# Federated Learning on Riemannian Manifolds: A Gradient-Free Projection-Based Approach

arXiv: https://arxiv.org/abs/2507.22855 · HTML: https://mlsanity.com/paper/2507.22855/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 48 references that are in the map.
- First posted on arXiv 30 Jul 2025.

## Builds on (most cited first, 30 of 32)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Federated Learning: Strategies for Improving Communication Efficiency](https://mlsanity.com/paper/1610.05492/) (2016)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [ZOO: Zeroth Order Optimization based Black-box Attacks to Deep Neural Networks without Training Substitute Models](https://mlsanity.com/paper/1708.03999/) (2017)
- [Local SGD Converges Fast and Communicates Little](https://mlsanity.com/paper/1805.09767/) (2018)
- [Can Decentralized Algorithms Outperform Centralized Algorithms? A Case Study for Decentralized Parallel Stochastic Gradient Descent](https://mlsanity.com/paper/1705.09056/) (2017)
- [Mime: Mimicking Centralized Stochastic Algorithms in Federated Learning](https://mlsanity.com/paper/2008.03606/) (2020)
- [D2: Decentralized Training over Decentralized Data](https://mlsanity.com/paper/1803.07068/) (2018)
- [AutoZOOM: Autoencoder-based Zeroth Order Optimization Method for Attacking Black-box Neural Networks](https://mlsanity.com/paper/1805.11770/) (2018)
- [Edge Artificial Intelligence for 6G: Vision, Enabling Technologies, and Applications](https://mlsanity.com/paper/2111.12444/) (2021)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Communication-Efficient Stochastic Zeroth-Order Optimization for Federated Learning](https://mlsanity.com/paper/2201.09531/) (2022)
- [A Unified Gradient Regularization Family for Adversarial Examples](https://mlsanity.com/paper/1511.06385/) (2015)
- [STEM: A Stochastic Two-Sided Momentum Algorithm Achieving Near-Optimal Sample and Communication Complexities for Federated Learning](https://mlsanity.com/paper/2106.10435/) (2021)
- [Decentralized Riemannian Gradient Descent on the Stiefel Manifold](https://mlsanity.com/paper/2102.07091/) (2021)
- [Zeroth-Order Algorithms for Stochastic Distributed Nonconvex Optimization](https://mlsanity.com/paper/2106.02958/) (2021)
- [Federated Learning on Riemannian Manifolds](https://mlsanity.com/paper/2206.05668/) (2022)
- [Tracking and regret bounds for online zeroth-order Euclidean and Riemannian optimisation](https://mlsanity.com/paper/2010.00211/) (2020)
- [Dissolving Constraints for Riemannian Optimization](https://mlsanity.com/paper/2203.10319/) (2022)
- [Decentralized projected Riemannian gradient method for smooth optimization on compact submanifolds](https://mlsanity.com/paper/2304.08241/) (2023)
- [Composite federated learning with heterogeneous data](https://mlsanity.com/paper/2309.01795/) (2023)
- [Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms](https://mlsanity.com/paper/2309.14506/) (2023)
- [From the Greene--Wu Convolution to Gradient Estimation over Riemannian Manifolds](https://mlsanity.com/paper/2108.07406/) (2021)
- [Problem-Parameter-Free Decentralized Nonconvex Stochastic Optimization](https://mlsanity.com/paper/2402.08821/) (2024)
- [Federated Learning on Riemannian Manifolds with Differential Privacy](https://mlsanity.com/paper/2404.10029/) (2024)
- [Riemannian Accelerated Zeroth-order Algorithm: Improved Robustness and Lower Query Complexity](https://mlsanity.com/paper/2405.05713/) (2024)
- [Nonconvex Federated Learning on Compact Smooth Submanifolds With Heterogeneous Data](https://mlsanity.com/paper/2406.08465/) (2024)
- [Stochastic optimization over proximally smooth sets](https://mlsanity.com/paper/2002.06309/) (2020)

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

- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)

## Abstract (arXiv)

Federated learning (FL) has emerged as a powerful paradigm for collaborative model training across distributed clients while preserving data privacy. However, existing FL algorithms predominantly focus on unconstrained optimization problems with exact gradient information, limiting its applicability in scenarios where only noisy function evaluations are accessible or where model parameters are constrained. To address these challenges, we propose a novel zeroth-order projection-based algorithm on Riemannian manifolds for FL. By leveraging the projection operator, we introduce a computationally efficient zeroth-order Riemannian gradient estimator. Unlike existing estimators, ours requires only a simple Euclidean random perturbation, eliminating the need to sample random vectors in the tangent space, thus reducing computational cost. Theoretically, we first prove the approximation properties of the estimator and then establish the sublinear convergence of the proposed algorithm, matching the rate of its first-order counterpart. Numerically, we first assess the efficiency of our estimator using kernel principal component analysis. Furthermore, we apply the proposed algorithm to two real-world scenarios: zeroth-order attacks on deep neural networks and low-rank neural network training to validate the theoretical findings.