# PAC-Bayesian Bound for the Conditional Value at Risk

arXiv: https://arxiv.org/abs/2006.14763 · HTML: https://mlsanity.com/paper/2006.14763/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jun 2020.

## Built on by (most cited first, 13 of 13)

- [On Empirical Risk Minimization with Dependent and Heavy-Tailed Data](https://mlsanity.com/paper/2109.02224/) (2021)
- [Learning via Wasserstein-Based High Probability Generalisation Bounds](https://mlsanity.com/paper/2306.04375/) (2023)
- [Group Distributionally Robust Dataset Distillation with Risk Minimization](https://mlsanity.com/paper/2402.04676/) (2024)
- [Online Estimation and Optimization of Utility-Based Shortfall Risk](https://mlsanity.com/paper/2111.08805/) (2021)
- [Comparing Comparators in Generalization Bounds](https://mlsanity.com/paper/2310.10534/) (2023)
- [STL Robustness Risk over Discrete-Time Stochastic Processes](https://mlsanity.com/paper/2104.01503/) (2021)
- [Sample-Based Bounds for Coherent Risk Measures: Applications to Policy Synthesis and Verification](https://mlsanity.com/paper/2204.09833/) (2022)
- [Concentration Bounds for Optimized Certainty Equivalent Risk Estimation](https://mlsanity.com/paper/2405.20933/) (2024)
- [Safe Collaborative Filtering](https://mlsanity.com/paper/2306.05292/) (2023)
- [Non-Vacuous Generalization Bounds: Can Rescaling Invariances Help?](https://mlsanity.com/paper/2509.26149/) (2025)
- [PAC-Bayesian Bounds on Constrained f-Entropic Risk Measures](https://mlsanity.com/paper/2510.11169/) (2025)
- [Instantiating Bayesian CVaR lower bounds in Interactive Decision Making Problems](https://mlsanity.com/paper/2604.12519/) (2026)
- [The Privacy Price of Tail-Risk Learning: Effective Tail Sample Size in Differentially Private CVaR Optimization](https://mlsanity.com/paper/2605.16219/) (2026)

## Abstract (arXiv)

Conditional Value at Risk (CVaR) is a family of "coherent risk measures" which generalize the traditional mathematical expectation. Widely used in mathematical finance, it is garnering increasing interest in machine learning, e.g., as an alternate approach to regularization, and as a means for ensuring fairness. This paper presents a generalization bound for learning algorithms that minimize the CVaR of the empirical loss. The bound is of PAC-Bayesian type and is guaranteed to be small when the empirical CVaR is small. We achieve this by reducing the problem of estimating CVaR to that of merely estimating an expectation. This then enables us, as a by-product, to obtain concentration inequalities for CVaR even when the random variable in question is unbounded.