Optimal Rates of (Locally) Differentially Private Heavy-tailed Multi-Armed Bandits

Youming Tao, Yulian Wu, Peng Zhao, Di Wang
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:1546-1574, 2022.

Abstract

In this paper we investigate the problem of stochastic multi-armed bandits (MAB) in the (local) differential privacy (DP/LDP) model. Unlike previous results that assume bounded/sub-Gaussian reward distributions, we focus on the setting where each arm’s reward distribution only has $(1+v)$-th moment with some $v\in (0, 1]$. In the first part, we study the problem in the central $\epsilon$-DP model. We first provide a near-optimal result by developing a private and robust Upper Confidence Bound (UCB) algorithm. Then, we improve the result via a private and robust version of the Successive Elimination (SE) algorithm. Finally, we establish the lower bound to show that the instance-dependent regret of our improved algorithm is optimal. In the second part, we study the problem in the $\epsilon$-LDP model. We propose an algorithm that can be seen as locally private and robust version of SE algorithm, which provably achieves (near) optimal rates for both instance-dependent and instance-independent regret. Our results reveal differences between the problem of private MAB with bounded/sub-Gaussian rewards and heavy-tailed rewards. To achieve these (near) optimal rates, we develop several new hard instances and private robust estimators as byproducts, which might be used to other related problems. Finally, experiments also support our theoretical findings and show the effectiveness of our algorithms.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-tao22a, title = { Optimal Rates of (Locally) Differentially Private Heavy-tailed Multi-Armed Bandits }, author = {Tao, Youming and Wu, Yulian and Zhao, Peng and Wang, Di}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {1546--1574}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/tao22a/tao22a.pdf}, url = {https://proceedings.mlr.press/v151/tao22a.html}, abstract = { In this paper we investigate the problem of stochastic multi-armed bandits (MAB) in the (local) differential privacy (DP/LDP) model. Unlike previous results that assume bounded/sub-Gaussian reward distributions, we focus on the setting where each arm’s reward distribution only has $(1+v)$-th moment with some $v\in (0, 1]$. In the first part, we study the problem in the central $\epsilon$-DP model. We first provide a near-optimal result by developing a private and robust Upper Confidence Bound (UCB) algorithm. Then, we improve the result via a private and robust version of the Successive Elimination (SE) algorithm. Finally, we establish the lower bound to show that the instance-dependent regret of our improved algorithm is optimal. In the second part, we study the problem in the $\epsilon$-LDP model. We propose an algorithm that can be seen as locally private and robust version of SE algorithm, which provably achieves (near) optimal rates for both instance-dependent and instance-independent regret. Our results reveal differences between the problem of private MAB with bounded/sub-Gaussian rewards and heavy-tailed rewards. To achieve these (near) optimal rates, we develop several new hard instances and private robust estimators as byproducts, which might be used to other related problems. Finally, experiments also support our theoretical findings and show the effectiveness of our algorithms. } }
Endnote
%0 Conference Paper %T Optimal Rates of (Locally) Differentially Private Heavy-tailed Multi-Armed Bandits %A Youming Tao %A Yulian Wu %A Peng Zhao %A Di Wang %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-tao22a %I PMLR %P 1546--1574 %U https://proceedings.mlr.press/v151/tao22a.html %V 151 %X In this paper we investigate the problem of stochastic multi-armed bandits (MAB) in the (local) differential privacy (DP/LDP) model. Unlike previous results that assume bounded/sub-Gaussian reward distributions, we focus on the setting where each arm’s reward distribution only has $(1+v)$-th moment with some $v\in (0, 1]$. In the first part, we study the problem in the central $\epsilon$-DP model. We first provide a near-optimal result by developing a private and robust Upper Confidence Bound (UCB) algorithm. Then, we improve the result via a private and robust version of the Successive Elimination (SE) algorithm. Finally, we establish the lower bound to show that the instance-dependent regret of our improved algorithm is optimal. In the second part, we study the problem in the $\epsilon$-LDP model. We propose an algorithm that can be seen as locally private and robust version of SE algorithm, which provably achieves (near) optimal rates for both instance-dependent and instance-independent regret. Our results reveal differences between the problem of private MAB with bounded/sub-Gaussian rewards and heavy-tailed rewards. To achieve these (near) optimal rates, we develop several new hard instances and private robust estimators as byproducts, which might be used to other related problems. Finally, experiments also support our theoretical findings and show the effectiveness of our algorithms.
APA
Tao, Y., Wu, Y., Zhao, P. & Wang, D.. (2022). Optimal Rates of (Locally) Differentially Private Heavy-tailed Multi-Armed Bandits . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:1546-1574 Available from https://proceedings.mlr.press/v151/tao22a.html.

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