# Noisy Linear Convergence of Stochastic Gradient Descent for CV@R Statistical Learning under Polyak-Lojasiewicz Conditions

arXiv: https://arxiv.org/abs/2012.07785 · HTML: https://mlsanity.com/paper/2012.07785/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2020.

## Built on by (most cited first, 6 of 6)

- [Non-convex Distributionally Robust Optimization: Non-asymptotic Analysis](https://mlsanity.com/paper/2110.12459/) (2021)
- [Risk-Aware Learning for Scalable Voltage Optimization in Distribution Grids](https://mlsanity.com/paper/2110.01490/) (2021)
- [Risk-Averse No-Regret Learning in Online Convex Games](https://mlsanity.com/paper/2203.08957/) (2022)
- [Federated Learning Under Restricted User Availability](https://mlsanity.com/paper/2309.14176/) (2023)
- [Distributionally Time-Varying Online Stochastic Optimization under Polyak-Lojasiewicz Condition with Application in Conditional Value-at-Risk Statistical Learning](https://mlsanity.com/paper/2309.09411/) (2023)
- [Uncertainty Principles in Risk-Aware Statistical Estimation](https://mlsanity.com/paper/2104.14283/) (2021)

## Abstract (arXiv)

Conditional Value-at-Risk (CV@R) is one of the most popular measures of risk, which has been recently considered as a performance criterion in supervised statistical learning, as it is related to desirable operational features in modern applications, such as safety, fairness, distributional robustness, and prediction error stability. However, due to its variational definition, CV@R is commonly believed to result in difficult optimization problems, even for smooth and strongly convex loss functions. We disprove this statement by establishing noisy (i.e., fixed-accuracy) linear convergence of stochastic gradient descent for sequential CV@R learning, for a large class of not necessarily strongly-convex (or even convex) loss functions satisfying a set-restricted Polyak-Lojasiewicz inequality. This class contains all smooth and strongly convex losses, confirming that classical problems, such as linear least squares regression, can be solved efficiently under the CV@R criterion, just as their risk-neutral versions. Our results are illustrated numerically on such a risk-aware ridge regression task, also verifying their validity in practice.