Thompson Sampling with Less Exploration is Fast and Optimal

Tianyuan Jin, Xianglin Yang, Xiaokui Xiao, Pan Xu
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:15239-15261, 2023.

Abstract

We propose $\epsilon$-Exploring Thompson Sampling ($\epsilon$-TS), a modified version of the Thompson Sampling (TS) algorithm for multi-armed bandits. In $\epsilon$-TS, arms are selected greedily based on empirical mean rewards with probability $1-\epsilon$, and based on posterior samples obtained from TS with probability $\epsilon$. Here, $\epsilon\in(0,1)$ is a user-defined constant. By reducing exploration, $\epsilon$-TS improves computational efficiency compared to TS while achieving better regret bounds. We establish that $\epsilon$-TS is both minimax optimal and asymptotically optimal for various popular reward distributions, including Gaussian, Bernoulli, Poisson, and Gamma. A key technical advancement in our analysis is the relaxation of the requirement for a stringent anti-concentration bound of the posterior distribution, which was necessary in recent analyses that achieved similar bounds. As a result, $\epsilon$-TS maintains the posterior update structure of TS while minimizing alterations, such as clipping the sampling distribution or solving the inverse of the Kullback-Leibler (KL) divergence between reward distributions, as done in previous work. Furthermore, our algorithm is as easy to implement as TS, but operates significantly faster due to reduced exploration. Empirical evaluations confirm the efficiency and optimality of $\epsilon$-TS.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-jin23b, title = {Thompson Sampling with Less Exploration is Fast and Optimal}, author = {Jin, Tianyuan and Yang, Xianglin and Xiao, Xiaokui and Xu, Pan}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {15239--15261}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/jin23b/jin23b.pdf}, url = {https://proceedings.mlr.press/v202/jin23b.html}, abstract = {We propose $\epsilon$-Exploring Thompson Sampling ($\epsilon$-TS), a modified version of the Thompson Sampling (TS) algorithm for multi-armed bandits. In $\epsilon$-TS, arms are selected greedily based on empirical mean rewards with probability $1-\epsilon$, and based on posterior samples obtained from TS with probability $\epsilon$. Here, $\epsilon\in(0,1)$ is a user-defined constant. By reducing exploration, $\epsilon$-TS improves computational efficiency compared to TS while achieving better regret bounds. We establish that $\epsilon$-TS is both minimax optimal and asymptotically optimal for various popular reward distributions, including Gaussian, Bernoulli, Poisson, and Gamma. A key technical advancement in our analysis is the relaxation of the requirement for a stringent anti-concentration bound of the posterior distribution, which was necessary in recent analyses that achieved similar bounds. As a result, $\epsilon$-TS maintains the posterior update structure of TS while minimizing alterations, such as clipping the sampling distribution or solving the inverse of the Kullback-Leibler (KL) divergence between reward distributions, as done in previous work. Furthermore, our algorithm is as easy to implement as TS, but operates significantly faster due to reduced exploration. Empirical evaluations confirm the efficiency and optimality of $\epsilon$-TS.} }
Endnote
%0 Conference Paper %T Thompson Sampling with Less Exploration is Fast and Optimal %A Tianyuan Jin %A Xianglin Yang %A Xiaokui Xiao %A Pan Xu %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-jin23b %I PMLR %P 15239--15261 %U https://proceedings.mlr.press/v202/jin23b.html %V 202 %X We propose $\epsilon$-Exploring Thompson Sampling ($\epsilon$-TS), a modified version of the Thompson Sampling (TS) algorithm for multi-armed bandits. In $\epsilon$-TS, arms are selected greedily based on empirical mean rewards with probability $1-\epsilon$, and based on posterior samples obtained from TS with probability $\epsilon$. Here, $\epsilon\in(0,1)$ is a user-defined constant. By reducing exploration, $\epsilon$-TS improves computational efficiency compared to TS while achieving better regret bounds. We establish that $\epsilon$-TS is both minimax optimal and asymptotically optimal for various popular reward distributions, including Gaussian, Bernoulli, Poisson, and Gamma. A key technical advancement in our analysis is the relaxation of the requirement for a stringent anti-concentration bound of the posterior distribution, which was necessary in recent analyses that achieved similar bounds. As a result, $\epsilon$-TS maintains the posterior update structure of TS while minimizing alterations, such as clipping the sampling distribution or solving the inverse of the Kullback-Leibler (KL) divergence between reward distributions, as done in previous work. Furthermore, our algorithm is as easy to implement as TS, but operates significantly faster due to reduced exploration. Empirical evaluations confirm the efficiency and optimality of $\epsilon$-TS.
APA
Jin, T., Yang, X., Xiao, X. & Xu, P.. (2023). Thompson Sampling with Less Exploration is Fast and Optimal. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:15239-15261 Available from https://proceedings.mlr.press/v202/jin23b.html.

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