Distributed second order methods with increasing number of working nodes

TL;DR

This paper extends the idling mechanism, previously used in first-order distributed optimization, to second-order methods, specifically applying it to the Distributed Quasi Newton method, and demonstrates that it maintains convergence properties while reducing computational costs.

Contribution

It introduces the application of the idling mechanism to distributed second-order methods, particularly the DQN, showing theoretical convergence and practical efficiency improvements.

Findings

DQN with idling achieves similar convergence rates as standard DQN.

The method reduces computational costs by allowing nodes to be idle.

Simulation results confirm the theoretical benefits and flexibility of the approach.

Abstract

Recently, an idling mechanism has been introduced in the context of distributed \emph{first order} methods for minimization of a sum of nodes' local convex costs over a generic, connected network. With the idling mechanism, each node $i$, at each iteration $k$, is active -- updates its solution estimate and exchanges messages with its network neighborhood -- with probability $p_k$, and it stays idle with probability $1-p_k$, while the activations are independent both across nodes and across iterations. In this paper, we demonstrate that the idling mechanism can be successfully incorporated in \emph{distributed second order methods} also. Specifically, we apply the idling mechanism to the recently proposed Distributed Quasi Newton method (DQN). We first show theoretically that, when $p_k$ grows to one across iterations in a controlled manner, DQN with idling exhibits very similar theoretical convergence and convergence rates properties as the standard DQN method, thus achieving the same order of convergence rate (R-linear) as the standard DQN, but with significantly cheaper updates. Simulation examples confirm the benefits of incorporating the idling mechanism, demonstrate the method's flexibility with respect to the choice of the $p_k$'s, and compare the proposed idling method with related algorithms from the literature.