Stability and Generalization of Push-Sum Based Decentralized Optimization over Directed Graphs

• URL: http://arxiv.org/abs/2602.20567v1
• Date: Tue, 24 Feb 2026 05:32:03 GMT
• Title: Stability and Generalization of Push-Sum Based Decentralized Optimization over Directed Graphs
• Authors: Yifei Liang, Yan Sun, Xiaochun Cao, Li Shen,
• Abstract summary: Push-based decentralized communication enables optimization over communication networks, where information exchange may be asymmetric.<n>We develop a unified uniform-stability framework for the Gradient Push (SGP) algorithm.<n>A key technical ingredient is an imbalance-aware generalization bound through two quantities.
• Score: 55.77845440440496
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Push-Sum-based decentralized learning enables optimization over directed communication networks, where information exchange may be asymmetric. While convergence properties of such methods are well understood, their finite-iteration stability and generalization behavior remain unclear due to structural bias induced by column-stochastic mixing and asymmetric error propagation. In this work, we develop a unified uniform-stability framework for the Stochastic Gradient Push (SGP) algorithm that captures the effect of directed topology. A key technical ingredient is an imbalance-aware consistency bound for Push-Sum, which controls consensus deviation through two quantities: the stationary distribution imbalance parameter $δ$ and the spectral gap $(1-λ)$ governing mixing speed. This decomposition enables us to disentangle statistical effects from topology-induced bias. We establish finite-iteration stability and optimization guarantees for both convex objectives and non-convex objectives satisfying the Polyak--Łojasiewicz condition. For convex problems, SGP attains excess generalization error of order $\tilde{\mathcal{O}}\!\left(\frac{1}{\sqrt{mn}}+\fracγ{δ(1-λ)}+γ\right)$ under step-size schedules, and we characterize the corresponding optimal early stopping time that minimizes this bound. For PŁ objectives, we obtain convex-like optimization and generalization rates with dominant dependence proportional to $κ\!\left(1+\frac{1}{δ(1-λ)}\right)$, revealing a multiplicative coupling between problem conditioning and directed communication topology. Our analysis clarifies when Push-Sum correction is necessary compared with standard decentralized SGD and quantifies how imbalance and mixing jointly shape the best attainable learning performance.

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