and receive messages to\/from others in communication rounds, so that they can exploit the task similarity and collaborate to expedite learning processes. The task of each agent is to find the best arm that maximizes the expected reward among her arm set. \n\nOur CoPE-KB formulation can handle different tasks in parallel and characterize the dependency of rewards on options, which provides a more general and flexible model for real-world applications. For example, in distributed recommendation systems~\\cite{distributed_recommendation_system}, different computing devices can face different tasks, and it is inefficient to learn the reward of each option individually. Instead, CoPE-KB enables us to directly learn the relationship between option features and user utilities, and exploit the similarity of such relationship among different tasks to accelerate learning. There are also many other applications, such as clinical trials~\\cite{clinical_trial}, where we conduct multiple clinical trials in parallel and utilize the common useful information to accelerate drug development, and neural architecture search~\\cite{neural_architecture_search}, where we simultaneously run different tests of neural architectures under different environmental setups to expedite search processes.\\yihan{Added two more applications here. Removed the application section}\n\n\n\n\n\nWe consider two important pure exploration settings under the CoPE-KB model, i.e., \\emph{Fixed-Confidence (FC)}, where agents aim to minimize the number of used samples under a given confidence, and  \\emph{Fixed-Budget (FB)}, where the goal is to minimize the error probability under a given sample budget. Note that due to the high dimension (possibly infinite) of the RKHS, \nit is highly non-trivial to simplify the burdensome computation and communication in the RKHS, and to derive theoretical bounds only dependent on the effective dimension of the kernelized feature space. \n\\longbo{add sentences about challenges: Despite being a more general model, we note that there are challenges.... }\\yihan{Added a sentence about challenges. Also added the formal computation costs in the following sentence}\nTo tackle the above challenges, we adopt efficient kernelized estimators and design novel algorithms $\\textsf{CoopKernelFC}$ and $\\textsf{CoopKernelFB}$ for the FC and FB settings, respectively, which only cost $\\textup{Poly}(nV)$ computation and communication complexity instead of $\\textup{Poly}(\\dim(\\mathcal{H}_K))$ as in \\cite{high_dimensional_ICML2021,neural_pure_exploration2021}, where $n$ is the number of arms, $V$ is the number of agents, and $\\mathcal{H}_K$ is the high-dimensional RKHS.  \nWe also establish matching upper and lower bounds in terms of sampling and communication complexity to demonstrate the optimality of our algorithms (within logarithmic factors).\n\nOur work distinguishes itself from prior CoPE works, e.g.,~\\cite{distributed2013,tao2019collaborative,top_collaborative2020}, in the following aspects: \n(i) Prior works~\\cite{distributed2013,tao2019collaborative,top_collaborative2020} only consider the classic MAB setting, \nwhile we adopt a high-dimensional RKHS to allow more general  real-world reward dependency on option features.\n(ii) Unlike \\cite{distributed2013,tao2019collaborative,top_collaborative2020} which restrict tasks (given arm sets and rewards) among agents to be the same, we allow different tasks for different agents, and explicitly quantify how task similarities impact learning acceleration. \n(iii) In lower bound analysis, prior works~\\cite{distributed2013,tao2019collaborative} mainly focus on a $2$-armed case, whereas we derive a novel lower bound analysis for general multi-armed cases with high-dimensional linear reward structures.\n\nMoreover, when reducing CoPE-KB  to prior CoPE with classic MAB setting (all agents are solving the same classic MAB task)~\\cite{distributed2013,tao2019collaborative}, our lower and upper bounds also match the existing state-of-the-art results in \\cite{tao2019collaborative}. \n\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe contributions of this paper are  summarized as follows\n\\begin{itemize}\n\t\\item We formulate a novel Collaborative Pure Exploration in Kernel Bandit (CoPE-KB) problem, which  models distributed multi-task decision making problems with general reward functions, and finds applications in many real-world online learning tasks, and study two settings of CoPE-KB, i.e., CoPE-KB with fixed-confidence (FC) and CoPE-KB with fixed-budget (FB). \n\t\n\t\\item For CoPE-KB with fixed-confidence (FC), we propose an algorithm $\\textsf{CoopKernelFC}$, which adopts an efficient kernelized estimator to significantly reduce computation and communication complexity from  existing $\\textup{Poly}(\\textup{dim}(\\mathcal{H}_K))$ to only $\\textup{Poly}(nV)$. \n\n\tWe derive matching upper and lower bounds of sample complexity $\\tilde{O}( \\frac{\\rho^*}{V} \\log \\delta^{-1})$ and communication rounds $O(\\log \\Delta^{-1}_{\\min})$. Here $\\rho^*$ is the problem hardness (defined in Section~\\ref{sec:fc_ub}), and $\\Delta^{-1}_{\\min}$ is the minimum reward gap. \\longbo{very briefly explain the notations}\\yihan{Explained it}\n\n\n\t\n\t\n\t\\item For CoPE-KB with fixed-budget (FB), we design an efficient algorithm $\\textsf{CoopKernelFB}$ with error probability $\\tilde{O} \\sbr{ \\exp \\sbr{ - \\frac{ T V }{ \\rho^*}} n^2 V}$ and communication rounds $O( \\log(\\omega(\\ti{\\mathcal{X}})) )$. A matching lower bound of communication rounds is also established to validate the communication optimality of $\\textsf{CoopKernelFB}$ (within double-logarithmic factors).  \\longbo{very briefly explain the notations}\\yihan{Explained them} Here $T$ is the sample budget, $\\ti{\\mathcal{X}}$ is the set of arms, and $\\omega(\\ti{\\mathcal{X}})$ is the principle dimension of data projections in $\\ti{\\mathcal{X}}$ to RKHS (defined in Section~\\ref{sec:def_omega}).\n\t\n\t \n\t\\item Our results explicitly quantify the impacts of task similarities on learning acceleration.\n\tOur novel analytical techniques, including data dimension decomposition, linear structured instance transformation and round-speedup induction, can be of independent interests and are applicable to other bandit problems.\n\\end{itemize}\n\nDue to space limit, we defer all the proofs to Appendix.\n\n\n\\section{Related Work}\nThis work falls in the literature of multi-armed bandits~\\cite{lai_robbins1985,thompson1933,UCB_auer2002,bandit_survey_bubeck2012,bandit_survey2020}. Here we mainly review three most related lines of research, i.e., collaborative pure exploration and kernel bandit.\n\n\\noindent\n\\textbf{Collaborative Pure Exploration (CoPE).}\nThe collaborative pure exploration literature is initiated by \\cite{distributed2013}, which considers the classic MAB and fully-collaborative settings, and designs fixed-confidence algorithms based on majority vote with upper bound analysis.  \n \\citet{tao2019collaborative} further develop a fixed-budget algorithm by calling conventional single-agent fixed-confidence algorithms, and completes the analysis of round-speedup lower bounds. \n\\citet{top_collaborative2020} extend the formulation of \\cite{distributed2013,tao2019collaborative} to the best $m$ arm identification problem, and designs fixed-confidence and fixed-budget algorithms with tight round-speedup upper and lower bounds, which give a strong separation between best arm identification and the extended best $m$ arm identification. \nOur CoPE-KB model encompasses the classic MAB and fully-collaborative settings in the above works~\\cite{distributed2013,tao2019collaborative,top_collaborative2020}, but faces unique challenges on computation and communication efficiency due to the high-dimensional reward structures. \n\n\n\n\n\\noindent\n\\textbf{Collaborative Regret Minimization.}\nThere are other works studying collaborative (distributed) bandit with the regret minimization objective.\n\\citet{rosenski2016multi,bistritz2018distributed,liu2010distributed} study the multi-player bandit with collisions motivated by cognitive radio networks, where multiple players simultaneously choose arms from the same set and receive no reward if more than one player choose the same arm (i.e., a collision happens). \n\\citet{bubeck2020coordination,bubeck2021cooperative} investigate a variant multi-player bandit problem where players cannot communicate but have access to shared randomness, and they propose algorithms that achieve nearly optimal regrets without collisions.\n\\citet{chakraborty2017coordinated} introduce another distributed bandit problem, where each agent decides either to pull an arm or to broadcast a message in order to maximize the total reward. \\citet{szorenyi2013gossip,korda2016distributed} adapt bandit algorithms to peer-to-peer networks, where the peers pick arms from the same set and can only communicate with a few random others along network links. \nThe above works consider different learning objectives and communication protocols from ours, and do not involve the challenges of simultaneously handling multiple different tasks and analyzing the relationship between communication rounds and learning speedup.\n\n\n\\noindent\n\\textbf{Kernel Bandit.}\nThere are a number of works for kernel bandit with the regret minimization objective. \n\\citet{GP_UCB_Srinivas2010} study the Gaussian process bandit problem with RKHS, which is the Bayesian version of kernel bandits, and designs an Upper Confidence Bound (UCB) style algorithm. \n\\citet{IGP_UCB2017} further improve the regret results of \\cite{GP_UCB_Srinivas2010} by constructing tighter kernelized confidence intervals.\n\\citet{kernelUCB_valko2013} consider kernel bandit from the frequentist perspective and provides an alternative regret analysis based on effective dimension. \n\\citet{krause2011contextual,multi_task_2017} study the multi-task kernel bandits, where the kernel function of RKHS is constituted by two compositions from task similarities and arm features. \n\\citet{dubey2020kernel} investigate the multi-agent kernel bandit with a local communication protocol,\nwith the learning objective being to reduce the average regret suffered by per agent. \nFor kernel bandit with the pure exploration objective, there are only two works~\\cite{high_dimensional_ICML2021,neural_pure_exploration2021} to our best knowledge. \\citet{high_dimensional_ICML2021} design a single-agent algorithm which uses a robust inverse propensity score estimator to reduce the sample complexity incurred by rounding procedures. \n\\citet{neural_pure_exploration2021} propose a variant of \\cite{high_dimensional_ICML2021} which applies neural networks to approximate nonlinear reward functions. \nAll of these works consider either regret minimization or single-agent setting, which largely differs from our problem, and they do not investigate the distributed decision making and (communication) round-speedup trade-off. Thus, their algorithms and analysis cannot be applied to solve our CoPE-KB problem. \n\n\n\n\n\\section{Collaborative Pure Exploration in Kernel Bandit (CoPE\\mbox{-}KB)}\n\n\\noindent\nIn this section, we present the formal formulation of the \\emph{Collaborative Pure Exploration in Kernel Bandit (CoPE-KB)}, and discuss the two important settings under CoPE-KB that will be investigated. \n\n\n\n\\noindent\n\\textbf{Agents and rewards.} There are $V$ agents $ [V]=\\{1, \\dots, V\\}$, who collaborate to solve different but possibly related instances (tasks) of the pure exploration in kernel bandit (PE-KB) problem. For each agent $v \\in [V]$, she is given a set of $n$ arms $\\mathcal{X}_v=\\{x_{v,1}, \\dots, x_{v,n}\\} \\subseteq \\mathbb{R}^{d_{\\mathcal{X}}}$, where $d_{\\mathcal{X}}$ is the dimension of arm feature vectors.\nThe expected reward of each arm $x \\in \\mathcal{X}_v$ is $f_v(x)$, where $f_v:\\mathcal{X}_v \\mapsto \\mathbb{R}$ is an unknown reward function. \nLet $\\mathcal{X}=\\cup_{v \\in [V]} \\mathcal{X}_v$.\nFollowing the literature in kernel bandits~\\cite{GP_UCB_Srinivas2010,krause2011contextual,kernelUCB_valko2013,multi_task_2017}, we assume that for any $v \\in [V]$, $f_v$ has a bounded norm in a Reproducing Kernel Hilbert Space (RKHS) specified by kernel $K_{\\mathcal{X}}: \\mathcal{X} \\times \\mathcal{X} \\mapsto \\mathbb{R}$ (see below for more details). \nAt each timestep $t$, each agent $v$ pulls an arm $x_{v,t} \\in \\mathcal{X}_v$ and observes a random reward $y_{v,t}=f(x_{v,t})+\\eta_{v,t}$, where $\\eta_{v,t}$ is an independent and zero-mean $1$-sub-Gaussian noise (without loss of generality).\\footnote{A random variable $\\eta$ is called $1$-sub-Gaussian if it satisfies that $\\mathbb{E}[\\exp(\\lambda \\eta-\\lambda \\mathbb{E}[\\eta])] \\leq \\exp(\\lambda^2\/2)$ for any $\\lambda \\in \\mathbb{R}$.} \nWe assume that the best arms $x_{v,*}=\\operatornamewithlimits{argmax}_{x \\in \\mathcal{X}_v} f_v(x)$ are unique for all $v \\in [V]$, which is a common assumption in the pure exploration literature, e.g.,~\\cite{even2006action,best_arm_COLT2010,kaufmann2016complexity,towards_COLT2017}. \n\n\n\n\\noindent\n\\textbf{Multi-Task Kernel Composition.}\nWe assume that the functions $f_v$ are parametric functionals of a global function $F:\\mathcal{X} \\times \\mathcal{Z} \\mapsto \\mathbb{R}$, which satisfies that,  for each agent $v \\in [V]$, there exists a task feature vector $z_v \\in \\mathcal{Z}$ such that\n\\begin{eqnarray}\nf_v(x)=F(x, z_v), \\ \\forall x \\in \\mathcal{X}_v. \\label{kernel-form}\n\\end{eqnarray}\nHere $\\mathcal{X}$ and $\\mathcal{Z}$ denote the arm feature space and task feature space, respectively.\nEq.~\\eqref{kernel-form} allows tasks to be different for agents, whereas prior CoPE works~\\cite{distributed2013,tao2019collaborative,top_collaborative2020} restrict the tasks ($\\mathcal{X}_v$ and $f_v$) to be the same for all agents $v \\in [V]$.\n\n\n\nDenote $\\ti{\\mathcal{X}}=\\mathcal{X} \\times \\mathcal{Z}$ and $\\ti{x}=(x, z_v)$.  \nAs a standard assumption in kernel bandits~\\cite{GP_UCB_Srinivas2010,krause2011contextual,multi_task_2017,dubey2020kernel}, we assume that $F$ has a bounded norm in a global RKHS $\\mathcal{H}_{K}$ specified by kernel $K:\\ti{\\mathcal{X}} \\times \\ti{\\mathcal{X}} \\mapsto \\mathbb{R}$, and \nthere exists a feature mapping $\\phi: \\ti{\\mathcal{X}} \\mapsto \\mathcal{H}_{K}$ and an unknown parameter $\\theta^* \\in \\mathcal{H}_{K}$ such that \n$$\nF(\\ti{x})= \\phi(\\ti{x})^\\top \\theta^*, \\  \\forall \\ti{x} \\in \\ti{\\mathcal{X}} ,\n\\quad\\text{and}\\quad K(\\ti{x}, \\ti{x}')= \\phi(\\ti{x})^\\top \\phi(\\ti{x}'), \\  \\forall \\ti{x}, \\ti{x}' \\in \\ti{\\mathcal{X}} .\n$$\nHere $\\|\\theta^*\\|:=\\sqrt{{\\theta^*}^\\top \\theta^*} \\leq B$ for some known constant $B>0$. \n$K:\\ti{\\mathcal{X}} \\times \\ti{\\mathcal{X}} \\mapsto \\mathbb{R}$ is a product composite kernel, which satisfies that for any $z,z' \\in \\mathcal{Z},  x,x' \\in \\mathcal{X}$, \n$$\nK((x, z),(x', z'))=K_{\\mathcal{Z}}(z,z') \\cdot K_{\\mathcal{X}}(x,x'),\n$$\nwhere $K_{\\mathcal{X}}$ is the arm feature kernel that depicts the feature structure of arms, and\n$K_{\\mathcal{Z}}$ is the task feature kernel that measures the similarity of functions $f_v$. \nFor example, in the fully-collaborative setting,\nall agents solve a common task, and we have that $z_v=1$ for all $v \\in [V]$, $K_{\\mathcal{Z}}(z,z')=1$ for all $z,z' \\in \\mathcal{Z}$, and $K=K_{\\mathcal{X}}$. On the contrary, if all tasks are different, then $\\textup{rank}(K_{\\mathcal{Z}})=V$. \n$K_{\\mathcal{Z}}$ allows us to characterize the  influences of task similarities ($1 \\leq \\textup{rank}(K_{\\mathcal{Z}}) \\leq V$) on learning. \n\n\n\n\\longbo{give a simple Bernoulli example, can use a figure}\\yihan{Added a figure}\nWe give a simple $2$-agent ($2$-task) illustrating example in Figure~\\ref{fig:example}. \nAgent 1 is given Items 1,2 with the expected rewards $\\mu_1,\\mu_2$, respectively, denoted by  $\\mathcal{X}_1=\\{x_{1,1},x_{1,2}\\}$. Agent 2 is given Items 2,3 with the expected rewards $\\mu_2,\\mu_3$, respectively, denoted by $\\mathcal{X}_2=\\{x_{2,1},x_{2,2}\\}$. Here $x_{1,2}=x_{2,1}$ is the same item. \nIn this case, $\\phi(\\ti{x}_{1,1})=[1,0,0]^\\top$, $\\phi(\\ti{x}_{1,2})=\\phi(\\ti{x}_{2,1})=[0,1,0]^\\top$,  $\\phi(\\ti{x}_{2,2})=[0,0,1]^\\top$, and $\\theta^*=[\\mu_1,\\mu_2,\\mu_3]^\\top$. \nThe two agents can share the learned information on the second dimension of $\\theta^*$ to accelerate learning processes.\n\n\\begin{wrapfigure}[13]{r}{0.5\\textwidth}\n\t\\centering    \n\t\\vspace*{-0.8em}\n\t\\includegraphics[width=0.5\\textwidth]{fig\/example.pdf}  \n\t\\caption{Illustrating example.}     \n\t\\label{fig:example}     \n\\end{wrapfigure}\n\n\n\n\nNote that the RKHS $\\mathcal{H}_{K}$ can have infinite dimensions, and any direct operation on $\\mathcal{H}_{K}$, e.g., the calculation of $\\phi(\\ti{x})$ and  explicit expression of the estimate of $\\theta^*$, is impracticable.\nIn this paper, all our algorithms only query the kernel function $K(\\cdot,\\cdot)$ instead of directly operating on $\\mathcal{H}_{K}$, and $\\phi(\\ti{x})$ and $\\theta^*$ are only used in our theoretical analysis, which is different from existing works, e.g., \\cite{high_dimensional_ICML2021,neural_pure_exploration2021}. \n\n\n\n\\noindent\n\\textbf{Communication.} \nFollowing the popular communication protocol in existing CoPE works~\\cite{distributed2013,tao2019collaborative,top_collaborative2020}, we allow these $V$ agents to exchange information via communication rounds, in which each agent can broadcast and receive messages from others. \nWhile we do not restrict the exact length of a message, for practical implementation it should be bounded by $O(n)$ bits. Here $n$ is the number of arms for each agent, and we consider the number of bits for representing a real number as a constant. \n\n\n\nIn the CoPE-KB problem, our goal is to design computation and communication efficient algorithms to coordinate different agents to simultaneously complete multiple tasks in collaboration and characterize how the take similarities impact the learning speedup. \n\n\n\n\n\\noindent\n\\textbf{Fixed-Confidence and Fixed-Budget.}\nWe consider two versions of the CoPE-KB problem, one with \\emph{fixed-confidence (FC)} and the other with \\emph{fixed-budget (FB)}. Specifically, in the FC setting, given a confidence parameter $\\delta \\in (0,1)$, the agents aim to identify $x_{v,*}$ for all $v \\in [V]$ with probability at least $1-\\delta$ and minimize the average number of samples used by each agent. \nIn the FB setting, on the other hand, the agents are given an overall $T\\cdot V$ sample budget ($T$ average samples per agent), and aim to use at most $T\\cdot V$ samples to identify $x_{v,*}$ for all $v \\in [V]$ and minimize the error probability.\nIn both FC and FB settings, agents are requested to minimize the number of communication rounds.\n\nTo evaluate the learning acceleration of our algorithms, following the CoPE literature, e.g., \\cite{distributed2013,tao2019collaborative,top_collaborative2020}, we also define the speedup metric of our algorithms.   \nFor a CoPE-KB instance $\\mathcal{I}$, let $T_{\\mathcal{A}_M,\\mathcal{I}}$ denote the average number of samples used by each agent in multi-agent algorithm $\\mathcal{A}_M$ to identify $x_{v,*}$ for all $v \\in [V]$, and let $T_{\\mathcal{A}_S,\\mathcal{I}}$ denote the average number of samples used by each task for a single-agent algorithm $\\mathcal{A}_S$ to sequentially (without communication) identify $x_{v,*}$ for all $v \\in [V]$.\nThen, the speedup of $\\mathcal{A}_M$ on instance $\\mathcal{I}$ is formally defined as \n\\begin{align}\n\\beta_{\\mathcal{A}_M,\\mathcal{I}}= \\inf_{\\mathcal{A}_S} \\frac{T_{\\mathcal{A}_S,\\mathcal{I}}}{T_{\\mathcal{A}_M,\\mathcal{I}}}. \\label{eq:def_beta}\n\\end{align}\n\n\nIt can be seen that $1 \\leq \\beta_{\\mathcal{A}_M,\\mathcal{I}} \\leq V$, where $\\beta_{\\mathcal{A}_M,\\mathcal{I}}=1$ for the case where all tasks are different and $\\beta_{\\mathcal{A}_M,\\mathcal{I}}$ can approach $V$ \nfor a fully-collaborative instance. \nBy  taking $T_{\\mathcal{A}_M,\\mathcal{I}}$ and $T_{\\mathcal{A}_S,\\mathcal{I}}$ as the smallest numbers of samples needed to meet the confidence constraint, the definition of $\\beta_{\\mathcal{A}_M,\\mathcal{I}}$ can be similarly defined for error probability results. \n\n\n\n\\yihan{Do we need to point out what conditions make our problem reduce to prior CoPE problem (formally)?}\n\\longbo{yes, we could }\\yihan{Pointed out it}\nIn particular, when all agents $v \\in [V]$ have the same arm set $\\mathcal{X}_v=\\mathcal{X}=\\{\\boldsymbol{e}_1,\\dots,\\boldsymbol{e}_n\\}$ (i.e., standard bases in $\\mathbb{R}^n$) and the same reward function $f_v(x)=f(x)=x^\\top \\theta^*$ for any $x \\in \\mathcal{X}$, all agents are solving a common classic MAB task, and then the task feature $\\mathcal{Z}=\\{1\\}$ and $K_{\\mathcal{Z}}(z,z')=1$ for any $z,z' \\in \\mathcal{Z}$. In this case, our CoPE-KB problem reduces to prior CoPE with classic MAB setting~\\cite{distributed2013,tao2019collaborative}.\n\n\n\n\n\n\n\n\n\n\\section{Fixed-Confidence CoPE-KB}\nWe start with the fixed-confidence (FC) setting and propose the $\\textsf{CoopKernelFC}$ algorithm.\nWe explicitly quantify how  task similarities impact learning acceleration, and provide sample complexity and round-speedup lower bounds  to demonstrate the  optimality of $\\textsf{CoopKernelFC}$.\n\n\n\\subsection{Algorithm $\\textsf{CoopKernelFC}$}\n\\yuko{Even if we write the algorithm first, it is quite difficult to understand the procedure. Even if we explain the pseudo-code line by line, it's hard to catch up.\nFirst, the reader needs to understand the estimation method, i.e., how to get  $\\hat{\\theta}_t$, which is required in Kernel bandit. We also need to explain idea on  choosing optimal $\\lambda^*$. We can explain that the arm selection ratio that minimizes the error in that estimate corresponds to $\\lambda^*$. However, I don't think most of the reviewers understand that this is the Kernel version of G-optimal design.  We may include the linear bandits method in the Appendix, or cite any linear bandit algorithms that use similar ideas for the reader to look at. Maybe RAGE in Fiez et al 2019 is the closest to our algorithm? We may put the section \"Kernelized Estimator\" before Algorithm details.}\\yuko{Do we need more comparison in terms of algorithm and sample complexity with \\cite{neural_pure_exploration2021}?}\n\n\\begin{algorithm}[t!]\n\t\\caption{Distributed Algorithm $\\textsf{CoopKernelFC}$: for Agent $v$ ($v\\in [V]$)} \\label{alg:kernel_bai}\n\t\\KwIn{$\\delta$, $\\ti{\\mathcal{X}}_1,\\dots,\\ti{\\mathcal{X}}_V$, $K(\\cdot, \\cdot): \\ti{\\mathcal{X}} \\times \\ti{\\mathcal{X}} \\mapsto \\mathbb{R} $, $B$,  rounding procedure $\\textsf{ROUND}_{\\varepsilon}(\\cdot,\\cdot)$ with approximation parameter $\\varepsilon$.}\n\t\\textbf{Initialization:} $\\mathcal{B}_{v'}^{(1)} \\leftarrow \\mathcal{X}_{v'}$ for all $v' \\in [V]$. $t \\leftarrow 1$ \\tcp*{initialize alive arm sets $\\mathcal{B}_{v'}^{(1)}$}\n\n\t\\While{$\\exists v' \\in [V], |\\mathcal{B}_{v'}^{(t)}| >1$} \n\t{\n\t\t\n\t\t$\\delta_t \\leftarrow \\frac{\\delta}{2 t^2}$\\;\n\t\t\tLet $\\lambda^*_t$ and $\\rho^*_t$ be the optimal solution and optimal value of \n\t\t\t$\\min \\limits_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\max \\limits_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_{v'}^{(t)}, v' \\in [V]}  \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{(\\xi_t I + \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top)^{-1}} $,\n\t\t\twhere $\\xi_t$ is a regularization parameter that satisfies Eq.