Convergence Rates of Distributed Nesterov-like Gradient Methods on Random Networks

TL;DR

This paper introduces accelerated distributed gradient methods for random networks that are resilient to link failures, computationally efficient, and have faster convergence rates than existing methods, demonstrated through theoretical analysis and simulations.

Contribution

The paper proposes novel Nesterov-like gradient algorithms for random networks, improving convergence rates over traditional distributed gradient methods and extending static network algorithms to stochastic settings.

Findings

Achieves convergence rates of O(log k/k) and O(1/k^2) for the proposed methods.

Outperforms standard distributed gradient methods with rates of (1/k^{2/3}).

Validated through simulations confirming theoretical convergence improvements.

Abstract

We consider distributed optimization in random networks where N nodes cooperatively minimize the sum \sum_{i=1}^N f_i(x) of their individual convex costs. Existing literature proposes distributed gradient-like methods that are computationally cheap and resilient to link failures, but have slow convergence rates. In this paper, we propose accelerated distributed gradient methods that: 1) are resilient to link failures; 2) computationally cheap; and 3) improve convergence rates over other gradient methods. We model the network by a sequence of independent, identically distributed random matrices {W(k)} drawn from the set of symmetric, stochastic matrices with positive diagonals. The network is connected on average and the cost functions are convex, differentiable, with Lipschitz continuous and bounded gradients. We design two distributed Nesterov-like gradient methods that modify the D-NG and D-NC methods that we proposed for static networks. We prove their convergence rates in terms of the expected optimality gap at the cost function. Let k and K be the number of per-node gradient evaluations and per-node communications, respectively. Then the modified D-NG achieves rates O(log k/k) and O(\log K/K), and the modified D-NC rates O(1/k^2) and O(1/K^{2-\xi}), where \xi>0 is arbitrarily small. For comparison, the standard distributed gradient method cannot do better than \Omega(1/k^{2/3}) and \Omega(1/K^{2/3}), on the same class of cost functions (even for static networks). Simulation examples illustrate our analytical findings.