Dimension-Free Decentralized Nonsmooth Nonconvex Stochastic Optimization

We investigate decentralized nonsmooth nonconvex stochastic optimization over a network of n nodes, with the goal of finding an (δ,ε)-Goldstein stationary point. The best existing algorithm achieves O(δ-1(ε-3+dε-1)) sample complexity and widetildeO(γ-1/2δ-1(ε-3+dε-1)) communication complexity, where d is the problem dimension and γ is the spectral gap of the communication matrix. However, the polynomial dependence on d can be a major bottleneck in high-dimensional regimes. In this paper, we propose a novel algorithm that achieves O(δ-1ε-3) sample complexity and widetildeO(γ-1/2δ-1ε-3) communication complexity. The primary technique is an elegant decentralized online-to-nonconvex conversion that reduces the original problem to a decentralized online convex optimization (D-OCO) problem. A key property of our conversion is that its consensus requirements can be inherited directly from the consensus of the underlying D-OCO decisions. In particular, this property enables us to establish an explicit connection between the dimension dependence and the consensus error, which in turn shows that the polynomial dependence on d can be removed with only logarithmic additional communication.