# Stochastic Subgradient Descent on a Generic Definable Function Converges to a Minimizer

It was previously shown by Davis and Drusvyatskiy that every Clarke critical point of a generic, semialgebraic (and more generally definable in an o-minimal structure), weakly convex function is lying on an active manifold and is either a local minimum or an active strict saddle. In the first part of this work, we show that when the weak convexity assumption fails a third type of point appears: a sharply repulsive critical point. Moreover, we show that the corresponding active manifolds satisfy the Verdier and the angle conditions which were introduced by us in our previous work. In the second part of this work, we show that, under a density-like assumption on the perturbation sequence, the stochastic subgradient descent (SGD) avoids sharply repulsive critical points with probability one. We show that such a density-like assumption could be obtained upon adding a small random perturbation (e.g. a nondegenerate Gaussian) at each iteration of the algorithm. These results, combined with our previous work on the avoidance of active strict saddles, show that the SGD on a generic definable (e.g. semialgebraic) function converges to a local minimum.

08/04/2021

In non-smooth stochastic optimization, we establish the non-convergence ...
02/16/2021

12/16/2019

### Active strict saddles in nonsmooth optimization

We introduce a geometrically transparent strict saddle property for nons...
01/16/2020

### Some convergent results for Backtracking Gradient Descent method on Banach spaces

Our main result concerns the following condition: Condition C. Let X ...
08/26/2021

### Subgradient methods near active manifolds: saddle point avoidance, local convergence, and asymptotic normality

Nonsmooth optimization problems arising in practice tend to exhibit bene...
06/08/2015

### Linear Convergence of the Randomized Feasible Descent Method Under the Weak Strong Convexity Assumption

In this paper we generalize the framework of the feasible descent method...
11/10/2021

### SGD Through the Lens of Kolmogorov Complexity

We prove that stochastic gradient descent (SGD) finds a solution that ac...