~\\eqref{eq:xi_t}\n\t\t\t \\label{line:min_max_lambda}  \\tcp*{compute the optimal sample allocation}\n\t\t\t$N^{(t)} \\leftarrow  \\left \\lceil 8 (2^t)^2 (1+\\varepsilon)^2 \\rho^*_t \\log \\sbr{ 2  n^2 V \/ \\delta_t}  \\right \\rceil $ \\label{line:compute_N_t}   \\tcp*{compute the number of required samples}\n\t\t\t$(\\ti{s}_1, \\dots, \\ti{s}_{N^{(t)}}) \\leftarrow \\textsf{ROUND}_{\\varepsilon}(\\lambda^*_t, N^{(t)})$\\;\n\t\t\t\\label{line:common_end}\n\t\t\tLet $\\ti{\\bs{s}}_v^{(t)}$ be the sub-sequence of $(\\ti{s}_1, \\dots, \\ti{s}_{N^{(t)}})$ which only contains the arms in $\\tilde{\\mathcal{X}}_v$\n\t\t\t\\label{line:individual_start} \\tcp*{generate the sample sequence for agent $v$}\n\t\t\tPull arms $\\ti{\\bs{s}}_v^{(t)}$ and observe random rewards $\\boldsymbol{y}_v^{(t)}$\\; \\label{line:sample_observe}\n\t\t\tBroadcast $\\{(N^{(t)}_{v,i}, \\bar{y}^{(t)}_{v,i}) \\}_{i \\in [n]}$, where $N^{(t)}_{v,i}$ is the number of samples and $\\bar{y}^{(t)}_{v,i}$ is the average observed reward on arm $\\tilde{x}_{v,i}$\\;\n\t\t\t\\label{line:broadcast}\n\t\t\tReceive $\\{(N^{(t)}_{v',i}, \\bar{y}^{(t)}_{v',i}) \\}_{i \\in [n]}$ from all other agents $v' \\in [V]\\setminus\\{v\\}$\\; \\label{line:receive}\n\t\t\tFor notational simplicity, we combine the subscripts $v', i$ in $\\ti{x}_{v',i}, N^{(t)}_{v',i}, \\bar{y}^{(t)}_{v',i}$ by using $\\ti{x}_{(v'-1)n+i},N^{(t)}_{(v'-1)n+i}, \\bar{y}^{(t)}_{(v'-1)n+i}$, respectively. Then,\n\t\t\t$k_t(\\ti{x}) \\leftarrow [\\sqrt{N^{(t)}_1} K(\\ti{x}, \\ti{x}_1), \\dots, \\sqrt{N^{(t)}_{nV}} K(\\ti{x}, \\ti{x}_{nV})]^\\top$ for any $\\ti{x} \\in \\ti{\\mathcal{X}}$. $K^{(t)} \\leftarrow [\\sqrt{N^{(t)}_i N^{(t)}_j} K(\\ti{x}_i,\\ti{x}_j)]_{i,j \\in [nV]}$. $\\bar{\\boldsymbol{y}}^{(t)} \\leftarrow [ \\sqrt{N^{(t)}_1} \\bar{y}^{(t)}_1, \\dots, \\sqrt{N^{(t)}_{nV}} \\bar{y}^{(t)}_{nV} ]^\\top$ \\tcp*{organize overall observation information}\n\t\t\t\\For{\\textup{all $v' \\in [V]$}}\n\t\t\t{\n\t\t\t\t$\\hat{\\Delta}_t(\\ti{x}_i, \\ti{x}_j) \\leftarrow (k_t(\\ti{x}_i)-k_t(\\ti{x}_j))^\\top  (K^{(t)}+ N^{(t)} \\xi_t I)^{-1} \\bar{\\boldsymbol{y}}^{(t)},\\ \\ \\forall \\ti{x}_i, \\ti{x}_j \\in \\mathcal{B}_{v'}^{(t)} $ \\label{line:estimated_reward_gap} \\tcp*{estimate the reward gap between $\\ti{x}_i$ and $\\ti{x}_j$}\n\t\t\t\t$\\mathcal{B}_{v'}^{(t+1)} \\leftarrow \\mathcal{B}_{v'}^{(t)} \\setminus \\{ \\ti{x} \\in \\mathcal{B}_{v'}^{(t)} | \\exists \\ti{x}' \\in \\mathcal{B}_{v'}^{(t)} : \\hat{\\Delta}_t(\\ti{x}', \\ti{x}) \\geq 2^{-t} \\}$\\label{line:individual_end} \\tcp*{discard sub-optimal arms}\n\t\t\t}\n\t\t$t \\leftarrow t+1$\\;\n\t}\nE\t\\Return $\\mathcal{B}_1^{(t)}, \\dots, \\mathcal{B}_V^{(t)}$\\;\n\\end{algorithm}\n\n\n\n\\subsubsection{Algorithm}\n\n$\\textsf{CoopKernelFC}$ has three key components: (i) maintain alive arm sets for all agents, (ii) perform sampling individually according to the globally optimal sample allocation, and (iii) exchange the distilled observation information to estimate reward gaps and eliminate sub-optimal arms, via efficient kernelized computation and communication schemes.\n\nThe procedure of $\\textsf{CoopKernelFC}$ (Algorithm~\\ref{alg:kernel_bai}) for each agent $v$ is as follows.\nAgent $v$ maintains alive arm sets $\\mathcal{B}_{v'}^{(t)}$ for all $v' \\in [V]$ by successively eliminating sub-optimal arms in each phase.\nIn phase $t$, she solves a global min-max optimization, which takes into account the objectives and available arm sets of all agents, to obtain the optimal sample allocation $\\lambda^*_t \\in \\triangle_{\\ti{\\mathcal{X}}}$ and optimal value $\\rho^*_t$ (Line~\\ref{line:min_max_lambda}). Here $\\triangle_{\\ti{\\mathcal{X}}}$ is the collection of all distributions on $\\mathcal{X}$. $\\xi_t$ is a regularization parameter such that \n\\begin{align}\n\t\\sqrt{\\xi_t} \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{X}}_v, v \\in [V]} \\| \\phi(\\ti{x}_i) - \\phi(\\ti{x}_j) \\|_{\\sbr{\\xi_t I + \\sum_{x \\in \\mathcal{X}} \\frac{1}{nV} \\phi(x) \\phi(x)^\\top }^{-1}}  \\leq \\frac{1}{ (1+\\varepsilon) B \\cdot 2^{t+1}} , \\label{eq:xi_t}\n\\end{align} \nwhich ensures the estimation bias for reward gap to be bounded by $2^{-(t+1)}$ and can be efficiently computed by kernelized transformation (specified in Section~\\ref{sec:computation_efficiency}).\nThen, agent $v$ uses $\\rho^*_t$ to compute the number of required samples $N^{(t)}$, which guarantees that the confidence radius of estimation for reward gaps is within $2^{-t}$ (Line~\\ref{line:compute_N_t}). \nIn algorithm $\\textsf{CoopKernelFC}$, we use a rounding procedure $\\textsf{ROUND}_{\\varepsilon}(\\lambda,N)$ with approximation parameter $\\epsilon$ from \\cite{high_dimensional_ICML2021,allen2021round}, which rounds the sample allocation $\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}$ into the integer numbers of samples $\\kappa \\in \\mathbb{N}^{|\\ti{\\mathcal{X}}|}$, such that $\\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\kappa_{\\ti{x}}=N$ and \n\\begin{eqnarray*}\n&&\\max \\limits_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_{v'}^{(t)}, v' \\in [V]}  \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{(N \\xi_t I + \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}}  \\kappa_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top)^{-1}} \\\\\n&&\\leq (1+\\epsilon) \\max \\limits_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_{v'}^{(t)}, v' \\in [V]}  \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{(N \\xi_t I + \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} N \\lambda_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top)^{-1}} .\n\\end{eqnarray*}\nBy calling $\\textsf{ROUND}_{\\varepsilon}(\\lambda^*_t,N^{(t)})$, agent $v$ generates an overall sample sequence $(\\ti{s}_1, \\dots, \\ti{s}_{N^{(t)}})$ according to $\\lambda^*_t$, and extracts a sub-sequence $\\ti{\\bs{s}}_v^{(t)}$ that only contains the arms in $\\tilde{\\mathcal{X}}_v$ to sample (Lines~\\ref{line:common_end}-\\ref{line:sample_observe}).\nAfter sampling, she only communicates the number of samples $N^{(t)}_{v,i}$  and average observed reward $\\bar{y}^{(t)}_{v,i}$ for each arm with other agents (Lines~\\ref{line:broadcast}-\\ref{line:receive}).\nWith the overall observation information, she estimates the reward gap $\\hat{\\Delta}_t(\\ti{x}_i, \\ti{x}_j)$ between any arm pair $\\ti{x}_i, \\ti{x}_j \\in \\mathcal{B}_{v'}^{(t)}$ for all $v' \\in [V]$ and discards sub-optimal arms (Lines~\\ref{line:estimated_reward_gap}-\\ref{line:individual_end}). \n\n\n\\subsubsection{Computation and Communication Efficiency.} \\label{sec:computation_efficiency}\nHere we explain the efficiency of $\\textsf{CoopKernelFC}$. Note that in CoPE-KB, due to its high-dimensional reward structures, directly using the empirical mean to estimate rewards will cause loose sample complexity, and naively calculating and transmitting infinite-dimensional parameter $\\theta^*$ will incur huge computation and communication costs. As a result, we cannot directly compute and communicate scalar empirical rewards as in prior CoPE with classic MAB works~\\cite{distributed2013,tao2019collaborative,top_collaborative2020}. \n\n\n\n\n\\noindent\n\\textbf{Computation Efficiency.}\n$\\textsf{CoopKernelFC}$ uses three efficient kernelized operations, i.e., optimization solver (Line~\\ref{line:min_max_lambda}), condition for regularization parameter $\\xi_t$ (Eq.~\\eqref{eq:xi_t}) and estimator of reward gaps (Line~\\ref{line:estimated_reward_gap}). Unlike prior kernel bandit algorithms~\\cite{high_dimensional_ICML2021,neural_pure_exploration2021} which explicitly compute $\\phi(\\tilde{x})$ and maintain the estimate of $\\theta^*$ on the infinite-dimensional RKHS, $\\textsf{CoopKernelFC}$ only queries kernel function $K(\\cdot,\\cdot)$ and significantly reduces the computation (memory) costs from $\\textup{Poly}(\\textup{dim}(\\mathcal{H}_K))$ to only $\\textup{Poly}(nV)$.\n\n\nBelow we give the formal expressions of these operations and defer their detailed derivation to Appendix~\\ref{apx:kernel_computation}.\n\n\n\\paragraph{Kernelized Estimator.}\nWe first introduce the kernelized estimator of reward gaps (Line~\\ref{line:estimated_reward_gap}).\nFollowing the standard estimation procedure in linear\/kernel bandits~\\cite{rage2019,peace2020,high_dimensional_ICML2021,neural_pure_exploration2021}, we consider the following regularized least square estimator of underlying reward parameter $\\theta^*$ \n\\begin{align*}\n\\hat{\\theta}_t = & \\sbr{N^{(t)} \\xi_t I + \\sum_{j=1}^{N^{(t)}}  \\phi(\\tilde{s}_j) \\phi(\\tilde{s}_j)^\\top}^{-1} \\sum_{j=1}^{N^{(t)}} \\phi(\\tilde{s}_j) y_j \n\\end{align*}\nNote that this form of $\\hat{\\theta}_t$ has $N^{(t)}$ terms in the summation, which are cumbersome to compute and communicate. Since the samples $(\\ti{s}_1, \\dots, \\ti{s}_{N^{(t)}})$ are composed by arms $\\ti{x}_1, \\dots, \\ti{x}_{nV}$, we merge repetitive computations for same arms in the summation and obtain (for notational simplicity, we combine the subscripts $v', i$ in $\\ti{x}_{v',i}, N^{(t)}_{v',i}, \\bar{y}^{(t)}_{v',i}$ by using $\\ti{x}_{(v'-1)n+i},N^{(t)}_{(v'-1)n+i}, \\bar{y}^{(t)}_{(v'-1)n+i}$, respectively)\n\\begin{align}\n\\hat{\\theta}_t \\overset{\\textup{(a)}}{=} & \\sbr{N^{(t)} \\xi_t I + \\sum_{i=1}^{nV} N^{(t)}_i \\phi(\\ti{x}_i) \\phi(\\ti{x}_i)^\\top}^{-1} \\sum_{i=1}^{nV} N^{(t)}_i \\phi(\\ti{x}_i) \\bar{y}^{(t)}_{i} \n\\nonumber\\\\\n\\overset{\\textup{(b)}}{=} & \\Phi_t^\\top \\sbr{N^{(t)} \\xi_t I + K^{(t)} }^{-1} \\bar{y}^{(t)}. \\label{eq:kernelized_estimator}\n\\end{align}\nHere $N^{(t)}_i$ is the number of samples and $\\bar{y}^{(t)}_i$ is the average observed reward on arm $\\ti{x}_i$ for any $i \\in [nV]$. $\\Phi_t=[\\sqrt{N^{(t)}_1} \\phi(\\ti{x}_1)^\\top; \\dots; \\sqrt{N^{(t)}_{nV}} \\phi(\\ti{x}_{nV})^\\top]$ is the empirically weighted feature vector, $K^{(t)} = \\Phi_t \\Phi_t^\\top = [\\sqrt{N^{(t)}_i N^{(t)}_j} K(\\ti{x}_i,\\ti{x}_j)]_{i,j \\in [nV]}$ is the kernel matrix, and $\\bar{y}^{(t)}=[ \\sqrt{N^{(t)}_1} \\bar{y}^{(t)}_1, \\dots, \\sqrt{N^{(t)}_{nV}} \\bar{y}^{(t)}_{nV} ]^\\top$ is the average observations.\nEquality (a) rearranges the summation according to different chosen arms, and (b) follows from kernel transformation.\n\nThen, by multiplying $\\sbr{ \\phi(\\ti{x}_i)- \\phi(\\ti{x}_j)}^\\top$, we  obtain the estimator of reward gaps $f(\\ti{x}_i)-f(\\ti{x}_j)$ as \n\\begin{align}\n\\hat{\\Delta}(\\ti{x}_i,\\ti{x}_j) \n=\\sbr{ \\phi(\\ti{x}_i)- \\phi(\\ti{x}_j)}^\\top \\hat{\\theta}_t = \\sbr{k_t(\\ti{x}_i)- k_t(\\ti{x}_j)}^\\top \\sbr{N^{(t)} \\xi_t I + K^{(t)} }^{-1}  \\bar{y}^{(t)} , \\label{eq:kernelized_hat_Delta}\n\\end{align}\nwhere $k_t(\\ti{x})=\\Phi_t \\phi(\\ti{x})=[\\sqrt{N^{(t)}_1} K(\\ti{x}, \\ti{x}_1), \\dots, \\sqrt{N^{(t)}_{nV}} K(\\ti{x}, \\ti{x}_{nV})]^\\top$ for any $\\ti{x} \\in \\ti{\\mathcal{X}}$. \nThis estimator not only transforms heavy operations on the infinite-dimensional RKHS to efficient ones that only query the kernel function, but also merges repetitive computations for same arms (equality (a)) and only requires calculations dependent on $nV$.\n\n\n\n\\noindent\n\\paragraph{Kernelized Optimization Solver\/Condition for Regularization Parameter.}\nNow we introduce the optimization solver (Line~\\ref{line:min_max_lambda}) and condition for regularization parameter $\\xi_t$ (Eq.~\\eqref{eq:xi_t}).\n\nFor the kernelized optimization solver, we solve the min-max optimization in Line~\\ref{line:min_max_lambda} by projected gradient descent, which follows the procedure in \\cite{high_dimensional_ICML2021}.\nSpecifically, let $A(\\xi,\\lambda) = \\xi I + \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top$ for any $\\xi>0, \\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}$. We define function $h(\\lambda)=\\max \\limits_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}, v \\in [V]}  \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{A(\\xi_t,\\lambda)^{-1}}$, and denote the optimal solution of $h(\\lambda)$ by $\\ti{x}^*_i(\\lambda), \\ti{x}^*_j(\\lambda)$. \nThen, the gradient of $h(\\lambda)$ is given by \n\\begin{align}\n[\\nabla_{\\lambda} h(\\lambda)]_{\\ti{x}}= - \\sbr{ \\sbr{\\phi(\\ti{x}^*_i(\\lambda))-\\phi(\\ti{x}^*_j(\\lambda))}^\\top A(\\xi_t,\\lambda)^{-1} \\phi(\\ti{x}) }^2, \\  \\forall \\ti{x} \\in \\ti{\\mathcal{X}} ,  \\label{eq:nabla_h_lambda}\n\\end{align}\nwhich can be efficiently calculated by the following kernel transformation\n\\begin{align}\n& \\sbr{\\phi(\\ti{x}^*_i(\\lambda))-\\phi(\\ti{x}^*_j(\\lambda))}^\\top A(\\xi_t,\\lambda)^{-1} \\phi(\\ti{x})\n\\nonumber\\\\\n= & \\xi_t^{-1} \\sbr{ K(\\ti{x}^*_i(\\lambda), \\ti{x})- K(\\ti{x}^*_j(\\lambda), \\ti{x})  -  \\sbr{k_{\\lambda}(\\ti{x}^*_i(\\lambda))-k_{\\lambda}(\\ti{x}^*_j(\\lambda))}^\\top \\sbr{\\xi_t I + K_{\\lambda} }^{-1} k_{\\lambda}(\\ti{x}) } , \\label{eq:kernelized_gradient}\n\\end{align}\nwhere $k_{\\lambda}(\\ti{x}) =[\\frac{1}{\\sqrt{\\lambda_1}} K(\\ti{x}, \\ti{x}_1), \\dots, \\frac{1}{\\lambda_{nV}} K(\\ti{x}, \\ti{x}_{nV})]^\\top$ and $K_{\\lambda_u}=[\\frac{1}{\\sqrt{\\lambda_i \\lambda_j}} K(\\ti{x}_i,\\ti{x}_j)]_{i,j \\in [nV]}$.\n\nFor condition Eq.~\\eqref{eq:xi_t} on the regularization parameter $\\xi_t$, we can transform it to\n\\begin{align}\n\\max_{\\begin{subarray}{c} \\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{X}}_v\\\\ v \\in [V] \\end{subarray}} \\sqrt{ \\sbr{K(\\ti{x}_i, \\ti{x}_i)+K(\\ti{x}_j, \\ti{x}_j)-2 K(\\ti{x}_i, \\ti{x}_j)} - \\| k_{\\lambda_u}(\\ti{x}_i)-k_{\\lambda_u}(\\ti{x}_j) \\|^2_{\\sbr{\\xi_t I + K_{\\lambda_u} }^{-1}} } \\leq \\frac{1}{ (1+\\varepsilon) B \\cdot 2^{t+1}} , \\label{eq:kernelized_condition_xi}\n\\end{align}\nwhere $\\lambda_u=[\\frac{1}{nV}, \\dots, \\frac{1}{nV}]^\\top$ is the uniform distribution on $\\ti{\\mathcal{X}}$,\n$k_{\\lambda_u}(\\ti{x}) =[\\frac{1}{\\sqrt{nV}} K(\\ti{x}, \\ti{x}_1), \\dots, \\frac{1}{\\sqrt{nV}} K(\\ti{x}, \\ti{x}_{nV})]^\\top$ and $K_{\\lambda_u}=[\\frac{1}{nV} K(\\ti{x}_i,\\ti{x}_j)]_{i,j \\in [nV]}$.\n\nBoth the kernelized optimization solver and condition for $\\xi_t$ avoid inefficient operations directly on infinite-dimensional RKHS by querying the kernel function, and only cost $\\textup{Poly}(nV)$ computation (memory) complexity (Eqs.~\\eqref{eq:kernelized_gradient},\\eqref{eq:kernelized_condition_xi} only contains scalar $K(\\ti{x}_i, \\ti{x}_j)$, $nV$-dimensional vector $k_{\\lambda}$ and $nV \\times nV$-dimensional matrix $K_{\\lambda}$).\n\n\n\n\n\\noindent\n\\textbf{Communication Efficiency.}\nBy taking advantage of the  kernelized estimator (Eq.~\\eqref{eq:kernelized_hat_Delta}), $\\textsf{CoopKernelFC}$ merges repetitive computations for the same arms and only transmits $nV$ scalar tuples  $\\{(N^{(t)}_{v,i}, \\bar{y}^{(t)}_{v,i}) \\}_{i \\in [n], v \\in [V]}$ among agents instead of transmitting all $N^{(t)}$ samples as in \\cite{dubey2020kernel}. This significantly reduces the communication cost from $O(N^{(t)})$ bits to $O(nV)$ bits (Lines~\\ref{line:broadcast}-\\ref{line:receive}).\n\n\n\\subsection{Theoretical performance of $\\textsf{CoopKernelFC}$} \\label{sec:fc_ub}\nDefine the problem hardness of identifying the best arms $\\ti{x}^*_{v}$ for all $v \\in [V]$ as \n\\begin{align}\n\\rho^*=\\min_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\max_{\\ti{x} \\in \\mathcal{X}_v, v \\in [V]} \\frac{  \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|^2_{(\\xi_* I + \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top)^{-1}} }{ \\sbr{f(\\ti{x}^*_{v})- f(\\ti{x})}^2 }, \\label{eq:rho_star}\n\\end{align}\nwhere $\\xi_*=\\min_{t \\geq 1} \\xi_t$. $\\rho^*$ is the information-theoretic lower bound of the CoPE-KB problem, which is adapted from linear\/kernel bandit pure exploration~\\cite{rage2019,peace2020,high_dimensional_ICML2021,neural_pure_exploration2021}.\n\\longbo{briefly state the information theoretic lower bound and give a ref}\\yihan{Added them}\nLet $S$ denote the per-agent sample complexity, i.e., average number of samples  used by each agent  in algorithm $\\textsf{CoopKernelFC}$. \n\nThe sample complexity and number of communication rounds of $\\textsf{CoopKernelFC}$ are as follows.\n\n\n\\begin{theorem}[Fixed-Confidence Upper Bound] \\label{thm:coop_kernel_bai_ub}\n\tWith probability at least $1-\\delta$, algorithm $\\textsf{CoopKernelFC}$ returns the correct answers $\\ti{x}^*_{v}$ for all $v \\in [V]$, with per-agent sample complexity\n\t\\begin{align*}\n\tS = O \\sbr{   \\frac{ \\rho^*}{V} \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ n V }{\\delta} } + \\log\\log \\Delta^{-1}_{\\min}  }   } \n\t\\end{align*}\n\tand communication rounds $O( \\log \\Delta^{-1}_{\\min} )$.\n\\end{theorem}\n\n\n\\noindent\n\\textbf{Remark 1.}\n$\\rho^*$ is comprised of two sources of problem hardness, one due to handling different tasks and the other due to distinguishing different arms (We will decompose the sample complexity into these two parts in Corollary~\\ref{corollary:ub_fc}(c)).\nWe see that the sample complexity of $\\textsf{CoopKernelFC}$ matches the lower bound (up to logarithmic factors). For fully-collaborative instances where single-agent algorithms~\\cite{rage2019,peace2020} have $\\tilde{O}(\\rho^* \\log \\delta^{-1})$ sample complexity, our $\\textsf{CoopKernelFC}$  achieves the maximum $V$-speedup (i.e., enjoys $\\tilde{O}(\\frac{\\rho^*}{V} \\log \\delta^{-1})$ sample complexity) using only logarithmic communication rounds. \n\\yuko{Should we mention the comparison with the result in pure exploration in kernel bandit \\cite{neural_pure_exploration2021} for the case where $V=1$ in addition to linear case \\cite{rage2019,peace2020}?}\n\n\\noindent\n\\textbf{Interpretation.}\nWe further interpret Theorem~\\ref{thm:coop_kernel_bai_ub} via standard expressive tools in kernel bandits~\\cite{GP_UCB_Srinivas2010,kernelUCB_valko2013,multi_task_2017}, i.e., effective dimension and maximum information gain, to characterize the relationship between sample complexity and data structures, and demonstrate how task similarity influences learning performance. \n\n\nTo this end, define the maximum information gain over all sample allocation $\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}$ as \n$$\n\\Upsilon = \\max_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\log\\det\\sbr{ I + {\\xi_*}^{-1} K_{\\lambda} } .\n$$\nDenote $\\lambda^* = \\operatornamewithlimits{argmax}_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\log\\det \\sbr{ I + {\\xi_*}^{-1} K_{\\lambda} } $ and $\\alpha_1 \\geq \\dots \\geq \\alpha_{nV}$ the eigenvalues of $K_{\\lambda^*}$, and define the effective dimension of $K_{\\lambda^*}$ as \n$$\nd_{\\textup{eff}}=\\min \\bigg\\{ j :j \\xi_*\\log(nV) \\geq \\sum_{i=j+1}^{nV} \\alpha_i \\bigg\\}. \n$$\nWe then have the following corollary. \n\\begin{corollary} \\label{corollary:ub_fc}\n\\longbo{make it look like Coro 2}\\yihan{Revised Corollary 1}\nThe per-agent sample complexity of algorithm $\\textsf{CoopKernelFC}$, denoted by $S$, can also be bounded as follows:\n\t\\begin{align*}\n\t&\\textup{(a) }  S=O \\sbr{   \\frac{ \\Upsilon }{\\Delta^2_{\\min} V} \\cdot  g(\\Delta_{\\min}, \\delta) },  \\text{ where $\\Upsilon$ is the maximum information gain.}  \n\t\\\\\n\t&\\textup{(b) }  S=O \\sbr{ \\frac{  d_{\\textup{eff}}  }{\\Delta^2_{\\min} V} \\cdot \\log \\sbr{ nV \\cdot \\sbr{ 1 + \\frac{ \\textup{Trace}(K_{\\lambda^*}) }{ \\xi_* d_{\\textup{eff}} } } }  g(\\Delta_{\\min}, \\delta) }, \\text{ where $d_{\\textup{eff}}$ is the effective dimension.} \n\t\\\\\n\t&\\textup{(c) }  S=O \\sbr{ \\frac{  \\textup{rank}(K_{z}) \\cdot \\textup{rank}(K_{x}) }{\\Delta^2_{\\min} V} \\cdot \\log \\sbr{  \\frac{\\textup{Trace}\\sbr{I + \\xi_*^{-1} K_{\\lambda^*}}}{\\textup{rank}(K_{\\lambda^*})} } g(\\Delta_{\\min}, \\delta) }.\n\t\\end{align*}\n\tHere $g(\\Delta_{\\min}, \\delta)=\\log \\Delta^{-1}_{\\min} \\sbr{ \\log \\sbr{ \\frac{ n V }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min} }$.\n\\end{corollary}\n\n\n\\noindent\n\\textbf{Remark 2.}\nCorollary~\\ref{corollary:ub_fc}(a) shows that, our sample complexity can be bounded by the maximum information gain of any sample allocation on $\\ti{\\mathcal{X}}$, which extends conventional information-gain-based results in regret minimization kernel bandits~\\cite{GP_UCB_Srinivas2010,IGP_UCB2017,dubey2020kernel} to the pure exploration setting in the view of experimental (allocation) design. \n\nIn terms of dimension dependency, it is demonstrated in Corollary~\\ref{corollary:ub_fc}(b) that our result only depends on the effective dimension of kernel representation, which is the number of principle directions that  data projections in RKHS spread. \n\nWe also provide a fundamental decomposition of sample complexity into two compositions from task similarities and arm features in Corollary~\\ref{corollary:ub_fc}(c), which shows  that the more  tasks are similar, the fewer samples we need for accomplishing all tasks. \nFor example, when tasks are the same (fully-collaborative), i.e., $\\textup{rank}(K_{z})=1$, each agent only spends a $\\frac{1}{V}$ fraction of samples used by single-agent algorithms~\\cite{high_dimensional_ICML2021,neural_pure_exploration2021}. Conversely, when the tasks are totally different, i.e., $\\textup{rank}(K_{z})=V$, no advantage can be attained by multi-agent deployments, since the information from neighboring agents is useless for solving local tasks.\n\n\n\n\n\\subsection{Lower Bound for Fixed-Confidence Setting} \\label{sec:fc_lb}\nWe now present lower bounds for the sample complexity and a round-speedup for fully-collaborative instances, using a novel measure transformation techniques. The  bounds validate   the optimality of $\\textsf{CoopKernelFC}$ in both sampling and communication.  \nSpecifically, Theorems \\ref{thm:fc_lb_sample} and \\ref{thm:fc_lb_round} below formally present our bounds. In the theorems, we refer to a distributed algorithm $\\mathcal{A}$ for CoPE-KB  as $\\delta$-correct, if it returns the correct answers $\\ti{x}^*_{v}$ for all $v \\in [V]$ with probability at least $1-\\delta$. \n\n\\begin{theorem}[Fixed-Confidence Sample Complexity Lower Bound] \\label{thm:fc_lb_sample}\n\tConsider the fixed-confidence collaborative pure exploration in kernel bandit problem with Gaussian noise $\\eta_{v,t}$. Given any $\\delta \\in (0,1)$, a $\\delta$-correct distributed algorithm $\\mathcal{A}$ must have per-agent sample complexity\n\t$\n\t\\Omega ( \\frac{\\rho^*}{V} \\log \\delta^{-1} )\n\t$.\n\\end{theorem}\n\n\\noindent\n\\textbf{Remark 3.}\nTheorem~\\ref{thm:fc_lb_sample} shows that even if the agents are allowed to share samples without limitation, each agent still requires at least $\\tilde{\\Omega}(\\frac{\\rho^*}{V})$ samples on average. Together with Theorem~\\ref{thm:coop_kernel_bai_ub}, one sees that $\\textsf{CoopKernelFC}$ is within logarithmic factors of the optimal sampling. \n\n\n\\begin{theorem}[Fixed-Confidence Round-Speedup Lower Bound] \\label{thm:fc_lb_round}\n\tThere exists a fully-collaborative instance of the fixed-confidence CoPE-KB problem with multi-armed and  linear reward structures, for which given any $\\delta \\in (0,1)$, a $\\delta$-correct and $\\beta$-speedup distributed algorithm $\\mathcal{A}$ must utilize\n\t$$\n\t\\Omega \\sbr{\\frac{\\log \\Delta^{-1}_{\\min}}{ \\log(1+\\frac{V}{\\beta})+\\log \\log \\Delta^{-1}_{\\min} } }\n\t$$\n\tcommunication rounds in expectation. In particular, when $\\beta=V$, $\\mathcal{A}$ must require\n\t$\n\t\\Omega ( \\frac{\\log \\Delta^{-1}_{\\min}}{\\log \\log \\Delta^{-1}_{\\min}} )\n\t$\n\tcommunication rounds in expectation.\n\\end{theorem}\n\n\\noindent\n\\textbf{Remark 4.}\nTheorem~\\ref{thm:fc_lb_round} exhibits that logarithmic communication rounds are indispensable for achieving the full speedup, which validates that  $\\textsf{CoopKernelFC}$ is near-optimal in communication.  