Risk level dependent Minimax Quantile lower bounds for Interactive Statistical Decision Making
Raghav Bongole, Amirreza Zamani, Tobias J. Oechtering, Mikael Skoglund
TL;DR
This work addresses tail risk in interactive statistical decision making by developing a delta-explicit minimax-quantile framework that complements traditional minimax risk. It introduces high-probability Fano and Le Cam tools tailored to ISDM and establishes a quantile-to-expectation conversion, along with a tight link between strict and lower minimax quantiles. The authors show that, for essentially all confidence levels, minimax and lower minimax quantiles coincide, enabling uniform, delta-explicit tail guarantees that translate into expectation bounds. They demonstrate the practicality of the approach by instantiating it on a two-armed Gaussian bandit, where the delta-explicit bound scales as $\sqrt{T\log(1/\delta)}$, matching optimal rates and showcasing the framework’s potential for broader interactive settings, including RL.
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.