Decentralized Nonsmooth Nonconvex Optimization with Client-Data Sampling

This paper studies decentralized nonsmooth nonconvex optimization with Lipschitz-continuous local functions. We propose an efficient stochastic first-order method with a novel client-data sampling strategy in which the mini-batch sizes of the stochastic gradients on all clients follow a multinomial distribution. We show that our method can find a $(δ,ε)$-Goldstein stationary point on each client with the best-known dependence on $ε$~and $δ$ in both computational and communication complexity. Moreover, our results tighten the upper complexity bounds with respect to the Lipschitz parameter, the variance upper bound, the number of clients, and the spectral gap. We further conduct numerical experiments to demonstrate the empirical advantages of the proposed method.