# naim

Published articles for naim.

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## Riemannian Lyapunov Framework: Optimization as Closed Loop Control on Riemannian Manifolds

DevFeed: [Riemannian Lyapunov Framework: Optimization as Closed Loop Control on Riemannian Manifolds](<https://devfeed.tech/articles/riemannian-lyapunov-framework-optimization-as-closed-loop-control-on-riemannian-manifolds-108463.md>)

Original publisher: [Read original article](<https://arxiv.org/abs/2601.22284>)

Author: Yixuan Wang, Omkar Sudhir Patil, Warren E. Dixon

Published: 2026-10-30T04:00:00Z

Content type: article

Language: en

Sources: [arXiv: Machine Learning](<https://devfeed.tech/sources/arxiv-machine-learning.md>)

Topics: [Operations research and optimization](<https://devfeed.tech/topics/operations-research-and-optimization.md>), [Algorithms](<https://devfeed.tech/topics/algorithms.md>)

Tags: [alignment](<https://devfeed.tech/tags/alignment.md>), [benchmarks](<https://devfeed.tech/tags/benchmarks.md>), [classification](<https://devfeed.tech/tags/classification.md>), [control](<https://devfeed.tech/tags/control.md>), [controller](<https://devfeed.tech/tags/controller.md>), [cs-lg](<https://devfeed.tech/tags/cs-lg.md>), [framework](<https://devfeed.tech/tags/framework.md>), [guarantees](<https://devfeed.tech/tags/guarantees.md>), [image-classification](<https://devfeed.tech/tags/image-classification.md>), [invariant](<https://devfeed.tech/tags/invariant.md>), [language](<https://devfeed.tech/tags/language.md>), [naim](<https://devfeed.tech/tags/naim.md>), [optimization](<https://devfeed.tech/tags/optimization.md>), [parameter](<https://devfeed.tech/tags/parameter.md>), [state](<https://devfeed.tech/tags/state.md>), [synthesis](<https://devfeed.tech/tags/synthesis.md>), [template](<https://devfeed.tech/tags/template.md>), [theoretical](<https://devfeed.tech/tags/theoretical.md>), [under](<https://devfeed.tech/tags/under.md>), [using](<https://devfeed.tech/tags/using.md>), [velocity](<https://devfeed.tech/tags/velocity.md>)

### AI overview

The Riemannian Lyapunov Framework treats first-order optimization as closed-loop control on a Riemannian parameter manifold. It represents an optimizer with a metric, internal-state transition, direction-field generator, and lifting parameter, recovering methods such as SGD, Adam, AdamW, Lion, and Shampoo. Its Normally Attracting Invariant Manifold family describes fast contraction of velocity state followed by slower descent. Under descent alignment and bounded stochastic disturbances, Riemannian backstepping yields a strict Lyapunov function and proves uniform ultimate boundedness; with an additional Polyak-Łojasiewicz condition, convergence is linear to a bounded neighborhood shaped by target-direction drift. Three instantiations are reported to match predictions across geometric diagnostics, image classification, and language modeling.

### Source excerpt

arXiv:2601.22284v2 Announce Type: replace Abstract: We introduce the Riemannian Lyapunov Framework (RLF), a unified framework that represents first order optimization as a closed loop controlled dynamical system on a Riemannian parameter manifold. Within this framework, optimizers are specified by a quadruple $(M, g, \Phi, \eta)$ consisting of a Riemannian metric, an internal state transition map, a direction field generator, and a lifting parameter. Classical and modern methods including SGD, Adam, AdamW, Lion and Shampoo are recovered as specific instantiations of this quadruple. The central structural object of the framework is a time indexed family of target submanifolds in the extended state space of parameters, velocities, and internal memory, which we call the Normally Attracting Invariant Manifold (NAIM) family. These submanifolds organize training dynamics into two timescales: a fast contraction of the velocity state onto the current target submanifold, followed by a slow descent along it. Using Riemannian backstepping, we construct a strict Lyapunov function on the extended state space and prove Uniform Ultimate Boundedness (UUB) of the optimization trajectory under a descent alignment assumption and bounded stochastic disturbances. Under the additional Polyak-Lojasiewicz (PL) condition, the Lyapunov function converges linearly to a bounded neighborhood of the optimum, defined as the drift of the target direction field between consecutive iterates after parallel transport. To validate the framework's generativity, we derive three concrete instantiations of the quadruple, each exercising a distinct design axis, and show on geometric diagnostics, image classification benchmarks, and language modeling that their empirical behavior matches the theoretical predictions. The framework casts optimizer design as controller synthesis, complementing heuristic perspectives by providing a constructive template with provable guarantees.