Improved Convergence Analysis of Topology Dependence in Decentralized SGD

2026-06-08

AI Summary

Existing theoretical analyses struggle to accurately characterize the influence of network topology on the convergence behavior of decentralized stochastic gradient descent (SGD), particularly suffering from a disconnect between theory and empirical observations in both homogeneous and heterogeneous data settings. This work moves beyond the conventional paradigm that relies solely on the spectral gap of the mixing matrix and, for the first time, incorporates the full spectrum of eigenvalues into the convergence analysis, thereby establishing a tighter theoretical framework. Leveraging spectral graph theory and eigenvalue decomposition, the proposed approach demonstrates that the entire set of eigenvalues collectively governs the convergence rate. Empirical results confirm that this refined framework more accurately predicts convergence behavior across diverse network topologies and substantially narrows the gap between theoretical predictions and experimental outcomes under homogeneous data distributions.

Abstract

Decentralized SGD is a fundamental algorithm in decentralized learning, although the influence of an underlying network topology on its convergence behavior is not yet fully understood. Existing convergence analyses have shown that topologies with a small spectral gap significantly deteriorate the convergence rate of Decentralized SGD in both homogeneous and heterogeneous cases. However, many prior papers have reported that indeed the choice of the topology has a significant experimental impact in the heterogeneous case, but has little experimental impact on training behavior in the homogeneous case. In this paper, we present a tighter convergence analysis of Decentralized SGD, offering a more precise understanding of how topologies affect the convergence rate than the prior analysis. Specifically, unlike existing convergence analyses that used only the spectral gap as a property of the topology, our novel analysis shows that all eigenvalues of the mixing matrix affect the convergence rate. Throughout the experiments, we carefully evaluated the convergence behavior of Decentralized SGD and demonstrated that our novel convergence analysis can more accurately describe the effect of topology on the convergence rate.

Problem

• Decentralized SGD
• network topology
• convergence analysis
• spectral gap
• heterogeneous data

Innovation

• Decentralized SGD
• convergence analysis
• network topology
• mixing matrix
• eigenvalues