Stochastic Weakly Convex Optimization Beyond Lipschitz Continuity

Abstract

This paper considers stochastic weakly convex optimization without the standard Lipschitz continuity assumption. Based on new adaptive regularization (stepsize) strategies, we show that a wide class of stochastic algorithms, including the stochastic subgradient method, preserve the $\mathcal{O}(1/\sqrt{K})$ convergence rate with constant failure rate. Our analyses rest on rather weak assumptions: the Lipschitz parameter can be either bounded by a general growth function of $\Vert x\Vert$ or locally estimated through independent random samples.