Stochastic Subgradient Descent Escapes Active Strict Saddles on Weakly Convex Functions
This addresses convergence guarantees in non-smooth optimization for machine learning, though it is incremental as it builds on existing theory for weakly convex functions.
The paper shows that stochastic subgradient descent (SGD) avoids active strict saddles in non-smooth stochastic optimization, establishing that under generic conditions for weakly convex functions, SGD converges to a local minimizer.
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.