Distributed Optimization over Time-varying Graphs with Imperfect Sharing of Information

Abstract

We study strongly convex distributed optimization problems where a set of agents are interested in solving a separable optimization problem collaboratively. In this paper, we propose and study a two time-scale decentralized gradient descent algorithm for a board class of lossy sharing of information over time-varying graphs. One time-scale fades out the (lossy) incoming information from neighboring agents, and one time-scale regulates the local loss functions' gradients. For strongly convex loss functions, with a proper choice of step-sizes, we show that the agents' estimates converge to the global optimal state at a rate of O T −1/2 . Another important contribution of this work is to provide novel tools to deal with diminishing average weights over time-varying graphs.

Almost Sure Convergence of Distributed Optimization with Imperfect Information Sharing

“…In this work, the communication between the agents is assumed to be imperfect. We adapt the general framework of the noisy sharing of information introduced in [18] as described below. Given the states x i (t) of agents i ∈ [n] at time t, we assume that each agent has access to an imperfect weighted average of its in-neighbours states, denoted by xi (t) given by xi…”

“…Regarding the local cost function, we assume that agent i ∈ [n] has access to the gradient ∇f i (x i (t)) of its local cost function f i (•) at each local decision variable x i (t) at time t. Inspired by [18], we present the update rule in this work as…”

“…Consequently, each agent receives an imperfect estimate of the intended messages from the neighboring agents. Various gradient descent algorithms with imperfect information sharing have been proposed [16][17][18], showing the convergence rates in L 2 sense for the diverse set of problem setups.…”

“…In this paper, we study the distributed convex optimization problems over time-varying networks with imperfect information sharing. We consider the two-time-scale gradient descent method studied in [18,22] to solve the optimization problem. One time-scale adjusts the (imperfect) incoming information from the neighboring agents, and one time-scale controls the local cost functions' gradients.…”

DIMIX: DIminishing MIXing for Sloppy Agents

“…In particular, this is used in the context of distributed averaging/consensus problem [12,23,24], where each node has an initial numerical value and aims at evaluating the average of all initial values using exchange of quantized information, over a fixed or a time-varying network. In the context of distributed optimization, various compression approaches have been introduced to mitigate the communication overhead [25][26][27][28][29].…”