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

TL;DR

The paper extends steepest descent to curved spaces by deriving convergence of Riemannian gradient descent under a geodesic Lipschitz condition and demonstrates a concrete $S^2$ example with $f(x,y,z)=z$ to illustrate $O(1/k)$ decay due to manifold geometry. It further develops adaptive momentum in infinite-dimensional Hilbert spaces, proving weak convergence to critical points via Opial's lemma and achieving an accelerated $O(1/k^2)$ rate with $\alpha_k=c/k$, $\beta_k=d/k$. Additionally, it analyzes stochastic steepest descent with non-Gaussian noise, showing $\mathbb{E}[f(x_k)] - f(x^*) = O(1/k^{\gamma-0.5})$ for $0.5<\gamma\le1$, supported by a quadratic example. Together, these results broaden gradient-based optimization to manifold and infinite-dimensional settings under generalized smoothness and heavy-tailed noise, with practical implications for geometry-aware machine learning and variational problems.

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.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Example 1
• Remark 1
• Lemma 1
• Theorem 2
• proof