Inexact subgradient methods for semialgebraic functions

TL;DR

This paper analyzes inexact subgradient methods for nonconvex semialgebraic functions, establishing convergence properties and fluctuation bounds in the presence of persistent errors, with implications for machine learning optimization.

Contribution

It provides the first comprehensive analysis of inexact subgradient methods for nonconvex semialgebraic functions, including convergence, fluctuation bounds, and complexity results under various step-size regimes.

Findings

Iterates fluctuate near the critical set within an $O(\epsilon^ ho)$ neighborhood.

Constant step-size enlarges fluctuation region but remains within $O(\epsilon^ ho)$.

New complexity bounds for averaged iterates in convex semialgebraic functions.

Abstract

Motivated by the extensive application of approximate gradients in machine learning and optimization, we investigate inexact subgradient methods subject to persistent additive errors. Within a nonconvex semialgebraic framework, assuming boundedness or coercivity, we establish that the method yields iterates that eventually fluctuate near the critical set at a proximity characterized by an $O(\epsilon^\rho)$ distance, where $\epsilon$ denotes the magnitude of subgradient evaluation errors, and $\rho$ encapsulates geometric characteristics of the underlying problem. Our analysis comprehensively addresses both vanishing and constant step-size regimes. Notably, the latter regime inherently enlarges the fluctuation region, yet this enlargement remains on the order of $\epsilon^\rho$. In the convex scenario, employing a universal error bound applicable to coercive semialgebraic functions, we derive novel complexity results concerning averaged iterates. Additionally, our study produces auxiliary results of independent interest, including descent-type lemmas for nonsmooth nonconvex functions and an invariance principle governing the behavior of algorithmic sequences under small-step limits.