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

Abstract

Abstract-We consider distributed optimization in random networks where nodes cooperatively minimize the sum 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 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 and be the number of per-node gradient evaluations and per-node communications, respectively. Then the modified D-NG achieves rates and , and the modified D-NC rates and , where is arbitrarily small. For comparison, the standard distributed gradient method cannot do better than and , on the same class of cost functions (even for static networks). Simulation examples illustrate our analytical findings.

Distributed Nesterov gradient methods for random networks: Convergence in probability and convergence rates

“…We can modify our methods and relax these prior knowledge requirements such that the methods still provable converge, at rates that are close to the ones presented in this paper; for details, we refer to [10].…”

“…In subsequent results, ξ denotes an arbitrarily small positive number. A proof of Theorem 1, as well as explicit constants in the established rates, can be found in [10]. Theorem 1 indicates that the convergence rates do not depend on the underlying random network statistics.…”

“…In other words, mD-NG, when compared with D-NG, introduces an additional per-node communication at each iteration k. This allows with mD-NG for structural robustness to random variations in W (k). In contrast with mD-NG, D-NG may diverge when the network is random; we refer to a companion journal paper [10] for details. This interesting difference between mD-NG and D-NG may be, in a certain sense, related to the difference between adapt-then-combine and combine-then-adapt methods studied in [11].…”

“…From Theorem 1, we can see that mD-NC achieves faster theoretical rates than mD-NG. Typically, in simulations, mD-NG actually converges faster for practical accuracies, see [10].…”

Convergence Rates for Distributed Stochastic Optimization Over Random Networks

“…We consider standard distributed stochastic gradient methods where at each time step, each node makes a weighted average of its own and its neighbors' solution estimates, and performs a step in the negative direction of its noisy local gradient. The underlying network is allowed to be randomly varying, similarly to, e.g., the models in [4]- [6]. More specifically, the network is modeled through a sequence of independent identically distributed (i.i.d.)…”