A Distribution Optimization Framework for Confidence Bounds of Risk Measures

Hao Liang, Zhi-Quan Luo
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:20677-20705, 2023.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-liang23c, title = {A Distribution Optimization Framework for Confidence Bounds of Risk Measures}, author = {Liang, Hao and Luo, Zhi-Quan}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {20677--20705}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/liang23c/liang23c.pdf}, url = {https://proceedings.mlr.press/v202/liang23c.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T A Distribution Optimization Framework for Confidence Bounds of Risk Measures %A Hao Liang %A Zhi-Quan Luo %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-liang23c %I PMLR %P 20677--20705 %U https://proceedings.mlr.press/v202/liang23c.html %V 202 %X 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.
APA
Liang, H. & Luo, Z.. (2023). A Distribution Optimization Framework for Confidence Bounds of Risk Measures. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:20677-20705 Available from https://proceedings.mlr.press/v202/liang23c.html.

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