Tight Regret Bounds for Single-pass Streaming Multi-armed Bandits

Chen Wang
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:35525-35547, 2023.

Abstract

Regret minimization in streaming multi-armed bandits (MABs) has been studied extensively, and recent work has shown that algorithms with $o(K)$ memory have to incur $\Omega(T^{2/3})$ regret, where $K$ and $T$ are the numbers of arms and trials. However, the previous best regret upper bound is still $O(K^{1/3} T^{2/3}\log^{1/3}(T))$, which is achieved by the simple uniform exploration algorithm. In this paper, we close this gap and complete the picture of regret minimization in single-pass streaming MABs. We first improve the regret lower bound to $\Omega(K^{1/3}T^{2/3})$ for algorithms with $o(K)$ memory. We then show that the $\log^{1/3}(T)$ factor is not necessary by designing algorithms with at most $O(\log^*(K))$-arm memory and achieve $O(K^{1/3}T^{2/3})$ expected regret based on streaming $\varepsilon$-best arm algorithms. We further tested the empirical performances of our algorithms on simulated MABs instances, where the proposed algorithms outperform the benchmark uniform exploration algorithm by a large margin and, on occasion, reduce the regret by up to 70%.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-wang23a, title = {Tight Regret Bounds for Single-pass Streaming Multi-armed Bandits}, author = {Wang, Chen}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {35525--35547}, 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/wang23a/wang23a.pdf}, url = {https://proceedings.mlr.press/v202/wang23a.html}, abstract = {Regret minimization in streaming multi-armed bandits (MABs) has been studied extensively, and recent work has shown that algorithms with $o(K)$ memory have to incur $\Omega(T^{2/3})$ regret, where $K$ and $T$ are the numbers of arms and trials. However, the previous best regret upper bound is still $O(K^{1/3} T^{2/3}\log^{1/3}(T))$, which is achieved by the simple uniform exploration algorithm. In this paper, we close this gap and complete the picture of regret minimization in single-pass streaming MABs. We first improve the regret lower bound to $\Omega(K^{1/3}T^{2/3})$ for algorithms with $o(K)$ memory. We then show that the $\log^{1/3}(T)$ factor is not necessary by designing algorithms with at most $O(\log^*(K))$-arm memory and achieve $O(K^{1/3}T^{2/3})$ expected regret based on streaming $\varepsilon$-best arm algorithms. We further tested the empirical performances of our algorithms on simulated MABs instances, where the proposed algorithms outperform the benchmark uniform exploration algorithm by a large margin and, on occasion, reduce the regret by up to 70%.} }
Endnote
%0 Conference Paper %T Tight Regret Bounds for Single-pass Streaming Multi-armed Bandits %A Chen Wang %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-wang23a %I PMLR %P 35525--35547 %U https://proceedings.mlr.press/v202/wang23a.html %V 202 %X Regret minimization in streaming multi-armed bandits (MABs) has been studied extensively, and recent work has shown that algorithms with $o(K)$ memory have to incur $\Omega(T^{2/3})$ regret, where $K$ and $T$ are the numbers of arms and trials. However, the previous best regret upper bound is still $O(K^{1/3} T^{2/3}\log^{1/3}(T))$, which is achieved by the simple uniform exploration algorithm. In this paper, we close this gap and complete the picture of regret minimization in single-pass streaming MABs. We first improve the regret lower bound to $\Omega(K^{1/3}T^{2/3})$ for algorithms with $o(K)$ memory. We then show that the $\log^{1/3}(T)$ factor is not necessary by designing algorithms with at most $O(\log^*(K))$-arm memory and achieve $O(K^{1/3}T^{2/3})$ expected regret based on streaming $\varepsilon$-best arm algorithms. We further tested the empirical performances of our algorithms on simulated MABs instances, where the proposed algorithms outperform the benchmark uniform exploration algorithm by a large margin and, on occasion, reduce the regret by up to 70%.
APA
Wang, C.. (2023). Tight Regret Bounds for Single-pass Streaming Multi-armed Bandits. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:35525-35547 Available from https://proceedings.mlr.press/v202/wang23a.html.

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