提出量化黎曼梯度跟踪算法，解决分布式优化通信瓶颈问题。
Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking
- 在紧凑子流形上用量化梯度更新，避开精确投影约束。
- 首次实现量化下O(1/K)收敛率，与无量化方法持平。
- 适合资源受限的分布式系统，如边缘计算场景。
本文研究在紧凑子流形上的去中心化优化问题，即由n个形成无向连通图的代理共同最小化一组光滑（可能非凸）局部函数之和。然而，分布式优化常受通信瓶颈制约。为此，我们提出量化黎曼梯度跟踪（Q-RGT）算法，代理使用量化梯度更新本地变量。量化噪声的引入使算法可规避准确黎曼投影算子（如重构）的限制，进一步提升迭代效率。据我们所知，这是首个在量化存在下达到O(1/K)收敛速率的算法，与无量化方法收敛速度一致。此外，我们显式推导了与量化层级相关联的去中心化一致性下界。数值实验表明，Q-RGT性能可媲美非量化方法，同时显著降低通信瓶颈与计算开销。
原文摘要 · Abstract (English)
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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