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We propose perturbed proximal algorithms that can provably escape strictsaddles for nonsmooth weakly convex functions. The main results are based on anovel characterization of $\epsilon$-approximate local minimum for nonsmoothfunctions, and recent developments on perturbed gradient methods for escapingsaddle points for smooth problems. Specifically, we show that under standardassumptions, the perturbed proximal point, perturbed proximal gradient andperturbed proximal linear algorithms find $\epsilon$-approximate local minimumfor nonsmooth weakly convex functions in $O(\epsilon^{-2}\log(d)^4)$iterations, where $d$ is the dimension of the problem.

Escaping Saddle Points for Nonsmooth Weakly Convex Functions via Perturbed Proximal Algorithms