Title: Risk level dependent Minimax Quantile lower bounds for Interactive Statistical Decision Making Show Chinese title

Title: 基于风险等级的交互式统计决策最小最大分位数下界

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. Show Chinese abstract

Abstract: 极小极大风险和遗憾关注期望，忽略了在安全关键的多臂老虎机和强化学习中至关重要的罕见故障。 极小极大分位数捕捉这些尾部。 先前工作的三个方向激发了这项研究：限制在非交互估计的极小极大分位数界限；专注于期望风险而非特定风险水平的分位数界限的统一交互分析；以及仍然缺乏针对一般交互协议的特定分位数工具包的高概率多臂老虎机界限。 为了填补这一空白，在交互统计决策框架内，我们开发了高概率的Fano和Le Cam工具，并推导出明确风险水平的极小极大分位数界限，包括分位数到期望的转换以及严格和较低极小极大分位数之间的紧密联系。 将这些结果应用于两臂高斯老虎机立即恢复了最优率界限。