Distributed Optimization with Quantized Gradient Descent

TL;DR

This paper introduces a distributed quantized gradient descent algorithm for networked optimization with limited bandwidth, combining gradient steps with finite-time quantized consensus to achieve convergence.

Contribution

It proposes a novel method that integrates quantized consensus with gradient descent for distributed optimization over directed graphs with limited communication.

Findings

Algorithm converges to a neighborhood of the optimal solution

Achieves linear convergence rate asymptotically

Effectively handles quantized communication constraints

Abstract

In this paper, we consider the unconstrained distributed optimization problem, in which the exchange of information in the network is captured by a directed graph topology, thus, nodes can only communicate with their neighbors. Additionally, in our problem, the communication channels among the nodes have limited bandwidth. In order to alleviate this limitation, quantized messages should be exchanged among the nodes. For solving this distributed optimization problem, we combine a gradient descent method with a distributed quantized consensus algorithm (which requires the nodes to exchange quantized messages and converges in a finite number of steps). Specifically, at every optimization step, each node (i) performs a gradient descent step (i.e., subtracts the scaled gradient from its current estimate), and (ii) performs a finite-time calculation of the quantized average of every node's estimate in the network. As a consequence, this algorithm approximately mimics the centralized gradient descent algorithm. We show that our algorithm asymptotically converges to a neighborhood of the optimal solution with linear convergence rate. The performance of the proposed algorithm is demonstrated via simple illustrative examples.