Asymptotically Optimal Best Arm Identification with Fixed-Budget under Differential Privacy

2026-10-03

AI Summary

This study addresses the fixed-budget best-arm identification problem under pure ε-differential privacy, tackling the fundamental challenge of balancing statistical efficiency with privacy preservation. We propose AO-Pri-BAI, an adaptive algorithm that leverages a Laplace tree mechanism to maintain private estimates and optimizes the sampling strategy via minimax interaction learning. Theoretically, we derive a privacy-aware transport exponent as an upper bound on error decay and, drawing upon information-theoretic principles, prove that the algorithm achieves an asymptotically optimal decay rate while satisfying strict differential privacy constraints. Empirical evaluations demonstrate that AO-Pri-BAI significantly outperforms existing baselines in non-asymptotic regimes.

Abstract

Best arm identification under differential privacy is a pure-exploration problem in which both statistical efficiency and privacy protection must be achieved simultaneously. We study fixed-budget best arm identification for bandits under pure $\epsilon$-differential privacy, where the learner must recommend an arm after a prescribed sampling budget while protecting the full transcript. We prove that the optimal exponential decay rate of the error probability is upper bounded by an instance-dependent privacy-aware transportation exponent that differs from the analogous quantity used to characterize the stopping time in fixed-confidence analysis by Jourdan and Azize [2025]. Guided by this exponent, we propose AO-Pri-BAI, an adaptive algorithm that maintains private running estimates through Laplace-tree mechanisms and learns a sampling design through a min--max interaction between hard alternatives and arm allocations. We prove that AO-Pri-BAI satisfies pure $\epsilon$-differential privacy. We also establish that the exponent of the failure probability of AO-Pri-BAI matches the privacy-aware benchmark. Numerical studies show that even in the non-asymptotic setting, AO-Pri-BAI outperforms benchmark algorithms on various instances, complementing the theoretical analyses.

Problem

• Best Arm Identification
• Differential Privacy
• Fixed-Budget
• Multi-Armed Bandits
• Pure Exploration

Innovation

• Best Arm Identification
• Differential Privacy
• Fixed-Budget
• Laplace-tree Mechanism
• Asymptotic Optimality