Quadratically Regularized Subgradient Methods for Weakly Convex Optimization with Weakly Convex Constraints
Abstract
Optimization models with non-convex constraints arise in many tasks in machine learning, e.g., learning with fairness constraints or Neyman-Pearson classification with non-convex loss. Although many efficient methods have been developed with theoretical convergence guarantees for non-convex unconstrained problems, it remains a challenge to design provably efficient algorithms for problems with non-convex functional constraints. This paper proposes a class of subgradient methods for constrained optimization where the objective function and the constraint functions are weakly convex and nonsmooth. Our methods solve a sequence of strongly convex subproblems, where a quadratic regularization term is added to both the objective function and each constraint function. Each subproblem can be solved by various algorithms for strongly convex optimization. Under a uniform Slater's condition, we establish the computation complexities of our methods for finding a nearly stationary point.
Cite
@article{arxiv.1908.01871,
title = {Quadratically Regularized Subgradient Methods for Weakly Convex Optimization with Weakly Convex Constraints},
author = {Runchao Ma and Qihang Lin and Tianbao Yang},
journal= {arXiv preprint arXiv:1908.01871},
year = {2023}
}
Comments
This article has been published in International Conference on Machine Learning (ICML), 2020. We didn't post the final version to arxiv soon after publication, which leads to the paper being cited under the old title and causes other confusion. We therefore update it in arxiv to avoid the issues of multiple versions