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arxiv: 2406.06408 · v2 · submitted 2024-06-10 · 📊 stat.ML · cs.CR· cs.LG· math.ST· stat.TH

Differentially Private Best-Arm Identification

Achraf Azize , Marc Jourdan , Aymen Al Marjani , Debabrota Basu This is my paper

Pith reviewed 2026-05-23 23:54 UTC · model grok-4.3

classification 📊 stat.ML cs.CRcs.LGmath.STstat.TH
keywords best-arm identificationdifferential privacysample complexitytop-two algorithmlocal DPglobal DPtransportation costsrandomised response
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The pith

Differentially private Top-Two algorithms achieve the asymptotic lower bound on sample complexity for best-arm identification up to multiplicative constants.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives lower bounds on the number of samples any fixed-confidence best-arm identification algorithm must use when it must also satisfy either local or global differential privacy. These bounds split into a high-privacy regime, where privacy parameters couple with total-variation distances between arms, and a low-privacy regime that recovers the classical non-private bounds. The authors then construct two private Top-Two variants: CTB-TT, which inserts a randomised-response mean estimator for local DP, and AdaP-TT*, which runs arm-dependent adaptive episodes and adds Laplace noise for global DP. By suitably adapting the transportation costs that govern arm allocation, AdaP-TT* matches its lower bound up to constants while CTB-TT is asymptotically optimal. The results matter for any data-sensitive sequential decision task, such as private clinical trials or user studies, that must still identify the best option with high probability.

Core claim

We derive lower bounds on the sample complexity of any δ-correct BAI algorithm that satisfies ε-global DP or ε-local DP; the bounds reveal two privacy regimes whose hardness is governed by a coupled privacy-Total-Variation quantity in the high-privacy case. We introduce CTB-TT, an ε-local-DP Top-Two algorithm that plugs in a randomised-response estimator of the means, and AdaP-TT*, an ε-global-DP Top-Two algorithm whose private mean estimator runs in arm-dependent adaptive episodes and adds Laplace noise. By adapting the transportation costs, the expected sample complexity of AdaP-TT* reaches the asymptotic lower bound up to multiplicative constants; CTB-TT is asymptotically optimal for ε-lD

What carries the argument

Top-Two sampling with private mean estimators (Randomised Response for local DP, adaptive Laplace for global DP) and adapted transportation costs that control allocation between the leader and challenger arms.

If this is right

  • In the high-privacy regime sample complexity depends on the coupled effect of privacy and novel total-variation information-theoretic quantities.
  • In the low-privacy regime the lower bounds reduce exactly to the classical non-private lower bounds.
  • CTB-TT is asymptotically optimal for ε-local DP by using a randomised-response mean estimator.
  • AdaP-TT* reaches the asymptotic lower bound up to multiplicative constants for ε-global DP by adapting transportation costs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same transportation-cost adaptation may extend to differentially private regret minimization or other sequential decision problems.
  • The high/low-privacy regime split suggests concrete thresholds on ε beyond which privacy imposes no extra asymptotic cost.
  • Empirical checks on non-Gaussian reward distributions would test whether the optimality claims survive when the private estimator variance assumption is only approximately met.

Load-bearing premise

The reward distributions admit a private mean estimator whose variance scales appropriately with the privacy budget ε, and the algorithm can execute arm-dependent adaptive episodes without violating the fixed-confidence guarantee.

What would settle it

Running AdaP-TT* or CTB-TT on standard Gaussian arms and observing that its expected sample complexity exceeds any fixed constant times the paper's lower bound would falsify the claimed near-optimality.

Figures

Figures reproduced from arXiv: 2406.06408 by Achraf Azize, Aymen Al Marjani, Debabrota Basu, Marc Jourdan.

Figure 1
Figure 1. Figure 1: Empirical stopping time τδ (mean ± std. over 1000 runs, δ = 10−2 ) with respect to the privacy budget ϵ for ϵ-local DP on Bernoulli instance µ1 (left) and µ2 (right). The shaded vertical line separates the two privacy regimes. taking the collected observations into account. In contrast to the classical batched setting where the batch size is fixed, the size of the resulting batches is adaptive and data-dep… view at source ↗
Figure 2
Figure 2. Figure 2: Empirical stopping time τδ (mean ± std. over 1000 runs) with respect to the privacy budget ϵ for ϵ-global DP on Bernoulli instance µ1 (left) and µ2 (right). The shaded vertical line separates the two privacy regimes. 5.2 Global DP We compare the performances of AdaP-TT, AdaP-TT⋆ and DP-SE for FC-BAI in different Bernoulli instances as in Sajed and Sheffet (2019). The first instance has means µ1 = (0.95, 0.… view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the stopping time τ (mean ± std. over 1000 runs) of CTB-TT and TTUCB with respect to the privacy budget ϵ for δ = 10−2 on different Bernoulli instances. The shaded vertical line separates the two privacy regimes. 59 [PITH_FULL_IMAGE:figures/full_fig_p059_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the stopping time τ (mean ± std. over 1000 runs) of Imp-AdaP-TT, AdaP-TT, DP-SE, and TTUCB with respect to the privacy budget ϵ for δ = 10−2 on different Bernoulli instances. The shaded vertical line separates the two privacy regimes. Both the x-axis and y-axis are in logarithmic scale. 60 [PITH_FULL_IMAGE:figures/full_fig_p060_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the stopping time τ (mean ± std. over 1000 runs) of AdaP-TT⋆ , AdaP-TT, DP-SE, and TTUCB with respect to the privacy budget ϵ for δ = 10−2 on different Bernoulli instances. The shaded vertical line separates the two privacy regimes. Only the x-axis is in logarithmic scale. 61 [PITH_FULL_IMAGE:figures/full_fig_p061_5.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies best-arm identification (BAI) in the fixed-confidence setting under both ε-local and ε-global differential privacy. It derives lower bounds on sample complexity that identify two regimes: a high-privacy regime whose hardness couples privacy with total-variation information quantities, and a low-privacy regime that recovers the classical non-private bounds. The authors introduce CTB-TT (local DP) that plugs in a Randomised-Response mean estimator and is claimed to be asymptotically optimal, and AdaP-TT* (global DP) that uses arm-dependent adaptive episodes with Laplace noise; by adapting transportation costs the expected sample complexity of AdaP-TT* is stated to match the lower bound up to multiplicative constants.

