Inexact proximal methods for weakly convex functions

TL;DR

This paper introduces inexact proximal methods for weakly convex functions, providing convergence guarantees and rates under certain conditions, even with errors in proximal computations.

Contribution

It develops a unified framework for inexact proximal methods applicable to weakly convex functions, extending convergence analysis to scenarios with computational errors.

Findings

Global convergence with convergence rates under KL property

Framework applicable to both inexact proximal point and gradient methods

Convergence guarantees for functions with Lipschitz continuous gradients

Abstract

This paper proposes and develops inexact proximal methods for finding stationary points of the sum of a smooth function and a nonsmooth weakly convex one, where an error is present in the calculation of the proximal mapping of the nonsmooth term. A general framework for finding zeros of a continuous mapping is derived from our previous paper on this subject to establish convergence properties of the inexact proximal point method when the smooth term is vanished and of the inexact proximal gradient method when the smooth term satisfies a descent condition. The inexact proximal point method achieves global convergence with constructive convergence rates when the Moreau envelope of the objective function satisfies the Kurdyka-Lojasiewicz (KL) property. Meanwhile, when the smooth term is twice continuously differentiable with a Lipschitz continuous gradient and a differentiable approximation of the objective function satisfies the KL property, the inexact proximal gradient method achieves the global convergence of iterates with constructive convergence rates.