Decentralized Optimization over Time-Varying Row-Stochastic Digraphs

TL;DR

This paper tackles decentralized optimization over time-varying broadcast networks where only row-stochastic mixing matrices are available. It introduces PULM, a memory-enhanced gossip protocol that achieves exact average consensus with exponential convergence, and PULM-DGD, a decentralized gradient method that attains a stationary point for smooth nonconvex objectives at rate $\mathcal{O}(\frac{\ln(T)}{T})$ in communication rounds. The results show exact convergence under TVBNs without out-degree information, extending decentralization to highly dynamic networks and packet-loss settings. This broadens the practical reliability and robustness of distributed learning and coordination in directed, time-varying topologies.

Abstract

Decentralized optimization over directed graphs is essential for applications such as robotic swarms, sensor networks, and distributed learning. In many practical scenarios, the underlying network takes the form of a Time-Varying Broadcast Network (TVBN), where only row-stochastic mixing matrices can be constructed due to the unavailability of out-degree information. Achieving exact convergence for decentralized optimization over TVBNs has remained a long-standing open problem, as the limiting distribution of time-varying row-stochastic mixing matrices depends on unpredictable future graph realizations, rendering standard bias-correction techniques infeasible. This paper develops the first decentralized optimization algorithm that achieves exact convergence using only time-varying row-stochastic matrices. We first propose PULM (Pull-with-Memory), a gossip protocol that achieves average consensus with exponential convergence by alternating between row-stochastic mixing and local adjustment steps. Building on PULM, we develop PULM-DGD, which converges to a stationary solution at a rate of $\mathcal{O}(\ln(T)/T)$ for smooth nonconvex objectives, where $T$ denotes the communication round. Our results significantly broaden the applicability of decentralized optimization to highly dynamic communication environments.

Figures (14)

• Figure 1: Left: A time-varying network with $n=4$ nodes experiencing unexpected network failure; "tx" and "rx" denote "transmission" and "reception," respectively. Middle and right: The corresponding column- and row-stochastic mixing matrices. The network failure results in an incorrect column-stochastic matrix due to sudden changes in out-degree information. However, a correct row-stochastic matrix can still be constructed since it relies only on in-degree information (i.e., the actually received messages), which remains accessible.
• Figure 2: Diffusion of the $1/n$ diagonal anchor under the adjust--gossip update. The adjust step anchors each diagonal entry $w_{jj}$ at $1/n$; the gossip step then diffuses this value down column $j$, pulling off-diagonal entries toward $1/n$. Cell colors indicate distance to $1/n$ (greener means closer), showing how the anchored mass spreads from the diagonal throughout each column, driving $W^{(k)}$ toward average consensus.
• Figure 3: Convergence of $W^{(k)}$. $W$ Error implies $\|W^{(k)}-n^{-1}\mathds{1}_n\mathds{1}_n^\top\|_F$. Details are in Appendix \ref{['app:performances-different-topologies']}.
• Figure 4: Performance comparison of push-family and PULM-family algorithms under packet loss on time-varying directed networks. In consensus, push denotes push-sum and pulm denotes PULM; in regression, push denotes push-DIGing and pulm denotes PULM-DGD. The packet-loss probability is $p_t$. See Appendix \ref{['sec:exp-app']} for full experimental settings.
• Figure 5: Effect of the inner communication rounds $R_k$ on the performance of Algorithm \ref{['alg:PWM-GD']}. Top row: logistic regression on synthetic data (loss and accuracy) and on real data (loss and accuracy) for $R_k\in\{1,3,5,10,20\}$. Bottom row: MNIST training (loss and accuracy) for $R_k\in\{1,3,5,7,10\}$, and CIFAR-10 training (loss and accuracy) for $R_k\in\{1,6,10,20\}$. Larger $R_k$ increases communication per outer iteration but yields only marginal changes in convergence behavior. See Appendix \ref{['sec:detail-diff-inner']} for experimental details.

Theorems & Definitions (28)

• Example 1: Random Radio Broadcast
• Example 2: Byzantine Attack
• Example 3: Packet Loss and Network Failure
• Definition 1: Compatible Mixing Matrices
• Proposition 1
• Proposition 2: Limiting Property
• proof
• Remark 1: Column-stochastic vs Row-stochastic matrices
• Theorem 1
• proof