A Distribution Optimization Framework for Confidence Bounds of Risk Measures
This work addresses the need for more precise risk assessment in decision-making, offering incremental improvements in confidence bounds for risk-sensitive applications.
The paper tackles the problem of improving confidence bounds for various risk measures by introducing a distribution optimization framework that uses Wasserstein or supremum distance-based transformations, resulting in consistently tighter bounds compared to previous methods.
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.