Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking

TL;DR

This paper introduces Q-RGT, a quantized gradient tracking algorithm for decentralized optimization on compact submanifolds, achieving efficient convergence with reduced communication overhead and handling quantization noise.

Contribution

It is the first to establish an $ ext{O}(1/K)$ convergence rate for quantized Riemannian decentralized optimization, surpassing previous limitations.

Findings

Q-RGT matches non-quantized methods in convergence speed.

The algorithm reduces communication and computational costs.

Explicit lower bounds on consensus with quantization levels are derived.

Abstract

This paper considers the problem of decentralized optimization on compact submanifolds, where a finite sum of smooth (possibly non-convex) local functions is minimized by $n$ agents forming an undirected and connected graph. However, the efficiency of distributed optimization is often hindered by communication bottlenecks. To mitigate this, we propose the Quantized Riemannian Gradient Tracking (Q-RGT) algorithm, where agents update their local variables using quantized gradients. The introduction of quantization noise allows our algorithm to bypass the constraints of the accurate Riemannian projection operator (such as retraction), further improving iterative efficiency. To the best of our knowledge, this is the first algorithm to achieve an $\mathcal{O}(1/K)$ convergence rate in the presence of quantization, matching the convergence rate of methods without quantization. Additionally, we explicitly derive lower bounds on decentralized consensus associated with a function of quantization levels. Numerical experiments demonstrate that Q-RGT performs comparably to non-quantized methods while reducing communication bottlenecks and computational overhead.