Moreover, when CoPE-KB  reduces\nto prior CoPE with classic MAB setting~\\cite{distributed2013,tao2019collaborative}, i.e., all agents are solving the same classic MAB task, our upper and lower bounds  (Theorems~\\ref{thm:coop_kernel_bai_ub} and \\ref{thm:fc_lb_round}) match the state-of-the-art results in \\cite{tao2019collaborative}. \n\n\n\n\n\n\\noindent\n\\textbf{Novel Analysis for Fixed-Confidence Round-Speedup Lower Bound.} \nWe highlight that our round-speedup lower bound for the FC setting analysis has the following novel aspects. (i) Unlike prior CoPE work~\\cite{tao2019collaborative} which focuses on a preliminary $2$-armed case without considering reward structures, we investigate multi-armed instances with high-dimensional linear reward structures.\n(ii) We develop a \\emph{linear structured progress lemma} (Lemma~\\ref{lemma:progress} in Appendix~\\ref{apx:fc_round_lb}), which effectively handles the challenges due to different possible sample allocation on multiple arms and derives the required communication rounds under linear reward structures. \n(iii) We propose \\emph{multi-armed measure transformation} and \\emph{linear structured instance transformation lemmas} (Lemmas~\\ref{lemma:kl_exchange},\\ref{lemma:dis_exchange_complete} in Appendix~\\ref{apx:fc_round_lb}), which bound the change of probability measures in instance transformation with multiple arms and high-dimensional linear rewards, and serve as basic analytical tools in our proof.\n\n\\begin{comment}\n\nNow we introduce an innovative analysis of round-speedup lower bound for the FC setting (Theorem~\\ref{thm:fc_lb_round}).\nUnlike prior CoPE work~\\cite{tao2019collaborative} which only analyzes a preliminary $2$-armed case with non-structured rewards in lower bound analysis,\nhere we investigate more general multi-armed case with high-dimensional linear rewards, and develop innovative techniques of measure (instance) transformation and experimental design to tackle the challenges incurred by different sample allocation on multiple arms and high-dimensional reward structures.\n\nThe lower bound is proved by induction and instance transformation as follows: we start from a basic instance parameterized by $\\theta^*_0$ with constant gaps, where it trivially holds that we need at least $0$ communication round and constant samples. Then, we switch to a harder instance parameterized by $\\theta^*_0\/\\kappa$, where $\\kappa>1$ is carefully chosen according to $\\beta$. Under this harder instance, we employ innovative measure transformation techniques to show that, a $\\delta$-correct and $\\beta$-speedup algorithm needs at least one more communication round and $\\kappa$ times samples to identify the correct answers. Repeating the above induction steps for $\\Omega \\sbr{\\frac{\\log \\Delta^{-1}_{\\min}}{ \\log(1+ V\/ \\beta)+\\log \\log \\Delta^{-1}_{\\min} } }$ times and the reward gap of instance will decrease to $\\Delta_{\\min}$. Then, we can conclude that a $\\delta$-correct and $\\beta$-speedup algorithm requires at least $\\Omega \\sbr{\\frac{\\log \\Delta^{-1}_{\\min}}{ \\log(1+ V\/ \\beta)+\\log \\log \\Delta^{-1}_{\\min} } }$ communication rounds.\n\n\n\\longbo{move the proof to appendix} \n\\yihan{Rewrote all proof outlines in this paper.}\n\n\n\\paragraph{Proof Procedure.} In the following, we describe the proof procedure of Theorem~\\ref{thm:fc_lb_round}.\n\nLet $\\mathcal{I}(\\mathcal{X}, \\theta^*)$ denote a fully-collaborative instance where all agents have the same action set $\\mathcal{X}$ and the same reward function $f(x)=\\phi(x)^\\top \\theta^*$, $\\forall x \\in \\mathcal{X}$.\nLet $\\rho(\\mathcal{X}, \\theta^*) =  \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{x \\in \\mathcal{X}} \\frac{\\| \\phi(x_*)-\\phi(x) \\|^2_{A(\\xi_*, \\lambda)^{-1} } }{ (f(x_*)-f(x))^2 } $ denote the harness of instance $\\mathcal{I}(\\mathcal{X}, \\theta^*)$.\nFor any integer $\\alpha \\geq 0$, let $\\mathcal{E}(\\alpha, T)$ and $\\mathcal{E}^{+1}(\\alpha, T)$ denote the events that algorithm $\\mathcal{A}$ uses at least $\\alpha$ and $\\alpha+1$ communication rounds with at most $T$ samples in the first $\\alpha$ rounds, respectively. \n\nWe first prove an important round progress lemma for linear reward structures, which shows that if $\\mathcal{A}$ uses at least $\\alpha$ communication rounds but only $\\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq}$ samples in the first $\\alpha$ rounds, then it is very likely that $\\mathcal{A}$ requires at least one more communication round for lack of observations.\n\n\nLet $\\mathcal{E}^*(\\alpha, T)$ denote the event that $\\mathcal{A}$ uses exactly $\\alpha$ communication rounds with at most $T$ samples in the first $\\alpha$ rounds.\nTo prove Lemma~\\ref{lemma:progress}, it suffices to prove that $ \\Pr[\\mathcal{E}^*(\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} )] \\leq 2\\delta + \\frac{1}{\\sqrt{q}}$.\nFor any sample allocation $\\lambda \\in \\triangle_{\\mathcal{X}}$ used by $\\mathcal{A}$, we can construct a counter instance $\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))$ with parameter  \n$$\n\\theta(\\lambda)=\\theta^*-\\frac{2(y_{j}^\\top \\theta^*) A(\\xi_*,\\lambda)^{-1} y_{j}}{y_{j}^\\top A(\\xi_*,\\lambda)^{-1} y_{j}},\n$$\nwhere $y_i=\\phi(x_*)-\\phi(x_i)$ for any $i \\in [n]$ and $j = \\operatornamewithlimits{argmax}_{i \\in [n]} \\frac{ y_i^\\top A(\\xi_*,\\lambda)^{-1} y_i }{ (y_i^\\top \\theta^*)^2 }$.\nUnder this counter instance $\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))$, $x_*$ is sub-optimal and the probability that $\\mathcal{A}$ returns $x_*$ under $\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))$ is bounded by $\\delta$.\n\nNow we show that with inadequate samples, $\\mathcal{A}$ cannot distinguish between $\\mathcal{I}(\\mathcal{X}, \\theta^*)$ and $\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))$, and the probability that $\\mathcal{A}$ returns $x_*$ under $\\mathcal{I}(\\mathcal{X}, \\theta^*)$ is bounded by $\\delta$ plus an instance distance term. Then, we can bound $\\mathcal{E}^*(\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq})$ by decomposing it into two cases that $\\mathcal{A}$ returns or does not return $x_*$. \nSpecifically, we decompose $\\mathcal{E}^*(\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq})$ as\n\\begin{align*}\n\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E}^*(\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq})  }=& \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E}^*(\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq}), \\textup{$\\mathcal{A}$ returns $x_*$} }  + \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E}^*(\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq}), \\textup{$\\mathcal{A}$ does not return $x_*$} }\n\\\\\n\\leq & \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))} \\mbr{ \\mathcal{E}^*(\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq}), \\textup{$\\mathcal{A}$ returns $x_*$} } + \\| \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}- \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))} \\|_{\\textup{TV}} + \\delta \n\\\\\n\\leq &  \\| \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}- \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))} \\|_{\\textup{TV}} + 2 \\delta \n\\\\\n\\overset{\\textup{(a)}}{\\leq} & \\sqrt{ V T_{\\mathcal{A}} \\cdot \\frac{ (y_j^\\top \\theta^*)^2 \\cdot y_j^\\top A(\\xi_*,\\lambda)^{-1} (\\sum_{i \\in [n]} \\lambda_i x_i x_i^\\top) A(\\xi_*,\\lambda)^{-1} y_j}{ (y_j^\\top A(\\xi_*,\\lambda)^{-1} y_j)^2 }    } + 2 \\delta\n\\nonumber \\\\\n\\leq & \\sqrt{ \\frac{ \\rho(\\mathcal{X}, \\theta^*)}{q}  \\cdot \\frac{ (y_j^\\top \\theta^*)^2 }{ y_j^\\top A(\\xi_*,\\lambda)^{-1} y_j } } + 2 \\delta\n\\nonumber \\\\\n\\overset{\\textup{(b)}}{\\leq} & \\frac{1}{\\sqrt{q}} + 2 \\delta ,\n\\end{align*}\nwhere (a) comes from the fact that $\\| \\mathcal{D}- \\mathcal{D}' \\|_{\\textup{TV}} \\leq \\sqrt{\\frac{1}{2} \\textup{KL}(\\mathcal{D} \\| \\mathcal{D}') }$ for any distribution $\\mathcal{D}, \\mathcal{D}'$ and the calculation of Gaussian Kullback\u2013Leibler divergence, and (b) is due to the definition of $\\rho(\\mathcal{X}, \\theta^*)$.\n\nNext, we present a key lemma of instance transformation, which implies that if $\\mathcal{A}$ uses at least $\\alpha+1$ communication rounds with at most $\\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq}$ samples in the first $\\alpha$ rounds under instance $\\mathcal{I}(\\mathcal{X}, \\theta^*)$, then when switching to the harder instance $\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)$, with high probability $\\mathcal{A}$ uses at most $\\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta}$ additional samples in the $(\\alpha+1)$-th round.\n\n\n\\begin{lemma}[Linear Structured Instance Transformation Lemma] \\label{lemma:dis_exchange}\n\tFor any integer $\\alpha \\geq 0$, $q \\geq 100$ and $\\kappa \\geq 1$, we have  \n\t\\begin{align*}\n\t \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\mathcal{E} \\sbr{\\alpha+1, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} + \\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta} } }\n\t\\geq  \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E}^{+1} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } } -  \\delta - \\delta(n,V,\\beta),\n\n\t\\end{align*}\n\twhere $\\delta(n,V,\\beta)$ is a second order term related to problem parameters.\n\\end{lemma}\n\nThe central idea behind the proof of Lemma~\\ref{lemma:dis_exchange} is that with inadequate $\\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq}$ samples, the difference of probabilities of any event between instances $\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)$ and $\\mathcal{I}(\\mathcal{X}, \\theta^*)$ can be bounded by a small instance distance term. Since $\\mathcal{A}$ is $\\beta$-speedup, the probability that $\\mathcal{A}$ costs more than $\\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta}$ samples in the $(\\alpha+1)$-th round under $\\mathcal{I}(\\mathcal{X}, \\theta^*)$ is bounded by $\\delta$. \nBy change of probability measures, we can obtain that the probability of this event\nunder $\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)$ is bounded by $\\delta$ plus a second order term $\\delta(n,V,\\beta)$, which gives Lemma~\\ref{lemma:dis_exchange}.\n\nSetting $\\kappa=\\sqrt{1+\\frac{Vq}{\\beta}}$, it holds that $\\frac{\\rho(\\mathcal{X}, \\theta\/ \\kappa)}{Vq}=\\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} + \\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta}$. Combining Lemmas~\\ref{lemma:progress},\\ref{lemma:dis_exchange}, we obtain an integrated instance transformation and round progress inequality\n\\begin{align}\n \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\mathcal{E} \\sbr{\\alpha+1, \\frac{\\rho(\\mathcal{X}, \\theta\/ \\kappa)}{Vq} } }\n\\geq  \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } } - 3 \\delta  - \\delta(n,V,\\beta) - \\frac{1}{\\sqrt{q}} \\label{eq:combine_induction}\n\\end{align}\n\nNow we conduct induction using Eq.~\\eqref{eq:combine_induction}.\nThe induction starts from a basic instance $\\mathcal{I}(\\mathcal{X}, \\theta_0)$ with constant gap $\\Delta_0$, where $\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta_0)} \\mbr{ \\mathcal{E} \\sbr{0, \\frac{ \\rho(\\mathcal{X}, \\theta_0)}{Vq} } }=1$ trivially holds. \nWe repeatedly apply Eq.~\\eqref{eq:combine_induction} to transform the instance from $\\mathcal{I}(\\mathcal{X}, \\theta_0)$ to $\\mathcal{I}(\\mathcal{X}, \\theta_0\/ \\kappa)$, and meanwhile the gap reduces from $\\Delta_0$ to $\\Delta_0 \/ \\kappa$. \nAfter \n$$\nt=\\Omega \\sbr{\\frac{\\log \\Delta^{-1}_{\\min}}{ \\log(1+\\frac{V}{\\beta})+\\log \\log \\Delta^{-1}_{\\min} } }\n$$\ninduction steps, the instance switches to  $\\mathcal{I}(\\mathcal{X}, \\theta_t)$ with desirable gap $\\Delta_{\\min}$. \nSumming up the probability deviations during instance transformation, we have\n\\begin{align*}\n\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta_t)} \\mbr{ \\mathcal{E} \\sbr{t, \\frac{ \\rho(\\mathcal{X}, \\theta^*)}{Vq} } }\n\\geq 1 - \\sbr{ 3 \\delta  + \\delta(n,V,\\beta) + \\frac{1}{\\sqrt{q}} } t .\n\\end{align*}\nBy carefully choosing the problem parameters $\\delta,n,V,\\beta$, we can obtain that $\\mathcal{E} \\sbr{t, \\frac{ \\rho(\\mathcal{X}, \\theta^*)}{Vq} }$ occurs with at least a constant probability, and thus $\\mathcal{A}$ uses at least $\\Omega \\sbr{\\frac{\\log \\Delta^{-1}_{\\min}}{ \\log(1+V\/ \\beta)+\\log \\log \\Delta^{-1}_{\\min} } }$ communication rounds in expectation.\n\n\\end{comment}\n\n\n\n\\section{Fixed-Budget CoPE-KB}\nWe now turn to the fixed-budget (FB) setting and design an efficient  algorithm $\\textsf{CoopKernelFB}$. \nWe also establish a fixed-budget round-speedup lower bound to validate its communication optimality.\n\n\\begin{algorithm}[t!]\n\t\\caption{Distributed Algorithm $\\textsf{CoopKernelFB}$: for Agent $v$ ($v\\in [V]$)} \\label{alg:kernel_bai_fb}\n\t\\KwIn{Per-agent budget $T$, $\\ti{\\mathcal{X}}_1,\\dots,\\ti{\\mathcal{X}}_V$, $K(\\cdot, \\cdot): \\ti{\\mathcal{X}} \\times \\ti{\\mathcal{X}} \\mapsto \\mathbb{R} $, regularization parameter $\\xi_*$,  rounding procedure $\\textsf{ROUND}_{\\varepsilon}(\\cdot,\\cdot)$ with  approximation parameter $\\varepsilon$.}\n\t\\textbf{Initialization:} $R \\leftarrow \\left \\lceil \\log_2(\\omega(\\ti{\\mathcal{X}})) \\right \\rceil$. $N \\leftarrow \\left \\lfloor TV\/R \\right \\rfloor$.\n\t$\\mathcal{B}_{v'}^{(1)} \\leftarrow \\mathcal{X}_{v'}$ for all $v' \\in [V]$. $t \\leftarrow 1$ \\tcp*{pre-determine the number of phases and the number of samples for each phase} \\label{line:fb_initialization}\n\t\\While{$t \\leq R$ \\textup{\\textbf{and}} $\\exists v' \\in [V], |\\mathcal{B}_{v'}^{(t)}| >1$}\n\t{\n\t\t\tLet $\\lambda^*_t$ and $\\rho^*_t$ be the optimal solution and optimal value of \n\t\t\t$\\min \\limits_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\max \\limits_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_{v'}^{(t)}, v' \\in [V]}  \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{(\\xi_* I + \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top)^{-1}} $ \\label{line:fb_min_max_opt} \\tcp*{compute the optimal sample allocation}\n\t\t\t$(\\ti{s}_1, \\dots, \\ti{s}_{N^{(t)}}) \\leftarrow \\textsf{ROUND}_{\\varepsilon}(\\lambda^*_t, N^{(t)})$\\; \\label{line:fb_round}\n\t\t\tLet $\\ti{\\bs{s}}_v^{(t)}$ be the sub-sequence of $(\\ti{s}_1, \\dots, \\ti{s}_{N})$ which only contains the arms in $\\tilde{\\mathcal{X}}_v$ \\tcp*{generate the sample sequence for agent $v$} \\label{line:fb_subsequence}\n\t\t\tPull arms $\\ti{\\bs{s}}_v^{(t)}$ and observe random rewards $\\boldsymbol{y}_v^{(t)}$\\;\n\t\t    Broadcast $\\{(N^{(t)}_{v,i}, \\bar{y}^{(t)}_{v,i}) \\}_{i \\in [n]}$\\; \\label{line:fb_broadcast}\n\t\t\tReceive $\\{(N^{(t)}_{v',i}, \\bar{y}^{(t)}_{v',i}) \\}_{i \\in [n]}$ from all other agents $v' \\in [V]\\setminus\\{v\\}$\\;  \\label{line:fb_receive}\n\t\t\t$k_t(\\ti{x}) \\leftarrow [\\sqrt{N^{(t)}_1} K(\\ti{x}, \\ti{x}_1), \\dots, \\sqrt{N^{(t)}_{nV}} K(\\ti{x}, \\ti{x}_{nV})]^\\top$ for any $\\ti{x} \\in \\ti{\\mathcal{X}}$. $K^{(t)} \\leftarrow [\\sqrt{N^{(t)}_i N^{(t)}_j} K(\\ti{x}_i,\\ti{x}_j)]_{i,j \\in [nV]}$. $\\bar{\\boldsymbol{y}}^{(t)} \\leftarrow [ \\sqrt{N^{(t)}_1} \\bar{y}^{(t)}_1, \\dots, \\sqrt{N^{(t)}_{nV}} \\bar{y}^{(t)}_{nV} ]^\\top$ \\tcp*{organize overall observation information}\n\t\t\t\\For{\\textup{all $v' \\in [V]$}}\n\t\t\t{\n\t\t\t\t$\\hat{f}_t(\\ti{x}) \\leftarrow k_t(\\ti{x})^\\top  (K^{(t)}+ N^{(t)} \\xi_* I)^{-1} \\bar{\\boldsymbol{y}}^{(t)}$ for all $\\ti{x} \\in \\mathcal{B}_{v'}^{(t)}$  \\label{line:fb_estimate_reward}\n\t\t\t\t\\tcp*{estimate the rewards of alive arms}\n\t\t\t\tSort all $\\ti{x} \\in \\mathcal{B}_{v'}^{(t)}$ by $\\hat{f}_t(\\ti{x})$ in decreasing order. Let $\\ti{x}_{(1)}, \\dots, \\ti{x}_{(|\\mathcal{B}_{v'}^{(t)}|)}$ denote the sorted arm sequence\\; \n\t\t\t\tLet $i_{t+1}$ be the largest index such that $\\omega( \\{ \\ti{x}_{(1)}, \\dots, \\ti{x}_{(i_{t+1})} \\} ) \\leq \\omega(\\mathcal{B}_{v'}^{(t)})\/2$\\; \\label{line:fb_elimination1}\n\t\t\t\t$\\mathcal{B}_{v'}^{(t+1)} \\leftarrow \\{ \\ti{x}_{(1)}, \\dots, \\ti{x}_{(i_{t+1})} \\}$ \\label{line:fb_elimination2} \\tcp*{cut down the alive arm set to half dimension}\n\t\t\t}\n\t\t$t \\leftarrow t+1$\\;\n\t}\n\t\\Return $\\mathcal{B}_1^{(t)}, \\dots, \\mathcal{B}_V^{(t)}$\\;\n\\end{algorithm}\n\n\\subsection{Algorithm $\\textsf{CoopKernelFB}$}\n\n\\subsubsection{Algorithm} \\label{sec:def_omega} \n$\\textsf{CoopKernelFB}$ consists of three key steps: (i)  pre-determine the numbers of phases and samples according to data dimension, (ii) maintain alive arm sets for all agents, plan a globally optimal sample allocation, (iii) communicate observation information and cut down alive arms to a half in the dimension sense. \n\n\nThe procedure of $\\textsf{CoopKernelFB}$ is given in  Algorithm~\\ref{alg:kernel_bai_fb}. \nDuring initialization, we determine the number of phases $R$ and the number of samples for each phase $N$ according to the principle dimension $\\omega(\\ti{\\mathcal{X}})$ (Line~\\ref{line:fb_initialization}), defined as: \n$$\n\\omega(\\ti{\\mathcal{S}}) = \\min \\limits_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\max \\limits_{\\ti{x}_i,\\ti{x}_j \\in \\ti{\\mathcal{S}}}  \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{(\\xi_* I + \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top)^{-1}}, \\quad \\forall \\,\\, \\ti{\\mathcal{S}} \\subseteq \\ti{\\mathcal{X}}\n$$\ni.e.,  the principle dimension of data projections in $\\ti{\\mathcal{S}}$ to the RKHS. \nIn each phase $t$, each agent $v$ maintains alive arm sets $\\mathcal{B}_{v'}^{(t)}$ for all agents $v' \\in [V]$, and solves an integrated optimization to obtain a globally optimal sample allocation $\\lambda^*_t$ (Line~\\ref{line:fb_min_max_opt}). \nThen, she generates a sample sequence $(\\ti{s}^{(t)}_1, \\dots, \\ti{s}^{(t)}_{N})$ according to $\\lambda^*_t$, and selects the sub-sequence $\\ti{\\bs{s}}_v^{(t)}$ that only contains her available arms to perform sampling (Lines~\\ref{line:fb_round}-\\ref{line:fb_subsequence}). \nDuring communication, she only sends and receives the number of samples $N^{(t)}_{v,i}$ and average observed reward $\\bar{y}^{(t)}_{v,i}$ for each arm to and from other agents (Lines~\\ref{line:fb_broadcast}-\\ref{line:fb_receive}).\nUsing the shared information, she estimates rewards of alive arms and only selects the best half of them in the dimension sense to enter the next phase (Lines~\\ref{line:fb_estimate_reward}-\\ref{line:fb_elimination2}). \n\n\n\n\\subsubsection{Computation and Communication Efficiency.}\n$\\textsf{CoopKernelFB}$ also adopts the efficient kernelized optimization solver (Eqs.~\\eqref{eq:nabla_h_lambda},\\eqref{eq:kernelized_gradient}) to solve the min-max optimization in Line~\\ref{line:fb_min_max_opt} and employs the kernelized estimator  (Eq.~\\eqref{eq:kernelized_estimator}) to estimate the rewards in Line~\\ref{line:fb_estimate_reward}. Moreover, \n$\\textsf{CoopKernelFB}$ only  spends  $\\textup{Poly}(nV)$ computation time and $O(nV)$-bit communication costs. \n\n\n\\subsection{Theoretical performance of $\\textsf{CoopKernelFB}$}\n\nWe present  the error probability of $\\textsf{CoopKernelFB}$ in the following theorem, where \n $\\lambda_u = \\frac{1}{nV} \\boldsymbol{1}$. \n\n\\begin{theorem}[Fixed-Budget Upper Bound] \\label{thm:kernel_bai_fb_ub}\n\tSuppose $\\omega( \\{ \\ti{x}^*_{v}, \\ti{x} \\} ) \\geq 1$ for any $\\ti{x} \\in \\mathcal{X}_v \\setminus \\{ \\ti{x}^*_{v} \\}, v \\in [V]$ and $T = \\Omega( \\rho^* \\log(\\omega(\\ti{\\mathcal{X}}) ) )$, and the regularization parameter $\\xi_*>0$ satisfies $\\sqrt{\\xi_*} \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{X}}_v, v \\in [V]} \\| \\phi(\\ti{x}_i) - \\phi(\\ti{x}_j) \\|_{A(\\xi_*, \\lambda_u)^{-1}}  \\leq \\frac{\\Delta_{\\min}}{ 2 (1+\\varepsilon) B }$. With  at most $T$ samples per agent, $\\textsf{CoopKernelFB}$  returns the correct answers $\\ti{x}^*_{v}$ for all $v \\in [V]$, with error probability \n\t$$\n\tErr = O \\sbr{ n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) \\cdot \\exp \\sbr{ - \\frac{ T V }{ \\rho^* \\log(\\omega(\\ti{\\mathcal{X}})) } } }\n\t$$\n\tand communication rounds $O( \\log(\\omega(\\ti{\\mathcal{X}})) )$.\n\\end{theorem}\n\n\n\\noindent\n\\textbf{Remark 5.}\nTheorem~\\ref{thm:kernel_bai_fb_ub} implies that, to guarantee an error probability  $\\delta$, $\\textsf{CoopKernelFB}$ only requires $O(\\frac{\\rho^* \\log(\\omega(\\ti{\\mathcal{X}}))}{V} \\log( \\frac{ n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) }{ \\delta } ) )$ samples, which matches the sample complexity lower bound  (Theorem~\\ref{thm:fc_lb_sample}) up to logarithmic factors.\nIn addition, $\\textsf{CoopKernelFB}$ attains the maximum $V$-speedup for fully-collaborative instances with only logarithmic communication rounds, which also matches the round-speedup lower bound (Theorem~\\ref{thm:fb_lb_round}) within double logarithmic factors.  \n\n\n\n\\noindent\n\\textbf{Technical Novelty in Error Probability Analysis.} \nOur analysis extends prior single-agent analysis~\\cite{peace2020} to the multi-agent setting. \nThe single-agent analysis in~\\cite{peace2020}  only uses a single universal Gaussian-process concentration bound. Instead, we establish novel estimate  concentrations and high probability events for each arm pair and each agent to handle the distributed environment, and build a connection between the principle dimension $\\omega( \\mathcal{B}_v^{(t)} )$ and problem hardness $\\rho^*$ via elimination rules (Lines~\\ref{line:fb_elimination1}-\\ref{line:fb_elimination2}) to guarantee the identification correctness.\n\n\n\n\\begin{comment}\n\n\\noindent\n\\emph{Proof Procedure.} \nBelow we describe the proof procedure of Theorem~\\ref{thm:kernel_bai_fb_ub}.\n\nFor any $t \\in [R]$, $\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}$, we construct the following high probability event\n$$\n\\mathcal{J}_{t,\\ti{x}_i,\\ti{x}_j} = \\lbr{ \\abr{ \\sbr{\\hat{f}_t(\\ti{x}_i)-\\hat{f}_t(\\ti{x}_j)} - \\sbr{f(\\ti{x}_i)-f(\\ti{x}_j)} } < \\Delta_{t,\\ti{x}_i,\\ti{x}_j} } .\n$$\nHere \n$\n\\Delta_{t,\\ti{x}_i,\\ti{x}_j} = \\inf_{\\Delta>0} \\lbr{ \\frac{\\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{A(\\xi_*,\\lambda^*_t)^{-1}} }{\\Delta^2} \\leq 2 \\rho^* } \n$ is carefully defined to control the error probability.\nUsing the Hoeffding inequality and  definitions of $N$ and $\\Delta_{t,\\ti{x}_i,\\ti{x}_j}$, we can obtain $\\Pr \\mbr{ \\neg \\mathcal{J}_{t,\\ti{x}_i,\\ti{x}_j} } = O \\sbr{ \\exp \\sbr{- \\frac{ TV }{ \\rho^* \\log(\\omega(\\ti{\\mathcal{X}})) } } }$.\nWe also define an overall high probability event $\\mathcal{J} = \\bigcap_{t \\in [R], \\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}}  \\mathcal{J}_{t,\\ti{x}_i,\\ti{x}_j}$.\nBy a union bound, we have $\\Pr \\mbr{ \\neg \\mathcal{J} } = O \\sbr{ n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) \\cdot \\exp \\sbr{ - \\frac{ T V }{ \\rho^* \\log(\\omega(\\ti{\\mathcal{X}})) } } }$.\n\nThen, to prove the error probability of $\\textsf{CoopKernelFB}$, it suffices to show that conditioning on $\\mathcal{J}$, the algorithm returns the correct answers $\\ti{x}^*_v$ for all $v \\in [V]$. Now we prove this by contradiction.\n\nSuppose that there exist a phase $t \\in [R]$ and an agent $ v \\in [V]$, such that the correct answer $\\ti{x}^*_v$ is eliminated by agent $v$ in phase $t$. \nThen, letting $\\mathcal{B'}_v^{(t)}=\\{ \\ti{x} \\in \\mathcal{B}_v^{(t)}: \\hat{f}_t(\\ti{x}) > \\hat{f}_t(\\ti{x}^*_{v}) \\}$ denote the set of arms in $\\mathcal{B}_v^{(t)}$ with higher estimated rewards than $\\ti{x}^*_v$,\nusing the elimination rules (Lines~\\ref{line:fb_elimination1},\\ref{line:fb_elimination2}), we have $\\omega( \\mathcal{B'}_v^{(t)} \\cup \\{ \\ti{x}^*_{v} \\} ) > \\frac{1}{2} \\omega( \\mathcal{B}_v^{(t)} )$.