New Riemannian Lyapunov Optimizers Offer Control-Theoretic Framework for Machine Learning

Global: New Riemannian Lyapunov Optimizers Offer Control-Theoretic Framework for Machine Learning

A team of machine learning researchers has introduced a novel family of optimization algorithms called Riemannian Lyapunov Optimizers (RLOs), according to a preprint posted on arXiv on January 2026. The work proposes a geometric framework that unifies classic optimizers and derives them systematically from control theory. The authors aim to improve stability and performance of large‑scale training.

Control-Theoretic Foundations

The paper reinterprets optimization as an extended‑state discrete‑time controlled dynamical system defined on a Riemannian parameter manifold. By treating the optimizer as a controller, the authors derive update rules from first principles rather than relying on heuristic adjustments.

Invariant Manifold Structure

Central to the approach is the identification of a Normally Attracting Invariant Manifold (NAIM), which organizes training dynamics into two distinct stages: an initial rapid alignment of a speed state to a target graph, followed by a controlled evolution that remains confined to the manifold.

Theoretical Guarantees

To certify convergence, the researchers construct a strict Lyapunov function that decreases along trajectories on the NAIM. This function provides a formal guarantee that the optimizer will converge to the desired manifold under the stated assumptions.

Optimizer Generator and Design

Leveraging the control‑theoretic formulation, the authors present an “optimizer generator” capable of reproducing classic algorithms such as SGD and Adam, while also enabling the systematic design of new RLOs with provable stability properties.

Experimental Results

Empirical validation includes geometric diagnostics and benchmark tests on large‑scale datasets. According to the abstract, the RLO‑based methods achieve state‑of‑the‑art performance compared with existing optimizers.

Implications for Machine Learning

The authors suggest that grounding optimizer design in control theory could bridge the gap between theoretical analysis and practical deployment, offering a unified language for future research in stable and effective optimization.

This report is based on information from arXiv, licensed under Academic Preprint / Open Access. Based on the abstract of the research paper. Full text available via ArXiv.