A nearly linearly convergent first-order method for nonsmooth functions with quadratic growth

TL;DR

This paper introduces a parameter-free first-order method that achieves nearly linear convergence for a broad class of nonsmooth, nonconvex functions with quadratic growth, extending classical smooth convex results.

Contribution

It presents a novel, parameter-free algorithm based on Goldstein's subgradient method that converges nearly linearly for nonsmooth functions with quadratic growth.

Findings

Applicable to max-of-smooth, decomposable, and semialgebraic functions

Achieves nearly linear convergence rate

Extends classical smooth convex convergence results to nonsmooth settings

Abstract

Classical results show that gradient descent converges linearly to minimizers of smooth strongly convex functions. A natural question is whether there exists a locally nearly linearly convergent method for nonsmooth functions with quadratic growth. This work designs such a method for a wide class of nonsmooth and nonconvex locally Lipschitz functions, including max-of-smooth, Shapiro's decomposable class, and generic semialgebraic functions. The algorithm is parameter-free and derives from Goldstein's conceptual subgradient method.