\nLet $\\ti{x}_0=\\operatornamewithlimits{argmax}_{\\ti{x} \\in \\mathcal{B'}_v^{(t)}} \\Delta_{\\ti{x}}$.\nIt holds that\n\\begin{align}\n & \\frac{\\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}_0) \\|_{A(\\xi_*,\\lambda^*_t)^{-1}} }{ 2 \\Delta_{\\ti{x}_0}^2 }\n\\! \\overset{\\textup{(c)}}{\\leq} \\! \\frac{ \\omega( \\mathcal{B}_v^{(t)} ) }{ 2 \\Delta_{\\ti{x}_0}^2 }\n\\nonumber\\\\\n\\! < & \\! \\frac{\\omega( \\mathcal{B'}_v^{(t)} \\cup \\{ \\ti{x}^*_{v} \\} )}{ \\Delta_{\\ti{x}_0}^2 }\n\\overset{\\textup{(d)}}{\\leq} \\! \\min_{\\lambda \\in \\triangle \\mathcal{X}} \\! \\max_{\\ti{x} \\in \\mathcal{B'}_v^{(t)} } \\!\\! \\frac{ \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|_{A(\\xi_*,\\lambda)^{-1}} }{ \\Delta_{\\ti{x}}^2 }\n\\! \\leq \\!\\! \\rho^*\\!\\! , \\label{eq:fb_ub_analysis}\n\\end{align}\nwhere (c) comes from the definitions of $\\omega(\\cdot)$ and $\\lambda^*_t$, and (d) is due to the definitions of $\\omega(\\cdot)$ and $\\ti{x}_0$.\nThen, combining Eq.~\\eqref{eq:fb_ub_analysis} and the definition of $\\Delta_{t,\\ti{x}^*_{v},\\ti{x}_0}$, we have $\\Delta_{t,\\ti{x}^*_{v},\\ti{x}_0} \\leq \\Delta_{\\ti{x}_0}$. Thus,\nconditioning on $\\mathcal{J}$, we have\n\\begin{align*}\n\\hat{f}_t(\\ti{x}^*_{v})-\\hat{f}_t(\\ti{x}_0) \\geq f(\\ti{x}^*_{v})-f(\\ti{x}_0) - \\Delta_{\\ti{x}_0} =0 ,\n\\end{align*}\nwhich contradicts the definition of $\\ti{x}_0$ that $\\ti{x}_0$ has a higher estimated reward than $\\ti{x}^*_{v}$.\nHence, for any $v \\in  [V], t \\in [R]$, the correct answer $\\ti{x}^*_{v}$ is never eliminated. \nAccording to the definition of $R$ and elimination rules, \nwe obtain that $\\textsf{CoopKernelFB}$ returns the correct answers $\\ti{x}^*_{v}$ for all $v \\in [V]$.\n\\end{comment}\n\n\n\\noindent\n\\textbf{Interpretation.}  \nSimilar to Corollary~\\ref{corollary:ub_fc}, we can also interpret the error probability result with the standard tools of maximum information gain and effective dimension in kernel bandits~~\\cite{GP_UCB_Srinivas2010,kernelUCB_valko2013,multi_task_2017}, and decompose the error probability into two compositions from task similarities and arm features.\n\n\\begin{corollary} \\label{corollary:ub_fb}\n\tThe error probability of algorithm $\\textsf{CoopKernelFC}$, denoted by $Err$, can also be bounded as follows: \n\t\\begin{align*}\n\t&\\textup{(a)} \\, Err=O \\sbr{ \\exp \\sbr{ - \\frac{ T V \\Delta^2_{\\min} }{ \\Upsilon \\log(\\omega(\\ti{\\mathcal{X}})) } } \\cdot n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) } \\text{, where $\\Upsilon$ is the maximum information gain.}\n\t\\\\\n\t&\\textup{(b)}\\,  Err=O \\sbr{ \\exp \\sbr{ - \\frac{ T V \\Delta^2_{\\min} }{ d_{\\textup{eff}} \\log \\sbr{  nV \\cdot \\sbr{ 1 + \\frac{ \\textup{Trace}(K_{\\lambda^*})  }{ \\xi_* d_{\\textup{eff}} } } } \\log(\\omega(\\ti{\\mathcal{X}})) } }  \\cdot n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) } \\text{, where $d_{\\textup{eff}}$ is the}\\\\\n\t& \\text{ effective dimension. }\n\t\\\\\n\t&\\textup{(c)} \\, Err=O \\sbr{ \\exp \\sbr{ - \\frac{ T V \\Delta^2_{\\min} }{ \\textup{rank}(K_{z}) \\cdot \\textup{rank}(K_{x}) \\log \\sbr{  \\frac{\\textup{Trace}\\sbr{I + \\xi_*^{-1} K_{\\lambda^*}}}{\\textup{rank}(K_{\\lambda^*})} } \\log(\\omega(\\ti{\\mathcal{X}})) } } \\cdot n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) } .\n\t\\end{align*}\n\\end{corollary}\n\n\\noindent\n\\textbf{Remark 6.}\nCorollaries~\\ref{corollary:ub_fb}(a), \\ref{corollary:ub_fb}(b) bound the error probability by maximum information gain and effective dimension, respectively, which capture essential structures of tasks and arm features and only depend on the effective dimension of the feature space of kernel representation.\n %\nFurthermore, we exhibit how task similarities influence the error probability performance in Corollary~\\ref{corollary:ub_fb}(c). For example, in the fully-collaborative case where $\\textup{rank}(K_{z})=1$, the error probability enjoys an exponential decay factor of $V$ compared to conventional single-agent results~\\cite{peace2020} (achieves a $V$-speedup). Conversely, when the tasks are totally different with $\\textup{rank}(K_{z})=V$, the error probability degenerates to conventional single-agent results~\\cite{peace2020}, since in this case information sharing brings no benefit.\n\n\n\n\\subsection{Lower Bound for Fixed-Budget Setting} \\label{sec:fb_lb}\n\nIn this subsection, we establish a round-speedup lower bound for the FB setting.\n\n\\begin{theorem}[Fixed-Budget Round-Speedup Lower Bound] \\label{thm:fb_lb_round}\n\tThere exists a fully-collaborative instance of the fixed-budget CoPE-KB problem with multi-armed and linear reward structures, for which given any $\\beta \\in [\\frac{V}{\\log(\\omega(\\ti{\\mathcal{X}}))}, V]$, a $\\beta$-speedup distributed algorithm $\\mathcal{A}$ must utilize\n\t$$\n\t\\Omega \\sbr{\\frac{ \\log(\\omega(\\ti{\\mathcal{X}})) }{ \\log(\\frac{V}{\\beta})+\\log \\log(\\omega(\\ti{\\mathcal{X}})) } }\n\t$$\n\tcommunication rounds in expectation. \n\tIn particular, when $\\beta=V$, $\\mathcal{A}$ must use \n\t$\n\t\\Omega ( \\frac{\\log(\\omega(\\ti{\\mathcal{X}})) }{ \\log \\log(\\omega(\\ti{\\mathcal{X}})) } )\n\t$\n\tcommunication rounds in expectation.\n\\end{theorem}\n\n\\noindent\n\\textbf{Remark 7.}\nTheorem~\\ref{thm:fb_lb_round} shows that under the FB setting, to achieve the full speedup, agents require at least logarithmic communication rounds with respect to the principle dimension $\\omega(\\ti{\\mathcal{X}})$, which validates the communication optimality of $\\textsf{CoopKernelFB}$.\nIn the degenerated case when all agents solve the same non-structured pure exploration problem, same as in prior classic MAB setting~\\cite{distributed2013,tao2019collaborative}, both our upper (Theorem~\\ref{thm:kernel_bai_fb_ub}) and lower (Theorem~\\ref{thm:fb_lb_round}) bounds match the  state-of-the-art results in \\cite{tao2019collaborative}.\n\n\n\n\n\n\n\n\n\n\\noindent\n\\textbf{Novel Analysis for Fixed-Budget Round-Speedup Lower Bound.} \nDifferent from the FC setting, here we borrow the proof idea of prior limited adaptivity work~ \\cite{learning_adaptivity2017} to establish a non-trivial lower bound analysis under Bayesian environments, and perform instance transformation by changing data dimension instead of tuning reward gaps. \nIn our analysis, we employ novel techniques to calculate the information entropy and support size of posterior reward distributions in order to build induction among different rounds and derive the required communication rounds.\n\n\n\\begin{comment}\n\\paragraph{Proof Procedure.}\nThe proof procedure is described as follows. \nWe construct a random instance $\\mathcal{D}_{d}^{\\Delta}$ with data dimension $d=\\omega(\\ti{\\mathcal{X}})$ and reward gap $\\Delta$, and conduct induction to transform instances and push forward communication rounds.\n\n\nFirst, we start from a basic step: by information-theoretical analysis, one can obtain that for some small enough constant $\\delta_1 \\in (0,1)$, there is no $1$-round $\\beta$-speedup algorithm that can use $O(\\frac{V \\log d}{\\beta \\Delta^2})$ samples and guarantee $\\delta_1$ error probability.\n\nThen, we show an induction step: for any $\\delta \\in (0,1)$, if there is no $(r-1)$-round $\\beta$-speedup algorithm that can use $O(\\frac{V \\log d}{\\beta \\Delta^2 (r-1)^2})$ samples and guarantee $\\delta$ error probability, then there is no $r$-round $\\beta$-speedup algorithm that can use $O(\\frac{V \\log d}{\\beta \\Delta^2 r^2})$ samples and guarantee $\\delta-o( \\frac{1}{r^2} )$ error probability, either.\nThis induction statement can be validated by the following contradiction: if there exists such a $r$-round $\\beta$-speedup algorithm $\\mathcal{A}_r$, by algorithmic and instance constructions, one can also obtain a $(r-1)$-round $\\beta$-speedup algorithm $\\mathcal{A}_{r-1}$ that satisfies the conditions in induction hypothesis under some well-chosen posterior instance $\\mathcal{D}_{\\tilde{d}}^{\\Delta}$ with smaller data dimension $\\tilde{d} \\leq d$. \n\nPerforming $r_*=\\frac{\\log d}{ \\log(\\frac{V}{\\beta})+\\log \\log d }$ induction steps and accumulating the error probability deviations, we can obtain that there is no $r_*$-round $\\beta$-speedup algorithm that can use $O(\\frac{V \\log d}{\\beta \\Delta^2 r_*^2})$ samples and guarantee $\\delta_1-\\sum_{r=2}^{r_*} o(\\frac{1}{r^2})$ error probability. \nLet $\\delta_*<\\delta_1-\\sum_{r=2}^{r_*} o(\\frac{1}{r^2})$.\nThen, any $\\beta$-speedup algorithm that can use $O(\\frac{V \\log d}{\\beta \\Delta^2 r_*^2})$ samples and guarantee $\\delta_*$ error probability requires at least $r_*$ communication rounds.\n\\end{comment}\n\n\n\n\n\n\n\n\\section{Experiments}\n\n\\begin{figure} [t!]\n\t\\centering     \n\t\\subfigure[FC, $K_{\\mathcal{Z}}=1$ (fully-collaborative)] {  \\label{fig:fc_fc}       \n\t\t\\includegraphics[width=0.25\\textwidth]{fig\/FC_Fully_Coop_var_gap.pdf} \n\t\t\\hspace*{1em}\n\t}   \n\t\\subfigure[FC, $1<K_{\\mathcal{Z}}<V$] {   \\label{fig:fc_mt}\n\t\t\\includegraphics[width=0.25\\textwidth]{fig\/FC_Multi_Task_var_gap.pdf} \n\t\t\\hspace*{1em}\n\t}   \n\t\\subfigure[FC, $K_{\\mathcal{Z}}=V$] {    \\label{fig:fc_dt}\n\t\t\\includegraphics[width=0.25\\textwidth]{fig\/FC_Diff_Task_var_gap.pdf} \n\t}   \n\t\\subfigure[FB, $K_{\\mathcal{Z}}=1$ (fully-collaborative)] {   \\label{fig:fb_fc}\n\t\t\\includegraphics[width=0.25\\textwidth]{fig\/FB_Fully_Coop_var_budget.pdf} \n\t\t\\hspace*{1em}\n\t}   \n\t\\subfigure[FB, $1<K_{\\mathcal{Z}}<V$] {     \\label{fig:fb_mt}\n\t\t\\includegraphics[width=0.25\\textwidth]{fig\/FB_Multi_Task_var_budget.pdf} \n\t\t\\hspace*{1em}\n\t}   \n\t\\subfigure[FB, $K_{\\mathcal{Z}}=V$] {    \\label{fig:fb_dt}\n\t\t\\includegraphics[width=0.25\\textwidth]{fig\/FB_Diff_Task_var_budget.pdf} \n\t}   \n\t\\caption{Experimental results for FC and FB settings.\\yihan{Updated the figures.}}\n\\end{figure} \n\nIn this section, we conduct experiments  to validate the empirical performance of our algorithms.\nIn our experiments, we set $V=5$, $d=4$, $n=6$, $\\delta=0.005$ and $\\phi(\\ti{x})=I \\ti{x}$ for any $\\ti{x} \\in \\ti{\\mathcal{X}}$. The entries of $\\theta^*$ form an arithmetic sequence that starts from $0.1$ and has the common difference $\\Delta_{\\min}$, i.e., $\\theta^*=[0.1,0.1+\\Delta_{\\min},\\dots,0.1+(d-1)\\Delta_{\\min}]^\\top$.\nFor the FC setting, we vary the gap $\\Delta_{\\min} \\in [0.1,0.8]$ to generate different instances (points), and run $50$ independent simulations to plot the average sample complexity with $95\\%$ confidence intervals.\nFor the FB setting, we change the budget $T \\in [7000, 300000]$ to obtain different instances, and perform $100$ independent runs to show the error probability across runs. The specific values of gap $\\Delta_{\\min}$ and budget $T$ can be seen in X-axis of the figures.\n\n\n\\noindent\n\\textbf{Fixed-Confidence.} \nIn the FC setting (Figures~\\ref{fig:fc_fc}-\\ref{fig:fc_dt}), we compare  $\\textsf{CoopKernelFC}$ with five baselines: $\\textsf{CoopKernel\\mbox{-}IndAlloc}$ is an ablation variant of $\\textsf{CoopKernelFC}$ which individually calculates sample allocations for different agents. $\\textsf{IndRAGE}$~\\cite{rage2019}, $\\textsf{IndALBA}$~\\cite{soare2014best} and $\\textsf{IndPolyALBA}$~\\cite{du2021combinatorial} are single-agent algorithms, which use $V$ copies of single-agent $\\textsf{RAGE}$~\\cite{rage2019}, $\\textsf{ALBA}$~\\cite{soare2014best} and $\\textsf{PolyALBA}$~\\cite{du2021combinatorial} policies to solve the $V$ tasks independently. $\\textsf{IndRAGE}\/V$ is a $V$-speedup baseline, which divides the sample complexity of the best single-agent algorithm $\\textsf{IndRAGE}$ by $V$.\nOne can see that $\\textsf{CoopKernelFC}$ achieves the best sample complexity in Figures~\\ref{fig:fc_fc},\\ref{fig:fc_mt}, which demonstrates the effectiveness of our sample allocation and cooperation scheme.\nMoreover, the empirical results also reflect the impacts of task similarities on learning speedup, and keep consistent with our theoretical analysis. Specifically, in the fully-collaborative case (Figure~\\ref{fig:fc_fc}), $\\textsf{CoopKernelFC}$ matches $\\textsf{IndRAGE}\\mbox{-}V$ since it attains the $V$ speedup; in the intermediate ($1<K_{\\mathcal{Z}}<V$) case (Figure~\\ref{fig:fc_mt}), the curve of $\\textsf{CoopKernelFC}$ lies between $\\textsf{IndRAGE}\/V$ and $\\textsf{IndRAGE}$, since it only achieves smaller than $V$ speedup due to the decrease of task similarity; in the totally-different-task ($K_{\\mathcal{Z}}=V$) case (Figure~\\ref{fig:fc_dt}), $\\textsf{CoopKernelFC}$ performs similar to the single-agent algorithm $\\textsf{IndRAGE}$, since information sharing among agents brings no advantage in this case.\n\n\n\\noindent\n\\textbf{Fixed-Budget.} \nIn the FB setting (Figures~\\ref{fig:fb_fc}-\\ref{fig:fb_dt}), we compare  $\\textsf{CoopKernelFB}$ with three baselines: $\\textsf{CoopKernelFB\\mbox{-}IndAlloc}$ is an ablation variant of $\\textsf{CoopKernelFB}$ where agents calculate and use different sample allocations. $\\textsf{IndPeaceFB}$~\\cite{peace2020} and $\\textsf{IndUniformFB}$ solve the $V$ tasks independently by calling $V$ copies of single-agent $\\textsf{PeaceFB}$~\\cite{peace2020} and uniform sampling policies, respectively.\nAs shown in Figures~\\ref{fig:fb_fc},\\ref{fig:fb_mt}, our $\\textsf{CoopKernelFB}$ enjoys a lower error probability than all other algorithms. \nIn addition, these empirical results also validate the influences of task similarities on learning performance, and match our theoretical analysis. Specifically, as the task similarity decreases in Figures~\\ref{fig:fb_fc} to \\ref{fig:fb_dt}, the error probability of $\\textsf{CoopKernelFB}$ gets closer  to that of single-agent $\\textsf{IndRAGE\\mbox{-}FB}$, due to the slow-down of its learning speedup. \n\n\n\\section{Conclusion}\nIn this paper, we propose a novel Collaborative Pure Exploration in Kernel Bandit (CoPE-KB) problem with Fixed-Confidence (FC) and Fixed-Budget (FB) settings. CoPE-KB aims to coordinate multiple agents to identify best arms with general reward functions. \nWe design two computation and communication efficient algorithms $\\textsf{CoopKernelFC}$ and $\\textsf{CoopKernelFB}$ based on novel kernelized estimators. \nMatching upper and lower bounds are established to demonstrate the statistical and communication optimality of our algorithms. \nOur theoretical results explicitly characterize the impacts of task similarities on learning speedup and avoid heavy dependency on the high dimension of the kernelized feature space.\nIn our analysis, we also develop novel analytical techniques, including data dimension decomposition, linear structured instance transformation and (communication) round-speedup induction, which are applicable to other bandit problems and can be of independent interests.\n\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\\section*{Appendix}\n\n\\section{Proofs for the Fixed-Confidence Setting}\n\n\\subsection{Kernelized Computations in Algorithm $\\textsf{CoopKernelFC}$} \\label{apx:kernel_computation}\n\\vspace*{0.4em}\n\\noindent\n\\textbf{Kernelized Condition for Regularization Parameter $\\xi_t$.}\nWe first introduce how to compute the condition Eq.~\\eqref{eq:xi_t} for regularization parameter $\\xi_t$ in Line~\\ref{line:min_max_lambda} of Algorithm~\\ref{alg:kernel_bai}.\n\nLet $\\Phi_{\\lambda}=[\\sqrt{\\lambda_1} \\phi(\\ti{x}_1)^\\top; \\dots; \\sqrt{\\lambda_{nV}} \\phi(\\ti{x}_{nV})^\\top]$ and $K_{\\lambda}=\\Phi_{\\lambda} \\Phi_{\\lambda}^\\top=[\\sqrt{\\lambda_i \\lambda_j} K(\\ti{x}_i,\\ti{x}_j)]_{i,j \\in [nV]}$ for any $\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}$. Let $k_{\\lambda}(\\ti{x}) =\\Phi_{\\lambda} \\phi(\\ti{x})=[\\sqrt{\\lambda_1} K(\\ti{x}, \\ti{x}_1), \\dots, \\sqrt{\\lambda_{nV}} K(\\ti{x}, \\ti{x}_{nV})]^\\top$ for any $\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}$, $\\ti{x} \\in \\ti{\\mathcal{X}}$.\nSince $\\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} } \\phi(\\ti{x}) = \\xi_t \\phi(\\ti{x}) + \\Phi_{\\lambda}^\\top k_{\\lambda}(\\ti{x})$ for any $\\ti{x} \\in \\ti{\\mathcal{X}}$, we have\n\\begin{align*}\n\\phi(\\ti{x}) = & \\xi_t \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1} \\phi(\\ti{x}) + \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1} \\Phi_{\\lambda}^\\top k_{\\lambda}(\\ti{x}) \n\\\\\n= & \\xi_t \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1} \\phi(\\ti{x}) + \\Phi_{\\lambda}^\\top \\sbr{\\xi_t I + K_{\\lambda} }^{-1} k_{\\lambda}(\\ti{x}) \n\\end{align*}\nThus,\n\\begin{align*}\n\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) =  \\xi_t \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1} \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)} + \\Phi_{\\lambda}^\\top \\sbr{\\xi_t I + K_{\\lambda} }^{-1} \\sbr{k_{\\lambda}(\\ti{x}_i) - k_{\\lambda}(\\ti{x}_j)}\n\\end{align*}\nMultiplying $\\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top$ on both sides, we have\n\\begin{align*}\n& \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)} \n\\\\\n= & \\xi_t \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1} \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)} + \\sbr{k_{\\lambda}(\\ti{x}_i)-k_{\\lambda}(\\ti{x}_j)}^\\top \\sbr{\\xi_t I + K_{\\lambda} }^{-1} \\sbr{k_{\\lambda}(\\ti{x}_i) - k_{\\lambda}(\\ti{x}_j)} \n\\end{align*}\nThus,\n\\begin{align*}\n& \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{\\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1}} \n\\\\\n= & \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1}  \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}\n\\\\\n= &\\xi_t^{-1}  \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top  \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)} \n- \\xi_t^{-1} \\sbr{k_{\\lambda}(\\ti{x}_i)-k_{\\lambda}(\\ti{x}_j)}^\\top \\sbr{\\xi_t I + K_{\\lambda} }^{-1} \\sbr{k_{\\lambda}(\\ti{x}_i) - k_{\\lambda}(\\ti{x}_j)}  \n\\\\\n= &\\xi_t^{-1} \\sbr{K(\\ti{x}_i, \\ti{x}_i)+K(\\ti{x}_j, \\ti{x}_j)-2 K(\\ti{x}_i, \\ti{x}_j)}\n- \\xi_t^{-1} \\sbr{k_{\\lambda}(\\ti{x}_i)-k_{\\lambda}(\\ti{x}_j)}^\\top \\sbr{\\xi_t I + K_{\\lambda} }^{-1} \\sbr{k_{\\lambda}(\\ti{x}_i) - k_{\\lambda}(\\ti{x}_j)}  \n\\\\\n= &\\xi_t^{-1} \\sbr{K(\\ti{x}_i, \\ti{x}_i)+K(\\ti{x}_j, \\ti{x}_j)-2 K(\\ti{x}_i, \\ti{x}_j)}\n- \\xi_t^{-1} \\| k_{\\lambda}(\\ti{x}_i)-k_{\\lambda}(\\ti{x}_j) \\|^2_{\\sbr{\\xi_t I + K_{\\lambda} }^{-1}}  \n\\end{align*}\n\nLet $\\lambda_u = \\frac{1}{nV} \\textbf{1}$ be the uniform distribution on $\\ti{\\mathcal{X}}$.\nThen, the condition Eq.~\\eqref{eq:xi_t} for regularization parameter $\\xi_t$\n$$\n\\sqrt{\\xi_t} \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{X}}_v, v \\in [V]} \\| \\phi(\\ti{x}_i) - \\phi(\\ti{x}_j) \\|_{\\sbr{\\xi_t I + \\sum_{x \\in \\mathcal{X}} \\frac{1}{nV} \\phi(x) \\phi(x)^\\top }^{-1}}  \\leq \\frac{1}{ (1+\\varepsilon) B \\cdot 2^{t+1}}\n$$\nis equivalent to the following efficient kernelized statement\n\\begin{align*}\n\\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{X}}_v, v \\in [V]} & \\sqrt{ \\sbr{K(\\ti{x}_i, \\ti{x}_i)+K(\\ti{x}_j, \\ti{x}_j)-2 K(\\ti{x}_i, \\ti{x}_j)}\n\t-  \\| k_{\\lambda_u}(\\ti{x}_i)-k_{\\lambda_u}(\\ti{x}_j) \\|^2_{\\sbr{\\xi_t I + K_{\\lambda_u} }^{-1}} } \\leq \\frac{1}{ (1+\\varepsilon) B \\cdot 2^{t+1}} .\n\\end{align*}\n\n\n\\vspace*{0.4em}\n\\noindent\n\\textbf{Kernelized Optimization Solver.} \nNow we present the efficient kernelized optimization solver for the following convex optimization in Line~\\ref{line:min_max_lambda} of Algorithm~\\ref{alg:kernel_bai}:\n\\begin{align}\n\\min \\limits_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\max \\limits_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}, v \\in [V]}  \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{A(\\xi_t,\\lambda)^{-1}} \\label{eq:min_max_opt} ,\n\\end{align}\nwhere $A(\\xi,\\lambda) = \\xi I + \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top$ for any $\\xi>0, \\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}$.\n\nDefine function $h(\\lambda)=\\max \\limits_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}, v \\in [V]}  \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{A(\\xi_t,\\lambda)^{-1}}$, and define $\\ti{x}^*_i(\\lambda)$, $\\ti{x}^*_j(\\lambda)$ as the optimal solution of $h(\\lambda)$.\nThen, the gradient of $h(\\lambda)$ with respect to $\\lambda$ is\n\\begin{align}\n[\\nabla_{\\lambda} h(\\lambda)]_{\\ti{x}}= - \\sbr{ \\sbr{\\phi(\\ti{x}^*_i(\\lambda))-\\phi(\\ti{x}^*_j(\\lambda))}^\\top A(\\xi_t,\\lambda)^{-1} \\phi(\\ti{x}) }^2, \\forall \\ti{x}\\in \\ti{\\mathcal{X}} . \\label{eq:gradient_h_lambda}\n\\end{align}\nNext, we show how to efficiently compute gradient $[\\nabla_{\\lambda} h(\\lambda)]_{\\ti{x}}$ with kernel function $K(\\cdot,\\cdot)$.\n\nSince $\\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} } \\phi(\\ti{x}) = \\xi_t \\phi(\\ti{x}) + \\Phi_{\\lambda}^\\top k_{\\lambda}(\\ti{x})$ for any $\\ti{x} \\in \\ti{\\mathcal{X}}$, we have\n\\begin{align*}\n\\phi(\\ti{x}) = & \\xi_t \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1} \\phi(\\ti{x}) + \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1} \\Phi_{\\lambda}^\\top k_{\\lambda}(\\ti{x}) \n\\\\\n= & \\xi_t \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1} \\phi(\\ti{x}) + \\Phi_{\\lambda}^\\top \\sbr{\\xi_t I + K_{\\lambda} }^{-1} k_{\\lambda}(\\ti{x}) \n\\end{align*}\nMultiplying $\\sbr{\\phi(\\ti{x}^*_i(\\lambda))-\\phi(\\ti{x}^*_j(\\lambda))}^\\top$ on both sides, we have\n\\begin{align*}\n& \\sbr{\\phi(\\ti{x}^*_i(\\lambda))-\\phi(\\ti{x}^*_j(\\lambda))}^\\top \\phi(\\ti{x}) \n\\\\\n= & \\xi_t \\sbr{\\phi(\\ti{x}^*_i(\\lambda))-\\phi(\\ti{x}^*_j(\\lambda))}^\\top \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1} \\phi(\\ti{x}) + \\sbr{k_{\\lambda}(\\ti{x}^*_i(\\lambda))-k_{\\lambda}(\\ti{x}^*_j(\\lambda))}^\\top \\sbr{\\xi_t I + K_{\\lambda} }^{-1} k_{\\lambda}(\\ti{x}) \n\\end{align*}\nThen,\n\\begin{align}\n& \\sbr{\\phi(\\ti{x}^*_i(\\lambda))-\\phi(\\ti{x}^*_j(\\lambda))}^\\top \\sbr{\\xi_t I + \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} }^{-1} \\phi(\\ti{x})\n\\nonumber\\\\\n= &\\xi_t^{-1}  \\sbr{\\phi(\\ti{x}^*_i(\\lambda))-\\phi(\\ti{x}^*_j(\\lambda))}^\\top \\phi(\\ti{x})  - \\xi_t^{-1} \\sbr{k_{\\lambda}(\\ti{x}^*_i(\\lambda))-k_{\\lambda}(\\ti{x}^*_j(\\lambda))}^\\top \\sbr{\\xi_t I + K_{\\lambda} }^{-1} k_{\\lambda}(\\ti{x})  \n\\nonumber\\\\\n= & \\xi_t^{-1} \\sbr{ K(\\ti{x}^*_i(\\lambda), \\ti{x})- K(\\ti{x}^*_j(\\lambda), \\ti{x})  -  \\sbr{k_{\\lambda}(\\ti{x}^*_i(\\lambda))-k_{\\lambda}(\\ti{x}^*_j(\\lambda))}^\\top \\sbr{\\xi_t I + K_{\\lambda} }^{-1} k_{\\lambda}(\\ti{x}) }\n\\label{eq:apx_kernelized_gradient}\n\\end{align}\nTherefore, we can compute gradient $\\nabla_{\\lambda} h(\\lambda)$ (Eq.~\\eqref{eq:gradient_h_lambda}) using the equivalent kernelized expression Eq.~\\eqref{eq:apx_kernelized_gradient}, and then the optimization (Eq.~\\eqref{eq:min_max_opt}) can be  efficiently solved by projected gradient descent.\n\n\n\n\\vspace*{0.4em}\n\\noindent\n\\textbf{Innovative Kernelized Estimator.}\nFinally, we explicate the innovative kernelized estimator of reward gaps in Line~\\ref{line:estimated_reward_gap} of Algorithm~\\ref{alg:kernel_bai}, which plays an important role in boosting the computation and communication efficiency.\n\nLet $\\hat{\\theta}_t$ denote the minimizer of the following regularized least square loss function:\n$$\n\\mathcal{L}(\\theta)= N^{(t)} \\xi_t \\| \\theta \\|^2 + \\sum_{j=1}^{N^{(t)}} (y_j - \\phi(\\ti{s}_j)^\\top \\theta)^2 .