# Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking

arXiv: https://arxiv.org/abs/2506.07351 · HTML: https://mlsanity.com/paper/2506.07351/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 46 references that are in the map.
- First posted on arXiv 9 Jun 2025.

## Builds on (most cited first, 30 of 30)

- [EXTRA: An Exact First-Order Algorithm for Decentralized Consensus Optimization](https://mlsanity.com/paper/1404.6264/) (2014)
- [On the Convergence of Decentralized Gradient Descent](https://mlsanity.com/paper/1310.7063/) (2013)
- [NEXT: In-Network Nonconvex Optimization](https://mlsanity.com/paper/1602.00591/) (2016)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Orthogonal Weight Normalization: Solution to Optimization over Multiple Dependent Stiefel Manifolds in Deep Neural Networks](https://mlsanity.com/paper/1709.06079/) (2017)
- [On orthogonality and learning recurrent networks with long term dependencies](https://mlsanity.com/paper/1702.00071/) (2017)
- [On the Linear Convergence of the ADMM in Decentralized Consensus Optimization](https://mlsanity.com/paper/1307.5561/) (2013)
- [Exact Diffusion for Distributed Optimization and Learning --- Part I: Algorithm Development](https://mlsanity.com/paper/1702.05122/) (2017)
- [A Linearly Convergent Algorithm for Decentralized Optimization: Sending Less Bits for Free!](https://mlsanity.com/paper/2011.01697/) (2020)
- [An Exact Quantized Decentralized Gradient Descent Algorithm](https://mlsanity.com/paper/1806.11536/) (2018)
- [Fast and accurate optimization on the orthogonal manifold without retraction](https://mlsanity.com/paper/2102.07432/) (2021)
- [Decentralized Riemannian Gradient Descent on the Stiefel Manifold](https://mlsanity.com/paper/2102.07091/) (2021)
- [Quantized Decentralized Stochastic Learning over Directed Graphs](https://mlsanity.com/paper/2002.09964/) (2020)
- [Linear Convergent Decentralized Optimization with Compression](https://mlsanity.com/paper/2007.00232/) (2020)
- [Decentralized Proximal Gradient Algorithms with Linear Convergence Rates](https://mlsanity.com/paper/1909.06479/) (2019)
- [DeEPCA: Decentralized Exact PCA with Linear Convergence Rate](https://mlsanity.com/paper/2102.03990/) (2021)
- [Infeasible Deterministic, Stochastic, and Variance-Reduction Algorithms for Optimization under Orthogonality Constraints](https://mlsanity.com/paper/2303.16510/) (2023)
- [Cloud K-SVD: A Collaborative Dictionary Learning Algorithm for Big, Distributed Data](https://mlsanity.com/paper/1412.7839/) (2014)
- [Distributed Linearized Alternating Direction Method of Multipliers for Composite Convex Consensus Optimization](https://mlsanity.com/paper/1512.08122/) (2015)
- [Riemannian conjugate gradient methods: General framework and specific algorithms with convergence analyses](https://mlsanity.com/paper/2112.02572/) (2021)
- [Decentralized Optimization Over the Stiefel Manifold by an Approximate Augmented Lagrangian Function](https://mlsanity.com/paper/2112.14949/) (2021)
- [Decentralized projected Riemannian gradient method for smooth optimization on compact submanifolds](https://mlsanity.com/paper/2304.08241/) (2023)
- [On the Local Linear Rate of Consensus on the Stiefel Manifold](https://mlsanity.com/paper/2101.09346/) (2021)
- [Quantized Distributed Gradient Tracking Algorithm with Linear Convergence in Directed Networks](https://mlsanity.com/paper/2104.03649/) (2021)
- [Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold](https://mlsanity.com/paper/2308.10547/) (2023)
- [Design and Analysis of Distributed Averaging with Quantized Communication](https://mlsanity.com/paper/1403.4696/) (2014)
- [On Centralized and Distributed Mirror Descent: Convergence Analysis Using Quadratic Constraints](https://mlsanity.com/paper/2105.14385/) (2021)
- [Decentralized Riemannian natural gradient methods with Kronecker-product approximations](https://mlsanity.com/paper/2303.09611/) (2023)
- [Distributed Riemannian Stochastic Gradient Tracking Algorithm on the Stiefel Manifold](https://mlsanity.com/paper/2405.16900/) (2024)
- [Improving the communication in decentralized manifold optimization through single-step consensus and compression](https://mlsanity.com/paper/2407.08904/) (2024)

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

- [Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions](https://mlsanity.com/paper/2607.20316/) (2026)

## Abstract (arXiv)

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 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.