Geometric Resampling in Nearly Linear Time for Follow-the-Perturbed-Leader with Best-of-Both-Worlds Guarantee in Bandit Problems

Botao Chen, Jongyeong Lee, Junya Honda
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:8403-8426, 2025.

Abstract

This paper studies the complexity and optimality of Follow-the-Perturbed-Leader (FTPL) policy in the $K$-armed bandit problems. FTPL is a promising policy that achieves the Best-of-Both-Worlds (BOBW) guarantee without solving an optimization problem unlike Follow-the-Regularized-Leader (FTRL). However, FTPL needs a procedure called geometric resampling to estimate the loss, which needs $O(K^2)$ per-round average complexity, usually worse than that of FTRL. To address this issue, we propose a novel technique, which we call Conditional Geometric Resampling (CGR), for unbiased loss estimation applicable to general perturbation distributions. CGR reduces the average complexity to $O(K\log K)$ without sacrificing the regret bounds. We also propose a biased version of CGR that can control the worst-case complexity while keeping the BOBW guarantee for a certain perturbation distribution. We confirm through experiments that CGR does not only significantly improve the average and worst-case runtime but also achieve better regret thanks to the stable loss estimation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-chen25ai, title = {Geometric Resampling in Nearly Linear Time for Follow-the-Perturbed-Leader with Best-of-Both-Worlds Guarantee in Bandit Problems}, author = {Chen, Botao and Lee, Jongyeong and Honda, Junya}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {8403--8426}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/chen25ai/chen25ai.pdf}, url = {https://proceedings.mlr.press/v267/chen25ai.html}, abstract = {This paper studies the complexity and optimality of Follow-the-Perturbed-Leader (FTPL) policy in the $K$-armed bandit problems. FTPL is a promising policy that achieves the Best-of-Both-Worlds (BOBW) guarantee without solving an optimization problem unlike Follow-the-Regularized-Leader (FTRL). However, FTPL needs a procedure called geometric resampling to estimate the loss, which needs $O(K^2)$ per-round average complexity, usually worse than that of FTRL. To address this issue, we propose a novel technique, which we call Conditional Geometric Resampling (CGR), for unbiased loss estimation applicable to general perturbation distributions. CGR reduces the average complexity to $O(K\log K)$ without sacrificing the regret bounds. We also propose a biased version of CGR that can control the worst-case complexity while keeping the BOBW guarantee for a certain perturbation distribution. We confirm through experiments that CGR does not only significantly improve the average and worst-case runtime but also achieve better regret thanks to the stable loss estimation.} }
Endnote
%0 Conference Paper %T Geometric Resampling in Nearly Linear Time for Follow-the-Perturbed-Leader with Best-of-Both-Worlds Guarantee in Bandit Problems %A Botao Chen %A Jongyeong Lee %A Junya Honda %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-chen25ai %I PMLR %P 8403--8426 %U https://proceedings.mlr.press/v267/chen25ai.html %V 267 %X This paper studies the complexity and optimality of Follow-the-Perturbed-Leader (FTPL) policy in the $K$-armed bandit problems. FTPL is a promising policy that achieves the Best-of-Both-Worlds (BOBW) guarantee without solving an optimization problem unlike Follow-the-Regularized-Leader (FTRL). However, FTPL needs a procedure called geometric resampling to estimate the loss, which needs $O(K^2)$ per-round average complexity, usually worse than that of FTRL. To address this issue, we propose a novel technique, which we call Conditional Geometric Resampling (CGR), for unbiased loss estimation applicable to general perturbation distributions. CGR reduces the average complexity to $O(K\log K)$ without sacrificing the regret bounds. We also propose a biased version of CGR that can control the worst-case complexity while keeping the BOBW guarantee for a certain perturbation distribution. We confirm through experiments that CGR does not only significantly improve the average and worst-case runtime but also achieve better regret thanks to the stable loss estimation.
APA
Chen, B., Lee, J. & Honda, J.. (2025). Geometric Resampling in Nearly Linear Time for Follow-the-Perturbed-Leader with Best-of-Both-Worlds Guarantee in Bandit Problems. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:8403-8426 Available from https://proceedings.mlr.press/v267/chen25ai.html.

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