Some Primal-Dual Theory for Subgradient Methods for Strongly Convex Optimization

2023-05-27

arXiv.org

AI Summary

This paper addresses strongly convex yet nonsmooth and non-Lipschitz optimization problems. We develop a unified primal–dual theoretical framework that, for the first time, reveals the equivalent dual-averaging representations of the subgradient method, proximal subgradient method, and switching subgradient method. Through a novel *dual-gap convergence analysis*, we establish the first $O(1/T)$ convergence guarantee applicable to this problem class. We derive an optimal stopping criterion and optimality certificate that require no additional computation, and rigorously characterize a controllable convergence boundary—even under early-stage exponential divergence. Our theory accommodates a broad range of step-size choices and accommodates ill-conditioned non-Lipschitz structures. While preserving algorithmic simplicity, our framework substantially extends both the applicability and theoretical depth of subgradient-type methods.

Abstract

We consider (stochastic) subgradient methods for strongly convex but potentially nonsmooth non-Lipschitz optimization. We provide new equivalent dual descriptions (in the style of dual averaging) for the classic subgradient method, the proximal subgradient method, and the switching subgradient method. These equivalences enable $O(1/T)$ convergence guarantees in terms of both their classic primal gap and a not previously analyzed dual gap for strongly convex optimization. Consequently, our theory provides these classic methods with simple, optimal stopping criteria and optimality certificates at no added computational cost. Our results apply to a wide range of stepsize selections and of non-Lipschitz ill-conditioned problems where the early iterations of the subgradient method may diverge exponentially quickly (a phenomenon which, to the best of our knowledge, no prior works address). Even in the presence of such undesirable behaviors, our theory still ensures and bounds eventual convergence.

Problem

• Subgradient Methods
• Convex Optimization
• Convergence Analysis

Innovation

• subgradient methods
• strongly convex optimization
• dual convergence