Minimax Quantile Lower Bounds for Interactive Statistical Decision Making with Privacy

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.

• The paper presents a δ-explicit minimax quantile framework that captures tail risks missed by traditional expectation-based analysis in interactive decision making.
• It extends interactive Fano’s and Le Cam’s methods to derive explicit lower bounds with scaling laws like log(1/δ)/n and √(T log(1/δ)), applicable to various tasks including bandits and Gaussian estimation.
• The integration of MI privacy constraints quantifies the cost of privacy as variance inflation, impacting quantile risk guarantees in both non-private and private settings.

Minimax Quantile Lower Bounds for Interactive Statistical Decision Making with Privacy

Context and Motivation

The paper "Minimax Quantile Lower Bounds for Interactive Statistical Decision Making with Privacy" (2606.23096) addresses foundational limitations of expectation-based minimax criteria in the analysis of statistical decision making, particularly in interactive settings. Standard minimax risk and regret provide worst-case expectation guarantees but fail to capture rare, high-loss events—an issue critical in reliability-sensitive and safety-critical applications. To overcome this, the authors introduce a $\delta$-explicit minimax-quantile framework for interactive statistical decision making (ISDM), extending tail-sensitive minimax lower bounds from passive estimation to adaptive, interactive protocols, and systematically accommodating privacy constraints, notably mutual-information (MI) privacy.

Formal Framework and Quantile Notions

The work adopts the ISDM formalism, encompassing both passive estimation and interactive tasks (e.g., bandits, RL). The key innovation is shifting from minimax expected loss:

$$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$

to tail-sensitive quantile lower bounds. The strict minimax quantile $\mathfrak{M}(\delta)$ specifies, for risk level $\delta\in(0,1]$, the smallest loss threshold $r$ such that, for any algorithm, at least $1-\delta$ fraction of runs guarantee $L(M,X) \leq r$ uniformly over models. The companion lower quantile $\mathfrak{M}_{-}(\delta)$ is defined for tail-probability formulations, facilitating converse arguments.

Structural results are established connecting these criteria: there is a conversion from quantiles to expectation, and equivalence between strict and lower minimax quantiles except for a countable set of $\delta$. This provides a robust bridge across expectation-based and quantile-based minimax theory.

Information-Theoretic Converse Tools

Two central high-probability lower-bound techniques are extended to the interactive decision making context:

• Interactive Fano's Method: A generalization yielding $\delta$-explicit minimax-quantile lower bounds via $$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$0-divergence and prior-based averaging, capturing the difficulty induced by large model classes.
• Interactive Le Cam's Method: A two-point reduction producing quantile lower bounds via TV/KL indistinguishability, leveraging uniform separation in loss and bounded divergence between induced laws.

Both methods yield explicit lower bounds of the form $$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$1 or $$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$2 (and $$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$3 for $$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$4-armed settings), with detailed quantile-risk curves rather than single-confidence values.

Incorporation of Privacy Constraints

A major contribution is the integration of MI privacy into the ISDM quantile framework. Privacy is treated as a restriction on the admissible decision class, with the MI constraint $$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$5 limiting information leakage about sensitive object $$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$6 via released outcome $$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$7 across models. For coordinatewise Gaussian privatization, a concrete two-point template is provided: the privacy mechanism enforces a minimum noise variance, and the quantile lower bound is inflated by a variance factor, making quantile degradation due to privacy explicit.

Applications and Numerical Results

The theoretical machinery is instantiated in several canonical scenarios:

• Gaussian Mean Estimation: Non-private quantile lower bounds for squared-error scale as $$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$8. Under coordinatewise Gaussian MI privacy, the lower bound is multiplicatively inflated by $$\mathfrak{M} = \inf_{ALG}\; \sup_{M} \mathbb{E}^{M,ALG}[L(M,X)]$$9, where $\mathfrak{M}(\delta)$0 is the privacy-induced noise floor.
• Two-Armed Gaussian Bandits: Classical non-private bounds scale as $\mathfrak{M}(\delta)$1 up to truncation by $\mathfrak{M}(\delta)$2. In the MI-private setting, quantile bounds inflate as $\mathfrak{M}(\delta)$3.
• $\mathfrak{M}(\delta)$4-Armed Bandits: The interactive Fano framework yields explicit scaling with $\mathfrak{M}(\delta)$5, capturing exploration cost across multiple arms.
• Comparative analysis underscores separation between non-interactive and interactive rates, the matching privacy-induced variance inflation in private settings, and truncation effects due to mean bounds.

Importantly, existing upper bounds for classical mean estimation under squared error match the quantile lower bounds herein. For bandit settings, tight upper bounds are available for bounded-reward stochastic environments, but matching algorithmic results for MI-private Gaussian settings remain an open area.

Implications and Future Directions

The research unifies minimax quantile theory for estimation, interaction, and privacy under the ISDM umbrella, allowing systematic derivation of tail-sensitive lower bounds in broad settings. Practically, the explicit incorporation of risk level and privacy budget facilitates informed design of reliable and private statistical algorithms, quantifying the exact cost of privacy as variance inflation in quantile regimes.

Theoretically, the approach paves the way for extensions to sharper multi-model lower bounds, broader interactive learning (RL, partial monitoring), and privacy notions beyond MI (e.g., differential privacy, local DP). The framework is amenable to algorithmic development targeting quantile guarantees under privacy, and further exploration of decision-estimation coefficients (DEC) in strict quantile regimes is warranted.

Conclusion

This paper delivers a robust, tail-sensitive minimax quantile theory for interactive statistical decision making incorporating privacy constraints. Structural equivalence results, versatile converse methods, and explicit private quantile bounds advance both the theory and practice of reliable interactive statistical learning. The unified ISDM quantile framework is positioned for broad extension across privacy and interactive learning paradigms.