\n$$\nLetting the derivative of $\\mathcal{L}(\\theta)$ equal to zero, we have \n\\begin{align*}\nN^{(t)} \\xi_t \\hat{\\theta}_t + \\sum_{j=1}^{N^{(t)}}  \\phi(x_j) \\phi(x_j)^\\top \\hat{\\theta}_t = \\sum_{j=1}^{N^{(t)}} \\phi(x_j) y_j .\n\\end{align*}\nRearranging the summation, we can obtain\n\\begin{align}\nN^{(t)} \\xi_t \\hat{\\theta}_t + \\sbr{\\sum_{i=1}^{nV} N^{(t)}_i \\phi(\\ti{x}_i) \\phi(\\ti{x}_i)^\\top} \\hat{\\theta}_t  = & \\sum_{i=1}^{nV} N^{(t)}_i \\phi(\\ti{x}_i) \\bar{y}^{(t)}_{i}, \\label{eq:derivative_zero}\n\\end{align}\nwhere $N^{(t)}_i$ is the number of samples and $\\bar{y}^{(t)}_i$ is the average observation on arm $\\ti{x}_i$ for any $i \\in [nV]$. \nLet $\\Phi_t=[\\sqrt{N^{(t)}_1} \\phi(\\ti{x}_1)^\\top; \\dots; \\sqrt{N^{(t)}_{nV}} \\phi(\\ti{x}_{nV})^\\top]$, $K^{(t)}=\\Phi_t \\Phi_t^\\top=[\\sqrt{N^{(t)}_i N^{(t)}_j} K(\\ti{x}_i,\\ti{x}_j)]_{i,j \\in [nV]}$ and $\\bar{y}^{(t)}=[ \\sqrt{N^{(t)}_1} \\bar{y}^{(t)}_1, \\dots, \\sqrt{N^{(t)}_{nV}} \\bar{y}^{(t)}_{nV} ]^\\top$.\nThen, we can write Eq.~\\eqref{eq:derivative_zero} as\n\\begin{align*}\n\\sbr{N^{(t)} \\xi_t I  + \\Phi_t^\\top \\Phi_t } \\hat{\\theta}_t = & \\Phi_t^\\top \\bar{y}^{(t)} .\n\\end{align*}\nSince $\\sbr{N^{(t)} \\xi_t I + \\Phi_t^\\top \\Phi_t } \\succ 0$ and $\\sbr{N^{(t)} \\xi_t I + \\Phi_t \\Phi_t^\\top } \\succ 0$,\n\\begin{align*}\n\\hat{\\theta}_t & = \\sbr{N^{(t)} \\xi_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\bar{y}^{(t)}\n\\\\\n& = \\Phi_t^\\top \\sbr{N^{(t)} \\xi_t I + \\Phi_t \\Phi_t^\\top }^{-1} \\bar{y}^{(t)} \n\\\\\n& = \\Phi_t^\\top \\sbr{N^{(t)} \\xi_t I + K^{(t)} }^{-1} \\bar{y}^{(t)} .\n\\end{align*}\nLet $k_t(\\ti{x})=\\Phi_t \\phi(\\ti{x})=[\\sqrt{N^{(t)}_1} K(\\ti{x}, \\ti{x}_1), \\dots, \\sqrt{N^{(t)}_{nV}} K(\\ti{x}, \\ti{x}_{nV})]^\\top$ for any $\\ti{x} \\in \\mathcal{X}$.\nThen, we obtain the efficient kernelized estimators of $f(\\ti{x}_i)$ and $f(\\ti{x}_i)-f(\\ti{x}_j)$ as \n\\begin{align*}\n\\hat{f}(\\ti{x}_i) & = \\phi(\\ti{x}_i)^\\top \\hat{\\theta}_t\n=  k_t(\\ti{x}_i)^\\top \\sbr{N^{(t)} \\xi_t I + K^{(t)} }^{-1}  \\bar{y}^{(t)} , \n\\\\\n\\hat{\\Delta}(\\ti{x}_i,\\ti{x}_j) \n& = \\sbr{k_t(\\ti{x}_i)- k_t(\\ti{x}_j)}^\\top \\sbr{N^{(t)} \\xi_t I + K^{(t)} }^{-1}  \\bar{y}^{(t)} . \n\\end{align*}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:coop_kernel_bai_ub}}\nOur proof of Theorem~\\ref{thm:coop_kernel_bai_ub} adapts the analysis procedure of \\cite{rage2019,peace2020} to the multi-agent setting.\n\nFor any $\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}$, let $\\Phi_{\\lambda}=[\\sqrt{\\lambda_1} \\phi(x^*_v)^\\top; \\dots; \\sqrt{\\lambda_{nV}} \\phi(\\ti{x}_{nV})^\\top]$ and $\\Phi^\\top_{\\lambda} \\Phi_{\\lambda}=\\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top$.\nIn order to prove Theorem~\\ref{thm:coop_kernel_bai_ub}, we first introduce the following lemmas.\n\n\\begin{lemma}[Concentration] \\label{lemma:concentration}\n\tDefining event\n\t\\begin{align*}\n\t\\mathcal{G} = \\Bigg\\{ & \\abr{ \\sbr{\\hat{f}_t(\\ti{x}_i)-\\hat{f}_t(\\ti{x}_j)} - \\sbr{f(\\ti{x}_i)-f(\\ti{x}_j)} }  \n\t< (1+\\varepsilon) \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_t I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} \\cdot \\\\& \\sbr{\\sqrt{ \\frac{2 \\log \\sbr{ 2 n^2 V \/ \\delta_t} }{N_t} } + \\sqrt{\\xi_t} \\cdot \\| \\theta^* \\|_2 } \\leq 2^{-t} , \\  \\forall \\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}, \\  \\forall v \\in [V], \\  \\forall t \\geq 1 \\Bigg\\} ,\n\t\\end{align*}\n\twe have \n\t\\begin{align*}\n\t\\Pr \\mbr{ \\mathcal{G} } \\geq 1 - \\delta .\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:concentration}]\n\tLet $\\hat{\\theta}_t$ be the regularized least square estimator of $\\theta^*$ with samples $x^*_v, \\dots, \\ti{x}_{N_t}$ and $\\gamma_t = N_t \\xi_t$. \n\t\n\tRecall that $\\Phi_t=[\\sqrt{N^{(t)}_1} \\phi(x^*_v)^\\top; \\dots; \\sqrt{N^{(t)}_{nV}} \\phi(\\ti{x}_{nV})^\\top]$ and $\\Phi_t^\\top \\Phi_t = \\sum_{i=1}^{nV} N^{(t)}_i \\phi(\\ti{x}_i) \\phi(\\ti{x}_i)^\\top$.\n\t\n\tIn addition, $\\bar{y}^{(t)}=[ \\sqrt{N^{(t)}_1} \\bar{y}^{(t)}_1, \\dots, \\sqrt{N^{(t)}_{nV}} \\bar{y}^{(t)}_{nV} ]^\\top$ and $\\hat{\\theta}_t = \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\bar{y}^{(t)}$.\n\t\n\tLet $\\bar{\\eta}^{(t)}=[ \\sqrt{N^{(t)}_1} \\bar{\\eta}^{(t)}_1, \\dots, \\sqrt{N^{(t)}_{nV}} \\bar{\\eta}^{(t)}_{nV} ]^\\top$, where $\\bar{\\eta}^{(t)}_i=\\bar{y}^{(t)}_i-\\phi(\\ti{x}_i)^\\top \\theta^*$ denote the average noise of the $N^{(t)}_i$ pulls on arm $\\ti{x}_{i}$ for any $i \\in [nV]$.\n\t\n\tThen,\n\t\\begin{align}\n\t& \\sbr{\\hat{f}_t(\\ti{x}_i)-\\hat{f}_t(\\ti{x}_j)} - \\sbr{f(\\ti{x}_i)-f(\\ti{x}_j)} \n\t\\nonumber\\\\\n\t= & \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{\\hat{\\theta}_t-\\theta^*}\n\t\\nonumber\\\\\n\t= & \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{ \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\bar{y}^{(t)} -\\theta^*}\n\t\\nonumber\\\\\n\t= & \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{ \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\sbr{ \\Phi_t \\theta^* + \\bar{\\eta}^{(t)}} -\\theta^*}\n\t\\nonumber\\\\\n\t= & \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{ \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\Phi_t \\theta^* + \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\bar{\\eta}^{(t)} - \\theta^*}\n\t\\nonumber\\\\\n\t= & \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\Big( \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\sbr{\\Phi_t^\\top \\Phi_t + \\gamma_t I} \\theta^* + \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\bar{\\eta}^{(t)} \\\\& - \\theta^*-  \\gamma_t  \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\theta^* \\Big)\n\t\\nonumber\\\\\n\t= & \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top  \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\bar{\\eta}^{(t)} -  \\gamma_t \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\theta^* \\label{eq:concentration_decompose}\n\t\\end{align}\n\tSince the mean of the first term is zero and its variance is bounded by \n\t\\begin{align*} \n\t& \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top  \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top  \\Phi_t \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}  \n\t\\\\ \n\t\\leq & \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top  \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t } \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}  \n\t\\\\ \n\t= & \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top  \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1}  \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)} \n\t\\\\ \n\t= & \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{\\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1}} ,\n\t\\end{align*}\n\tusing the Hoeffding inequality, we have\n\tthat with probability at least $1-\\frac{\\delta_t}{n^2 V}$,\n\t\\begin{align*}\n\t\\abr{ \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top  \\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\eta_v^{(t)} } < \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1}} \\sqrt{ 2 \\log \\sbr{ \\frac{2 n^2 V}{\\delta_t} } }\n\t\\end{align*}\n\tThus, with probability at least $1-\\frac{\\delta_t}{n^2 V}$,\n\t\\begin{align*}\n\t& \\abr{ \\sbr{\\hat{f}_t(\\ti{x}_i)-\\hat{f}_t(\\ti{x}_j)} - \\sbr{f(\\ti{x}_i)-f(\\ti{x}_j)} }\n\t\\\\\n\t< & \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1}} \\sqrt{ 2 \\log \\sbr{ \\frac{2 n^2 V}{\\delta_t} } } + \\gamma_t \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1}}  \\| \\theta^* \\|_{\\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1}} \n\t\\\\\n\t\\leq & \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1}} \\sqrt{ 2 \\log \\sbr{ \\frac{2 n^2 V}{\\delta_t} } } + \\sqrt{\\gamma_t} \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\gamma_t I + \\Phi_t^\\top \\Phi_t }^{-1}} \\| \\theta^* \\|_2 \n\t\\\\\n\t\\overset{\\textup{(a)}}{\\leq} &  \\frac{ (1+\\varepsilon) \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_t I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} }{ \\sqrt{N_t} } \\sqrt{ 2 \\log \\sbr{ \\frac{2 n^2 V}{\\delta_t} } } \\\\& + \\sqrt{\\xi_t N_t} \\cdot \\frac{ (1+\\varepsilon) \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_t I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} }{ \\sqrt{N_t} }  \\cdot \\| \\theta^* \\|_2 \n\t\\\\\n\t= &   (1+\\varepsilon) \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_t I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}}  \\sqrt{ \\frac{2 \\log \\sbr{ 2 n^2 V \/ \\delta_t} }{N_t} } \\\\& + (1+\\varepsilon) \\sqrt{\\xi_t} \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_t I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}}  \\| \\theta^* \\|_2 \n\t\\\\\n\t\\leq &   (1+\\varepsilon)  \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{B}}^{(t)}_v, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_t I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}}  \\sqrt{ \\frac{2 \\log \\sbr{ 2 n^2 V \/ \\delta_t} }{N_t} } \\\\& + (1+\\varepsilon)  \\sqrt{\\xi_t} \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{B}}^{(t)}_v, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_t I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} \\| \\theta^* \\|_2 ,\n\t\\end{align*}\n\twhere (a) is due to the rounding procedure.\n\t\n\tAccording to the choice of $\\xi_t$, it holds that\n\t\\begin{align*}\n\t& (1+\\varepsilon) \\sqrt{\\xi_t}  \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{B}}^{(t)}_v, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\xi_t I + \\Phi_{\\lambda^*_t}^\\top \\Phi_{\\lambda^*_t} }^{-1}}  \\| \\theta^* \\|_2\n\t\\\\\n\t\\leq & (1+\\varepsilon) \\sqrt{\\xi_t}  \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{B}}^{(t)}_v, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\xi_t I + \\Phi_{\\lambda_u}^\\top \\Phi_{\\lambda_u} }^{-1}}  \\| \\theta^* \\|_2\n\t\\\\\n\t\\leq & (1+\\varepsilon) \\sqrt{\\xi_t}  \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{X}}_v, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\xi_t I + \\Phi_{\\lambda_u}^\\top \\Phi_{\\lambda_u} }^{-1}} \\cdot B\n\n\n\t\\\\\n\t\\leq & \\frac{1}{2^{t+1}} .\n\t\\end{align*}\n\tThus, with probability at least $1-\\frac{\\delta_t}{n^2 V}$,\n\t\\begin{align*}\n\t& \\abr{ \\sbr{\\hat{f}_t(\\ti{x}_i)-\\hat{f}_t(\\ti{x}_j)} - \\sbr{f(\\ti{x}_i)-f(\\ti{x}_j)} }\n\t\\\\\n\t< & (1+\\varepsilon)  \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{B}}^{(t)}_v, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_t I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}}  \\sqrt{ \\frac{2 \\log \\sbr{ 2 n^2 V \/ \\delta_t} }{N_t} }  + \\frac{1}{2^{t+1}}\n\t\\\\\n\t= &   \\sqrt{ \\frac{2 (1+\\varepsilon)^2 \\rho^*_t \\log \\sbr{ 2  n^2 V \/ \\delta_t} }{N_t} } + \\frac{1}{2^{t+1}}\n\t\\\\\n\t\\leq & \\frac{1}{2^{t+1}} + \\frac{1}{2^{t+1}}\n\t\\\\\n\t= & \\frac{1}{2^t}\n\t\\end{align*}\n\t\n\t\n\tBy a union bound over arms $\\ti{x}_i,\\ti{x}_j$, agent $v$ and phase $t$, we have that\n\t\\begin{align*}\n\t\\Pr \\mbr{ \\mathcal{G} } \\geq 1 - \\delta .\n\t\\end{align*}\n\\end{proof}\n\n\nFor any $t>1$ and $v \\in [V]$, let $\\mathcal{S}_v^{(t)}=\\{ \\ti{x} \\in \\ti{\\mathcal{X}}_v: f(\\ti{x}^*_{v})- f(\\ti{x}) \\leq  2^{-t+2} \\}$.\n\\begin{lemma} \\label{lemma:induction}\n\tAssume that event $\\mathcal{G}$ occurs. Then, for any phase $t>1$ and agent $v \\in [V]$, we have that $\\ti{x}^*_{v} \\in \\mathcal{B}_v^{(t)}$ and $\\mathcal{B}_v^{(t)} \\subseteq \\mathcal{S}_v^{(t)}$.\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:induction}]\n\tWe prove the first statement by induction.\n\t\n\tTo begin, for any $v \\in [V]$, $\\ti{x}^*_{v} \\in \\mathcal{B}_v^{(1)}$ trivially holds.\n\t\n\tSuppose that $\\ti{x}^*_{v} \\in \\mathcal{B}_v^{(t)}$ holds for any $v \\in [V]$, and there exists some $v' \\in [V]$ such that $\\ti{x}^*_{v'} \\notin \\mathcal{B}_{v'}^{(t+1)}$. According to the elimination rule of algorithm $\\textsf{CoopKernelFC}$, we have that these exists some $\\ti{x}' \\in \\mathcal{B}_v^{(t)}$ such that \n\t$$\n\t\\hat{f}_t(\\ti{x}')-\\hat{f}_t(\\ti{x}^*_{v'}) \\geq 2^{-t} .\n\t$$\n\tUsing Lemma~\\ref{lemma:concentration}, we have\n\t$$\n\tf(\\ti{x}')-f(\\ti{x}^*_{v'}) > \\hat{f}_t(\\ti{x}')-\\hat{f}_t(\\ti{x}^*_{v'})-2^{-t} \\geq 0 ,\n\t$$\n\twhich contradicts the definition of $\\ti{x}^*_{v'}$.\n\tThus, we have that for any $v \\in [V]$, $\\ti{x}^*_{v} \\in \\mathcal{B}_v^{(t+1)}$, which completes the proof of the first statement.\n\t\n\tNow, we prove the second statement also by induction.\n\t\n\tTo begin, we prove that for any $v \\in [V]$, $\\mathcal{B}_v^{(2)} \\subseteq \\mathcal{S}_v^{(2)}$.\n\tSuppose that there exists some $v' \\in [V]$ such that $\\mathcal{B}_{v'}^{(2)} \\subsetneq \\mathcal{S}_{v'}^{(2)}$. Then, there exists some $\\ti{x}' \\in \\mathcal{B}_{v'}^{(2)}$ such that $f(\\ti{x}^*_{v'})- f(\\ti{x}') > 2^{-2+2} = 1$.\n\tUsing Lemma~\\ref{lemma:concentration}, we have that at the phase $t=1$,\n\t\\begin{align*}\n\t\\hat{f}_t(\\ti{x}^*_{v'}) - \\hat{f}_t(\\ti{x}') \\geq f(\\ti{x}^*_{v'})- f(\\ti{x}') - 2^{-1} > 1- 2^{-1}=2^{-1},\n\t\\end{align*}\n\twhich implies that $\\ti{x}'$ should have been eliminated in phase $t=1$ and gives a contradiction.\n\t\n\tSuppose that $\\mathcal{B}_v^{(t)} \\subseteq \\mathcal{S}_v^{(t)}$ ($t > 1$) holds for any $v \\in [V]$, and there exists some $v' \\in [V]$ such that $\\mathcal{B}_{v'}^{(t+1)} \\subsetneq \\mathcal{S}_{v'}^{(t+1)}$. \n\tThen, there exists some $\\ti{x}' \\in \\mathcal{B}_{v'}^{(t+1)}$ such that $f(\\ti{x}^*_{v'})- f(\\ti{x}') > 2^{-(t+1)+2}=2 \\cdot 2^{-t} $.\n\tUsing Lemma~\\ref{lemma:concentration}, we have that at the phase $t$,\n\t\\begin{align*}\n\t\\hat{f}_t(\\ti{x}^*_{v'}) - \\hat{f}_t(\\ti{x}') \\geq f(\\ti{x}^*_{v'})- f(\\ti{x}') - 2^{-t} > 2 \\cdot 2^{-t} - 2^{-t}=2^{-t},\n\t\\end{align*}\n\twhich implies that $\\ti{x}'$ should have been eliminated in phase $t$ and gives a contradiction. Thus, we complete the proof of Lemma~\\ref{lemma:induction}.\n\\end{proof}\n\n\nNow we prove Theorem~\\ref{thm:coop_kernel_bai_ub}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:coop_kernel_bai_ub}]\n\tWe first prove the correctness.\n\t\n\tLet $t^*=\\left \\lceil \\log_2 \\Delta^{-1}_{\\min} \\right \\rceil + 1$ be the index of the last phase of algorithm $\\textsf{CoopKernelFC}$. \n\tAccording to Lemma~\\ref{lemma:induction}, when $t = t^*$, $\\mathcal{B}_v^{(t)}=\\{ \\ti{x}^*_{v} \\}$ holds for any $v \\in [V]$, and thus algorithm $\\textsf{CoopKernelFC}$ returns the correct answer $\\ti{x}^*_{v}$ for all $v \\in [V]$.\n\t\n\tNext, we prove the sample complexity. \n\t\n\t\n\tIn algorithm $\\textsf{CoopKernelFC}$, the computation of $\\lambda^*_t, \\rho^*_t$ and $N_t$ is the same for all agents, and each agent $v$ just generates partial samples that belong to his arm set $\\ti{\\mathcal{X}}_v$ from the total $N_t$ samples. Hence, to bound the overall sample complexity, it suffices to bound $\\sum_{t=1}^{t^*} N_t$, and then we can obtain the per-agent sample complexity by dividing $V$.\n\tLet $\\varepsilon=0.1$. We have\n\t\\begin{align*}\n\t& \\sum_{t=1}^{t^*} N_t \n\t\\\\\n\t= & \\sum_{t=1}^{t^*} \\sbr{8 (2^t)^2 (1+\\varepsilon)^2 \\rho^*_t \\log \\sbr{ \\frac{2  n^2 V }{\\delta_t} } +1 }\n\t\\\\\n\t= & \\sum_{t=2}^{t^*} 8 (2^t)^2 \\sbr{2^{-t+2}}^2 (1+\\varepsilon)^2 \\frac{ \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{\\sbr{\\xi_t I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }{ \\sbr{2^{-t+2}}^2 }  \\log \\sbr{ \\frac{4 V n^2 t^2 }{\\delta} } \\\\&  + N_1 + t^*\n\t\\\\\n\t\\leq & \\sum_{t=2}^{t^*} \\sbr{128 (1+\\varepsilon)^2 \\frac{ \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x} \\in \\mathcal{B}_v^{(t)}, v \\in [V]} \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|^2_{\\sbr{\\xi_t I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}}  }{ \\sbr{2^{-t+2}}^2 } \\log \\sbr{ \\frac{4 V n^2 (t^*)^2 }{\\delta} }  } + N_1 + t^*\n\t\\\\\n\t\\leq & \\sum_{t=2}^{t^*} \\sbr{128 (1+\\varepsilon)^2 \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x} \\in \\mathcal{B}_v^{(t)}, v \\in [V]} \\frac{  \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|^2_{\\sbr{\\xi_t I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }{ \\sbr{f(\\ti{x}^*_{v})- f(\\ti{x})}^2 } \\log \\sbr{ \\frac{4 V n^2 (t^*)^2 }{\\delta} }  } + N_1 + t^*\n\t\\\\\n\t\\leq & \\sum_{t=2}^{t^*} \\sbr{128 (1+\\varepsilon)^2 \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x} \\in \\ti{\\mathcal{X}}_v, v \\in [V]} \\frac{  \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|^2_{\\sbr{\\xi_* I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }{ \\sbr{f(\\ti{x}^*_{v})- f(\\ti{x})}^2 } \\log \\sbr{ \\frac{4 V n^2 (t^*)^2 }{\\delta} }  } + N_1 + t^*\n\t\\\\\n\t\\leq & t^* \\cdot \\sbr{128 (1+\\varepsilon)^2 \\rho^* \\log \\sbr{ \\frac{4 V n^2 (t^*)^2 }{\\delta} }  } + N_1 + t^*\n\t\\\\\n\t= & O \\sbr{   \\rho^* \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   }\n\t\\end{align*}\n\tThus, the per-agent sample complexity is bounded by\n\t\\begin{align*}\n\tO \\sbr{   \\frac{ \\rho^*}{V} \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   } .\n\t\\end{align*}\n\tSince algorithm $\\textsf{CoopKernelFC}$ has at most $t^*=\\left \\lceil \\log_2 \\Delta^{-1}_{\\min} \\right \\rceil + 1$ phases, the number of communication rounds is bounded by $O(\\log \\Delta^{-1}_{\\min})$.\n\\end{proof}\n\n\\subsection{Proof of Corollary~\\ref{corollary:ub_fc}}\n\n\\begin{proof}[Proof of Corollary~\\ref{corollary:ub_fc}]\nRecall that $K_{\\lambda}=\\Phi_{\\lambda} \\Phi_{\\lambda}^\\top$ and $\\lambda^* = \\operatornamewithlimits{argmax}_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\log\\det \\sbr{ I + {\\xi_*}^{-1} K_{\\lambda} }$.\nWe have that $\\log\\det \\sbr{ I + {\\xi_*}^{-1} K_{\\lambda} }=\\log\\det \\sbr{ I + {\\xi_*}^{-1} \\Phi_{\\lambda}^\\top \\Phi_{\\lambda} } =\\log \\det \\sbr{  I + {\\xi_*}^{-1} \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}} } \\lambda_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }$. \nThen,\n\\begin{align*}\n\\rho^* = & \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x} \\in \\ti{\\mathcal{X}}_v, v \\in [V]} \\frac{  \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|^2_{\\sbr{\\xi_* I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }{ \\sbr{f(\\ti{x}^*_{v})- f(\\ti{x})}^2 }\n\\\\\n\\leq & \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x} \\in \\ti{\\mathcal{X}}_v, v \\in [V]} \\frac{  \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|^2_{\\sbr{\\xi_* I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }{ \\Delta^2_{\\min} }\n\\\\\n= & \\frac{1}{ \\Delta^2_{\\min} } \\cdot \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x} \\in \\ti{\\mathcal{X}}_v, v \\in [V]}   \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|^2_{\\sbr{\\xi_* I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} \n\\\\\n\\leq & \\frac{1}{ \\Delta^2_{\\min} } \\cdot \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}}   \\sbr{ 2 \\max_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\| \\phi(\\ti{x}) \\|_{\\sbr{\\xi_* I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }^2 \n\\\\\n= &  \\frac{4}{ \\Delta^2_{\\min} } \\cdot \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x} \\in \\ti{\\mathcal{X}}}  \\| \\phi(\\ti{x}) \\|^2_{\\sbr{  \\xi_* I +  \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}}} \\lambda_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }^{-1}}\n\\\\\n= &  \\frac{4}{ \\Delta^2_{\\min} } \\cdot \\max_{\\ti{x} \\in \\ti{\\mathcal{X}}}  \\| \\phi(\\ti{x}) \\|^2_{\\sbr{  \\xi_* I +  \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }^{-1}}\n\\\\\n\\overset{\\textup{(b)}}{=} &  \\frac{4}{ \\Delta^2_{\\min} } \\cdot  \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}}  \\| \\phi(\\ti{x}) \\|^2_{\\sbr{  \\xi_* I +  \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }^{-1}} ,\n\\end{align*}\nwhere (b) is due to Lemma~\\ref{lemma:lambda_*_max_sum}.\n\n\nSince $\\lambda^*_{\\ti{x}}  \\| \\phi(\\ti{x}) \\|^2_{\\sbr{  \\xi_* I +  \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }^{-1}} \\leq 1$ for any $\\ti{x} \\in \\mathcal{X}$,\n\\begin{align*}\n& \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}}  \\| \\phi(\\ti{x}) \\|^2_{\\sbr{  \\xi_* I +  \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }^{-1}}\n\\\\\n\\leq & 2 \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\log \\sbr{ 1 + \\lambda^*_{\\ti{x}}  \\| \\phi(\\ti{x}) \\|^2_{\\sbr{  \\xi_* I +  \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }^{-1}} }\n\\\\\n\\overset{\\textup{(c)}}{\\leq} & 2 \\log \\frac{\\det \\sbr{ \\xi_* I +  \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top  }}{ \\det \\sbr{ \\xi_* I } }\n\\\\\n= & 2 \\log \\det \\sbr{  I + \\xi_*^{-1} \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top  }\n\\\\\n= & 2 \\log \\det \\sbr{  I + \\xi_*^{-1} K_{\\lambda^*} } ,\n\\end{align*}\nwhere (c) comes from Lemma~\\ref{lemma:sum_ln_det}.\n\nThus, we have\n\\begin{align}\n\\rho^* = & \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x} \\in \\ti{\\mathcal{X}}_v, v \\in [V]} \\frac{  \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|^2_{\\sbr{\\xi_* I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }{ \\sbr{f(\\ti{x}^*_{v})- f(\\ti{x})}^2 }\n\\nonumber\\\\\n\\leq &  \\frac{4}{ \\Delta^2_{\\min} } \\cdot  \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}}  \\| \\phi(\\ti{x}) \\|^2_{\\sbr{  \\xi_* I +  \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }^{-1}}\n\\nonumber\\\\\n\\leq & \\frac{ 8 }{ \\Delta^2_{\\min} } \\cdot  \\log \\det \\sbr{  I + \\xi_*^{-1} K_{\\lambda^*} }   \\label{eq:ln_det_K_lambda}\n\\end{align}\n\nIn the following, we interpret the term $\\log \\det \\sbr{  I + \\xi_*^{-1} K_{\\lambda^*} }$ using two standard expressive tools, i.e., maximum information gain and effective dimension, respectively.\n\n\\vspace*{0.4em}\n\\noindent\n\\textbf{Maximum Information Gain.}\nRecall that the maximum information gain over all sample allocation $\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}$ is defined as\n$$\n\\Upsilon = \\max_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\log\\det\\sbr{ I + {\\xi_*}^{-1} K_{\\lambda} } .\n$$\nThen, using Eq.~\\eqref{eq:ln_det_K_lambda} and the definitions of $\\lambda_*$, the per-agent sample complexity is bounded by\n\\begin{align*}\n& O \\sbr{   \\frac{ \\rho^*}{V} \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   } \n\\\\\n= & O \\sbr{   \\frac{ \\log\\det\\sbr{ I + {\\xi_*}^{-1} K_{\\lambda_*} } }{\\Delta^2_{\\min} V} \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   } \n\\\\\n= & O \\sbr{   \\frac{ \\Upsilon}{\\Delta^2_{\\min} V} \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   }\n\\end{align*}\n\n\\vspace*{0.4em}\n\\noindent\n\\textbf{Effective Dimension.}\nRecall that $\\alpha_1 \\geq \\dots \\geq \\alpha_{nV}$ denote the eigenvalues of $K_{\\lambda^*}$ in decreasing order. The effective dimension of $K_{\\lambda^*}$ is defined as\n$$\nd_{\\textup{eff}}=\\min \\bigg\\{ j :j \\xi_*\\log(nV) \\geq \\sum_{i=j+1}^{nV} \\alpha_i \\bigg\\} ,\n$$\nand it holds that $d_{\\textup{eff}} \\xi_* \\log(nV) \\geq \\sum_{i=d_{\\textup{eff}}+1}^{nV} \\alpha_i$. \n\nLet $\\varepsilon=d_{\\textup{eff}} \\xi_* \\log(nV)-\\sum_{i=d_{\\textup{eff}}+1}^{nV} \\alpha_i $, and thus $\\varepsilon \\leq d_{\\textup{eff}} \\xi_* \\log(nV)$.\nThen, we have $\\sum_{i=1}^{d_{\\textup{eff}}} \\alpha_i = \\textup{Trace}(K_{\\lambda^*}) - \\sum_{i=d_{\\textup{eff}}+1}^{nV} \\alpha_i = \\textup{Trace}(K_{\\lambda^*}) - d_{\\textup{eff}} \\xi_* \\log(nV) + \\varepsilon$ and $\\sum_{i=d_{\\textup{eff}}+1}^{nV} \\alpha_i=d_{\\textup{eff}} \\xi_* \\log(nV)-\\varepsilon$.