Delay-Tolerant Augmented-Consensus-based Distributed Directed Optimization

TL;DR

The paper addresses distributed optimization over directed graphs with heterogeneous, bounded delays by introducing the DTAC-ADDOPT algorithm, which leverages an augmented consensus framework to guarantee convergence to the global optimum of $F(\mathbf{z})=\sum_i f_i(\mathbf{z})$ under constant step-sizes. The core method extends ADD-OPT with an augmented matrix $\overline{C}$ to accommodate delays up to $\overline{\tau}$, ensuring linear convergence provided $0<\alpha<\min\{\alpha_3, 1/[n(\overline{\tau}+1)l]\}$. The authors establish a structured convergence proof (Lemma lem_ts) via a contraction mapping on augmented variables and validate performance through academic and real-data simulations, including MNIST-based distributed logistic regression. The results show that delay-tolerant optimization on digraphs is feasible in practice, with trade-offs where larger delays require smaller step-sizes and potentially slower convergence, yet the method remains robust to heterogeneous link delays. Practical impact lies in enabling reliable distributed learning and optimization in networks with non-negligible and varying communication latencies.

Abstract

Distributed optimization finds applications in large-scale machine learning, data processing and classification over multi-agent networks. In real-world scenarios, the communication network of agents may encounter latency that may affect the convergence of the optimization protocol. This paper addresses the case where the information exchange among the agents (computing nodes) over data-transmission channels (links) might be subject to communication time-delays, which is not well addressed in the existing literature. Our proposed algorithm improves the state-of-the-art by handling heterogeneous and arbitrary but bounded and fixed (time-invariant) delays over general strongly-connected directed networks. Arguments from matrix theory, algebraic graph theory, and augmented consensus formulation are applied to prove the convergence to the optimal value. Simulations are provided to verify the results and compare the performance with some existing delay-free algorithms.

Figures (4)

• Figure 1: This figure shows the decay of the optimization residual (average error) under time-delays over (left) a static ER network and (right) a dynamic ER network. As it is clear from the figures, the algorithm converges under time-delays. There are some oscillation in the decay of the right figure, which is due to change in the network topology.
• Figure 2: This figure shows the decay of the optimization residual (mean-square-error) subject to different values of time-delays over an ER network. The left figure shows the residual decay for $\alpha=0.001$ and the right figure for $\alpha=0.005$. As it is clear from the figure, for large value of $\overline{\tau}$ the residual decay becomes unstable and loses convergence.
• Figure 3: This figure shows a sample set of images of hand-written numbers from $0$ to $9$ taken from the MNIST data set. This data set is used for image classification via the optimization objective \ref{['eq_F_mnist']} and \ref{['eq_fij_regression']}.
• Figure 4: This simulation presents different distributed techniques over the exponential graph (given in the right figure) to optimize the objective function \ref{['eq_F_mnist']} and \ref{['eq_fij_regression']}. Note that only the proposed DTAC-ADDOPT is simulated subject to information-exchange delays, and the other techniques are simulated in the absence of delays.

Theorems & Definitions (18)

• Lemma 1
• proof
• Corollary 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• Lemma 5
• proof
• Lemma 6