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Optimal Best Arm Identification under Differential Privacy

Marc Jourdan, Achraf Azize

TL;DR

This work tackles fixed‑confidence Best Arm Identification under ε‑global DP for Bernoulli bandits. It introduces a new information divergence d_ε that blends KL and TV costs under privacy, derives a tight non‑asymptotic lower bound on sample complexity, and pairs a private geometric estimator with a GLR stopping rule. Building on this, the authors propose a Top Two sampling rule (DPTT) whose asymptotic performance matches the lower bound up to a constant strictly less than 8, for any privacy budget ε. The approach yields δ‑correct, ε‑DP BAI with strong theoretical guarantees and superior empirical performance across privacy regimes, suggesting practical utility for privacy‑sensitive adaptive experiments. Potential extensions include other distribution families, structure, and alternative DP variants.

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 $ε$.

Optimal Best Arm Identification under Differential Privacy

TL;DR

This work tackles fixed‑confidence Best Arm Identification under ε‑global DP for Bernoulli bandits. It introduces a new information divergence d_ε that blends KL and TV costs under privacy, derives a tight non‑asymptotic lower bound on sample complexity, and pairs a private geometric estimator with a GLR stopping rule. Building on this, the authors propose a Top Two sampling rule (DPTT) whose asymptotic performance matches the lower bound up to a constant strictly less than 8, for any privacy budget ε. The approach yields δ‑correct, ε‑DP BAI with strong theoretical guarantees and superior empirical performance across privacy regimes, suggesting practical utility for privacy‑sensitive adaptive experiments. Potential extensions include other distribution families, structure, and alternative DP variants.

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 . First, we provide a tighter lower bound on the expected sample complexity of any -correct and -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 . 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 , we show an asymptotic upper bound on its expected sample complexity that matches our lower bound to a multiplicative constant smaller than . Our algorithm outperforms existing -correct and -global DP BAI algorithms for different values of .
Paper Structure (75 sections, 68 theorems, 398 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 75 sections, 68 theorems, 398 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background: Best Arm Identification under Differential Privacy
  3. Lower Bound on the Expected Sample Complexity
  4. Generalized Likelihood Ratio Stopping Rule
  5. Top Two Sampling Rule
  6. Experiments
  7. Conclusion
  8. Outline
  9. Notation
  10. Related Work and Extended Discussion
  11. Related Work
  12. Gap in the Literature.
  13. Extended Discussions
  14. Limitations of Theorem \ref{['thm:asymptotic_upper_bound']}
  15. Beyond Bernoulli Distributions
  16. ...and 60 more sections

Key Result

Theorem 1

Let $f: \mathcal{X} \rightarrow \mathbb{R}^d$ be an algorithm with sensitivity $s(f) \mathrel{\triangleq} \underset{\substack{\mathcal{D}, \mathcal{D'} \text{ s.t }d_{\text{Ham}}(\mathcal{D} , \mathcal{D'}) = 1}}{\max} \left\|f(\mathcal{D}) - f(\mathcal{D'})\right\|_1$, where $\left\|\cdot\right\|_1

Figures (2)

  • Figure 1: Empirical stopping time $\tau_{\epsilon,\delta}$ (mean $\pm 2$ std) for $\delta=10^{-2}$ with respect to the privacy budget $\epsilon$ on Bernoulli instances (a) $\mu_{1}$ and (b) $\mu_{2}$. The vertical line separates the two privacy regimes.
  • Figure 2: Empirical stopping time $\tau_{\epsilon,\delta}$ (mean $\pm 2$ std. over 1000 runs) for $\delta=10^{-2}$ with respect to the privacy budget $\epsilon$ for $\epsilon$-global DP on Bernoulli instances $\mu_{3}$, $\mu_{4}$, $\mu_{5}$ and $\mu_{6}$ (top left to bottom right). The shaded vertical line separates the two privacy regimes.

Theorems & Definitions (126)

  • Definition 1: $\epsilon$-DP Dwork_Calibration
  • Theorem 1: Laplace mechanism, Theorem 3.6 dwork2014algorithmic
  • Theorem 2
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • Theorem 6
  • Theorem 7
  • proof
  • Definition 2: The policy
  • ...and 116 more