Stochastic set-valued optimization and its application to robust learning

TL;DR

This paper introduces a stochastic set-valued optimization framework for robust machine learning, utilizing hyperbox sets and multi-gradient algorithms to improve robustness against distributional shifts.

Contribution

It develops a novel set-valued optimization approach with hyperbox sets, incorporating subquantiles and superquantiles for enhanced robustness in machine learning.

Findings

Improved robustness and reduced variability under distributional shift.

Effective trade-offs capturing tail behaviors of loss distributions.

Competitive accuracy maintained with enhanced robustness.

Abstract

In this paper, we develop a stochastic set-valued optimization (SVO) framework tailored for robust machine learning. In the SVO setting, each decision variable is mapped to a set of objective values, and optimality is defined via set relations. We focus on SVO problems with hyperbox sets, which can be reformulated as multi-objective optimization (MOO) problems with finitely many objectives and serve as a foundation for representing or approximating more general mapped sets. Two special cases of hyperbox-valued optimization (HVO) are interval-valued (IVO) and rectangle-valued (RVO) optimization. We construct stochastic IVO/RVO formulations that incorporate subquantiles and superquantiles into the objective functions of the MOO reformulations, providing a new characterization for subquantiles. These formulations provide interpretable trade-offs by capturing both lower- and upper-tail behaviors of loss distributions, thereby going beyond standard empirical risk minimization and classical robust models. To solve the resulting multi-objective problems, we adopt stochastic multi-gradient algorithms and select a Pareto knee solution. In numerical experiments, the proposed algorithms with this selection strategy exhibit improved robustness and reduced variability across test replications under distributional shift compared with empirical risk minimization, while maintaining competitive accuracy.