Quantized Distributed Nonconvex Optimization Algorithms with Linear Convergence under the Polyak--${\L}$ojasiewicz Condition

TL;DR

This paper introduces quantized distributed algorithms for nonconvex optimization that achieve linear convergence under the Polyak--Łojasiewicz condition, even with low data rates, by combining encoding schemes with gradient tracking.

Contribution

It proposes two novel quantized distributed algorithms that ensure linear convergence under the PL condition without requiring convexity, and demonstrates their efficiency with low data rates.

Findings

Algorithms achieve linear convergence to a global optimum.

Larger quantization levels accelerate convergence.

Low data rates are sufficient for guaranteed convergence.

Abstract

This paper considers distributed optimization for minimizing the average of local nonconvex cost functions, by using local information exchange over undirected communication networks. To reduce the required communication capacity, we introduce an encoder--decoder scheme. By integrating them with distributed gradient tracking and proportional integral algorithms, respectively, we then propose two quantized distributed nonconvex optimization algorithms. Assuming the global cost function satisfies the Polyak--{\L}ojasiewicz condition, which does not require the global cost function to be convex and the global minimizer is not necessarily unique, we show that our proposed algorithms linearly converge to a global optimal point and that larger quantization level leads to faster convergence speed. Moreover, we show that a low data rate is sufficient to guarantee linear convergence when the algorithm parameters are properly chosen. The theoretical results are illustrated by numerical examples.