Stochastic Subgradient Descent Escapes Active Strict Saddles on Weakly Convex Functions

TL;DR

This paper proves that stochastic subgradient descent (SGD) avoids certain saddle points called active strict saddles on weakly convex functions, ensuring convergence to local minima under generic conditions.

Contribution

It establishes the non-convergence of SGD to active strict saddles on weakly convex functions and introduces new geometric conditions for analysis.

Findings

SGD avoids active strict saddles on weakly convex functions

Under generic conditions, SGD converges to local minima

New projection formula for Whitney stratifiable functions

Abstract

In non-smooth stochastic optimization, we establish the non-convergence of the stochastic subgradient descent (SGD) to the critical points recently called active strict saddles by Davis and Drusvyatskiy. Such points lie on a manifold $M$ where the function $f$ has a direction of second-order negative curvature. Off this manifold, the norm of the Clarke subdifferential of $f$ is lower-bounded. We require two conditions on $f$. The first assumption is a Verdier stratification condition, which is a refinement of the popular Whitney stratification. It allows us to establish a reinforced version of the projection formula of Bolte \emph{et.al.} for Whitney stratifiable functions, and which is of independent interest. The second assumption, termed the angle condition, allows to control the distance of the iterates to $M$. When $f$ is weakly convex, our assumptions are generic. Consequently, generically in the class of definable weakly convex functions, the SGD converges to a local minimizer.