Row-stochastic matrices can provably outperform doubly stochastic matrices in decentralized learning

TL;DR

This paper demonstrates that row-stochastic matrices can outperform doubly stochastic matrices in decentralized learning by analyzing convergence in a weighted Hilbert space, revealing new geometric insights.

Contribution

The authors develop a weighted Hilbert-space framework to compare stochastic matrices, showing conditions where row-stochastic matrices converge faster despite smaller spectral gaps.

Findings

Row-stochastic matrices can have tighter convergence guarantees than doubly stochastic matrices.

The weighted Hilbert-space analysis reveals additional penalty terms affecting convergence.

Topology conditions are derived to ensure row-stochastic matrices outperform doubly stochastic matrices.

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 yield the same expected descent direction for the 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 Euclidean analysis. In this geometry, the row-stochastic matrix becomes self-adjoint whereas the doubly stochastic one does not, creating additional 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.