aiqbench analysis
This research demonstrates that in decentralized learning scenarios with heterogeneous node weights, row-stochastic matrices can achieve strictly faster convergence than doubly stochastic alternatives by minimizing consensus error penalties inherent to standard Euclidean analysis. By establishing a weighted Hilbert-space framework, the study provides rigorous topology conditions and design guidelines for optimizing network structures where spectral gaps alone are insufficient predictors of performance.
Row-Stochastic Matrices Can Provably Outperform Doubly Stochastic Matrices in Decentralized Learning
arXiv:2511.19513v4 Announce Type: replace Abstract: Decentralized learning often involves a weighted global loss with heterogeneous node weights $\lambda$. We revisit two natural strategies for incorporating these weights: (i) embedding them into the local losses to retain a uniform weight (and thus a doubly stochastic matrix), and (ii) keeping the original losses while employing a $\lambda$-induced row-stochastic matrix. Although prior work shows that both strategies target the same $\lambda$-weighted global loss, it remains unclear whether the Euclidean-space guarantees are tight and what fundamentally differentiates their behaviors. To clarify this, we develop a weighted Hilbert-space framework $L^2(\lambda;\mathbb{R}^d)$ and obtain convergence rates that are strictly tighter than those from standard Euclidean analysis. In this geometry, the row-stochastic matrix becomes \emph{self-adjoint} whereas the doubly stochastic one does not, creating additional \emph{penalty terms} that amplify consensus error, thereby slowing convergence. Consequently, the difference in convergence arises not only from spectral gaps but also from these penalty terms. We then derive sufficient conditions under which the row-stochastic design converges faster even with a smaller spectral gap. Finally, by using a Rayleigh-quotient and Loewner-order eigenvalue comparison, we further obtain topology conditions that guarantee this advantage and yield practical topology-design guidelines.
Abstract:Decentralized learning often involves a weighted global loss with heterogeneous node weights $\lambda$. We revisit two natural strategies for incorporating these weights: (i) embedding them into the local losses to retain a uniform weight (and thus a doubly stochastic matrix), and (ii) keeping the original losses while employing a $\lambda$-induced row-stochastic matrix. Although prior work shows that both strategies target the same $\lambda$-weighted global loss, it remains unclear whether the Euclidean-space guarantees are tight and what fundamentally differentiates their behaviors. To clarify this, we develop a weighted Hilbert-space framework $L^2(\lambda;\mathbb{R}^d)$ and obtain convergence rates that are strictly tighter than those from standard Euclidean analysis. In this geometry, the row-stochastic matrix becomes \emph{self-adjoint} whereas the doubly stochastic one does not, creating additional \emph{penalty terms} that amplify consensus error, thereby slowing convergence. Consequently, the difference in convergence arises not only from spectral gaps but also from these penalty terms. We then derive sufficient conditions under which the row-stochastic design converges faster even with a smaller spectral gap. Finally, by using a Rayleigh-quotient and Loewner-order eigenvalue comparison, we further obtain topology conditions that guarantee this advantage and yield practical topology-design guidelines.
Machine Learning (cs.LG)
arXiv:2511.19513 [cs.LG]
(or arXiv:2511.19513v4 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2511.19513 Focus to learn more arXiv-issued DOI via DataCite
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