\n\\begin{align*}\n& \\log \\det \\sbr{  I + \\xi_*^{-1} K_{\\lambda^*} } \n\\\\\n= & \\log \\sbr{  \\Pi_{i=1}^{nV} \\sbr{1 + \\xi_*^{-1} \\alpha_i} }\n\\\\\n= & \\log \\sbr{  \\Pi_{i=1}^{d_{\\textup{eff}}} \\sbr{1 + \\xi_*^{-1} \\alpha_i} \\cdot \\Pi_{i=d_{\\textup{eff}}+1}^{nV} \\sbr{1 + \\xi_*^{-1} \\alpha_i} }\n\\\\\n\\leq & \\log \\sbr{ \\sbr{  1 + \\xi_*^{-1} \\cdot \\frac{\\textup{Trace}(K_{\\lambda^*}) - d_{\\textup{eff}} \\xi_* \\log(nV) + \\varepsilon}{d_{\\textup{eff}}}}^{d_{\\textup{eff}}} \\sbr{  1 + \\xi_*^{-1} \\cdot \\frac{d_{\\textup{eff}} \\xi_* \\log(nV)-\\varepsilon}{nV-d_{\\textup{eff}}}}^{nV-d_{\\textup{eff}}} }\n\\\\\n\\leq & {d_{\\textup{eff}}} \\log \\sbr{ 1 + \\xi_*^{-1} \\cdot \\frac{\\textup{Trace}(K_{\\lambda^*}) - d_{\\textup{eff}} \\xi_* \\log(nV) + \\varepsilon}{d_{\\textup{eff}}} } + \\log \\sbr{  1 +   \\frac{d_{\\textup{eff}}  \\log(nV)}{nV-d_{\\textup{eff}}}}^{nV-d_{\\textup{eff}}} \n\\\\\n= & {d_{\\textup{eff}}} \\log \\sbr{ 1 + \\xi_*^{-1} \\cdot \\frac{\\textup{Trace}(K_{\\lambda^*}) - d_{\\textup{eff}} \\xi_* \\log(nV) + \\varepsilon}{d_{\\textup{eff}}} } + \\log \\sbr{  1 +   \\frac{d_{\\textup{eff}}  \\log(nV-d_{\\textup{eff}} + d_{\\textup{eff}})}{nV-d_{\\textup{eff}}} }^{nV-d_{\\textup{eff}}} \n\\\\\n\\overset{\\textup{(d)}}{\\leq} & {d_{\\textup{eff}}} \\log \\sbr{ 1 + \\xi_*^{-1} \\cdot \\frac{\\textup{Trace}(K_{\\lambda^*}) - d_{\\textup{eff}} \\xi_* \\log(nV) + \\varepsilon}{d_{\\textup{eff}}} } + \\log \\sbr{  1 +   \\frac{d_{\\textup{eff}}  \\log(nV + d_{\\textup{eff}})}{nV} }^{nV} \n\\\\\n= & {d_{\\textup{eff}}} \\log \\sbr{ 1 + \\xi_*^{-1} \\cdot \\frac{\\textup{Trace}(K_{\\lambda^*}) - d_{\\textup{eff}} \\xi_* \\log(nV) + \\varepsilon}{d_{\\textup{eff}}} } + nV \\log \\sbr{  1 +   \\frac{d_{\\textup{eff}}  \\log(nV + d_{\\textup{eff}})}{nV} }\n\\\\\n\\leq & {d_{\\textup{eff}}} \\log \\sbr{ 1 + \\frac{ \\textup{Trace}(K_{\\lambda^*})  }{ \\xi_* d_{\\textup{eff}} } } +  d_{\\textup{eff}}  \\log(nV + d_{\\textup{eff}}) \n\\\\\n\\leq & {d_{\\textup{eff}}} \\log \\sbr{ 2 nV \\cdot \\sbr{ 1 + \\frac{ \\textup{Trace}(K_{\\lambda^*})  }{ \\xi_* d_{\\textup{eff}} } } },\n\\end{align*}\nwhere inequality (d) is due to that $\\sbr{  1 +   \\frac{d_{\\textup{eff}}  \\log(x + d_{\\textup{eff}})}{x} }^{x} $ is monotonically increasing with respect to $x \\geq 1$.\n\nThen, using Eq.~\\eqref{eq:ln_det_K_lambda}, the per-agent sample complexity is bounded by\n\\begin{align*}\n& O \\sbr{   \\frac{ \\rho^*}{V} \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   } \n\\\\\n= & O \\sbr{   \\frac{ \\log\\det\\sbr{ I + {\\xi_*}^{-1} K_{\\lambda_*} } }{\\Delta^2_{\\min} V} \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   } \n\\\\\n= & O \\sbr{   \\frac{ d_{\\textup{eff}} }{\\Delta^2_{\\min} V} \\cdot  \\log \\sbr{ nV \\cdot \\sbr{ 1 + \\frac{ \\textup{Trace}(K_{\\lambda^*})  }{ \\xi_* d_{\\textup{eff}} } } } \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   } \n\\end{align*}\n\n\\vspace*{0.4em}\n\\noindent\n\\textbf{Decomposition.}\nLet $K=[K(\\ti{x}_i,\\ti{x}_j)]_{i,j \\in [nV]}$, $K_z=[K_z(z_v,z_{v'})]_{v,v' \\in [V]}$ and $K_x=[K_x(x_i,x_j)]_{i,j \\in [nV]}$. \nSince kernel function $K(\\dot,\\dot)$ is a Hadamard composition of $K_z(\\dot,\\dot)$ and $K_x(\\dot,\\dot)$, it holds that $\\textup{rank}(K_{\\lambda^*}) = \\textup{rank}(K) \\leq \\textup{rank}(K_z) \\cdot \\textup{rank}(K_x)$.\n\n\\begin{align*}\n& \\log \\det \\sbr{  I + \\xi_*^{-1} K_{\\lambda^*} } \n\\\\\n= & \\log \\sbr{  \\Pi_{i=1}^{nV} \\sbr{1 + \\xi_*^{-1} \\alpha_i} }\n\\\\\n= & \\log \\sbr{  \\Pi_{i=1}^{\\textup{rank}(K_{\\lambda^*})} \\sbr{1 + \\xi_*^{-1} \\alpha_i} }\n\\\\\n\\leq & \\log \\sbr{  \\frac{ \\sum_{i=1}^{\\textup{rank}(K_{\\lambda^*})} \\sbr{1 + \\xi_*^{-1} \\alpha_i}  }{\\textup{rank}(K_{\\lambda^*})} }^{\\textup{rank}(K_{\\lambda^*})}\n\\\\\n= & \\textup{rank}(K_{\\lambda^*}) \\log \\sbr{ \\frac{ \\sum_{i=1}^{\\textup{rank}(K_{\\lambda^*})} \\sbr{1 + \\xi_*^{-1} \\alpha_i} }{\\textup{rank}(K_{\\lambda^*})} }\n\\\\\n\\leq & \\textup{rank}(K_z) \\cdot \\textup{rank}(K_x) \\log \\sbr{  \\frac{\\textup{Trace}\\sbr{I + \\xi_*^{-1} K_{\\lambda^*}}}{\\textup{rank}(K_{\\lambda^*})} }\n\\end{align*}\n\nThen, using Eq.~\\eqref{eq:ln_det_K_lambda}, the per-agent sample complexity is bounded by\n\\begin{align*}\n& O \\sbr{   \\frac{ \\rho^*}{V} \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   } \n\\\\\n= & O \\sbr{   \\frac{ \\log\\det\\sbr{ I + {\\xi_*}^{-1} K_{\\lambda_*} } }{\\Delta^2_{\\min} V} \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   } \n\\\\\n= & O \\sbr{   \\frac{ \\textup{rank}(K_z) \\cdot \\textup{rank}(K_x) }{\\Delta^2_{\\min} V} \\cdot  \\log \\sbr{  \\frac{\\textup{Trace}\\sbr{I + \\xi_*^{-1} K_{\\lambda^*}}}{\\textup{rank}(K_{\\lambda^*})} } \\cdot  \\log \\Delta^{-1}_{\\min} \\sbr{\\log \\sbr{ \\frac{ V n  }{\\delta} } + \\log \\log \\Delta^{-1}_{\\min}  }   } \n\\end{align*}\n\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thm:fc_lb_sample}}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:fc_lb_sample}]\n\tOur proof of Theorem~\\ref{thm:fc_lb_sample} adapts the analysis procedure of \\cite{rage2019} to the multi-agent setting.\n\t\n\n\tSuppose that $\\mathcal{A}$ is a $\\delta$-correct algorithm for CoPE-KB.\n\tFor any $i \\in [nV]$, let $\\nu_{\\theta^*,i}=\\mathcal{N}(\\phi(\\ti{x}_i)^\\top \\theta^*, 1)$ denote the reward distribution of arm $\\ti{x}_i$, and $T_i$ denote the number of times arm $\\ti{x}_i$ is pulled by algorithm $\\mathcal{A}$. \n\tLet $\\Theta=\\{\\theta \\in \\mathcal{H}_K: \\exists v,\\ \\exists \\ti{x} \\in \\ti{\\mathcal{X}}_v \\setminus \\{\\ti{x}^*_v\\} , \\  \\sbr{\\phi(\\ti{x}^*_v)-\\phi(\\ti{x})}^\\top \\theta < 0 \\}$.\n\t\n\tSince $\\mathcal{A}$ is $\\delta$-correct, according to the ``Change of Distribution'' lemma (Lemma A.3) in \\cite{kaufmann2016complexity}, we have that for any $\\theta \\in \\Theta$,\n\t\\begin{align*}\n\t\\sum_{i=1}^{nV} \\mathbb{E}[T_i] \\cdot \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta,i}) \\geq \\log(1\/2.4 \\delta) .\n\t\\end{align*}\n\tThus, we have\n\t\\begin{align*}\n\t\\min_{\\theta \\in \\Theta} \\sum_{i=1}^{nV} \\mathbb{E}[T_i] \\cdot \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta,i}) \\geq \\log(1\/2.4 \\delta) .\n\t\\end{align*}\n\tLet $\\boldsymbol{t}^*$ be the optimal solution of the following optimization problem\n\t\\begin{align*}\n\t& \\hspace*{0.3em} \\min \\sum_{i=1}^{nV} t_i\n\t\\\\\n\ts.t. \\  & \\min_{\\theta \\in \\mathcal{H}_K} \\sum_{i=1}^{nV} t_i \\cdot \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta,i}) \\geq \\log(1\/2.4 \\delta) .\n\t\\end{align*}\n\tThen, we have\n\t\\begin{align*}\n\t\\min_{\\theta \\in \\mathcal{H}_K} \\sum_{i=1}^{nV} \\frac{t^*_i}{\\sum_{j=1}^{nV} t^*_j} \\cdot \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta,i}) \\geq \\frac{\\log(1\/2.4 \\delta)}{\\sum_{j=1}^{nV} t^*_j} \\geq \\frac{\\log(1\/2.4 \\delta)}{\\sum_{j=1}^{nV} \\mathbb{E}[T_j] } .\n\t\\end{align*}\n\tSince $\\sum_{i=1}^{nV} \\frac{t^*_i}{\\sum_{j=1}^{nV} t^*_j}=1$,\n\t\\begin{align*}\n\t\\max_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\min_{\\theta \\in \\Theta} \\sum_{i=1}^{nV} \\lambda_i \\cdot \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta,i}) \\geq \\frac{\\log(1\/2.4 \\delta)}{\\sum_{i=1}^{nV} \\mathbb{E}[T_i] } .\n\t\\end{align*}\n\tThus, we have\n\t\\begin{align}\n\t\\sum_{i=1}^{nV} \\mathbb{E}[T_i] \\geq & \\frac{\\log(1\/2.4 \\delta)}{ \\max_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\min_{\\theta \\in \\Theta} \\sum_{i=1}^{nV} \\lambda_i \\cdot \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta,i}) } \n\t\\nonumber \\\\\n\t= & \\log(1\/2.4 \\delta) \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\theta \\in \\Theta} \\frac{1}{  \\sum_{i=1}^{nV} \\lambda_i \\cdot \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta,i}) } \\label{eq:sum_E_T_i}\n\t\\end{align}\n\tFor $\\lambda \\in \\triangle_{\\mathcal{X}}$, let $A(\\xi_*, \\lambda)=\\xi_* I + \\sum_{i=1}^{nV} \\lambda_i \\phi(\\ti{x}_i) \\phi(\\ti{x}_i)^\\top $.\n\tFor any $\\lambda \\in \\triangle_{\\mathcal{X}}$, $v \\in [V]$, $j \\in [nV]$ such that $\\ti{x}_j \\in \\ti{\\mathcal{X}}_v \\setminus \\{x^*_v\\}$, define\n\t\\begin{align*}\n\t\\theta_j(\\lambda)=\\theta^*-\\frac{(2 (\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top \\theta^*) \\cdot A(\\xi_*, \\lambda)^{-1} (\\phi(x^*_v)-\\phi(\\ti{x}_j))}{(\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top A(\\xi_*, \\lambda)^{-1} (\\phi(x^*_v)-\\phi(\\ti{x}_j))}\n\t\\end{align*}\n\tHere $(\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top \\theta_j(\\lambda)= - (\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top \\theta^* <0$, and thus $\\theta_j(\\lambda) \\in \\Theta$.\n\t\n\t\n\tThe KL-divergence between $\\nu_{\\theta^*,i}$ and $\\nu_{\\theta_j(\\lambda),i}$ is \n\t\\begin{align*}\n\t& \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta_j(\\lambda),i}) \n\t\\\\\n\t= & \\frac{1}{2} \\sbr{ \\phi(\\ti{x}_i)^\\top ( \\theta^* - \\theta_j(\\lambda) ) }^2\n\t\\\\\n\t= & \\frac{2 (\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top \\theta^*)^2 (\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top \\cdot A(\\xi_*, \\lambda)^{-1} \\phi(\\ti{x}_i) \\phi(\\ti{x}_i)^\\top A(\\xi_*, \\lambda)^{-1} (\\phi(x^*_v)-\\phi(\\ti{x}_j))}{ \\sbr{(\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top A(\\xi_*, \\lambda)^{-1} (\\phi(x^*_v)-\\phi(\\ti{x}_j))}^2 },\n\t\\end{align*}\n\tand thus,\n\t\\begin{align}\n\t& \\sum_{i=1}^{nV} \\lambda_i \\cdot \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta_j(\\lambda),i}) \n\t\\nonumber\\\\\n\t= & \\frac{2 (\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top \\theta^*)^2 (\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top \\cdot A(\\xi_*, \\lambda)^{-1} (\\sum_{i=1}^{nV} \\lambda_i \\phi(\\ti{x}_i) \\phi(\\ti{x}_i)^\\top) A(\\xi_*, \\lambda)^{-1} (\\phi(x^*_v)-\\phi(\\ti{x}_j))}{ \\sbr{(\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top A(\\xi_*, \\lambda)^{-1} (\\phi(x^*_v)-\\phi(\\ti{x}_j))}^2 }\n\t\\nonumber\\\\\n\t\\leq & \\frac{2 (\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top \\theta^*)^2}{ (\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top A(\\xi_*, \\lambda)^{-1} (\\phi(x^*_v)-\\phi(\\ti{x}_j)) }\n\t\\nonumber\\\\\n\t= & \\frac{2 (f(x^*_v)-f(\\ti{x}_j))^2}{ (\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top A(\\xi_*, \\lambda)^{-1} (\\phi(x^*_v)-\\phi(\\ti{x}_j)) } \\label{eq:lb_kl_computation}\n\t\\end{align}\n\t\n\tLet $\\mathcal{J}=\\{ j \\in [nV]:  \\ti{x}_j \\neq x^*_v, \\forall v \\in [V] \\}$ denote the set of indices of all sub-optimal arms.\n\tThen, plugging the above Eq.~\\eqref{eq:lb_kl_computation} into Eq.~\\eqref{eq:sum_E_T_i}, we have \n\t\\begin{align*}\n\t\\sum_{i=1}^{nV} \\mathbb{E}[T_i] \\geq & \\log(1\/2.4 \\delta) \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\theta \\in \\Theta} \\frac{1}{  \\sum_{i=1}^{nV} \\lambda_i \\cdot \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta,i}) }\n\t\\\\\n\t\\geq & \\log(1\/2.4 \\delta) \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{j \\in \\mathcal{J}} \\frac{1}{  \\sum_{i=1}^{nV} \\lambda_i \\cdot \\textup{KL}(\\nu_{\\theta^*,i}, \\nu_{\\theta_j(\\lambda),i}) }\n\t\\\\\n\t\\geq & \\log(1\/2.4 \\delta) \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x}_j \\in \\ti{\\mathcal{X}}_v \\setminus \\{x^*_v\\}, v \\in [V]} \\frac{  (\\phi(x^*_v)-\\phi(\\ti{x}_j))^\\top A(\\xi_*, \\lambda)^{-1} (\\phi(x^*_v)-\\phi(\\ti{x}_j))  }{  2 (f(x^*_v)-f(\\ti{x}_j))^2 }\n\t\t\\\\\n\t= & \\frac{1}{2} \\log(1\/2.4 \\delta) \\rho^*,\n\t\\end{align*}\n\twhich completes the proof of Theorem~\\ref{thm:fc_lb_sample}.\n\t\n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:fc_lb_round}} \\label{apx:fc_round_lb}\n\n\nOur proof of Theorem~\\ref{thm:fc_lb_round} generalizes the $2$-armed lower bound analysis in \\cite{tao2019collaborative} to the multi-armed case with linear reward structures. \nWe first introduce some notations and definitions.\n\nConsider a fully-collaborative instance $\\mathcal{I}(\\mathcal{X}, \\theta^*)$ of the CoPE-KB problem, where $\\ti{\\mathcal{X}}=\\mathcal{X}=\\mathcal{X}_v, f=f_v$ for all $v \\in [V]$ and $\\phi(x_i)^\\top \\theta^*$ is equal for all $x_i \\neq x_*$. Let $\\Delta=\\Delta_{\\min}=(\\phi(x_*)-\\phi(x_i))^\\top \\theta^*$ and $c=\\frac{\\phi(x_i)^\\top \\theta^*}{\\Delta}$ for any $x_i \\neq x_*$, where $c>0$ is a constant. Then, we have that $\\frac{ \\phi(x_*)^\\top \\theta^*}{\\Delta}=\\frac{\\phi(x_i)^\\top \\theta^*+\\Delta}{\\Delta}=1+c$.\n\nFor any integer $\\alpha \\geq 0$, let $\\mathcal{E}(\\alpha, T)$ be the event that $\\mathcal{A}$ uses at least $\\alpha$ communication rounds and at most $T$ samples before the end of the $\\alpha$-th round, and let $\\mathcal{E}^{+1}(\\alpha, T)$ be the event that $\\mathcal{A}$ uses at least $\\alpha+1$ communication rounds and at most $T$ samples before the end of the $\\alpha$-th round. \nLet $T_{\\mathcal{A}}$ and $T_{\\mathcal{A}, x_i}$ denote the expected number of samples used by $\\mathcal{A}$, and the expected number of samples used on arm $x_i$ by $\\mathcal{A}$, respectively.\nLet $\\lambda$ be the sample allocation of $\\mathcal{A}$, i.e., $\\lambda_i=\\frac{T_{\\mathcal{A}, x_i}}{T_{\\mathcal{A}}}$. \nLet $\\rho(\\mathcal{X}, \\theta^*) = \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{x \\in \\mathcal{X} \\setminus \\{x_*\\}} \\frac{\\| \\phi(x_*)-\\phi(x) \\|^2_{(\\xi_*I+\\sum_{x \\in \\mathcal{X}} \\lambda_x \\phi(x) \\phi(x)^\\top)^{-1} } }{ (f(x_*)-f(x))^2 } $ and  $d(\\mathcal{X})=\\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{x \\in \\mathcal{X} \\setminus \\{x_*\\}}\\| \\phi(x_*)-\\phi(x) \\|^2_{(\\xi_*I+\\sum_{x \\in \\mathcal{X}} \\lambda_x \\phi(x) \\phi(x)^\\top)^{-1} }$. Then, we have $\\rho(\\mathcal{X}, \\theta^*)=\\frac{d(\\mathcal{X})}{\\Delta^2} $.\n\nIn order to prove Theorem~\\ref{thm:fc_lb_round}, we first prove the following lemmas.\n\n\n\\begin{restatable}[Linear Structured Progress Lemma]{lemma}{lemmaProgress} \\label{lemma:progress}\n\tFor any integer $\\alpha \\geq 0$ and any $q \\geq 1$, we have  \n\t$$\n\t\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E}^{+1}\\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} }  }\n\t\\geq \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } } - 2\\delta - \\frac{1}{\\sqrt{q}} .\n\t$$\n\\end{restatable}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:progress}]\n\tLet $\\mathcal{F}$ be the event that $\\mathcal{A}$ uses exactly $\\alpha$ communication rounds and at most $\\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq}$ samples before the end of the $\\alpha$-th round. \n\tThen, we have \n\t$$\n\t\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E}^{+1} \\sbr{ \\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } }\n\t\\geq \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } } - \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{F} } .\n\t$$\n\tThus, to prove Lemma~\\ref{lemma:progress}, it suffices to prove \n\t\\begin{align}\n\t\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{F} } \\leq 2\\delta + \\frac{1}{\\sqrt{q}} . \\label{eq:event_H}\n\t\\end{align}\n\tWe can decompose $\\mathcal{F}$ as \n\t\\begin{align}\n\t\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{F}  }=& \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{F}, \\textup{$\\mathcal{A}$ returns $x_*$} }  + \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{F}, \\textup{$\\mathcal{A}$ does not return $x_*$} }\n\t\\nonumber\\\\\n\t=& \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{F}, \\textup{$\\mathcal{A}$ returns $x_*$} }  + \\delta \\label{eq:decompose_cF}\n\t\\end{align}\n\t\n\tLet $y_i=\\phi(x_*)-\\phi(x_i)$ for any $i \\in [n]$. Let $\\theta(\\xi_*,\\lambda)=\\theta^*-\\frac{2(y_j^\\top \\theta^*) A(\\xi_*,\\lambda)^{-1} y_j}{y_j^\\top A(\\xi_*,\\lambda)^{-1} y_j}$, where $A(\\xi_*,\\lambda)=\\xi_*I + \\sum_{i=1}^n \\lambda_i \\phi(x_i) \\phi(x_i)^\\top$ and $j = \\operatornamewithlimits{argmax}_{i \\in [n]} \\frac{ y_i^\\top A(\\xi_*,\\lambda)^{-1} y_i }{ (y_i^\\top \\theta^*)^2 }$. \n\tLet $\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))$ denote the instance where the underlying parameter is $\\theta(\\lambda)$. \n\tUnder $\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))$, it holds that $y_j^\\top \\theta(\\lambda)=-y_j^\\top \\theta^*<0$, and thus $x_*$ is sub-optimal.\n\tLet $\\mathcal{D}_{\\mathcal{I}}$ denote the product distribution of instance $\\mathcal{I}$ with at most $\\frac{\\rho(\\mathcal{X}, \\theta^*)}{q}$  samples over all agents.\n\t\n\tUsing the Pinsker's inequality (Lemma~\\ref{lemma:pinsker_inequality}) and Gaussian KL-divergence computation, we have \n\t\\begin{align} \n\t& \\| \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}- \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))} \\|_{\\textup{TV}} \n\t\\nonumber \\\\ \n\t\\leq & \\sqrt{\\frac{1}{2} \\textup{KL} \\sbr{ \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\| \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))} } }  \n\t\\nonumber \\\\\n\t\\leq & \\sqrt{\\frac{1}{4} \\sum_{i \\in [n]} (\\phi(x_i)^\\top (\\theta^*-\\theta(\\lambda)))^2 \\cdot \\lambda_i T_{\\mathcal{A}} }  \n\t\\nonumber \\\\\n\t= & \\sqrt{\\frac{1}{4} \\sum_{i \\in [n]} \\frac{4(y_j^\\top \\theta^*)^2 \\cdot y_j^\\top A(\\xi_*,\\lambda)^{-1}  \\phi(x_i) \\phi(x_i)^\\top A(\\xi_*,\\lambda)^{-1} y_j}{ (y_j^\\top A(\\xi_*,\\lambda)^{-1} y_j)^2 } \\cdot \\lambda_i T_{\\mathcal{A}} }\n\t\\nonumber \\\\\n\t= & \\sqrt{ T_{\\mathcal{A}}  \\frac{ (y_j^\\top \\theta^*)^2 \\cdot y_j^\\top A(\\xi_*,\\lambda)^{-1} (\\sum_{i \\in [n]} \\lambda_i \\phi(x_i) \\phi(x_i)^\\top) A(\\xi_*,\\lambda)^{-1} y_j}{ (y_j^\\top A(\\xi_*,\\lambda)^{-1} y_j)^2 }    }\n\t\\nonumber \\\\\n\t\\leq & \\sqrt{ T_{\\mathcal{A}}  \\frac{ (y_j^\\top \\theta^*)^2 \\cdot y_j^\\top A(\\xi_*,\\lambda)^{-1} (\\xi_* I+ \\sum_{i \\in [n]} \\lambda_i \\phi(x_i) \\phi(x_i)^\\top) A(\\xi_*,\\lambda)^{-1} y_j}{ (y_j^\\top A(\\xi_*,\\lambda)^{-1} y_j)^2 }    }\n\t\\nonumber \\\\\n\t= & \\sqrt{ T_{\\mathcal{A}} \\cdot \\frac{ (y_j^\\top \\theta^*)^2 }{ y_j^\\top A(\\xi_*,\\lambda)^{-1} y_j } }\n\t\\nonumber \\\\\n\t\\leq & \\sqrt{ \\frac{ \\rho(\\mathcal{X}, \\theta^*)}{q}  \\cdot \\frac{ 1 }{ \\frac{ y_j^\\top A(\\xi_*,\\lambda)^{-1} y_j }{(y_j^\\top \\theta^*)^2} } }\n\t\\nonumber \\\\\n\t\\leq & \\sqrt{ \\frac{ \\rho(\\mathcal{X}, \\theta^*)}{q}  \\cdot \\frac{ 1 }{ \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\frac{ y_j^\\top A(\\xi_*,\\lambda)^{-1} y_j }{(y_j^\\top \\theta^*)^2} } }\n\t\\nonumber \\\\\n\t= & \\frac{1}{\\sqrt{q}} \\label{eq:compute_kl}\n\t\\end{align}\n\tSince $x_*$ is sub-optimal under $\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))$, using the measure change technique, we have\n\t\\begin{align*}\n\t\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{F}, \\textup{$\\mathcal{A}$ returns $x_*$} } \\leq & \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))} \\mbr{ \\mathcal{F}, \\textup{$\\mathcal{A}$ returns $x_*$} }\n\t\\\\ & + \\| \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}- \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta(\\lambda))} \\|_{\\textup{TV}}\n\t\\\\\n\t\\leq & \\delta + \\frac{1}{\\sqrt{q}} \n\t\\end{align*}\n\tPlugging the above equality into Eq.~\\eqref{eq:decompose_cF}, we have\n\t\\begin{align*}\n\t\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{F}  } \n\t\\leq  2 \\delta + \\frac{1}{\\sqrt{q}} ,\n\t\\end{align*}\n\twhich completes the proof of Lemma~\\ref{lemma:progress}.\n\\end{proof}\n\n\n\nLet $\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)$ denote the instance where the underlying parameter is $\\theta\/ \\kappa$. Under $\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)$, the reward gap is $(\\phi(x_*)-\\phi(x_i))^\\top \\theta\/ \\kappa=\\frac{1}{\\kappa} \\Delta$ and the sample complexity is $\\rho(\\mathcal{X}, \\theta\/ \\kappa)=\\frac{\\kappa^2 d(\\mathcal{X})}{\\Delta^2}$. \nLet $\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}$ and $\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}$ denote the product distributions of instances $\\mathcal{I}(\\mathcal{X}, \\theta^*)$ and $\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)$ with $T_{\\mathcal{A}}$ samples, respectively.\n\n\\begin{lemma}[Multi-armed Measure Transformation Lemma] \\label{lemma:kl_exchange}\n\tSuppose that algorithm $\\mathcal{A}$ uses $T_{\\mathcal{A}}=\\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\zeta}$ samples over all agents on instance $\\mathcal{I}(\\mathcal{X}, \\theta^*)$, where $\\zeta \\geq 100$.  Then, for any event $E$ on $\\mathcal{I}(\\mathcal{X}, \\theta^*)$ and any $Q \\geq \\zeta$, we have \n\t\\begin{align*}\n\t\\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{E} \\leq & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E}  \\cdot \\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n Q) }{\\zeta}}  } + \\frac{1}{n Q^2} .\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\n\tFor any $i \\in [n]$, let $Z_{i,1}, \\dots, Z_{i,T_{\\mathcal{A}, x_i}}$ denote the observed $T_{\\mathcal{A}, x_i}$ samples on arm $x_i$, and define event \n\t$$\n\tL_i=\\lbr{ \\sum_{t=1}^{T_{\\mathcal{A}, x_i}} Z_{i,t} \\geq T_{\\mathcal{A}, x_i} \\cdot  \\phi(x_i)^\\top \\theta\/ \\kappa + \\frac{z}{  \\phi(x_i)^\\top \\theta\/ \\kappa- \\phi(x_i)^\\top \\theta^* } } ,\n\t$$\n\twhere $z \\geq 0$ is a parameter specified later. We also define event $L=\\cap_{i \\in [n]} L_i$.