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

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 quan-tiles. Instantiating these results for the two-armed Gaussian bandit immediately recovers optimal-rate bounds. Index T erms-- information theory, learning theory 1. INTRODUCTION The concept of minimax risk has become a staple in learning theory and statistics [1-3]. In interactive or online learning settings, the minimax risk is often replaced by its counterpart, the minimax regret.