Topology-aware Generalization of Decentralized SGD
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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 𝒪(m/N+1/m+λ^2)-stable in expectation in the non-convex non-smooth setting, where N is the total sample size of the whole system, m is the worker number, and 1-λ is the spectral gap that measures the connectivity of the communication topology. These results then deliver an 𝒪(1/N+((m^-1λ^2)^α/2+ m^-α)/N^1-α/2) in-average generalization bound, which is non-vacuous even when λ 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 has a positive correlation 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.
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