arXiv Machine Learning July 9, 2026 By Zavier Li

Optimization Geometrodynamics: A Framework for Dynamic Geometric Optimization

Read the original on arXiv Machine Learning →

arXiv:2607. 06723v1 Announce Type: cross Abstract: Most gradient-based optimization methods move parameters through a fixed background geometry, even when their internal states implicitly define changing notions of length, curvature, and preconditioning.

Summary generated by The Flow from the publisher's feed. The full article lives at arXiv Machine Learning.

benchmarks

arXiv Machine Learning

Jul 9

Geometric--Nongeometric Optimizer Calculus: A Modular Language for Reachable Gradient Methods

arXiv:2607. 07206v1 Announce Type: new Abstract: Adaptive optimizers mix several mechanisms: a metric or preconditioner maps gradients to descent directions, while estimation, memory, step-size control, constraints, stochasticity, target modification, and discretization determine which directions are available and how they are used.

By Zavier Li

benchmarks

arXiv Machine Learning

Jul 9

Restricted Dynamic Geometric Complexity: Certificates for Structured Preconditioning

arXiv:2607. 07204v1 Announce Type: cross Abstract: Optimization geometrodynamics views optimizer state as evolving geometry.

By Zavier Li

benchmarks

arXiv Machine Learning

Jul 14

Restricted Dynamic Geometric Complexity: Path-Space Reduction and M\"obius--Jacobi Response

arXiv:2607. 07204v2 Announce Type: replace-cross Abstract: Structured preconditioners restrict optimization to a small family of positive metrics, but endpoint condition-number reachability does not measure the geometric effort required to reach a useful metric.

By Zavier Li

arXiv Machine Learning

Jul 16

Smooth Quasar-Convex Optimization with Constraints

arXiv:2510. 01943v3 Announce Type: replace-cross Abstract: Quasar-convex functions form a broad nonconvex class with applications to linear dynamical systems, generalized linear models, and Riemannian optimization, among others.

By David Mart\'inez-Rubio

arXiv Machine Learning

Jul 27

Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers

arXiv:2607. 22004v1 Announce Type: new Abstract: Energy natural gradient descent (ENGD) aligns parameter updates with the curvature of an underlying function-space energy, but existing formulations assume an unconstrained Euclidean parameter domain.

By Zhangyong Liang, Huanhuan Gao

benchmarks

arXiv Machine Learning

Jul 14

Optimization Geometrodynamics: Variational Reduction and Interaction Curvature

arXiv:2607. 06723v2 Announce Type: replace-cross Abstract: Adaptive optimizers carry hidden states that change how visible gradients become parameter motion.

By Zavier Li

Related stories

Geometric--Nongeometric Optimizer Calculus: A Modular Language for Reachable Gradient Methods

Restricted Dynamic Geometric Complexity: Certificates for Structured Preconditioning

Restricted Dynamic Geometric Complexity: Path-Space Reduction and M\"obius--Jacobi Response

Smooth Quasar-Convex Optimization with Constraints

Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers

Optimization Geometrodynamics: Variational Reduction and Interaction Curvature