Stochastic Subgradient Method Converges on Tame Functions

Abstract

This work considers the question: what convergence guarantees does the stochastic subgradient method have in the absence of smoothness and convexity? We prove that the stochastic subgradient method, on any semialgebraic locally Lipschitz function, produces limit points that are all first-order stationary. More generally, our result applies to any function with a Whitney stratifiable graph. In particular, this work endows the stochastic subgradient method, and its proximal extension, with rigorous convergence guarantees for a wide class of problems arising in data science-including all popular deep learning architectures.

Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning

“…Note also that virtually all deep network architectures used in applications are actually definable, see e.g. [24]. (b) Despite our efforts we do not see any means to obtain Corollary 6 easily.…”

“…Proof : Using the chain rule characterization, all proofs boil down to providing a chain rule with the Clarke subdifferential for each of the above mentioned situation. We refer to [48] for convex, Clarke and prox regular functions, [24] for tame functions.…”

“…On a more applied side, conservative fields allow to analyze fundamental modern numerical algorithms in machine learning or numerical analysis based on automatic differentiation [49,28] and decomposition [17,24] in a nonsmooth context. Automatic differentiation is indeed proved to yield conservative fields which allows in turn to study discrete stochastic algorithms that are massively used to train AI systems.…”

Subgradient Methods for Sharp Weakly Convex Functions

“…Our current work sits within the broader scope of analyzing subgradient and proximal methods for weakly convex problems [9, 11, 13-16, 25, 26]; see also the recent survey [12]. In particular, the paper [9] proves a global sublinear rate of convergence, in terms of a natural stationarity measure, of a (stochastic) subgradient method on any weakly convex function. In contrast, here we are interested in subgradient methods that are locally linearly convergent under the additional sharpness assumption.…”

Gradient Sampling Methods for Nonsmooth Optimization

“…Thus, one might hope to prove convergence results for a GS algorithm (with predetermined stepsizes rather than line searches) that parallel convergence theory for stochastic gradient methods. Recent work by Davis, Drusvyatskiy, Kakade and Lee [DDKL18] gives convergence results for stochastic subgradient methods on a broad class of problems.…”