Distributed gradient-based optimization in the presence of dependent aperiodic communication

TL;DR

This paper analyzes the convergence of distributed gradient algorithms under dependent, aperiodic communication failures, introducing new stochastic conditions that ensure convergence despite unreliable and non-independent communication patterns.

Contribution

It establishes convergence guarantees for distributed gradient methods with dependent communication delays, introducing the concept of stochastically strongly connected networks and related stochastic dominance conditions.

Findings

Convergence is guaranteed if AoI processes are stochastically dominated by a finite first moment variable.

Introduces stochastically strongly connected (SSC) networks for time-varying communication.

Shows that under certain mixing conditions, AoI processes have finite p-th moments, ensuring convergence.

Abstract

Iterative distributed optimization algorithms involve multiple agents that communicate with each other, over time, in order to minimize/maximize a global objective. In the presence of unreliable communication networks, the Age-of-Information (AoI), which measures the freshness of data received, may be large and hence hinder algorithmic convergence. In this paper, we study the convergence of general distributed gradient-based optimization algorithms in the presence of communication that neither happens periodically nor at stochastically independent points in time. We show that convergence is guaranteed provided the random variables associated with the AoI processes are stochastically dominated by a random variable with finite first moment. This improves on previous requirements of boundedness of more than the first moment. We then introduce stochastically strongly connected (SSC) networks, a new stochastic form of strong connectedness for time-varying networks. We show: If for any $p \ge0$ the processes that describe the success of communication between agents in a SSC network are $\alpha$-mixing with $n^{p-1}\alpha(n)$ summable, then the associated AoI processes are stochastically dominated by a random variable with finite $p$-th moment. In combination with our first contribution, this implies that distributed stochastic gradient descend converges in the presence of AoI, if $\alpha(n)$ is summable.