We present a distribution optimization framework that significantly improves confidence bounds for various risk measures compared to previous methods. Our framework encompasses popular risk measures such as the entropic risk measure, conditional value at risk (CVaR), spectral risk measure, distortion risk measure, equivalent certainty, and rank-dependent expected utility, which are well established in risk-sensitive decision-making literature. To achieve this, we introduce two estimation schemes based on concentration bounds derived from the empirical distribution, specifically using either the Wasserstein distance or the supremum distance. Unlike traditional approaches that add or subtract a confidence radius from the empirical risk measures, our proposed schemes evaluate a specific transformation of the empirical distribution based on the distance. Consequently, our confidence bounds consistently yield tighter results compared to previous methods. We further verify the efficacy of the proposed framework by providing tighter problem-dependent regret bound for the CVaR bandit.

翻译：我们提出一种分布优化框架，相较于现有方法显著提升了各类风险测度的置信界限。该框架涵盖风险敏感决策文献中广泛使用的多种风险测度，包括熵风险测度、条件风险价值（CVaR）、谱风险测度、扭曲风险测度、等价确定性及秩相关期望效用。为此，我们引入基于经验分布浓度界限的两种估计方案，具体采用Wasserstein距离或上确界距离。与传统方法在经验风险测度上加减置信半径不同，本方案基于距离度量对经验分布实施特定变换。因此，我们的置信界限始终比现有方法更紧凑。我们进一步通过为CVaR赌博机问题提供更优的问题依赖遗憾界，验证了所提框架的有效性。