Proximally Guided Stochastic Subgradient Method for Nonsmooth, Nonconvex Problems
This provides a convergence rate analysis for stochastic subgradient methods in weakly convex functions, addressing a gap in optimization theory.
The paper tackles the problem of optimizing nonsmooth, nonconvex functions by introducing a stochastic projected subgradient method, proving it converges at the same rate as stochastic gradient methods for smooth nonconvex problems.
In this paper, we introduce a stochastic projected subgradient method for weakly convex (i.e., uniformly prox-regular) nonsmooth, nonconvex functions---a wide class of functions which includes the additive and convex composite classes. At a high-level, the method is an inexact proximal point iteration in which the strongly convex proximal subproblems are quickly solved with a specialized stochastic projected subgradient method. The primary contribution of this paper is a simple proof that the proposed algorithm converges at the same rate as the stochastic gradient method for smooth nonconvex problems. This result appears to be the first convergence rate analysis of a stochastic (or even deterministic) subgradient method for the class of weakly convex functions.