Optimal Best Arm Identification under Differential Privacy

2025-10-20

AI Summary

This paper investigates the fixed-confidence best-arm identification (BAI) problem for Bernoulli bandits under global differential privacy (DP), aiming to close the substantial gap between existing sample-complexity upper and lower bounds. To this end, we introduce a novel information-theoretic privacy-aware divergence—replacing the standard KL divergence—to derive a tighter DP-specific lower bound. Our method integrates a transportation-cost-based stopping rule, arm-dependent geometric batching for private mean estimation, and Top Two sampling. Theoretically, our algorithm achieves an asymptotic sample-complexity upper bound that matches the new lower bound within a constant factor of 8, establishing the first constant-factor tightness for ε-DP BAI under arbitrary privacy budgets ε > 0. Empirical results demonstrate consistent superiority over existing DP-BAI algorithms across diverse ε values.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Privacy-enhancing technologies
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Best Arm Identification (BAI) algorithms are deployed in data-sensitive applications, such as adaptive clinical trials or user studies. Driven by the privacy concerns of these applications, we study the problem of fixed-confidence BAI under global Differential Privacy (DP) for Bernoulli distributions. While numerous asymptotically optimal BAI algorithms exist in the non-private setting, a significant gap remains between the best lower and upper bounds in the global DP setting. This work reduces this gap to a small multiplicative constant, for any privacy budget $ε$. First, we provide a tighter lower bound on the expected sample complexity of any $δ$-correct and $ε$-global DP strategy. Our lower bound replaces the Kullback-Leibler (KL) divergence in the transportation cost used by the non-private characteristic time with a new information-theoretic quantity that optimally trades off between the KL divergence and the Total Variation distance scaled by $ε$. Second, we introduce a stopping rule based on these transportation costs and a private estimator of the means computed using an arm-dependent geometric batching. En route to proving the correctness of our stopping rule, we derive concentration results of independent interest for the Laplace distribution and for the sum of Bernoulli and Laplace distributions. Third, we propose a Top Two sampling rule based on these transportation costs. For any budget $ε$, we show an asymptotic upper bound on its expected sample complexity that matches our lower bound to a multiplicative constant smaller than $8$. Our algorithm outperforms existing $δ$-correct and $ε$-global DP BAI algorithms for different values of $ε$.

Problem

• Closing the sample complexity gap for differentially private best arm identification
• Developing asymptotically optimal algorithms under global differential privacy constraints
• Providing tight bounds for private BAI using novel information-theoretic quantities

Innovation

• Tighter lower bound using KL divergence and Total Variation trade-off
• Private estimator with arm-dependent geometric batching stopping rule
• Top Two sampling rule matching lower bound within constant factor