\n\tThen, \n\t\\begin{align*}\n\t\\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{E} \\leq & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{E, L} + \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{\\neg L}\n\t\\end{align*}\n\t\n\tUsing the measure change technique, we bound the term $\\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{E, L}$ as\n\t\\begin{align*}\n\t& \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{E, L} \n\t\\\\\n\t= & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L} \\cdot \\exp \\sbr{ -\\frac{1}{2} \\sum_{i \\in [n]} \\sum_{t=1}^{T_{\\mathcal{A}, x_i}} \\sbr{ (Z_{i,t} - \\phi(x_i)^\\top \\theta\/ \\kappa)^2 - (Z_{i,t} - \\phi(x_i)^\\top \\theta^*)^2  }  } \n\t\\\\\n\t= & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L} \\cdot \\exp \\sbr{ -\\frac{1}{2} \\sum_{i \\in [n]} \\sum_{t=1}^{T_{\\mathcal{A}, x_i}} \\sbr{ (\\phi(x_i)^\\top \\theta\/ \\kappa)^2 - ( \\phi(x_i)^\\top \\theta^*)^2 -2 Z_{i,t} (\\phi(x_i)^\\top \\theta\/ \\kappa- \\phi(x_i)^\\top \\theta^* )  }  } \n\t\\\\\n\t= & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L} \\cdot \\exp \\Bigg( -\\frac{1}{2} \\sum_{i \\in [n]} \\Bigg( \\sbr{(\\phi(x_i)^\\top \\theta\/ \\kappa)^2 - ( \\phi(x_i)^\\top \\theta^*)^2} \\cdot T_{\\mathcal{A}, x_i}  \\\\& -2  \\sbr{\\phi(x_i)^\\top \\theta\/ \\kappa- \\phi(x_i)^\\top \\theta^* } \\cdot \\sum_{t=1}^{T_{\\mathcal{A}, x_i}} Z_{i,t}  \\Bigg)  \\Bigg)\n\t\\\\\n\t\\leq & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L}  \\cdot \\exp \\Bigg( -\\frac{1}{2} \\sum_{i \\in [n]} \\Bigg( \\sbr{(\\phi(x_i)^\\top \\theta\/ \\kappa)^2 - ( \\phi(x_i)^\\top \\theta^*)^2} \\cdot T_{\\mathcal{A}, x_i} \\\\&   -2  \\sbr{\\phi(x_i)^\\top \\theta\/ \\kappa- \\phi(x_i)^\\top \\theta^* } \\cdot \\sbr{ T_{\\mathcal{A}, x_i} \\cdot  \\phi(x_i)^\\top \\theta\/ \\kappa + \\frac{z}{  \\phi(x_i)^\\top \\theta\/ \\kappa- \\phi(x_i)^\\top \\theta^* }  }  \\Bigg)  \\Bigg)\n\t\\\\\n\t= & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L}  \\cdot \\exp \\Bigg( -\\frac{1}{2} \\sum_{i \\in [n]} \\Bigg( \\sbr{(\\phi(x_i)^\\top \\theta\/ \\kappa)^2 - ( \\phi(x_i)^\\top \\theta^*)^2} \\cdot T_{\\mathcal{A}, x_i} \\\\&     -2  \\sbr{\\phi(x_i)^\\top \\theta\/ \\kappa- \\phi(x_i)^\\top \\theta^* } \\cdot T_{\\mathcal{A}, x_i} \\cdot  \\phi(x_i)^\\top \\theta\/ \\kappa -2 z    \\Bigg)  \\Bigg)\n\t\\\\\n\t= & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L}  \\cdot \\exp \\Bigg( -\\frac{1}{2} \\sum_{i \\in [n]} \\Bigg( \\sbr{\\phi(x_i)^\\top \\theta\/ \\kappa- \\phi(x_i)^\\top \\theta^* } \\Big(\\phi(x_i)^\\top \\theta\/ \\kappa \\cdot T_{\\mathcal{A}, x_i} +  \\phi(x_i)^\\top \\theta^* \\cdot T_{\\mathcal{A}, x_i} \\\\& -2 \\cdot \\phi(x_i)^\\top \\theta\/ \\kappa\\cdot T_{\\mathcal{A}, x_i} \\Big) -2 z    \\Bigg)  \\Bigg)\n\t\\\\\n\t= & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L}  \\cdot \\exp \\Bigg( -\\frac{1}{2} \\sum_{i \\in [n]} \\Bigg( -\\sbr{\\phi(x_i)^\\top \\theta\/ \\kappa- \\phi(x_i)^\\top \\theta^* }^2 \\cdot T_{\\mathcal{A}, x_i} -2 z    \\Bigg)  \\Bigg)\n\t\\\\\n\t= & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L}  \\cdot \\exp \\sbr{ \\sum_{i \\in [n]} \\sbr{\\frac{1}{2} \\sbr{\\phi(x_i)^\\top \\theta\/ \\kappa -  \\phi(x_i)^\\top \\theta^*}^2 \\cdot T_{\\mathcal{A}, x_i} + z  }   }\n\t\\\\\n\t= & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L}  \\cdot \\exp \\sbr{ \\sum_{i \\in [n]} \\sbr{\\frac{1}{2}  (1-\\frac{1}{\\kappa})^2 (\\phi(x_i)^\\top \\theta^*)^2  \\cdot \\frac{  d(\\mathcal{X})}{ \\Delta^2 \\zeta } \\cdot \\lambda_i  + z  }   }\n\t\\\\\n\t\\leq  & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L}  \\cdot \\exp \\sbr{ \\frac{(1+c)^2}{2 }  \\cdot  \\frac{  d(\\mathcal{X})}{ \\zeta } + n z     }\n\t\\end{align*}\n\t\n\tNext, using the Chernoff-Hoeffding inequality, we bound the second term as\n\t\\begin{align*}\n\t\\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{\\neg L} \\leq & \\sum_{i \\in [n]} \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{\\neg L_i}\n\t\\\\\n\t\\leq & \\sum_{i \\in [n]} \\exp \\sbr{-2 \\frac{z^2}{  (\\phi(x_i)^\\top \\theta\/ \\kappa- \\phi(x_i)^\\top \\theta^*)^2 } \\cdot  \\frac{\\zeta}{\\lambda_i \\rho(\\mathcal{X}, \\theta^*)} }\n\t\\\\\n\t= & \\sum_{i \\in [n]} \\exp \\sbr{-2 \\frac{z^2}{ (1-\\frac{1}{\\kappa})^2 (\\phi(x_i)^\\top \\theta^*)^2  } \\cdot  \\frac{\\Delta^2 \\zeta}{\\lambda_i d(\\mathcal{X})} }\n\t\\\\\n\t\\leq & \\sum_{i \\in [n]} \\exp \\sbr{ - \\frac{2 }{  (1+c)^2  } \\cdot  \\frac{ z^2 \\zeta}{  d(\\mathcal{X})} }\n\t\\\\\n\t= & n \\cdot \\exp \\sbr{ - \\frac{2 }{  (1+c)^2  } \\cdot  \\frac{ z^2 \\zeta}{  d(\\mathcal{X})} }\n\t\\end{align*}\n\tThus, \n\t\\begin{align*}\n\t\\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{E} \\leq & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{E, L} + \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{\\neg L}\n\t\\\\\n\t\\leq & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L}  \\cdot \\exp \\sbr{ \\frac{(1+c)^2}{2 }  \\cdot  \\frac{  d(\\mathcal{X})}{ \\zeta } + n z     } + n \\cdot \\exp \\sbr{ - \\frac{2 }{  (1+c)^2  } \\cdot  \\frac{ z^2 \\zeta}{  d(\\mathcal{X})} }\n\t\\end{align*}\n\tLet $z=\\sqrt{\\frac{(1+c)^2 d(\\mathcal{X}) \\log (n Q) }{ \\zeta}}$. Then, we have\n\t\\begin{align*}\n\t\\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}} \\mbr{E} \\leq & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E, L}  \\cdot \\exp \\sbr{  \\frac{(1+c)^2}{2 }  \\cdot  \\frac{  d(\\mathcal{X})}{ \\zeta } + n \\sqrt{\\frac{(1+c)^2 d(\\mathcal{X}) \\log (n Q) }{ \\zeta}}  } \\\\&+ n \\cdot \\exp \\sbr{ - 2 \\log (n Q)  } \n\t\\\\\n\t\\leq & \\Pr_{\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}} \\mbr{E}  \\cdot \\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n Q) }{\\zeta}}  } + \\frac{1}{n Q^2}\n\t\\end{align*}\n\\end{proof}\n\n\n\n\\begin{lemma}[Linear Structured Instance Transformation Lemma] \\label{lemma:dis_exchange_complete}\n\tFor any integer $\\alpha \\geq 0$, $q \\geq 100$ and $\\kappa \\geq 1$, we have  \n\t\\begin{align*}\n\t\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\mathcal{E} \\sbr{\\alpha+1, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} + \\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta} } }\n\t\\!\\! \\geq & \\!\\!\\! \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E}^{+1} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } } -  \\delta  - \\sqrt{\\frac{(1+c)^2}{4} \\cdot \\frac{  d(\\mathcal{X}) }{ q }  } \\\\& - \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} - \\frac{1}{n V}\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\n\tLet $\\ell=\\{ Z_{i,1}, \\dots, Z_{i,T_{\\mathcal{A}, x_i}} \\}_{i \\in [n]}$ denote the $T_{\\mathcal{A}}$ samples of algorithm $\\mathcal{A}$ on instance $\\mathcal{I}(\\mathcal{X}, \\theta^*)$. Let $\\mathcal{S}$ denote the set of all possible $\\ell$, conditioned on which $\\mathcal{E}^{+1}(\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} )$ holds. Then, we have\n\t$$\n\t\\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\ell=s } = \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E}^{+1} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } }\n\t$$\n\tFor any agent $v \\in [V]$, let $\\mathcal{K}_v$ be the event that agent $v$ uses more than $\\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta}$ samples during the $(\\alpha+1)$-st round. \n\tConditioned on $s \\in \\mathcal{S}$, $\\mathcal{K}_v$ only depends on the samples of agent $v$ during the $(\\alpha+1)$-st round, and is independent of other agents. \n\t\n\tUsing the facts that $\\mathcal{A}$ is $\\delta$-correct and $\\beta$-speedup and all agents have the same performance on fully-collaborative instances, we have\n\t\\begin{align*}\n\t\\delta \\geq & \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\text{$\\mathcal{A}$ uses more than $\\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta}$} }\n\t\\\\\n\t\\geq & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\ell=s } \\cdot \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_1 \\vee \\dots \\vee \\mathcal{K}_V | \\ell=s }\n\t\\\\\n\t= & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\ell=s } \\cdot \\sbr{ 1- \\prod_{v \\in [V]} \\sbr{ 1- \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s } } }  \n\t\\\\\n\t= & \\mathcal{E}^{+1} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } - \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\ell=s } \\cdot \\prod_{v \\in [V]} \\sbr{ 1- \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s } } \n\t\\end{align*}\n\tRearranging the above inequality, we have\n\t\\begin{align} \\label{eq:dis_exchange_delta}\n\t\\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\ell=s } \\cdot \\prod_{v \\in [V]} \\sbr{ 1- \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s } } \\geq & \\mathcal{E}^{+1} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } -  \\delta\n\t\\end{align}\n\t\n\tUsing Lemma~\\ref{lemma:kl_exchange} with $\\zeta=\\beta$ and $Q=V$, we have\n\t\\begin{align*}\n\t& \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\mathcal{E} \\sbr{\\alpha+1, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} + \\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta} } }\n\t\\\\\n\t\\geq & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\ell=s } \\cdot \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\neg \\mathcal{K}_1 \\wedge \\dots  \\wedge \\neg \\mathcal{K}_V | \\ell=s }\n\t\\\\\n\t= & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\ell=s } \\cdot \\prod_{v \\in [V]} \\sbr{ 1- \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\mathcal{K}_i | \\ell=s } }\n\t\\\\\n\t\\geq & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\ell=s } \\cdot \\\\& \\prod_{v \\in [V]} \\max \\lbr{  1- \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s } \\cdot \\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } - \\frac{1}{n V^2}  , 0}\n\t\\\\\n\t\\geq & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\ell=s } \\cdot \\\\& \\sbr{ \\prod_{v \\in [V]} \\max \\lbr{  1- \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s } \\cdot \\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  }, 0} - \\frac{1}{n V}}\n\t\\\\\n\t= & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\ell=s } \\cdot  \\Bigg(  \\prod_{v \\in [V]} \\max \\Bigg(  1-  \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s } \\\\& - \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s } \\cdot \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1}, 0\\Bigg) - \\frac{1}{n V}\\Bigg)\n\t\\\\\n\t\\overset{\\textup{(e)}}{\\geq} & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\ell=s } \\cdot \\\\& \\sbr{ \\prod_{v \\in [V]} \\sbr{  1-  \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s }  } - \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} - \\frac{1}{n V}}\n\t\\\\\n\t\\geq & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\ell=s } \\cdot \\\\& \\prod_{v \\in [V]} \\sbr{  1-  \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s }  } - \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} - \\frac{1}{n V} \n\t\\\\\n\t\\geq & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\ell=s } \\cdot  \\prod_{v \\in [V]} \\sbr{  1-  \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s }  } \\\\ & - \\sum_{s \\in \\mathcal{S}} \\abr{ \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\ell=s } - \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\ell=s } } - \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} - \\frac{1}{n V} \n\t\\end{align*}\n\twhere (e) comes from Lemma~\\ref{lemma:prod}.\n\n\t\n\tUsing the Pinsker's inequality (Lemma~\\ref{lemma:pinsker_inequality}), we have\n\n\t\\begin{align} \n\t& \\sum_{s \\in \\mathcal{S}} \\abr{ \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\ell=s } - \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\ell=s } } \n\t\\\\\n\t\\leq &  \\sqrt{\\frac{1}{2} \\textup{KL}(\\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)}, \\mathcal{D}_{\\mathcal{I}(\\mathcal{X}, \\theta^*)}) } \n\t\\nonumber \\\\\n\t\\leq &  \\sqrt{\\frac{1}{4} \\sum_{i \\in [n]}  \\sbr{\\phi(x_i)^\\top \\theta\/ \\kappa -  \\phi(x_i)^\\top \\theta^*}^2 \\cdot \\lambda_i T_{\\mathcal{A}}  }  \n\t\\nonumber \\\\\n\t= & \\sqrt{\\frac{1}{4} \\sum_{i \\in [n]} \\sbr{ (1-\\frac{1}{\\kappa})^2 (\\phi(x_i)^\\top \\theta^*)^2 \\cdot  \\frac{  d(\\mathcal{X})}{ \\Delta^2 q } \\lambda_i }  }\n\t\\nonumber \\\\\n\t\\leq & \\sqrt{\\frac{(1+c)^2}{4} \\cdot \\frac{  d(\\mathcal{X}) }{ q }  } \\label{eq:prob_diff}\n\t\\end{align}\n\tThus, using Eqs.~\\eqref{eq:dis_exchange_delta},\\eqref{eq:prob_diff}, we have\n\t\\begin{align*}\n\t& \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\mathcal{E} \\sbr{\\alpha+1, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} + \\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta} } }\n\t\\\\\n\t\\geq & \\sum_{s \\in \\mathcal{S}} \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\ell=s } \\cdot  \\prod_{v \\in [V]} \\sbr{  1-  \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{K}_i | \\ell=s }  }  - \\sqrt{\\frac{(1+c)^2}{4} \\cdot \\frac{  d(\\mathcal{X}) }{ q }  } \\\\&- \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} - \\frac{1}{n V}\n\t\\\\\n\t\\geq & \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E}^{+1} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } } -  \\delta  - \\sqrt{\\frac{(1+c)^2}{4} \\cdot \\frac{  d(\\mathcal{X}) }{ q }  } \\\\& - \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} - \\frac{1}{n V}\n\t\\end{align*}\n\\end{proof}\n\n\nNow we prove Theorem~\\ref{thm:fc_lb_round}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:fc_lb_round}]\n\tCombining Lemmas~\\ref{lemma:progress},\\ref{lemma:dis_exchange_complete}, we have\n\t\\begin{align*}\n\t& \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\mathcal{E} \\sbr{\\alpha+1, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} + \\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta} } }\n\t\\\\\n\t\\geq & \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E}^{+1} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } } -  \\delta  - \\sqrt{\\frac{(1+c)^2}{4} \\cdot \\frac{  d(\\mathcal{X}) }{ q }  } \\\\&- \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} - \\frac{1}{n V}\n\t\\\\\n\t\\geq & \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } } - 3 \\delta  - \\sqrt{  \\frac{ (1+c)^2 d(\\mathcal{X}) }{ q }  }  \\\\& - \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} - \\frac{1}{n V}\n\t\\end{align*}\n\tLet $\\kappa=\\sqrt{1+\\frac{Vq}{\\beta}}$. Then, we have\n\t\\begin{align}\n\t& \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\mathcal{E} \\sbr{\\alpha+1, \\frac{ \\rho(\\mathcal{X}, \\theta\/ \\kappa) }{Vq} } }\n\t\\nonumber \\\\\n\t= & \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\mathcal{E} \\sbr{\\alpha+1, \\frac{\\kappa^2 \\cdot \\rho(\\mathcal{X}, \\theta^*)}{Vq} } }\n\t\\nonumber \\\\\n\t= & \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta\/ \\kappa)} \\mbr{ \\mathcal{E} \\sbr{\\alpha+1, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} + \\frac{\\rho(\\mathcal{X}, \\theta^*)}{\\beta} } }\n\t\\nonumber \\\\\n\t\\geq & \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E} \\sbr{\\alpha, \\frac{\\rho(\\mathcal{X}, \\theta^*)}{Vq} } } - 3 \\delta  - \\sqrt{  \\frac{ (1+c)^2 d(\\mathcal{X}) }{ q }  } \\nonumber\\\\& - \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} - \\frac{1}{n V} \\label{eq:round_forward}\n\t\\end{align}\n\t\n\tLet $\\mathcal{I}(\\mathcal{X}, \\theta_0)$ be the basic instance of induction, where the reward gap is $\\Delta_0= (\\phi(x_*) - \\phi(x_i))^\\top \\theta_0 = 1$ for any $i \\in [n]$.\n\tLet $t_0$ be the largest integer such that \n\t$$\n\t\\Delta \\cdot \\sbr{ 1+\\frac{V q}{\\beta} }^{\\frac{t_0}{2}} \\leq 1 ,\n\t$$\n\twhere $q=1000 t_0^2$.\n\tThen, we have\n\t$$\n\tt_0=\\Omega \\sbr{\\frac{\\log (\\frac{1}{\\Delta})}{ \\log(1+\\frac{V}{\\beta})+\\log \\log(\\frac{1}{\\Delta}) } }\n\t$$\n\t\n\tStarting from $\\mathcal{I}(\\mathcal{X}, \\theta_0)$, we repeatedly apply Eq.~\\eqref{eq:round_forward} for $t_0$ times to switch to $\\mathcal{I}(\\mathcal{X}, \\theta^*)$ where the reward gap is $\\Delta$.\n\tSince $\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta_0)} \\mbr{ \\mathcal{E} \\sbr{0, \\frac{ \\rho(\\mathcal{X}, \\theta_0)}{Vq} } }=1$, by induction, we have\n\t\\begin{align*}\n\t& \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E} \\sbr{t_0, \\frac{ \\rho(\\mathcal{X}, \\theta^*)}{Vq} } }\n\t\\\\\n\t\\geq & 1 - \\sbr{ 3 \\delta  + \\sqrt{  \\frac{ (1+c)^2 d(\\mathcal{X}) }{ q }  } + \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} + \\frac{1}{n V} } \\cdot t_0\n\t\\end{align*}\n\tWhen $\\delta=O(\\frac{1}{nV})$ and $q, V, \\beta$ are large enough, we can have\n\t\\begin{align*}\n\t\\sbr{ 3 \\delta  + \\sqrt{  \\frac{ (1+c)^2 d(\\mathcal{X}) }{ q }  } + \\sbr{\\exp \\sbr{  (1+c)^2 d(\\mathcal{X}) n  \\sqrt{\\frac{  \\log (n V) }{\\beta}}  } -1} + \\frac{1}{n V} } \\cdot t_0 \\leq \\frac{1}{2}\n\t\\end{align*}\n\tand then \n\t$$\n\t\\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E} \\sbr{t_0, \\frac{ \\rho(\\mathcal{X}, \\theta^*)}{Vq} } } \\geq \\frac{1}{2} .\n\t$$\n\tThus,\n\t\\begin{align*}\n\t& \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\text{$\\mathcal{A}$ uses $\\Omega \\sbr{\\frac{\\log (\\frac{1}{\\Delta})}{ \\log(1+\\frac{V}{\\beta})+\\log \\log(\\frac{1}{\\Delta}) } }$ communication rounds} }\n\t\\\\\n\t\\geq & \\Pr_{\\mathcal{I}(\\mathcal{X}, \\theta^*)} \\mbr{ \\mathcal{E} \\sbr{t_0, \\frac{ \\rho(\\mathcal{X}, \\theta^*)}{Vq} } }\n\t\\\\\n\t\\geq & \\frac{1}{2},\n\t\\end{align*}\n\twhich completes the proof of Theorem~\\ref{thm:fc_lb_round}.\t\n\\end{proof}\n\n\n\\section{Proofs for the Fixed-Budget Setting}\n\n\\subsection{Proof of Theorem~\\ref{thm:kernel_bai_fb_ub}}\n\n\\begin{proof} [Proof of Theorem~\\ref{thm:kernel_bai_fb_ub}]\n\tOur proof of Theorem~\\ref{thm:kernel_bai_fb_ub} adapts the error probability analysis in \\cite{peace2020} to the multi-agent setting.\n\t\n\tSince the number of samples used over all agents in each phase is $N=\\left \\lfloor TV\/R \\right \\rfloor$, the total number of samples used by algorithm $\\textsf{CoopKernelFB}$ is at most $TV$ and the total number of samples used per agent is at most $T$.\n\t\n\tNow we prove the error probability upper bound.\n\t\n\tRecall that for any $\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}$, $\\Phi_{\\lambda}=[\\sqrt{\\lambda_1} \\phi(x^*_v)^\\top; \\dots; \\sqrt{\\lambda_{nV}} \\phi(\\ti{x}_{nV})^\\top]$ and $\\Phi^\\top_{\\lambda} \\Phi_{\\lambda}=\\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top$.\n\n\tLet $\\gamma_*=\\xi_* N$.\n\t\n\tFor any $\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}, v \\in [V], t \\in [R]$, define \n\t$$\n\t\\Delta_{t,\\ti{x}_i,\\ti{x}_j} = \\inf_{\\Delta>0} \\lbr{ \\frac{\\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{(\\xi_* I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} }{\\Delta^2} \\leq 2 \\rho^* }\n\t$$ \n\tand event\n\t$$\n\t\\mathcal{J}_{t,\\ti{x}_i,\\ti{x}_j} = \\lbr{ \\abr{ \\sbr{\\hat{f}_t(\\ti{x}_i)-\\hat{f}_t(\\ti{x}_j)} - \\sbr{f(\\ti{x}_i)-f(\\ti{x}_j)} } < \\Delta_{t,\\ti{x}_i,\\ti{x}_j} } .\n\t$$\n\tIn the following, we prove $\\Pr \\mbr{ \\neg \\mathcal{J}_{t,\\ti{x}_i,\\ti{x}_j} } \\leq 2 \\exp \\sbr{- \\frac{ N }{ 2 (1+\\varepsilon) \\rho^*  } }$.\n\t\n\tSimilar to the analysis procedure of Lemma~\\ref{lemma:concentration}, we have\n\t\\begin{align*}\n\t& \\sbr{\\hat{f}_t(\\ti{x}_i)-\\hat{f}_t(\\ti{x}_j)} - \\sbr{f(\\ti{x}_i)-f(\\ti{x}_j)} \n\t\\\\\n\t= & \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top  \\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\bar{\\eta}^{(t)} -  \\gamma_* \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1} \\theta^*  ,\n\t\\end{align*}\n\twhere the mean of the first term is zero and its variance is bounded by \n\t$$\n\t\\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|^2_{\\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1}} \\leq \\frac{ (1+\\varepsilon) \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_* I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} }{ N } .\n\t$$\n\tUsing the Hoeffding inequality, we have\n\t\\begin{align*}\n\t& \\Pr \\mbr{ \\abr{ \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top  \\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\eta_v^{(t)} } \\geq \\frac{1}{2} \\Delta_{t,\\ti{x}_i,\\ti{x}_j} }\n\t\\\\\n\t\\leq & 2 \\exp \\sbr{- 2 \\frac{ \\frac{1}{4} \\Delta_{t,\\ti{x}_i,\\ti{x}_j}^2 }{ \\frac{ (1+\\varepsilon) \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_* I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} }{ N } }  }\n\t\\\\\n\t\\leq & 2 \\exp \\sbr{- \\frac{1}{2} \\frac{ N }{ \\frac{ (1+\\varepsilon) \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_* I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} }{ \\Delta_{t,\\ti{x}_i,\\ti{x}_j}^2 } }  }\n\t\\\\\n\t\\leq & 2 \\exp \\sbr{- \\frac{ N }{ 2 (1+\\varepsilon) \\rho^* } }\n\t\\end{align*}\n\tThus, with probability at least $1- 2 \\exp \\sbr{- \\frac{ N }{ 2 (1+\\varepsilon) \\rho^* } }$, we have \n\t$$\n\t\\abr{ \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top  \\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\eta_v^{(t)} } < \\frac{1}{2} \\Delta_{t,\\ti{x}_i,\\ti{x}_j} .\n\t$$\n\t%\n\tRecall that $\\xi_*$ satisfies $(1+\\varepsilon) \\sqrt{\\xi_*}  \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{X}}_v, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\xi_* I + \\Phi_{\\lambda_u}^\\top \\Phi_{\\lambda_u} }^{-1}} B\n\t\\leq \\frac{1}{2} \\Delta_{t,\\ti{x}_i,\\ti{x}_j}$. \n\tThen, we bound the regularization (bias) term. \n\t\\begin{align*}\n\t& \\abr{ \\gamma_* \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1} \\theta^* }\n\t\\\\\n\t\\leq & \\gamma_* \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1}}  \\| \\theta^* \\|_{\\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1}} \n\t\\\\\n\t\\leq & \\sqrt{\\gamma_*} \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1}} \\| \\theta^* \\|_2 \n\t\\\\\n\t\\leq & \\sqrt{\\xi_* N} \\cdot \\frac{ (1+\\varepsilon) \\cdot \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_* I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} }{ \\sqrt{N} }  \\cdot \\| \\theta^* \\|_2 \n\t\\\\\n\t\\leq & (1+\\varepsilon) \\sqrt{\\xi_*}  \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{B}}^{(t)}_v, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\xi_* I + \\Phi_{\\lambda^*_t}^\\top \\Phi_{\\lambda^*_t} }^{-1}}  \\| \\theta^* \\|_2\n\t\\\\\n\t\\leq & (1+\\varepsilon) \\sqrt{\\xi_*}  \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{B}}^{(t)}_v, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\xi_* I + \\Phi_{\\lambda_u}^\\top \\Phi_{\\lambda_u} }^{-1}}  \\| \\theta^* \\|_2\n\t\\\\\n\t\\leq & (1+\\varepsilon) \\sqrt{\\xi_*}  \\max_{\\ti{x}_i,\\ti{x}_j \\in \\tilde{\\mathcal{X}}_v, v \\in [V]} \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\xi_* I + \\Phi_{\\lambda_u}^\\top \\Phi_{\\lambda_u} }^{-1}} \\cdot B\n\t\\\\\n\t\\leq & \\frac{1}{2} \\Delta_{t,\\ti{x}_i,\\ti{x}_j}\n\t\\end{align*}\n\tThus, with probability at least $1- 2 \\exp \\sbr{- \\frac{ N }{ 2 (1+\\varepsilon) \\rho^* } }$, we have \n\t\\begin{align*}\n\t& \\abr{ \\sbr{\\hat{f}_t(\\ti{x}_i)-\\hat{f}_t(\\ti{x}_j)} - \\sbr{f(\\ti{x}_i)-f(\\ti{x}_j)} }\n\t\\\\\n\t\\leq & \\abr{ \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top  \\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1} \\Phi_t^\\top \\eta_v^{(t)} } +  \\abr{\\gamma_* \\sbr{\\phi(\\ti{x}_i)-\\phi(\\ti{x}_j)}^\\top \\sbr{\\gamma_* I + \\Phi_t^\\top \\Phi_t }^{-1} \\theta^*} \n\t\\\\\n\t< & \\Delta_{t,\\ti{x}_i,\\ti{x}_j},\n\t\\end{align*}\n\twhich completes the proof of $\\Pr \\mbr{ \\neg \\mathcal{J}_{t,\\ti{x}_i,\\ti{x}_j} } \\leq 2 \\exp \\sbr{- \\frac{ N }{ 2 (1+\\varepsilon) \\rho^* } }$.\n\t\n\tDefine event\n\t$$\n\t\\mathcal{J} = \\lbr{ \\abr{ \\sbr{\\hat{f}_t(\\ti{x}_i)-\\hat{f}_t(\\ti{x}_j)} - \\sbr{f(\\ti{x}_i)-f(\\ti{x}_j)} } < \\Delta_{t,\\ti{x}_i,\\ti{x}_j}, \\  \\forall \\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}, \\forall v \\in [V], \\forall t \\in [R] } ,\n\t$$\n\t\n\tBy a union bound over $\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}$, $v \\in [V]$ and $t \\in [R]$, we have \n\t\\begin{align*}\n\t\\Pr \\mbr{ \\neg \\mathcal{J} } \\leq & 2 n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) \\cdot \\exp \\sbr{- \\frac{ N }{ 2 (1+\\varepsilon) \\rho^* } } \n\t\\\\\n\t= & O \\sbr{ n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) \\cdot \\exp \\sbr{- \\frac{ TV }{ \\rho^* \\log(\\omega(\\ti{\\mathcal{X}})) } } }\n\t\\end{align*}\n\t\n\tIn order to prove Theorem~\\ref{thm:kernel_bai_fb_ub}, it suffices to prove that conditioning on $\\mathcal{J}$, algorithm $\\textsf{CoopKernelFB}$ returns the correct answers $\\ti{x}^*_{v}$ for all $v \\in [V]$.\n\t\n\tSuppose that there exist $v \\in [V]$ and $t \\in [R]$ such that $\\ti{x}^*_{v}$ is eliminated in phase $t$.