Generalized Steepest Descent Methods on Riemannian Manifolds and Hilbert Spaces: Convergence Analysis and Stochastic Extensions

TL;DR

This paper extends steepest descent methods to Riemannian manifolds and Hilbert spaces, analyzing convergence and stochastic variants, with practical implications for optimization in complex geometric settings.

Contribution

It introduces a generalized steepest descent framework on Riemannian manifolds and Hilbert spaces, including convergence analysis and stochastic extensions with non-Gaussian noise.

Findings

Convergence of steepest descent on Riemannian manifolds under generalized smoothness.

Achieved $O(1/k^2)$ convergence rate with adaptive momentum in Hilbert spaces.

Guaranteed convergence of stochastic steepest descent with non-Gaussian noise.

Abstract

Optimization techniques are at the core of many scientific and engineering disciplines. The steepest descent methods play a foundational role in this area. In this paper we studied a generalized steepest descent method on Riemannian manifolds, leveraging the geometric structure of manifolds to extend optimization techniques beyond Euclidean spaces. The convergence analysis under generalized smoothness conditions of the steepest descent method is studied along with an illustrative example. We also explore adaptive steepest descent with momentum in infinite-dimensional Hilbert spaces, focusing on the interplay of step size adaptation, momentum decay, and weak convergence properties. Also, we arrived at a convergence accuracy of $O(\frac{1}{k^2})$. Finally, studied some stochastic steepest descent under non-Gaussian noise, where bounded higher-order moments replace Gaussian assumptions, leading to a guaranteed convergence, which is illustrated by an example.