# A Distribution Optimization Framework for Confidence Bounds of Risk Measures

arXiv: https://arxiv.org/abs/2306.07059 · HTML: https://mlsanity.com/paper/2306.07059/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 37 references that are in the map.
- First posted on arXiv 12 Jun 2023.

## Builds on (most cited first, 8 of 8)

- [Being Optimistic to Be Conservative: Quickly Learning a CVaR Policy](https://mlsanity.com/paper/1911.01546/) (2019)
- [Universal Off-Policy Evaluation](https://mlsanity.com/paper/2104.12820/) (2021)
- [Off-Policy Risk Assessment in Contextual Bandits](https://mlsanity.com/paper/2104.08977/) (2021)
- [Optimal Thompson Sampling strategies for support-aware CVaR bandits](https://mlsanity.com/paper/2012.05754/) (2020)
- [A Wasserstein distance approach for concentration of empirical risk estimates](https://mlsanity.com/paper/1902.10709/) (2019)
- [Concentration bounds for empirical conditional value-at-risk: The unbounded case](https://mlsanity.com/paper/1808.01739/) (2018)
- [Estimation of Spectral Risk Measures](https://mlsanity.com/paper/1912.10398/) (2019)
- [Regret Bounds for Risk-sensitive Reinforcement Learning with Lipschitz Dynamic Risk Measures](https://mlsanity.com/paper/2306.02399/) (2023)

## Built on by (most cited first, 2 of 2)

- [Risk-sensitive Bandits: Arm Mixture Optimality and Regret-efficient Algorithms](https://mlsanity.com/paper/2503.08896/) (2025)
- [Variance-Optimal Arm Selection: Misallocation Minimization and Best Arm Identification](https://mlsanity.com/paper/2505.11985/) (2025)

## Abstract (arXiv)

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.