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

We consider distributed optimization in random networks where N nodes cooperatively minimize the sum \sum{i=1}^N fi(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/k²⁾ 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.