Stochastic subgradient descent on weakly convex functions escapes active strict saddles

Résumé

It was established in the 90's by Pemantle, Brandière and Duflo that stochastic gradient descent (SGD) escapes saddle points of smooth function. Knowing that critical points of a typical/generic smooth function are either local minima or saddle points, we can interpret this result as a generic convergence to local minima of SGD. The purpose of this talk is to extend such a result to the class of non-smooth, weakly-convex functions. We will first investigate the generic properties of non-smooth, semi-algebraic (or more generally definable in an o-minimal structure) functions. As recently shown by Davis and Drusvyatskiy, active strict saddles form a generic type of critical points in this class. Second, we will show that on weakly-convex functions SGD avoids active strict saddles with probability one. As a consequence, "generically" on a weakly-convex, semi-algebraic function, SGD converges to a local minimum.