On the convergence result of the gradient-push algorithm on directed graphs with constant stepsize

TL;DR

The paper analyzes the gradient-push algorithm for distributed optimization over directed graphs with a constant stepsize $α>0$. Under $L$-smooth local costs and a $eta$-strongly convex total cost, it proves exponential convergence to an $O(α)$-neighborhood of the optimizer, provided $α$ lies below a computable bound and the iterates remain bounded. It derives coupled contraction bounds for consensus error and the distance to the optimizer, establishing uniform boundedness and explicit convergence rates that depend on the graph's push-sum parameters and the smoothness/strong convexity constants. Numerical experiments in convex and nonconvex local-cost settings corroborate the theory and illustrate that the gradient-push can outperform Push-DIGing in certain regimes, with a hybrid GP–PD strategy offering robust performance. This work advances practical, provable guarantees for constant-stepsize distributed optimization on directed graphs, enabling efficient, communication-conscious implementations.

Abstract

Distributed optimization has recieved a lot of interest due to its wide applications in various fields. It consists of multiple agents that connected by a graph and optimize a total cost in a collaborative way. Often in the applications, the graph of the agents is given by a directed graph. The gradient-push algorithm is a fundamental method for distributed optimization for which the agents are connected by a directed graph. Despite of its wide usage in the literatures, its convergence property has not been established well for the important case that the stepsize is constant and the domain is the entire space. This work proves that the gradient-push algorithm with stepsize $α>0$ converges exponentially fast to an $O(α)$-neighborhood of the optimizer if the stepsize $α$ is less than a specific value. For the result, we assume that each cost is smooth and the total cost is strongly convex. Numerical experiments are provided to support the theoretical convergence result. \textcolor{black}{We also present a numerical test showing that the gradient-push algorithm may approach a small neighborhood of the minimizer faster than the Push-DIGing algorithm which is a variant of the gradient-push algorithm involving the communication of the gradient informations of the agents.

Figures (4)

• Figure 1: Performance of \ref{['eq-2-2']} for the convex local cost case.
• Figure 2: Performance of \ref{['eq-2-2']} for the nonconvex local cost case.
• Figure 3: Graphs of $\sum\limits_{i=1}^n\|z_i(t)-x^*\|^2$ with respect to the iteration $t$, for the gradient-push (GP) and the Push-DIGing (PD) algorithms with various stepsizes.
• Figure 4: Graphs of $\sum\limits_{i=1}^n\|z_i(t)-x^*\|^2$ with respect to time iteration (left) and communication cost (right) for the gradient-push(GP), Push-DIGing(PD) algorithms, and Algorithm 1.

Theorems & Definitions (25)

• Lemma 2.1
• proof
• Lemma 2.2: XSKK
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• Theorem 3.3
• Lemma 4.1