Nonconvex Federated Learning on Compact Smooth Submanifolds With Heterogeneous Data

TL;DR

This paper introduces a federated learning algorithm for manifold optimization problems that efficiently handles heterogeneous data, converges to a near-optimal solution, and reduces computational and communication costs.

Contribution

It proposes a novel algorithm combining stochastic Riemannian gradients and manifold projections for federated learning on smooth submanifolds with theoretical convergence guarantees.

Findings

Algorithm converges sub-linearly to a neighborhood of a first-order optimal solution.

Significantly reduces computational overhead compared to existing methods.

Achieves lower communication costs while maintaining accuracy.

Abstract

Many machine learning tasks, such as principal component analysis and low-rank matrix completion, give rise to manifold optimization problems. Although there is a large body of work studying the design and analysis of algorithms for manifold optimization in the centralized setting, there are currently very few works addressing the federated setting. In this paper, we consider nonconvex federated learning over a compact smooth submanifold in the setting of heterogeneous client data. We propose an algorithm that leverages stochastic Riemannian gradients and a manifold projection operator to improve computational efficiency, uses local updates to improve communication efficiency, and avoids client drift. Theoretically, we show that our proposed algorithm converges sub-linearly to a neighborhood of a first-order optimal solution by using a novel analysis that jointly exploits the manifold structure and properties of the loss functions. Numerical experiments demonstrate that our algorithm has significantly smaller computational and communication overhead than existing methods.