Risk-Based Robust Statistical Learning by Stochastic Difference-of-Convex Value-Function Optimization

• 成果类型: Article
• 署名作者: Liu, Junyi; Pang, Jong-Shi
• 署名单位: Tsinghua University; University of Southern California
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2021.2248
• 发表日期: 2023
• 页码: 397-414
• 关键词: value-at-risk; regression

摘要

This paper proposes the use of a variant of the conditional value-at-risk (CVaR) risk measure, called the interval conditional value-at-risk (In-CVaR), for the treatment of outliers in statistical learning by excluding the risks associated with the left and right tails of the loss. The risk-based robust learning task is to minimize the In-CVaR risk measure of a random functional that is the composite of a piecewise affine loss function with a potentially nonsmooth difference-of-convex statistical learningmodel. With the optimization formula of CVaR, the objective function of the minimization problem is the difference of two convex functions each being the optimal objective value of a univariate convex stochastic program. An algorithm that combines sequential sampling and convexification is developed, and its subsequential almost-sure convergence to a critical point is established. Numerical experiments demonstrate the effectiveness of the In-CVaR-based estimator computed by the sampling-based algorithm for robust regression and classification. Overall, this research extends the traditional approaches for treating outliers by allowing nonsmooth and nonconvex statistical learning models, employing a population risk-based objective, and applying a sampling-based algorithm with the stationarity guarantee for solving the resulting nonconvex and nonsmooth stochastic program.