Statistical Learning with Conditional Value at Risk

TL;DR

This paper introduces a risk-averse learning framework optimizing for conditional value-at-risk (CVaR) using stochastic gradient descent, with theoretical guarantees and practical effectiveness demonstrated through experiments.

Contribution

It develops CVaR-based learning algorithms that require only i.i.d. samples and provides convergence and generalization guarantees for convex and nonconvex loss functions.

Findings

Algorithms effectively minimize CVaR in experiments

Convergence rate of O(1/√n) for convex losses

Generalization bounds for nonconvex losses

Abstract

We propose a risk-averse statistical learning framework wherein the performance of a learning algorithm is evaluated by the conditional value-at-risk (CVaR) of losses rather than the expected loss. We devise algorithms based on stochastic gradient descent for this framework. While existing studies of CVaR optimization require direct access to the underlying distribution, our algorithms make a weaker assumption that only i.i.d.\ samples are given. For convex and Lipschitz loss functions, we show that our algorithm has $O(1/\sqrt{n})$-convergence to the optimal CVaR, where $n$ is the number of samples. For nonconvex and smooth loss functions, we show a generalization bound on CVaR. By conducting numerical experiments on various machine learning tasks, we demonstrate that our algorithms effectively minimize CVaR compared with other baseline algorithms.