Minimax Quantile Lower Bounds for Interactive Statistical Decision Making with Privacy

• URL: http://arxiv.org/abs/2606.23096v1
• Date: Mon, 22 Jun 2026 09:41:48 GMT
• Title: Minimax Quantile Lower Bounds for Interactive Statistical Decision Making with Privacy
• Abstract summary: Minimax risk and regret are expectation-based criteria and do not capture rare but consequential failures.<n>We develop a $$-explicit minimax-quantile theory for interactive statistical decision making.
• Score: 38.10483249859454
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Minimax risk and regret are expectation-based criteria and do not capture rare but consequential failures. To address this concern, we develop a $δ$-explicit minimax-quantile theory for interactive statistical decision making (ISDM). We first provide structural relations between minimax quantiles, lower minimax quantiles, and minimax risk. This includes a quantile-to-expectation conversion and an equivalence between strict and lower minimax quantiles outside a countable set of confidence levels. We then derive two converse tools for ISDM: a high-probability interactive Fano's method and a high-probability interactive Le Cam's method. Then, we show that mutual-information (MI) privacy can be handled in the same framework by restricting the admissible decision class. For coordinatewise Gaussian privatization, we derive a two-point template that isolates the privacy-induced variance inflation. We instantiate this template for Gaussian mean estimation, and use the same two-point strategy directly for two-armed Gaussian bandits. We then derive a minimax quantile lower bound for the $K$-armed Gaussian bandit problem, showing that the interactive Fano method captures the exploration cost over multiple possible best arms. The resulting lower bounds are explicit in the confidence level $δ$ and in the privacy budget for the private problems. They yield $\log(1/δ)/n$ scaling for squared-error Gaussian mean estimation, $\sqrt{T\log(1/δ)}$ scaling for two-armed bounded-mean Gaussian bandits, and $\sqrt{KT\log(1/δ)}$-type scaling for the $K$-armed bandits, with privacy appearing through a Gaussian variance-inflation factor for the private problems.

Related papers

• Pure-DP Statistical Query Release at the Conjectured Square-Root Rate [0.0]
  Nikolov and Ullman asked whether k statistical queries on a universe of size T can be released under pure differential privacy with expected worst-coordinate error at the square-root rate suggested by known lower bounds.<n>We prove their conjectured upper bound.<n>A companion Lean 4 development machine-checks the finite construction, pure privacy after deterministic decoding, and the displayed all-regimes upper bound.
  arXiv  Detail & Related papers  (2026-07-22T17:54:46Z)
• Adaptive Confidence Intervals in Efron's Gaussian Two-Groups Model [28.636700996382558]
  Motivated by noise-oblivious nature of adversaries, we study confidence intervals for the null location parameter $$ in Efron's Gaussian two-groups fraction model.<n>We characterize the minimax-optimal length among confidence intervals with a prescribed coverage level uniformly over the unknown proportion contamination and all noise-oblivious adversaries.
  arXiv  Detail & Related papers  (2026-04-28T22:40:54Z)
• Fundamental Limitations of Favorable Privacy-Utility Guarantees for DP-SGD [7.787109481104569]
  We analyze Differentially Private Gradient Descent (DP-SGD) in the $f$differential privacy framework.<n>We prove that enforcing a small separation imposes a strict lower bound on the noise multiplier $$, which directly limits the achievable utility.<n>Our experiments confirm that the noise levels implied by this bound leads to significant accuracy degradation at realistic training settings.
  arXiv  Detail & Related papers  (2026-01-15T09:50:36Z)
• Beyond Covariance Matrix: The Statistical Complexity of Private Linear Regression [66.93988594607842]
  Under privacy constraints, the complexity of private linear regression is emphnot captured by the usual covariance matrix.<n>We introduce an Information-Weighted Regression method that attains the optimal rates.<n> Notably, our results demonstrate that joint privacy comes at almost no additional cost.
  arXiv  Detail & Related papers  (2025-02-18T18:35:24Z)
• Fixed-Budget Differentially Private Best Arm Identification [62.36929749450298]
  We study best arm identification (BAI) in linear bandits in the fixed-budget regime under differential privacy constraints. We derive a minimax lower bound on the error probability, and demonstrate that the lower and the upper bounds decay exponentially in $T$.
  arXiv  Detail & Related papers  (2024-01-17T09:23:25Z)
• Contextual Combinatorial Bandits with Probabilistically Triggered Arms [55.9237004478033]
  We study contextual bandits with probabilistically triggered arms (C$2$MAB-T) under a variety of smoothness conditions. Under the triggering modulated (TPM) condition, we devise the C$2$-UC-T algorithm and derive a regret bound $tildeO(dsqrtT)$.
  arXiv  Detail & Related papers  (2023-03-30T02:51:00Z)
• Batch-Size Independent Regret Bounds for Combinatorial Semi-Bandits with Probabilistically Triggered Arms or Independent Arms [59.8188496313214]
  We study the semi-bandits (CMAB) and focus on reducing the dependency of the batch-size $K$ in the regret bound. First, for the setting of CMAB with probabilistically triggered arms (CMAB-T), we propose a BCUCB-T algorithm with variance-aware confidence intervals. Second, for the setting of non-triggering CMAB with independent arms, we propose a SESCB algorithm which leverages on the non-triggering version of the TPVM condition.
  arXiv  Detail & Related papers  (2022-08-31T13:09:39Z)
• Causal Bandits for Linear Structural Equation Models [58.2875460517691]
  This paper studies the problem of designing an optimal sequence of interventions in a causal graphical model. It is assumed that the graph's structure is known and has $N$ nodes. Two algorithms are proposed for the frequentist (UCB-based) and Bayesian settings.
  arXiv  Detail & Related papers  (2022-08-26T16:21:31Z)
• New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma [10.176795938619417]
  We prove new lower bounds for statistical estimation tasks under the constraint of $(varepsilon, delta)$-differential privacy. We show that estimating the Frobenius norm requires $Omega(d2)$ samples, and in spectral norm requires $Omega(d3/2)$ samples, both matching upper bounds up to logarithmic factors.
  arXiv  Detail & Related papers  (2022-05-17T17:55:10Z)
• Robust Estimation of Discrete Distributions under Local Differential Privacy [1.52292571922932]
  We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint. We show that combining the two constraints leads to a minimax estimation rate of $epsilonsqrtd/alpha2 k+sqrtd2/alpha2 kn$ up to a $sqrtlog (1/epsilon)$ factor.
  arXiv  Detail & Related papers  (2022-02-14T15:59:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.