Dimension-Free Decentralized Nonsmooth Nonconvex Stochastic Optimization

Abstract: 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 $O(δ<sup>{-1}(ε<sup>{-3}+dε<sup>{-1}))$ sample complexity and $$\widetilde{O}(γ<sup>{-1/2}δ<sup>{-1}(ε<sup>{-3}+dε<sup>{-1}))$$ 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 $O(δ<sup>{-1}ε<sup>{-3})$ sample complexity and $\widetilde{O}(γ<sup>{-1/2}δ<sup>{-1}ε<sup>{-3})$ 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.