Title: Optimal Best Arm Identification under Differential Privacy
Venue: NeurIPS 2025
Accepted: yes
Keywords: Bandit Algorithms

Differential Privacy

Best Arm Identification

Fixed-Confidence

Bernoulli Distribution
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 $\epsilon$. First, we provide a tighter lower bound on the expected sample complexity of any $\delta$-correct and $\epsilon$-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 $\epsilon$. 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 $\epsilon$, 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 $\delta$-correct and $\epsilon$-global DP BAI algorithms for different values of $\epsilon$.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors study the problem of Best Arm Identification (BAI) with fixed confidence $\\delta$ under global $\\epsilon$-DP, assuming Bernoulli bandits. In this setting, they improve upon a previously established lower bound and propose an asymptotically optimal algorithm that matches the new lower bound, leaving only a small multiplicative factor gap between the bounds. The proposed algorithm (DP Top-Two Sampling with new sampling rule) achieves better sample complexity than previously established algorithms, requiring only about twice as many samples as the non-private BAI in a low privacy regime.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is technically solid, well written, and builds upon a series of previous works. The Related Work section in the appendix provides a comprehensive overview of the literature on private bandits.\n\nThe paper proves a new lower bound for the BAI problem. Even though the tradeoff between KL and TV distances has already been introduced for regret minimization, the BAI setting required a novel proof technique. The authors propose a new sampling rule for the DP Top-Two algorithm, achieving provably asymptotically optimal sample complexity and demonstrating better numerical results than alternative methods.\n\nEven though this might be of limited interest to the broader DP community, I believe it will be a valuable contribution to DP bandit research. Therefore, I support the acceptance of the paper.\n\n**Weaknesses:**\n\n1) The setting is quite limited, as it focuses on BAI with a fixed confidence level, \\epsilon-DP, and only Bernoulli bandits.\n\n2) The problem is only solved asymptotically; a gap still remains, and it does not shrink in the limit to the non-private case.\n\n3) I do not find the paper’s motivation for private BAI convincing. While I do not doubt the importance of the problem, I believe it could be presented better, possibly with more examples.\n\n4) The presentation would benefit from polishing. For instance, the abbreviation \"GLR\" is used in the contributions section but is only defined in full in Section 4.\n\nQUESTIONS:\nI have a few questions regarding numerical experiments:\n\n1) How would Figure 1 change for different values of $\\delta$? \n\n2) Could you quantify the boundary between the low- and high-privacy regimes shown in Figure 1?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies fixed-confidence best-arm identification under global differential privacy for Bernoulli bandits. It develops a new information-theoretic lower bound on sample complexity, refining previous KL/TV-based transportation cost characterizations, and proposes an algorithm that achieves this lower bound to within a small constant factor.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n- **Strong theoretical contributions**: Developing novel concentration inequalities for Laplace distributions and mixed Bernoulli-Laplace sums, the `d_epsilon` divergence is elegant and leads to provably tighter lower bounds.\n- **Near-optimal performance**: The elimination of \"observation forgetting\" is clever, while it introduces more Laplace noise, the logarithmic growth is well-controlled and preserves more signal. The geometric batching approach is elegant.\n- **Empirical support**: Good experimental results showing DP-TT outperforms existing methods across different privacy regimes.\n\n**Weaknesses:**\n\n- **Limited scope**: Focused only on Bernoulli rewards; no extension to other sub-Gaussian families.\n- **Presentation density**: Some sections (e.g., signed `d_epsilon` definitions) are hard to parse; clearer intuition would help.\n- **Practical considerations**: Little discussion on how to set ε or interpret privacy-utility tradeoffs in real applications.\n\nQUESTIONS:\n1. Theorem 6 relies on a unique best arm. In practice, many arms could have nearly identical means. How does DP-TT behave in these near-tie scenarios, and would the stopping rule or estimator become unstable?  \n2. Experiment section describes the experiments on Bernoulli means but does not explore systematically how the sample complexity changes for a wider range of K, δ, or ε. Could the authors expand on how to design experiments to stress-test DP-TT, and share best practices for tuning these hyperparameters?  \n3. The conclusion mentions a possible extension to other DP frameworks, but no technical discussion is provided. Could the authors outline what would need to change in DP-TT or the lower bound if moving to (ε,δ)-DP or Rényi DP?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors consider a bandit problem with $K$ arms generating a Bernoulli distributed random variable when picked. The goal is to find the arm with highest corresponding mean $\\mu \\in (0,1)^K$, whilst adhering to an $\\epsilon$-differential privacy guarantee, in the sense that the arm identification algorithm's output should not reveal any information about the underlying sample.\n\nFormally, an algorithm consists of a stopping time $\\tau$, an arm recommendation $\\tilde{a}$ and a sequence of arm recommendations up until $\\tau$; $(a_1,\\dots,a_\\tau)$. Given two sequences of Hamming distance 1-apart arm draws $R,R'$, \n$$P( (T,\\tilde{a},(a_1,\\dots,a_T) \\in A | R) \\leq e^\\epsilon P( (T,\\tilde{a},(a_1,\\dots,a_T) \\in A | R').$$\n\nWithin this setup, the authors:\n* Provide a lower bound on the expected stopping time of a approximately corrected and $\\epsilon$-DP algorithm.\n* They provide a $\\epsilon$-DP stopping rule that matches the lower bound up to a constant factor.\n\nSTRENGTHS AND WEAKNESSES:\nAs a main strength: I believe the authors provide a novel and technically sound contribution to bandit problems under privacy, improving over existing bounds in the literature.\n\nThe main weakness of the paper I see is its setting: considering only Bernoulli samples is quite limiting. I would like to see more discussion on why this limited setting is studied and which parts do no hold if one has for example a Gaussian BAI problem. \n\nMinor comments:\n* The grammar and punctuation could be improved in certain places.\n* I would suggest (1) and (2) not to be given *inside* of Theorem 2 but before it.\n* Similarly, Theorem 3, a lot is defined inside of the theorem statement, unnecessarily so? I would suggest the authors use the additional space to properly introduce and unpack the necessary quantities for Theorem 3 (and 2).\n* I would avoid the name \"global DP\" personally; it seems that the setting is close to what is commonly called central DP in the i.i.d. setting.\n* The choice of the $\\delta$ notation is somewhat unfortunate given the differential privacy definition in full generality involving $\\epsilon$ and $\\delta$.\n* The notation $\\cF^K$ is poorly explained and seems superfluous; as it is a set that is characterized by $\\mu$.\n\nQUESTIONS:\nDo the authors think that the results hold under $(\\epsilon,\\delta)$-differential privacy as well? I.e. can the lower bound be shown to hold for the more general differential privacy definition?\n\nLIMITATIONS:\nThe main limitation of the paper is its setting; which only covers Bernoulli observations.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
The paper describes a fixed-confidence Best Arm Identification (BAI) setup under epsilon-global Differential Privacy (DP) for Bernoulli bandits. The paper presents a lower bound and a corresponding algorithm, DP-TT, which closes the gap between the two to within a small constant factor.

STRENGTHS AND WEAKNESSES:
Strengths:
* Adoption of DP framwork to a new domain with signifincally better guarantees

Weaknesses:
* Still a constant factor in the bound
* Currently focusing only on Bernoulli

QUESTIONS:
First, I have to note that I do not work in this field, I tried my best to read around and understand the work.  

* For the constant factor in the upper bound, is there a way to improve the allocation strategy to help the sample complexity further? Can it similarly be problem independent?  
* Current derivation appears quite dependent on Bernoulli distributions, is there a way to generalise the setup to cover other types of bandits? What would be the main blocker in such extension? 
* It appears that not forgetting samples here leads to improved sample complexity. As the same time, it feels that this statefullness can also be a problem for perhaps slightly different threat models since data keeps on being accumulated. I wonder what other trade-offs appear here e.g. memory?
* How does one pick (n, b) in practice? What would sample counts actually look like in practical scenarios?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-05