Convergence Rates of Decentralized Gradient Methods over Cluster Networks

Abstract

We present an analysis for the performance of decentralized consensus-based gradient (DCG) methods for solving optimization problems over a cluster network of nodes. This type of network is composed of a number of densely connected clusters with a sparse connection between them. Decentralized algorithms over cluster networks have been observed to constitute two-time-scale dynamics, where information within any cluster is mixed much faster than the one across clusters. Based on this observation, we present a novel analysis to study the convergence of the DCG methods over cluster networks. In particular, we show that these methods converge at a rate ln(T )/T and only scale with the number of clusters, which is relatively small to the size of the network. Our result improves the existing analysis, where these methods are shown to scale with the size of the network. The key technique in our analysis is to consider a novel Lyapunov function that captures the impact of multiple time-scale dynamics on the convergence of this method. We also illustrate our theoretical results by a number of numerical simulations using DCG methods over different cluster networks.

Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems

“…Proper choices of these constants will also help us to derive the convergence rates of (2). Similar approach has been used in different settings of two-time-scale methods, see for example [53,66]. We conclude this section by introducing two assumptions for our analysis studied later.…”

“…Other Settings. We also want to mention some related literature in game theory [39,40,41,42,43], two-time-scale stochastic approximation [44,45,46,47,48,49,50,51,52,53], reinforcement learning [54,55,56,57,58], two-time-scale optimization [59,60], and decentralized optimization [61,62,63,64,65,66,67]. These works study different variants of two-time-scale methods mostly for solving a single optimization problem, and often aim to find global optimality (or fixed points) using different structure of the underlying problems (e.g., Markov structure in stochastic games and reinforcement learning or strong monotonicity in stochastic approximation).…”