Differentially Private Episodic Reinforcement Learning with Heavy-tailed Rewards

Yulian Wu, Xingyu Zhou, Sayak Ray Chowdhury, Di Wang
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:37880-37918, 2023.

Abstract

In this paper we study the problem of (finite horizon tabular) Markov decision processes (MDPs) with heavy-tailed rewards under the constraint of differential privacy (DP). Compared with the previous studies for private reinforcement learning that typically assume rewards are sampled from some bounded or sub-Gaussian distributions to ensure DP, we consider the setting where reward distributions have only finite $(1+v)$-th moments with some $v \in (0,1]$. By resorting to robust mean estimators for rewards, we first propose two frameworks for heavy-tailed MDPs, i.e., one is for value iteration and another is for policy optimization. Under each framework, we consider both joint differential privacy (JDP) and local differential privacy (LDP) models. Based on our frameworks, we provide regret upper bounds for both JDP and LDP cases, and show that the moment of distributions and privacy budget have significant impact on regrets. Finally, we establish a lower bound of regret minimization for heavy-tailed MDPs in JDP model by reducing it to the instance-independent lower bound of heavy-tailed multi-armed bandits in DP model. We also show the lower bound for the problem in LDP by adopting some private minimax methods. Our results reveal that there are fundamental differences between the problem of private RL with sub-Gaussian and that with heavy-tailed rewards.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-wu23aa, title = {Differentially Private Episodic Reinforcement Learning with Heavy-tailed Rewards}, author = {Wu, Yulian and Zhou, Xingyu and Ray Chowdhury, Sayak and Wang, Di}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {37880--37918}, 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/wu23aa/wu23aa.pdf}, url = {https://proceedings.mlr.press/v202/wu23aa.html}, abstract = {In this paper we study the problem of (finite horizon tabular) Markov decision processes (MDPs) with heavy-tailed rewards under the constraint of differential privacy (DP). Compared with the previous studies for private reinforcement learning that typically assume rewards are sampled from some bounded or sub-Gaussian distributions to ensure DP, we consider the setting where reward distributions have only finite $(1+v)$-th moments with some $v \in (0,1]$. By resorting to robust mean estimators for rewards, we first propose two frameworks for heavy-tailed MDPs, i.e., one is for value iteration and another is for policy optimization. Under each framework, we consider both joint differential privacy (JDP) and local differential privacy (LDP) models. Based on our frameworks, we provide regret upper bounds for both JDP and LDP cases, and show that the moment of distributions and privacy budget have significant impact on regrets. Finally, we establish a lower bound of regret minimization for heavy-tailed MDPs in JDP model by reducing it to the instance-independent lower bound of heavy-tailed multi-armed bandits in DP model. We also show the lower bound for the problem in LDP by adopting some private minimax methods. Our results reveal that there are fundamental differences between the problem of private RL with sub-Gaussian and that with heavy-tailed rewards.} }
Endnote
%0 Conference Paper %T Differentially Private Episodic Reinforcement Learning with Heavy-tailed Rewards %A Yulian Wu %A Xingyu Zhou %A Sayak Ray Chowdhury %A Di 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-wu23aa %I PMLR %P 37880--37918 %U https://proceedings.mlr.press/v202/wu23aa.html %V 202 %X In this paper we study the problem of (finite horizon tabular) Markov decision processes (MDPs) with heavy-tailed rewards under the constraint of differential privacy (DP). Compared with the previous studies for private reinforcement learning that typically assume rewards are sampled from some bounded or sub-Gaussian distributions to ensure DP, we consider the setting where reward distributions have only finite $(1+v)$-th moments with some $v \in (0,1]$. By resorting to robust mean estimators for rewards, we first propose two frameworks for heavy-tailed MDPs, i.e., one is for value iteration and another is for policy optimization. Under each framework, we consider both joint differential privacy (JDP) and local differential privacy (LDP) models. Based on our frameworks, we provide regret upper bounds for both JDP and LDP cases, and show that the moment of distributions and privacy budget have significant impact on regrets. Finally, we establish a lower bound of regret minimization for heavy-tailed MDPs in JDP model by reducing it to the instance-independent lower bound of heavy-tailed multi-armed bandits in DP model. We also show the lower bound for the problem in LDP by adopting some private minimax methods. Our results reveal that there are fundamental differences between the problem of private RL with sub-Gaussian and that with heavy-tailed rewards.
APA
Wu, Y., Zhou, X., Ray Chowdhury, S. & Wang, D.. (2023). Differentially Private Episodic Reinforcement Learning with Heavy-tailed Rewards. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:37880-37918 Available from https://proceedings.mlr.press/v202/wu23aa.html.

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