Efficient Optimization with Orthogonality Constraint: a Randomized Riemannian Submanifold Method

TL;DR

This paper introduces a scalable Riemannian optimization method for orthogonality-constrained problems by using random submanifold updates, reducing computational costs and maintaining convergence guarantees.

Contribution

It proposes a novel randomized submanifold approach for Riemannian optimization, improving scalability for large-scale orthogonality-constrained problems with theoretical convergence analysis.

Findings

Reduces per-iteration complexity significantly.

Provides convergence guarantees for nonconvex and stochastic settings.

Demonstrates effectiveness across various problems in experiments.

Abstract

Optimization with orthogonality constraints frequently arises in various fields such as machine learning. Riemannian optimization offers a powerful framework for solving these problems by equipping the constraint set with a Riemannian manifold structure and performing optimization intrinsically on the manifold. This approach typically involves computing a search direction in the tangent space and updating variables via a retraction operation. However, as the size of the variables increases, the computational cost of the retraction can become prohibitively high, limiting the applicability of Riemannian optimization to large-scale problems. To address this challenge and enhance scalability, we propose a novel approach that restricts each update on a random submanifold, thereby significantly reducing the per-iteration complexity. We introduce two sampling strategies for selecting the random submanifolds and theoretically analyze the convergence of the proposed methods. We provide convergence results for general nonconvex functions and functions that satisfy Riemannian Polyak-Lojasiewicz condition as well as for stochastic optimization settings. Additionally, we demonstrate how our approach can be generalized to quotient manifolds derived from the orthogonal manifold. Extensive experiments verify the benefits of the proposed method, across a wide variety of problems.