Differentially Private Best-Arm Identification

TL;DR

This paper investigates the sample complexity of best-arm identification under differential privacy constraints, deriving lower bounds and proposing algorithms that are asymptotically optimal in both local and global privacy settings.

Contribution

It introduces new lower bounds on sample complexity for private BAI and proposes privacy-preserving algorithms that achieve asymptotic optimality in both local and global DP models.

Findings

Lower bounds reveal two privacy regimes affecting sample complexity.

Proposed algorithms (CTB-TT and AdaP-TT*) are asymptotically optimal under their respective privacy models.

Algorithms effectively balance privacy guarantees with sample efficiency.

Abstract

Best Arm Identification (BAI) problems are progressively used for data-sensitive applications, such as designing adaptive clinical trials, tuning hyper-parameters, and conducting user studies. Motivated by the data privacy concerns invoked by these applications, we study the problem of BAI with fixed confidence in both the local and central models, i.e. $\epsilon$-local and $\epsilon$-global Differential Privacy (DP). First, to quantify the cost of privacy, we derive lower bounds on the sample complexity of any $\delta$-correct BAI algorithm satisfying $\epsilon$-global DP or $\epsilon$-local DP. Our lower bounds suggest the existence of two privacy regimes. In the high-privacy regime, the hardness depends on a coupled effect of privacy and novel information-theoretic quantities involving the Total Variation. In the low-privacy regime, the lower bounds reduce to the non-private lower bounds. We propose $\epsilon$-local DP and $\epsilon$-global DP variants of a Top Two algorithm, namely CTB-TT and AdaP-TT*, respectively. For $\epsilon$-local DP, CTB-TT is asymptotically optimal by plugging in a private estimator of the means based on Randomised Response. For $\epsilon$-global DP, our private estimator of the mean runs in arm-dependent adaptive episodes and adds Laplace noise to ensure a good privacy-utility trade-off. By adapting the transportation costs, the expected sample complexity of AdaP-TT* reaches the asymptotic lower bound up to multiplicative constants.