Distributed Optimization with Gradient Tracking over Heterogeneous Delay-Prone Directed Networks

TL;DR

The paper tackles distributed optimization over directed graphs with heterogeneous, time-invariant communication delays. It proposes R-ADD-OPT, a gradient-tracking algorithm that embeds a robustified ratio-consensus protocol within ADD-OPT and augments the network to explicitly model delays. The authors prove exponential convergence to the unique optimum $z^*$ under a gradient step-size $\alpha$ lying in a delay-dependent interval $\alpha \in (0, \bar{\alpha})$, with $\bar{\alpha}$ computable from the maximum delay $\bar{\tau}$. They also show that the augmented-delay framework preserves the necessary contraction properties via the matrix $\Xi$ and the consensus vector $\boldsymbol{\pi}$, and provide numerical experiments on a 5-node network illustrating the impact of delays on convergence rate and validating the step-size guidance. The work advances distributed optimization in asynchronous, directed settings and suggests avenues for handling time-varying delays.

Abstract

In this paper, we address the distributed optimization problem over unidirectional networks with possibly time-invariant heterogeneous bounded transmission delays. In particular, we propose a modified version of the Accelerated Distributed Directed OPTimization (ADD-OPT) algorithm, herein called Robustified ADD-OPT (R-ADD-OPT), which is able to solve the distributed optimization problem, even when the communication links suffer from heterogeneous but bounded transmission delays. We show that if the gradient step-size of the R-ADD-OPT algorithm is within a certain range, which also depends on the maximum time delay in the network, then the nodes are guaranteed to converge to the optimal solution of the distributed optimization problem. The range of the gradient step-size that guarantees convergence can be computed a priori based on the maximum time delay in the network.

Figures (4)

• Figure 1: Example of the augmented digraph with fixed transmission delays.
• Figure 2: Digraph consisting of five agents.
• Figure 3: Spectral radius with alpha step-size value for different delays.
• Figure 4: Residual for different upper bound on the delays in the network $\bar{\tau}$. Time-varying delays are within the bound $0\leq \tau_{k}^{ji} \leq \bar{\tau}^{\star}$.

Theorems & Definitions (13)

• Lemma 1: Xi et. al. xi2017add
• Lemma 2: Nedic et. al. nedic2014distributed
• Lemma 3: Xi et. al. xin2019distributed
• Lemma 4
• Lemma 5: Gradient step-size for convergence, Xi et. al. xi2017add
• Theorem 1
• proof
• proof : Proof of Lemma \ref{['lem:sigma']}
• proof : Proof of Lemma \ref{['lem:Xi_linear_relation']}
• proof : Proof of Lemma \ref{['lem:step-size-range']}