Topology-aware Generalization of Decentralized SGD

Abstract

This paper studies the algorithmic stability and generalizability of decentralized stochastic gradient descent (D-SGD). We prove that the consensus model learned by D-SGD is $\mathcal{O}{(N^{-1}+m^{-1} +\lambda^2)}$-stable in expectation in the non-convex non-smooth setting, where $N$ is the total sample size, $m$ is the worker number, and $1+\lambda$ is the spectral gap that measures the connectivity of the communication topology. These results then deliver an $\mathcal{O}{(N^{-(1+\alpha)/2}+ m^{-(1+\alpha)/2}+\lambda^{1+\alpha} + \phi_{\mathcal{S}})}$ in-average generalization bound, which is non-vacuous even when $\lambda$ is closed to $1$, in contrast to vacuous as suggested by existing literature on the projected version of D-SGD. Our theory indicates that the generalizability of D-SGD is positively correlated with the spectral gap, and can explain why consensus control in initial training phase can ensure better generalization. Experiments of VGG-11 and ResNet-18 on CIFAR-10, CIFAR-100 and Tiny-ImageNet justify our theory. To our best knowledge, this is the first work on the topology-aware generalization of vanilla D-SGD. Code is available at https://github.com/Raiden-Zhu/Generalization-of-DSGD.