A Distribution Optimization Framework for Confidence Bounds of Risk
  Measures
Hao Liang Zhi-quan Luo
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.