Distributed gradient-based optimization in the presence of dependent aperiodic communication
This work addresses convergence challenges in distributed optimization for systems with dependent aperiodic communication, such as unreliable networks, offering incremental improvements over existing theoretical bounds.
The paper tackles the problem of distributed gradient-based optimization under unreliable, non-periodic, and dependent communication by establishing convergence conditions based on stochastic dominance of Age-of-Information (AoI) processes with finite first moments, improving on prior boundedness requirements. It introduces stochastically strongly connected (SSC) networks and shows that if communication success processes are α-mixing with summability conditions, AoI processes are dominated by variables with finite p-th moments, implying convergence for distributed stochastic gradient descent when α(n) is summable.
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 $α$-mixing with $n^{p-1}α(n)$ 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 $α(n)$ is summable.