Proximal methods avoid active strict saddles of weakly convex functions

TL;DR

This paper introduces a strict saddle property for nonsmooth functions, ensuring proximal algorithms converge to local minima in weakly convex problems, which is applicable to many practical semi-algebraic optimization scenarios.

Contribution

The paper defines a strict saddle property for nonsmooth functions and proves that proximal algorithms avoid saddle points under this condition in weakly convex problems.

Findings

Proximal algorithms converge to local minima when the strict saddle property holds.

The strict saddle property is shown to be generic in semi-algebraic optimization problems.

The introduced property provides a geometric understanding of convergence behavior in nonsmooth optimization.

Abstract

We introduce a geometrically transparent strict saddle property for nonsmooth functions. This property guarantees that simple proximal algorithms on weakly convex problems converge only to local minimizers, when randomly initialized. We argue that the strict saddle property may be a realistic assumption in applications, since it provably holds for generic semi-algebraic optimization problems.