Minimax Quantile Lower Bounds for Interactive Statistical Decision Making with Privacy

2026-06-22

AI Summary

Traditional minimax risk and regret fail to adequately capture rare yet severe failure events and lack characterization of performance limits under confidence level δ and privacy constraints. This work proposes a δ-explicit minimax quantile framework, integrating high-probability interactive Fano and Le Cam methods with mutual information-based privacy constraints to develop a high-probability converse tool tailored for privacy-preserving settings. The study establishes the first explicit lower bounds that directly depend on both δ and the privacy budget: for Gaussian mean estimation, the error scales as log(1/δ)/n; for two-armed bandits, the regret scales as √(T log(1/δ)), and for K-armed bandits, as √(KT log(1/δ)). The impact of privacy manifests as an inflation factor in the Gaussian variance.

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.

Problem

• minimax quantile
• interactive statistical decision making
• privacy
• high-probability bounds
• rare but consequential failures

Innovation

• minimax quantile
• interactive statistical decision making
• high-probability lower bounds
• privacy-preserving inference
• Gaussian bandits