# Removing Data Heterogeneity Influence Enhances Network Topology Dependence of Decentralized SGD

We consider decentralized stochastic optimization problems where a network of agents each owns a local cost function cooperate to find a minimizer of the global-averaged cost. A widely studied decentralized algorithm for this problem is D-SGD in which each node applies a stochastic gradient descent step, then averages its estimate with its neighbors. D-SGD is attractive due to its efficient single-iteration communication and can achieve linear speedup in convergence (in terms of the network size). However, D-SGD is very sensitive to the network topology. For smooth objective functions, the transient stage (which measures how fast the algorithm can reach the linear speedup stage) of D-SGD is on the order of O(n/(1-β)^2) and O(n^3/(1-β)^4) for strongly convex and generally convex cost functions, respectively, where 1-β∈ (0,1) is a topology-dependent quantity that approaches 0 for a large and sparse network. Hence, D-SGD suffers from slow convergence for large and sparse networks. In this work, we study the non-asymptotic convergence property of the D^2/Exact-diffusion algorithm. By eliminating the influence of data heterogeneity between nodes, D^2/Exact-diffusion is shown to have an enhanced transient stage that are on the order of O(n/(1-β)) and O(n^3/(1-β)^2) for strongly convex and generally convex cost functions, respectively. Moreover, we provide a lower bound of the transient stage of D-SGD under homogeneous data distributions, which coincides with the transient stage of D^2/Exact-diffusion in the strongly-convex setting. These results show that removing the influence of data heterogeneity can ameliorate the network topology dependence of D-SGD. Compared with existing decentralized algorithms bounds, D^2/Exact-diffusion is least sensitive to network topology.

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