Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking

• URL: http://arxiv.org/abs/2506.07351v1
• Date: Mon, 09 Jun 2025 01:57:25 GMT
• Title: Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking
• Authors: Jun Chen, Lina Liu, Tianyi Zhu, Yong Liu, Guang Dai, Yunliang Jiang, Ivor W. Tsang,
• Abstract summary: This paper considers the problem of decentralized optimization on compact submanifolds.<n>We propose an algorithm where agents update variables using quantized variables.<n>To the best of our knowledge, this is the first algorithm to achieve an $mathcalO (1/K)$ convergence rate in the presence of quantization.
• Score: 45.147301546565316
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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