\n\tDefine\n\t$$\n\t\\mathcal{B'}_v^{(t)}=\\{ \\ti{x} \\in \\mathcal{B}_v^{(t)}: \\hat{f}_t(\\ti{x}) > \\hat{f}_t(\\ti{x}^*_{v}) \\},\n\t$$\n\twhich denotes the subset of arms that are ranked before $\\ti{x}^*_{v}$ by the estimated rewards in $\\mathcal{B}_v^{(t)}$. According to the elimination rule, we have\n\t\\begin{align}\n\t\\omega( \\mathcal{B'}_v^{(t)} \\cup \\{ \\ti{x}^*_{v} \\} ) > \\frac{1}{2} \\omega( \\mathcal{B}_v^{(t)} ) = \\frac{1}{2} \\max_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}}  \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_* I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} \\label{eq:error_prob_dimension}\n\t\\end{align}\n\tDefine $\\ti{x}_0=\\operatornamewithlimits{argmax}_{\\ti{x} \\in \\mathcal{B'}_v^{(t)}} \\Delta_{\\ti{x}}$. We have\n\t\\begin{align*}\n\t& \\frac{1}{2}  \\frac{\\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}_0) \\|_{(\\xi_* I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} }{ \\Delta_{\\ti{x}_0}^2 }\n\t\\\\\n\t\\leq & \\frac{1}{2} \\max_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B}_v^{(t)}}  \\frac{\\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{(\\xi_* I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} }{ \\Delta_{\\ti{x}_0}^2 }\n\t\\\\\n\t\\overset{\\textup{(f)}}{<} & \\frac{\\omega( \\mathcal{B'}_v^{(t)} \\cup \\{ \\ti{x}^*_{v} \\} )}{ \\Delta_{\\ti{x}_0}^2 }\n\t\\\\\n\t\\overset{\\textup{(g)}}{=} & \\min_{\\lambda \\in \\triangle \\mathcal{X}} \\max_{\\ti{x}_i,\\ti{x}_j \\in \\mathcal{B'}_v^{(t)} \\cup \\{ \\ti{x}^*_{v} \\} } \\frac{ \\| \\phi(\\ti{x}_i)-\\phi(\\ti{x}_j) \\|_{\\sbr{\\xi_* I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }{ \\Delta_{\\ti{x}_0}^2 }\n\t\\\\\n\t\\overset{\\textup{(h)}}{\\leq} & \\min_{\\lambda \\in \\triangle \\mathcal{X}} \\max_{\\ti{x} \\in \\mathcal{B'}_v^{(t)} } \\frac{ \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|_{\\sbr{\\xi_* I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }{ \\Delta_{\\ti{x}}^2 }\n\t\\\\\n\t\\leq & \\min_{\\lambda \\in \\triangle \\mathcal{X}} \\max_{\\ti{x} \\in \\mathcal{X}_v \\setminus\\{ \\ti{x}^*_{v} \\}, v \\in [V] } \\frac{ \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|_{\\sbr{\\xi_* I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }{ \\Delta_{\\ti{x}}^2 }\n\t\\\\\n\t= & \\rho^*,\n\t\\end{align*}\n\twhere (f) can be obtained by dividing Eq.~\\eqref{eq:error_prob_dimension} by $\\Delta_{\\ti{x}_0}^2$, (g) comes from the definition of $\\omega(\\cdot)$, and (h) is due to the definition of $\\ti{x}_0$.\n\t\n\tAccording to the definition\n\t$$\n\t\\Delta_{t,\\ti{x}^*_{v},\\ti{x}_0} = \\inf_{\\Delta>0} \\lbr{ \\frac{\\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}_0) \\|^2_{(\\xi_* I + \\Phi^\\top_{\\lambda^*_t} \\Phi_{\\lambda^*_t} )^{-1}} }{\\Delta^2} \\leq 2 \\rho^* },\n\t$$\n\twe have $\\Delta_{t,\\ti{x}^*_{v},\\ti{x}_0} \\leq \\Delta_{\\ti{x}_0}$. \n\t\n\tConditioning on $\\mathcal{J}$, we have $\\abr{ \\sbr{\\hat{f}_t(\\ti{x}^*_{v})-\\hat{f}_t(\\ti{x}_0)} - \\sbr{f(\\ti{x}^*_{v})-f(\\ti{x}_0)} } < \\Delta_{t,\\ti{x}^*_{v},\\ti{x}_0} \\leq \\Delta_{\\ti{x}_0}$. \n\tThen, we have\n\t\\begin{align*}\n\t\\hat{f}_t(\\ti{x}^*_{v})-\\hat{f}_t(\\ti{x}_0) > \\sbr{f(\\ti{x}^*_{v})-f(\\ti{x}_0)} - \\Delta_{\\ti{x}_0} =0,\n\t\\end{align*}\n\twhich contradicts the definition of $\\ti{x}_0$ that satisfies $\\hat{f}_t(\\ti{x}_0) > \\hat{f}_t(\\ti{x}^*_{v})$.\n\tThus, for any $v \\in [V]$, $\\ti{x}^*_{v}$ will never be eliminated. \n\t\n\tSince $\\omega( \\{ \\ti{x}^*_{v}, \\ti{x} \\} ) \\geq 1$ for any $\\ti{x} \\in \\mathcal{X}_v \\setminus \\{ \\ti{x}^*_{v} \\}, v \\in [V]$ and $R=\\left \\lceil \\log_2(\\omega(\\ti{\\mathcal{X}})) \\right \\rceil \\geq \\left \\lceil \\log_2(\\omega(\\mathcal{X}_v)) \\right \\rceil $ for any $v \\in [V]$, we have that conditioning on $\\mathcal{J}$, algorithm $\\textsf{CoopKernelFB}$ returns the correct answers $\\ti{x}^*_{v}$ for all $v \\in [V]$.\n\t\n\tFor communication rounds, since algorithm $\\textsf{CoopKernelFB}$ has at most $R=\\left \\lceil \\log_2(\\omega(\\ti{\\mathcal{X}})) \\right \\rceil$ phases, the number of communication rounds is bounded by $O(\\log (\\omega(\\ti{\\mathcal{X}})))$.\n\\end{proof}\n\n\\subsection{Proof of Corollary~\\ref{corollary:ub_fb}}\n\n\\begin{proof}[Proof of Corollary~\\ref{corollary:ub_fb}]\nFollowing the analysis procedure of Corollary~\\ref{corollary:ub_fc}, we have\n\\begin{align*}\n\\rho^* = & \\min_{\\lambda \\in \\triangle_{\\mathcal{X}}} \\max_{\\ti{x} \\in \\ti{\\mathcal{X}}_v, v \\in [V]} \\frac{  \\| \\phi(\\ti{x}^*_{v})-\\phi(\\ti{x}) \\|^2_{\\sbr{\\xi_* I + \\Phi^\\top_{\\lambda} \\Phi_{\\lambda} }^{-1}} }{ \\sbr{f(\\ti{x}^*_{v})- f(\\ti{x})}^2 }\n\\nonumber\\\\\n\\leq & \\frac{ 8 }{ \\Delta^2_{\\min} } \\cdot  \\log \\det \\sbr{  I + \\xi_*^{-1} K_{\\lambda^*} }\n\\end{align*}\n\n\\vspace*{0.4em}\n\\noindent\n\\textbf{Maximum Information Gain.}\nRecall that the maximum information gain over all sample allocation $\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}$ is defined as\n$$\n\\Upsilon = \\max_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\log\\det\\sbr{ I + {\\xi_*}^{-1} K_{\\lambda} } .\n$$\nThen, the error probability is bounded by\n\\begin{align*}\n& O \\sbr{ n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) \\cdot \\exp \\sbr{- \\frac{ TV }{ \\rho^* \\log(\\omega(\\ti{\\mathcal{X}})) } } }\n\\\\\n= & O \\sbr{ n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) \\cdot \\exp \\sbr{- \\frac{ TV \\Delta^2_{\\min} }{ \\Upsilon \\log(\\omega(\\ti{\\mathcal{X}})) } } }\n\\end{align*}\n\n\\vspace*{0.4em}\n\\noindent\n\\textbf{Effective Dimension.}\nRecall that $\\alpha_1 \\geq \\dots \\geq \\alpha_{nV}$ denote the eigenvalues of $K_{\\lambda^*}$ in decreasing order. The effective dimension of $K_{\\lambda^*}$ is defined as\n$$\nd_{\\textup{eff}}=\\min \\bigg\\{ j :j \\xi_*\\log(nV) \\geq \\sum_{i=j+1}^{nV} \\alpha_i \\bigg\\} ,\n$$\nand we have\n\\begin{align*}\n\\log \\det \\sbr{  I + \\xi_*^{-1} K_{\\lambda^*} } \n\\leq  {d_{\\textup{eff}}} \\log \\sbr{ 2 nV \\cdot \\sbr{ 1 + \\frac{ \\textup{Trace}(K_{\\lambda^*})  }{ \\xi_* d_{\\textup{eff}} } } } .\n\\end{align*}\nThen, the error probability is bounded by\n\\begin{align*}\n& O \\sbr{ n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) \\cdot \\exp \\sbr{- \\frac{ TV }{ \\rho^* \\log(\\omega(\\ti{\\mathcal{X}})) } } }\n\\\\\n= & O \\sbr{ n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) \\cdot \\exp \\sbr{- \\frac{ TV \\Delta^2_{\\min} }{ d_{\\textup{eff}} \\log \\sbr{  nV \\cdot \\sbr{ 1 + \\frac{ \\textup{Trace}(K_{\\lambda^*})  }{ \\xi_* d_{\\textup{eff}} } } } \\log(\\omega(\\ti{\\mathcal{X}})) } } }\n\\end{align*}\n\n\\vspace*{0.4em}\n\\noindent\n\\textbf{Decomposition.}\nRecall that $K=[K(\\ti{x}_i,\\ti{x}_j)]_{i,j \\in [nV]}$, $K_z=[K_z(z_v,z_{v'})]_{v,v' \\in [V]}$, $K_x=[K_x(x_i,x_j)]_{i,j \\in [nV]}$, and $\\textup{rank}(K_{\\lambda^*}) = \\textup{rank}(K) \\leq \\textup{rank}(K_z) \\cdot \\textup{rank}(K_x)$. \nWe have\n\\begin{align*}\n\\log \\det \\sbr{  I + \\xi_*^{-1} K_{\\lambda^*} } \n\\leq  \\textup{rank}(K_z) \\cdot \\textup{rank}(K_x) \\log \\sbr{  \\frac{\\textup{Trace}\\sbr{I + \\xi_*^{-1} K_{\\lambda^*}}}{\\textup{rank}(K_{\\lambda^*})} }\n\\end{align*}\nThen, the error probability is bounded by\n\\begin{align*}\n& O \\sbr{ n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) \\cdot \\exp \\sbr{- \\frac{ TV }{ \\rho^* \\log(\\omega(\\ti{\\mathcal{X}})) } } }\n\\\\\n= & O \\sbr{ n^2 V \\log(\\omega(\\ti{\\mathcal{X}})) \\cdot \\exp \\sbr{- \\frac{ TV \\Delta^2_{\\min} }{ \\textup{rank}(K_z) \\cdot \\textup{rank}(K_x) \\log \\sbr{  \\frac{\\textup{Trace}\\sbr{I + \\xi_*^{-1} K_{\\lambda^*}}}{\\textup{rank}(K_{\\lambda^*})} } \\log(\\omega(\\ti{\\mathcal{X}})) } } }\n\\end{align*}\n\nTherefore, we complete the proof of Corollary~\\ref{corollary:ub_fb}.\n\\end{proof}\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:fb_lb_round}}\nOur proof of Theorem~\\ref{thm:fb_lb_round} follows the analysis procedure in \\cite{learning_adaptivity2017}.\n\nWe first introduce some definitions in information theory.\nFor any random variable $A$, let $H(A)$ denote the Shannon entropy of $A$.\nIf $A$ is uniformly distributed on its support, $H(A)=\\log |A|$.\nFor any $p \\in (0,1)$, let $H_2(p)=-p \\log_2 p - (1-p) \\log_2 (1-p)$ denote the binary entropy, and $H_2(p)=H(A)$ for $A \\sim \\textup{Bernoulli}(p)$.\nFor any random variables $A, B$, let $H(A;B)=H(A)-H(A|B)=H(B)-H(B|A)$ denote the mutual information of $A$ and $B$.\n\n\nConsider the following fully-collaborative instance $\\mathcal{D}_{d}^{\\Delta, p}$: for a uniformly drawn index $i^* \\in [d]$, $\\theta^*_{i^*}=p+\\Delta$ and $\\theta^*_{j}=p$ for all $j \\neq i^*$. The arm set $\\mathcal{X}=\\{x \\in \\{0,1\\}^d: \\sum_{i=1}^{d} x(i)=1 \\}$, and the feature mapping $\\phi(x)=I x$ for all $x \\in \\mathcal{X}$.\nUnder instance $\\mathcal{D}_{d}^{\\Delta, p}$, we have $\\omega(\\ti{\\mathcal{X}})=\\omega(\\mathcal{X})=d$.\n\nLet $\\delta_*>0$ be a small constant that we specify later.\nThere exists a single-agent algorithm $\\mathcal{A}_S$ (e.g., \\cite{peace2020}) that uses at most $T$ samples and guarantees $O(\\log d \\cdot \\exp(-\\frac{\\Delta^2 T}{ d \\log d}))$ error probability on instance $\\mathcal{D}_{d}^{\\Delta, p}$ for any $d>1$.\nRestricting the error probability to the constant $\\delta_*$, we have that for any $d>1$, $\\mathcal{A}_S$ uses at most $T={O}(\\frac{d}{\\Delta^2} \\log d \\cdot \\log \\log d)$ samples to guarantee $\\delta_*$ error probability on instance $\\mathcal{D}_{d}^{\\Delta, p}$.\n\nLet $\\alpha=\\frac{V}{\\beta}$ and $1 \\leq \\alpha \\leq \\log d$.\nAccording to the definition of speedup, a $\\frac{V}{\\alpha}$-speedup distributed algorithm $\\mathcal{A}$ must satisfy that for any $d>1$, $\\mathcal{A}$ uses at most $T=O(\\frac{\\frac{n}{\\Delta^2}}{\\frac{V}{\\alpha}} \\cdot V \\log d \\cdot \\log \\log d) = O(\\frac{\\alpha d}{\\Delta^2} \\log d \\cdot \\log \\log d)$ samples over all agents and guarantees $\\delta_*$ error probability on instance $\\mathcal{D}_{d}^{\\Delta, p}$.\n\n\\vspace*{0.5em}\n\\noindent\n\\textbf{Main Proof.}\nNow, in order to prove Theorem~\\ref{thm:fb_lb_round}, we first prove the following Lemma~\\ref{lemma:lb_fb_main_lemma}, which relaxes the sample budget within logarithmic factors.\n\\begin{lemma} \\label{lemma:lb_fb_main_lemma}\n\tFor any $d>1$,\n\tany distributed algorithm $\\mathcal{A}$ that can use at most $O(\\frac{\\alpha d (\\log \\alpha + \\log \\log d)^2}{\\Delta^2 (\\log d)^2} )$ samples over all agents and guarantee $\\delta_*$ error probability on instance $\\mathcal{D}_{d}^{\\Delta, p}$ needs $\\Omega(\\frac{\\log d}{\\log \\alpha + \\log \\log d})$ communication rounds.\n\\end{lemma}\n\n\n\n\n\nBelow we prove Lemma~\\ref{lemma:lb_fb_main_lemma} by induction (Lemmas~\\ref{lemma:lb_fb_basic},\\ref{lemma:lb_fb_induction}). \n\n\\begin{lemma}[Basic Step] \\label{lemma:lb_fb_basic}\n\tFor any $d>1$ and $1\\leq \\alpha \\leq \\log d$, there is no $1$-round algorithm $\\mathcal{A}_1$ that can use $O(\\frac{\\alpha d}{\\Delta^2})$ samples and guarantee $\\delta_1$ error probability for some constant $\\delta_1 \\in (0,1)$ on instance $\\mathcal{D}_{d}^{\\Delta, p}$.\n\\end{lemma}\n\\yihan{Remark: The tight lower bound of the number of samples is $\\Omega(\\frac{\\alpha d}{\\Delta^2} \\log d)$.}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:lb_fb_basic}]\n\tLet $I$ denote the random variable of index $i^*$. \n\tSince $I$ is uniformly distributed on $[d]$, $H(I)=\\log d$.\n\tLet $S$ denote the sample profile of $\\mathcal{A}_1$ on instance $\\mathcal{D}_{d}^{\\Delta, p}$. According to the definitions of $\\mathcal{X}$ and $\\theta^*$, for instance $\\mathcal{D}_{d}^{\\Delta, p}$, identifying the best arm is equivalent to identifying $I$. Suppose that $\\mathcal{A}_1$ returns the best arm with error probability $\\delta_1$. Using the Fano's inequality (Lemma~\\ref{lemma:fano}), we have \n\t\\begin{align}\n\tH(I|S) \\leq H_2(\\delta_1) +\\delta_1 \\log d \\label{eq:lb_fb_basic_H}\n\t\\end{align}\n\tUsing Lemma~\\ref{lemma:H(I;S)}, we have \n\t\\begin{align*}\n\tH(I|S) = & H(I) - H(I;S)\n\t\\\\\n\t= & \\log d - O( \\frac{\\alpha d}{\\Delta^2} \\cdot \\frac{\\Delta^2}{d} )\n\t\\\\\n\t= & \\log d - O(\\alpha)\n\t\\end{align*}\n\tThen, for some small enough constant $\\delta_1 \\in (0,1)$, Eq.~\\eqref{eq:lb_fb_basic_H} cannot hold. Thus, for any $d>1$ and $1\\leq \\alpha \\leq \\log d$,  there is no $1$-round algorithm $\\mathcal{A}_1$ that can use $O(\\frac{\\alpha d}{\\Delta^2})$ samples and guarantee $\\delta_1$ error probability on instance $\\mathcal{D}_{d}^{\\Delta, p}$.\n\\end{proof}\n\n\\begin{lemma}[Induction Step] \\label{lemma:lb_fb_induction}\n\tSuppose that $1\\leq \\alpha \\leq \\log d$ and $\\delta \\in (0,1)$. If for any $d>1$, there is no $(r-1)$-round algorithm $\\mathcal{A}_{r-1}$ that can use $O(\\frac{\\alpha d}{\\Delta^2 (r-1)^2})$ samples and guarantee $\\delta$ error probability on instance $\\mathcal{D}_{d}^{\\Delta, p}$, then for any $d>1$, there is no $r$-round algorithm $\\mathcal{A}_r$ that can use $O(\\frac{\\alpha d}{\\Delta^2 r^2})$ samples and guarantee $\\delta-O(\\frac{1}{r^2})$ error probability on instance $\\mathcal{D}_{d}^{\\Delta, p}$. \n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:lb_fb_induction}]\n\tWe prove this lemma by contradiction. Suppose that  for some $d>1$, there exists an $r$-round algorithm $\\mathcal{A}_{r}$ that can use $O(\\frac{\\alpha d}{\\Delta^2 r^2})$ samples and guarantee $\\delta$ error probability on instance $\\mathcal{D}_{d}^{\\Delta, p}$.\n\tIn the following, we show how to construct an $(r-1)$-round algorithm $\\mathcal{A}_{r-1}$ that can use $O(\\frac{\\alpha d}{\\Delta^2 (r-1)^2})$ samples and guarantee at most $\\delta+O(\\frac{1}{r^2})$ error probability on instance $\\mathcal{D}_{\\tilde{d}}^{\\Delta, p}$ for some $\\tilde{d}$.\n\t\n\tLet $S_1$ denote the sample profile of $\\mathcal{A}_r$ in the first round on instance $\\mathcal{D}_{d}^{\\Delta, p}$.\n\tSince the number of samples of $\\mathcal{A}_{r}$ is bounded by $O(\\frac{\\alpha d}{\\Delta^2 r^2})$, using Lemma~\\ref{lemma:H(I;S)}, we have\n\t\\begin{align*}\n\tH(I|S_1) = & H(I) - H(I;S_1)\n\t\\\\\n\t= & \\log d - O( \\frac{\\alpha d}{\\Delta^2 r^2}  \\cdot \\frac{\\Delta^2}{d} )\n\t\\\\\n\t= & \\log d - O( \\frac{\\alpha}{r^2} )\n\t\\end{align*}\n\t\n\tLet $\\mathcal{D}_{d|S_1}^{\\Delta, p}$ denote the posterior of $\\mathcal{D}_{d}^{\\Delta, p}$ after observing the sample profile $S_1$. \n\t\n\t\n\tUsing Lemma~\\ref{lemma:tvd_uniform} on random variable $I|S_1$ with parameters $\\gamma=O( \\frac{\\alpha}{r^2})$ and $\\epsilon=o(\\frac{1}{r^2})$, we can write $\\mathcal{D}_{d|S_1}^{\\Delta, p}$ as a convex combination of distributions $\\mathcal{Q}_0, \\mathcal{Q}_1, \\dots, \\mathcal{Q}_{\\ell}$, i.e., $\\mathcal{D}_{d|S_1}^{\\Delta, p}=\\sum_{j=0}^{\\ell} q_j \\mathcal{Q}_j$ such that $q_0=o(\\frac{1}{r^2})$, and for any $j \\geq 1$,\n\t\\begin{align*}\n\t|\\textup{supp}(\\mathcal{Q}_j)| \\geq & \\exp\\sbr{\\log d-\\frac{\\gamma}{\\epsilon} }\n\t\\\\ \n\t= & \\exp\\sbr{\\log d-o(\\alpha) }\n\t\\\\\n\t= & \\frac{d}{e^{o(\\alpha)}} ,\n\t\\end{align*}\n\tand\n\t\\begin{align*}\n\t\\|\\mathcal{Q}_j-\\mathcal{U}_j\\|_{\\textup{TV}} = o(\\frac{1}{r^2}),\n\t\\end{align*}\n\twhere $\\mathcal{U}_j$ is the uniform distribution on $\\textup{supp}(\\mathcal{Q}_j)$. \n\t\n\tSince\n\t\\begin{align*}\n\t\\Pr[\\mathcal{A}_{r} \\textup{ has an error}|\\mathcal{D}_{d|S_1}^{\\Delta, p}] = \\sum_{j=0}^{\\ell} q_j \\Pr[\\mathcal{A}_{r} \\textup{ has an error}|\\mathcal{Q}_j] \\leq \\delta ,\n\t\\end{align*}\n\tusing an average argument, there exists a distribution $\\mathcal{Q}_j$ for some $j\\geq 1$ such that\n\t\\begin{align*}\n\t\\Pr[\\mathcal{A}_{r} \\textup{ has an error}|\\mathcal{Q}_j] \\leq \\delta .\n\t\\end{align*}\n\tLet $\\tilde{d}=|\\textup{supp}(\\mathcal{Q}_j)|\\geq \\frac{d}{e^{o(\\alpha)}}$. \n\tSince $\\|\\mathcal{Q}_j-\\mathcal{U}_j\\|_{\\textup{TV}} = o(\\frac{1}{r^2})$ and $\\mathcal{U}_j$ is equivalent to $\\mathcal{D}_{\\tilde{d}}^{\\Delta, p}$, we have\n\t\\begin{align*}\n\t\\Pr[\\mathcal{A}_{r} \\textup{ has an error}|\\mathcal{D}_{\\tilde{d}}^{\\Delta, p}] \\leq \\delta + o(\\frac{1}{r^2}) .\n\t\\end{align*}\n\t\n\tUnder instance $\\mathcal{D}_{\\tilde{d}}^{\\Delta, p}$, the sample budget for $(r-1)$-round algorithms is $O(\\frac{\\alpha \\tilde{d}}{\\Delta^2 (r-1)^2})$, and it holds that\n\t\\begin{align*}\n\tO(\\frac{\\alpha \\tilde{d}}{\\Delta^2 (r-1)^2}) \\geq  O(\\frac{\\alpha \\frac{d}{e^{o(\\alpha)}} }{\\Delta^2 (r-1)^2}) \n\t\\geq  O(\\frac{\\alpha d^{1-o(1)} }{\\Delta^2 (r-1)^2}) \n\t\\geq  O(\\frac{\\alpha d }{\\Delta^2 r^2})\n\t\\end{align*}\n\t\n\t\n\t\n\t\n\tThen, we can construct an $(r-1)$-round algorithm $\\mathcal{A}_{r-1}$ using algorithm $\\mathcal{A}_{r}$ from the second round. The constructed $\\mathcal{A}_{r-1}$ uses at most $O(\\frac{\\alpha d}{\\Delta^2 r^2} ) \\leq O(\\frac{\\alpha \\tilde{d}}{\\Delta^2 (r-1)^2})$ samples and guarantees $\\delta + o(\\frac{1}{r^2})$ error probability. \n\t\n\tThe specific procedure of $\\mathcal{A}_{r-1}$ is as follows: if $\\mathcal{A}_{r}$ samples some dimension $i \\in \\textup{supp}(\\mathcal{Q}_j)$, then $\\mathcal{A}_{r-1}$ also samples $i$; otherwise, if $\\mathcal{A}_{r}$ samples some dimension $i \\in [d] \\setminus \\textup{supp}(\\mathcal{Q}_j)$, then $\\mathcal{A}_{r-1}$ samples $\\textup{Bernoulli}(p)$ and feeds the outcome to $\\mathcal{A}_{r}$. Finally, if $\\mathcal{A}_{r}$ returns some dimension $i \\in \\textup{supp}(\\mathcal{Q}_j)$, then $\\mathcal{A}_{r-1}$ also returns $i$; otherwise, if $\\mathcal{A}_{r}$ returns some dimension $i \\in [d] \\setminus \\textup{supp}(\\mathcal{Q}_j)$, then $\\mathcal{A}_{r-1}$ returns an arbitrary dimension in $[d] \\setminus \\textup{supp}(\\mathcal{Q}_j)$ (the error case).\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:lb_fb_main_lemma}]\nLet $r_*=\\frac{\\log d}{\\log \\alpha + \\log\\log d}$.\nCombining Lemmas~\\ref{lemma:lb_fb_basic},\\ref{lemma:lb_fb_induction}, we obtain that there is no $r_*$-round algorithm $\\mathcal{A}$ that can use $O(\\frac{\\alpha d}{\\Delta^2 r_*^2})$ samples and guarantee $\\delta_1-\\sum_{r=2}^{r_*} o(\\frac{1}{r^2})$ error probability on instance $\\mathcal{D}_{d}^{\\Delta, p}$ for any $d>1$.\n\n\n\nLet $\\delta_*<\\delta_1-\\sum_{r=2}^{r_*} o(\\frac{1}{r^2})$.\nThus, for any $d>1$ and $1\\leq \\alpha \\leq \\log d$, any distributed algorithm $\\mathcal{A}$ that can use $O(\\frac{\\alpha d}{\\Delta^2 r_*^2})$ samples and guarantee $\\delta_*$ error probability on instance $\\mathcal{D}_{d}^{\\Delta, p}$ must cost \n$\nr_*=\\Omega\\sbr{\\frac{\\log d}{\\log (\\frac{V}{\\beta}) + \\log\\log d}}\n$\ncommunication rounds.\n\\end{proof}\n\nTherefore, for any $d>1$ and $\\frac{V}{\\log d} \\leq \\beta \\leq V$, a $\\beta$-speedup distributed algorithm $\\mathcal{A}$ needs at least \n$\n\\Omega\\sbr{\\frac{\\log \\omega(\\ti{\\mathcal{X}})}{\\log (\\frac{V}{\\beta}) + \\log\\log \\omega(\\ti{\\mathcal{X}})}}\n$\ncommunication rounds under instance $\\mathcal{D}_{d}^{\\Delta, p}$, which completes the proof of Theorem~\\ref{thm:fb_lb_round}.\n\n\n\\section{Technical Tools}\n\n\\begin{lemma}[Lemma 15 in \\cite{high_dimensional_ICML2021}] \\label{lemma:lambda_*_max_sum}\n\tFor $\\lambda^* =\\operatornamewithlimits{argmax}_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\log\\det \\sbr{ I + {\\xi_*}^{-1} \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}} } \\lambda_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }$, we have\n\t\\begin{align*}\n\t\t\\max_{\\ti{x} \\in \\ti{\\mathcal{X}}}  \\| \\phi(\\ti{x}) \\|^2_{\\sbr{  \\xi_* I +  \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }^{-1}} = \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}}  \\| \\phi(\\ti{x}) \\|^2_{\\sbr{  \\xi_* I +  \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }^{-1}} \n\t\\end{align*}\n\\end{lemma}\n\n\n\\begin{lemma} \\label{lemma:sum_ln_det}\n\tFor $\\lambda^* =\\operatornamewithlimits{argmax}_{\\lambda \\in \\triangle_{\\ti{\\mathcal{X}}}} \\log\\det \\sbr{ I + {\\xi_*}^{-1} \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}} } \\lambda_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }$, we have\n\t\\begin{align*}\n\t\\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\log \\sbr{ 1 + \\lambda^*_{\\ti{x}}  \\| \\phi(\\ti{x}) \\|^2_{\\sbr{  \\xi_* I +  \\sum_{\\ti{x}' \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}'} \\phi(\\ti{x}') \\phi(\\ti{x}')^\\top }^{-1}} }\n\t\\leq \\log \\frac{\\det \\sbr{ \\xi_* I +  \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top  }}{ \\det \\sbr{ \\xi_* I } } \n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lemma:sum_ln_det}]\n\tFor any $j \\in [nV]$, let $M_j=\\det \\sbr{ \\xi_* I +  \\sum_{i \\in [j]} \\lambda^*_i \\phi(\\ti{x}_i) \\phi(\\ti{x}_i)^\\top}$.\n\t\\begin{align*}\n\t& \\det \\sbr{ \\xi_* I +  \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top  } \n\t\\\\\n\t= & \\det \\sbr{ \\xi_* I +  \\sum_{i \\in [nV-1]} \\lambda^*_i \\phi(\\ti{x}_i) \\phi(\\ti{x}_i)^\\top + \\lambda^*_{nV} \\phi(\\ti{x}_{nV}) \\phi(\\ti{x}_{nV})^\\top  } \n\t\\\\\n\t= & \\det \\sbr{M_{nV-1}} \\det \\sbr{ I + \\lambda^*_{nV} \\cdot M^{-\\frac{1}{2}}_{nV-1} \\phi(\\ti{x}_{nV}) \\sbr{M^{-\\frac{1}{2}}_{nV-1} \\phi(\\ti{x}_{nV})}^\\top } \n\t\\\\\n\t= & \\det \\sbr{M_{nV-1}} \\det \\sbr{ I + \\lambda^*_{nV} \\cdot \\phi(\\ti{x}_{nV})^\\top M^{-1}_{nV-1}  \\phi(\\ti{x}_{nV}) } \n\t\\\\\n\t= & \\det \\sbr{M_{nV-1}}  \\sbr{ 1 + \\lambda^*_{nV}  \\|\\phi(\\ti{x}_{nV})\\|^2_{M^{-1}_{nV-1}}  } \n\t\\\\\n\t= & \\det \\sbr{\\xi_* I} \\prod_{i=1}^{nV}  \\sbr{ 1 + \\lambda^*_{i}  \\|\\phi(\\ti{x}_{i})\\|^2_{M^{-1}_{i-1}}  } \n\t\\end{align*}\n\tThus,\n\t\\begin{align*}\n\t\\frac{\\det \\sbr{ \\xi_* I +  \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top  } }{\\det \\sbr{\\xi_* I} }\n\t=\\prod_{i=1}^{nV} \\sbr{ 1 + \\lambda^*_{i}  \\|\\phi(\\ti{x}_{i})\\|^2_{M^{-1}_{i-1}}  }  \n\t\\end{align*}\n\tTaking logarithm on both sides, we have\n\t\\begin{align*}\n\t& \\log \\frac{\\det \\sbr{ \\xi_* I +  \\sum_{\\ti{x} \\in \\ti{\\mathcal{X}}} \\lambda^*_{\\ti{x}} \\phi(\\ti{x}) \\phi(\\ti{x})^\\top  } }{\\det \\sbr{\\xi_* I} }\n\t\\\\\n\t= & \\sum_{i=1}^{nV} \\log \\sbr{ 1 + \\lambda^*_{i}  \\|\\phi(\\ti{x}_{i})\\|^2_{M^{-1}_{i-1}}  } \n\t\\\\\n\t\\geq & \\sum_{i=1}^{nV} \\log \\sbr{ 1 + \\lambda^*_{i}  \\|\\phi(\\ti{x}_{i})\\|^2_{M^{-1}_{nV}}  } ,\n\t\\end{align*}\n\twhich completes the proof of Lemma~\\ref{lemma:sum_ln_det}.\n\\end{proof}\n\n\n\\begin{lemma}[Pinsker's inequality] \\label{lemma:pinsker_inequality}\n\tIf $P$ and $Q$ are two probability distributions on a measurable space $(X, \\Sigma)$, then for any measurable event $A \\in \\Sigma$, it holds that \n\t\\begin{align*}\n\t\\abr{P(A)-Q(A)} \\leq \\sqrt{\\frac{1}{2} \\textup{KL}(P \\| Q)}.\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{lemma}[Lemma~29 in \\cite{tao2019collaborative}] \\label{lemma:prod}\n\tFor any $\\gamma_1, \\dots, \\gamma_K \\in [0,1]$ and $x \\geq 0$, it holds that\n\t\\begin{align*}\n\t\\prod_{i=1}^{K} \\max \\{1-\\gamma_i-\\gamma_i x, \\  0\\} \\geq \t\\prod_{i=1}^{K} (1-\\gamma_i) - x\n\t\\end{align*}\n\\end{lemma}\n\n\n\n\\begin{lemma}[Fano's Inequality] \\label{lemma:fano}\n\tLet $A, B$ be random variables and $f$ be a function that given $A$ predicts a value for $B$. If $\\Pr(f(A) \\neq B) \\leq \\delta$, then $H(B | A) \\leq H_2(\\delta) + \\delta \\cdot \\log |B|$.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lemma:H(I;S)}\n\tFor the instance $\\mathcal{D}_{d}^{\\Delta, p}$ with sample profile $\\mathcal{S}$, we have $H(I;S)=O( |S| \\cdot \\frac{\\Delta^2}{d} )$.\n\\end{lemma}\nThe proof of Lemma~\\ref{lemma:H(I;S)} follows the same analysis procedure of Lemma 7 in \\cite{learning_adaptivity2017}. \n\n\n\\begin{lemma}[Lemma 8 in \\cite{learning_adaptivity2017}]\\label{lemma:tvd_uniform}\n\tLet $A \\sim \\mathcal{D}$ be a random variable on $[d]$ with $H(A) \\geq \\log d - \\gamma$ for some $\\gamma \\geq 1$. For any\n\t$\\varepsilon > \\exp (-\\gamma)$, there exists $\\ell + 1$ distributions $\\psi_0, \\psi_1, . . . , \\psi_{\\ell}$ on $[d]$ along with $\\ell + 1$ probabilities $p_0, p_1, \\dots, p_{\\ell}$ ($\\sum_{i=0}^{\\ell} p_i=1$) for some $\\ell = O(\\gamma\/\\varepsilon^3)$ such that $\\mathcal{D}=\\sum_{i=1}^{\\ell} p_i \\psi_i$, $p_0=O(\\varepsilon)$, and for any $i \\geq 1$,\n\t\\begin{enumerate}[1.]\n\t\\item $\\log |\\textup{supp}(\\psi_i)| \\geq \\log d - \\gamma\/\\varepsilon.$\n\t\\item $\\|\\psi_i-\\mathcal{U}_i\\|_{\\textup{TV}} = O(\\varepsilon)$ where $\\mathcal{U}_i$ denotes the uniform distribution on $\\textup{supp}(\\psi_i)$.\n\t\\end{enumerate}\n\\end{lemma}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}