Significance. If the lower-bound derivations and the matching upper bounds hold, the work supplies the first asymptotic optimality results for differentially private BAI. The explicit separation into high- and low-privacy regimes and the adaptation of transportation costs to the private setting are technically useful extensions of the non-private literature. The reliance on standard mechanisms (Randomised Response, Laplace) keeps the algorithmic contribution clean and reproducible.

major comments (2)
  1. [Abstract / lower-bound derivation] Abstract and lower-bound section: the statement that the high-privacy regime hardness 'depends on a coupled effect of privacy and novel information-theoretic quantities involving the Total Variation' is load-bearing for the two-regime claim; the precise functional form of this coupling (e.g., the exact combination of KL, TV, and ε) must be exhibited and shown to be tight.
  2. [AdaP-TT* analysis] AdaP-TT* description: the claim that 'by adapting the transportation costs, the expected sample complexity of AdaP-TT* reaches the asymptotic lower bound up to multiplicative constants' requires an explicit theorem that states both the adapted cost and the resulting multiplicative factor; without that theorem the optimality statement cannot be verified.
minor comments (2)
  1. [Abstract] Notation for the two privacy regimes should be introduced once and used consistently; the abstract switches between 'high-privacy regime' and 'low-privacy regime' without a formal definition.
  2. [AdaP-TT* algorithm] The description of the arm-dependent adaptive episodes for the global-DP estimator should clarify whether the privacy budget is allocated per episode or globally, and how the fixed-confidence guarantee is preserved.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments below. Both points identify places where the manuscript would benefit from greater explicitness; we will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract / lower-bound derivation] Abstract and lower-bound section: the statement that the high-privacy regime hardness 'depends on a coupled effect of privacy and novel information-theoretic quantities involving the Total Variation' is load-bearing for the two-regime claim; the precise functional form of this coupling (e.g., the exact combination of KL, TV, and ε) must be exhibited and shown to be tight.

    Authors: We agree that the precise functional form should be stated explicitly for the two-regime claim to be verifiable from the abstract and early sections. The lower bounds appear in Theorems 3.1 (global DP) and 3.2 (local DP). In the high-privacy regime the sample-complexity lower bound takes the form Ω(max_i 1/(TV(μ_i,μ*)·g(ε)) + non-private term), where g(ε) is a privacy-dependent factor that replaces the usual KL divergence; the low-privacy regime recovers the classical KL-based bound once ε exceeds a threshold depending on the TV distances. We will revise the abstract to include this functional form and add a short paragraph immediately after the theorem statements that isolates the coupling term and notes that the bound is tight because the matching upper bounds (Theorems 4.1 and 5.3) attain the same expression up to constants. revision: yes

  2. Referee: [AdaP-TT* analysis] AdaP-TT* description: the claim that 'by adapting the transportation costs, the expected sample complexity of AdaP-TT* reaches the asymptotic lower bound up to multiplicative constants' requires an explicit theorem that states both the adapted cost and the resulting multiplicative factor; without that theorem the optimality statement cannot be verified.

    Authors: We accept the request for an explicit theorem. Section 5 defines the privacy-adjusted transportation cost C_ε(μ) obtained by replacing the usual stopping threshold with a Laplace-noise-adjusted version that accounts for the arm-dependent episode lengths. Theorem 5.3 already shows that the expected sample complexity of AdaP-TT* is at most 4·C_ε(μ)·(lower-bound expression), but the constant 4 and the exact definition of C_ε are currently only implicit in the proof. We will extract these two objects into a self-contained theorem statement placed immediately before the complexity analysis, making both the adapted cost and the multiplicative factor (which remains 4) fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper first derives independent information-theoretic lower bounds on sample complexity for ε-DP best-arm identification (both local and global models), identifying two privacy regimes via Total Variation quantities that recover non-private bounds at low privacy. It then constructs CTB-TT and AdaP-TT* by inserting standard mechanisms (Randomised Response, Laplace noise) into an existing Top-Two template and adapting transportation costs. Asymptotic optimality is claimed by direct matching to the separately derived lower bounds up to constants; no step reduces a prediction to a fitted quantity defined inside the paper, no self-definition of quantities, and no load-bearing self-citation chain is required for the central claims. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not introduce new free parameters, invented entities, or non-standard axioms beyond the usual definitions of ε-DP and δ-correct BAI; the work relies on standard concentration and information-theoretic tools from the bandit and DP literatures.

pith-pipeline@v0.9.0 · 5812 in / 1076 out tokens · 17829 ms · 2026-05-23T23:54:49.104728+00:00 · methodology

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