Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties

TL;DR

This paper introduces a globally-accelerated Riemannian optimization method that handles constraints via a geometric penalty, achieving near-optimal convergence rates without assuming iterates stay in a pre-defined set.

Contribution

It develops a novel accelerated Riemannian inexact proximal point algorithm that ensures global convergence and handles constraints, addressing an open problem in the field.

Findings

Achieves convergence rates similar to Nesterov's accelerated gradient descent with geometric penalties.

Enables enforcement of constraints within a compact set during optimization.

Introduces an inexact proximal step implementation using first-order methods in Riemannian balls.

Abstract

We propose a globally-accelerated, first-order method for the optimization of smooth and (strongly or not) geodesically-convex functions in a wide class of Hadamard manifolds. We achieve the same convergence rates as Nesterov's accelerated gradient descent, up to a multiplicative geometric penalty and log factors. Crucially, we can enforce our method to stay within a compact set we define. Prior fully accelerated works \emph{resort to assuming} that the iterates of their algorithms stay in some pre-specified compact set, except for two previous methods of limited applicability. For our manifolds, this solves the open question in [KY22] about obtaining global general acceleration without iterates assumptively staying in the feasible set. In our solution, we design an accelerated Riemannian inexact proximal point algorithm, which is a result that was unknown even with exact access to the proximal operator, and is of independent interest. For smooth functions, we show we can implement the prox step inexactly with first-order methods in Riemannian balls of certain diameter that is enough for global accelerated optimization.