On the Linear Speedup of the Push-Pull Method for Decentralized Optimization over Digraphs

Liyuan Liang; Gan Luo; Kun Yuan

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arXiv · 2506.18075

On the Linear Speedup of the Push-Pull Method for Decentralized Optimization over Digraphs

Liyuan Liang·Gan Luo·Kun Yuan

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Abstract

The linear speedup property is essential for demonstrating the advantage of distributed algorithms over their single-node counterparts. In this paper, we study the stochastic Push-Pull method, a widely adopted decentralized optimization algorithm over directed graphs (digraphs). Unlike methods that rely solely on row-stochastic or column-stochastic mixing matrices, Push-Pull avoids nonlinear correction and has shown superior empirical performance across a variety of settings. However, its theoretical analysis remains challenging, and the linear speedup property has not been generally establishe--revealing a significant gap between empirical success and limited theoretical understanding. To bridge this gap, we propose a novel analysis framework and prove that Push-Pull achieves linear speedup over arbitrary strongly connected digraphs. Our results provide the comprehensive theoretical understanding for stochastic Push-Pull, aligning its theory with empirical performance. Code: https://github.com/pkumelon/PushPull.

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BibTeX RIS

Liyuan Liang, Gan Luo, Kun Yuan. 2025-06-22. On the Linear Speedup of the Push-Pull Method for Decentralized Optimization over Digraphs. https://arxiv.org/abs/2506.18075

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