Risk level dependent Minimax Quantile lower bounds for Interactive Statistical Decision Making

2025-10-07

AI Summary

In safety-critical interactive decision-making, rare failure events—often overlooked by expectation-based risk measures—pose significant reliability concerns. Method: We propose the first minimax quantile risk analysis framework applicable to general interactive protocols. Unlike prior work focusing on expected risk or non-interactive quantile bounds, we develop a risk-level-dependent minimax quantile lower bound theory, introduce high-probability Fano and Le Cam information-theoretic tools tailored for interactive settings, and establish convertibility between quantile and expected risks. Contributions/Results: We prove, for the first time, the equivalence between strict and weak minimax quantile lower bounds. Instantiated on the two-armed Gaussian bandit, our derived quantile risk bound achieves the optimal convergence rate, demonstrating both tightness and broad applicability of the framework.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Sequential Decision Making
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Minimax risk and regret focus on expectation, missing rare failures critical in safety-critical bandits and reinforcement learning. Minimax quantiles capture these tails. Three strands of prior work motivate this study: minimax-quantile bounds restricted to non-interactive estimation; unified interactive analyses that focus on expected risk rather than risk level specific quantile bounds; and high-probability bandit bounds that still lack a quantile-specific toolkit for general interactive protocols. To close this gap, within the interactive statistical decision making framework, we develop high-probability Fano and Le Cam tools and derive risk level explicit minimax-quantile bounds, including a quantile-to-expectation conversion and a tight link between strict and lower minimax quantiles. Instantiating these results for the two-armed Gaussian bandit immediately recovers optimal-rate bounds.

Problem

• Address rare failures in safety-critical interactive decision making
• Develop quantile-specific bounds for general interactive protocols
• Establish explicit minimax-quantile relationships for risk analysis

Innovation

• Developed high-probability Fano and Le Cam tools
• Derived risk level explicit minimax-quantile bounds
• Established quantile-to-expectation conversion for bandits