Convergence and Complexity Guarantee for Inexact First-order Riemannian Optimization Algorithms

TL;DR

This paper provides convergence guarantees and complexity analysis for inexact Riemannian gradient descent algorithms in nonconvex constrained optimization, demonstrating their effectiveness through theoretical results and numerical experiments.

Contribution

It introduces a general framework (tBMM) for analyzing inexact Riemannian optimization algorithms and establishes convergence and complexity bounds under broad conditions.

Findings

tBMM converges to an $psilon$-stationary point within $O(psilon^{-2})$ iterations.

The analysis applies to classical Riemannian algorithms including inexact RGD and proximal methods.

Numerical experiments show tBMM outperforms existing methods on various Riemannian constrained problems.

Abstract

We analyze inexact Riemannian gradient descent (RGD) where Riemannian gradients and retractions are inexactly (and cheaply) computed. Our focus is on understanding when inexact RGD converges and what is the complexity in the general nonconvex and constrained setting. We answer these questions in a general framework of tangential Block Majorization-Minimization (tBMM). We establish that tBMM converges to an $\epsilon$-stationary point within $O(\epsilon^{-2})$ iterations. Under a mild assumption, the results still hold when the subproblem is solved inexactly in each iteration provided the total optimality gap is bounded. Our general analysis applies to a wide range of classical algorithms with Riemannian constraints including inexact RGD and proximal gradient method on Stiefel manifolds. We numerically validate that tBMM shows improved performance over existing methods when applied to various problems, including nonnegative tensor decomposition with Riemannian constraints, regularized nonnegative matrix factorization, and low-rank matrix recovery problems.