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Register a new userOptimal Rates of (Locally) Differentially Private Heavy-tailed Multi-Armed Bandits
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Youming Tao
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this paper we study the problem of stochastic multi-armed bandits (MAB) in the (local) differential privacy (DP/LDP) model. Unlike the previous results which need to assume bounded reward distributions, here we mainly focus on the case the reward distribution of each arm only has $(1+v)$-th moment with some $vin (0, 1]$. 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 show that the instance-dependent regret bound of our improved algorithm is optimal by showing its lower bound. In the second part of the paper, we study the problem in the $epsilon$-LDP model. We propose an algorithm which could be seen as locally private and robust version of the SE algorithm, and show it could achieve (near) optimal rates for both instance-dependent and instance-independent regrets. All of the above results can also reveal the differences between the problem of private MAB with bounded 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 could be used to other related problems. Finally, experimental results also support our theoretical analysis and show the effectiveness of our algorithms.
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We give an $(varepsilon,delta)$-differentially private algorithm for the multi-armed bandit (MAB) problem in the shuffle model with a distribution-dependent regret of $Oleft(left(sum_{ain [k]:Delta_a>0}frac{log T}{Delta_a}right)+frac{ksqrt{logfrac{1}{delta}}log T}{varepsilon}right)$, and a distribution-independent regret of $Oleft(sqrt{kTlog T}+frac{ksqrt{logfrac{1}{delta}}log T}{varepsilon}right)$, where $T$ is the number of rounds, $Delta_a$ is the suboptimality gap of the arm $a$, and $k$ is the total number of arms. Our upper bound almost matches the regret of the best known algorithms for the centralized model, and significantly outperforms the best known algorithm in the local model.
We study locally differentially private (LDP) bandits learning in this paper. First, we propose simple black-box reduction frameworks that can solve a large family of context-free bandits learning problems with LDP guarantee. Based on our frameworks, we can improve previous best results for private bandits learning with one-point feedback, such as private Bandits Convex Optimization, and obtain the first result for Bandits Convex Optimization (BCO) with multi-point feedback under LDP. LDP guarantee and black-box nature make our frameworks more attractive in real applications compared with previous specifically designed and relatively weaker differentially private (DP) context-free bandits algorithms. Further, we extend our $(varepsilon, delta)$-LDP algorithm to Generalized Linear Bandits, which enjoys a sub-linear regret $tilde{O}(T^{3/4}/varepsilon)$ and is conjectured to be nearly optimal. Note that given the existing $Omega(T)$ lower bound for DP contextual linear bandits (Shariff & Sheffe, 2018), our result shows a fundamental difference between LDP and DP contextual bandits learning.
In this paper, we study Combinatorial Semi-Bandits (CSB) that is an extension of classic Multi-Armed Bandits (MAB) under Differential Privacy (DP) and stronger Local Differential Privacy (LDP) setting. Since the server receives more information from users in CSB, it usually causes additional dependence on the dimension of data, which is a notorious side-effect for privacy preserving learning. However for CSB under two common smoothness assumptions cite{kveton2015tight,chen2016combinatorial}, we show it is possible to remove this side-effect. In detail, for $B_{infty}$-bounded smooth CSB under either $varepsilon$-LDP or $varepsilon$-DP, we prove the optimal regret bound is $Theta(frac{mB^2_{infty}ln T } {Deltaepsilon^2})$ or $tilde{Theta}(frac{mB^2_{infty}ln T} { Deltaepsilon})$ respectively, where $T$ is time period, $Delta$ is the gap of rewards and $m$ is the number of base arms, by proposing novel algorithms and matching lower bounds. For $B_1$-bounded smooth CSB under $varepsilon$-DP, we also prove the optimal regret bound is $tilde{Theta}(frac{mKB^2_1ln T} {Deltaepsilon})$ with both upper bound and lower bound, where $K$ is the maximum number of feedback in each round. All above results nearly match corresponding non-private optimal rates, which imply there is no additional price for (locally) differentially private CSB in above common settings.
We study stochastic convex optimization with heavy-tailed data under the constraint of differential privacy. Most prior work on this problem is restricted to the case where the loss function is Lipschitz. Instead, as introduced by Wang, Xiao, Devadas, and Xu, we study general convex loss functions with the assumption that the distribution of gradients has bounded $k$-th moments. We provide improved upper bounds on the excess population risk under approximate differential privacy of $tilde{O}left(sqrt{frac{d}{n}}+left(frac{d}{epsilon n}right)^{frac{k-1}{k}}right)$ and $tilde{O}left(frac{d}{n}+left(frac{d}{epsilon n}right)^{frac{2k-2}{k}}right)$ for convex and strongly convex loss functions, respectively. We also prove nearly-matching lower bounds under the constraint of pure differential privacy, giving strong evidence that our bounds are tight.
We study the best-arm identification problem in multi-armed bandits with stochastic, potentially private rewards, when the goal is to identify the arm with the highest quantile at a fixed, prescribed level. First, we propose a (non-private) successive elimination algorithm for strictly optimal best-arm identification, we show that our algorithm is $delta$-PAC and we characterize its sample complexity. Further, we provide a lower bound on the expected number of pulls, showing that the proposed algorithm is essentially optimal up to logarithmic factors. Both upper and lower complexity bounds depend on a special definition of the associated suboptimality gap, designed in particular for the quantile bandit problem, as we show when the gap approaches zero, best-arm identification is impossible. Second, motivated by applications where the rewards are private, we provide a differentially private successive elimination algorithm whose sample complexity is finite even for distributions with infinite support-size, and we characterize its sample complexity. Our algorithms do not require prior knowledge of either the suboptimality gap or other statistical information related to the bandit problem at hand.
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