On the Oracle Complexity of a Riemannian Inexact Augmented Lagrangian Method for Riemannian Nonsmooth Composite Problems

TL;DR

This paper establishes the first known oracle complexity bounds for a Riemannian inexact augmented Lagrangian method applied to nonsmooth composite problems, showing it can find an approximate stationary point efficiently.

Contribution

It provides the first oracle complexity analysis for a Riemannian inexact augmented Lagrangian method with classical dual updates, achieving optimal complexity bounds.

Findings

RiAL method finds an $ ext{ε}$-stationary point with $ ext{O}( ext{ε}^{-3})$ oracle calls.

Using classical dual stepsize is key to the method's high efficiency.

Numerical results confirm the importance of the dual stepsize for performance.

Abstract

In this paper, we establish for the first time the oracle complexity of a Riemannian inexact augmented Lagrangian (RiAL) method with the classical dual update for solving a class of Riemannian nonsmooth composite problems. By using the Riemannian gradient descent method with a specified stopping criterion for solving the inner subproblem, we show that the RiAL method can find an $\varepsilon$-stationary point of the considered problem with $\mathcal{O}(\varepsilon^{-3})$ calls to the first-order oracle. This achieves the best oracle complexity known to date. Numerical results demonstrate that the use of the classical dual stepsize is crucial to the high efficiency of the RiAL method.