# Improving the communication in decentralized manifold optimization through single-step consensus and compression

arXiv: https://arxiv.org/abs/2407.08904 · HTML: https://mlsanity.com/paper/2407.08904/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2024.

## Built on by (most cited first, 1 of 1)

- [Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking](https://mlsanity.com/paper/2506.07351/) (2025)

## Abstract (arXiv)

We are concerned with decentralized optimization over a compact submanifold, where the loss functions of local datasets are defined by their respective local datasets. A key challenge in decentralized optimization is mitigating the communication bottleneck, which primarily involves two strategies: achieving consensus and applying communication compression. Existing projection/retraction-type algorithms rely on multi-step consensus to attain both consensus and optimality. Due to the nonconvex nature of the manifold constraint, it remains an open question whether the requirement for multi-step consensus can be reduced to single-step consensus. We address this question by carefully elaborating on the smoothness structure and the asymptotic 1-Lipschitz continuity associated with the manifold constraint. Furthermore, we integrate these insights with a communication compression strategy to propose a communication-efficient gradient algorithm for decentralized manifold optimization problems, significantly reducing per-iteration communication costs. Additionally, we establish an iteration complexity of O(ε-1) to find an ε-stationary point, which matches the complexity in the Euclidean setting. Numerical experiments demonstrate the efficiency of the proposed method in comparison to state-of-the-art approaches.