  We investigate the finite-time analysis of finding
($\delta,\epsilon$)-stationary points for nonsmooth nonconvex objectives in
decentralized stochastic optimization. A set of agents aim at minimizing a
global function using only their local information by interacting over a
network. We present a novel algorithm, called Multi Epoch Decentralized Online
Learning (ME-DOL), for which we establish the sample complexity in various
settings. First, using a recently proposed online-to-nonconvex technique, we
show that our algorithm recovers the optimal convergence rate of smooth
nonconvex objectives. We then extend our analysis to the nonsmooth setting,
building on properties of randomized smoothing and Goldstein-subdifferential
sets. We establish the sample complexity of $O(\delta^{-1}\epsilon^{-3})$,
which to the best of our knowledge is the first finite-time guarantee for
decentralized nonsmooth nonconvex stochastic optimization in the first-order
setting (without weak-convexity), matching its optimal centralized counterpart.
We further prove the same rate for the zero-order oracle setting without using
variance reduction.
