Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization

TL;DR

This paper demonstrates that the subgradient method converges only to local minima for a broad class of nonsmooth functions, using a geometric interpretation and new regularity conditions in nonsmooth analysis.

Contribution

It introduces a geometric framework interpreting nonsmooth dynamics as Riemannian gradient flows on submanifolds, extending stratification conditions and convergence analysis.

Findings

Subgradient method converges only to local minima on definable Lipschitz functions.

New regularity conditions in nonsmooth analysis parallel classical stratification conditions.

Extended stochastic process techniques to analyze nonsmooth optimization dynamics.

Abstract

We show that the subgradient method converges only to local minimizers when applied to generic Lipschitz continuous and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, the argument we present is appealingly transparent: we interpret the nonsmooth dynamics as an approximate Riemannian gradient method on a certain distinguished submanifold that captures the nonsmooth activity of the function. In the process, we develop new regularity conditions in nonsmooth analysis that parallel the stratification conditions of Whitney, Kuo, and Verdier and extend stochastic processes techniques of Pemantle.