Global Convergence and Error Propagation in Neural Gradient Flows: A Riemannian Optimization Framework
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918526319591424 |
|---|---|
| author | Zheng, Shixin Wang, Yiwei Yang, Haizhao |
| author_facet | Zheng, Shixin Wang, Yiwei Yang, Haizhao |
| contents | We develop a geometric convergence theory for neural-network optimization within the minimizing movement scheme (MMS) framework. Reformulating each neural MMS step as a minimization over the set of increments in a Hilbert space, we show that under a $C^2$ network with locally non-degenerate Jacobian this increment set is a boundaryless smooth embedded submanifold, on which a natural preconditioned (Gauss--Newton-type) gradient flow in parameter space induces exactly the Riemannian gradient flow. Under a strict interior-localization condition and an explicit data condition, the reached sublevel set is geodesically convex and the subproblem objective is geodesically strongly convex on it; both the continuous Riemannian gradient flow and its discrete companion via the exponential map converge linearly to the unique subproblem minimizer. Propagating finite-time inner-solver inexactness and neural-approximation error through the MMS iterations yields a uniform function-space tracking bound and an explicit trajectory budget, so the inexact neural iterates converge to an $O(δ)$-neighborhood of the global minimum. Numerical experiments on nonlinear regression and a small-scale latent-diffusion testbed indicate that the Gauss--Newton-type inner solver achieves smaller trajectory errors with substantially fewer inner iterations than first-order baselines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27779 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global Convergence and Error Propagation in Neural Gradient Flows: A Riemannian Optimization Framework Zheng, Shixin Wang, Yiwei Yang, Haizhao Optimization and Control Numerical Analysis We develop a geometric convergence theory for neural-network optimization within the minimizing movement scheme (MMS) framework. Reformulating each neural MMS step as a minimization over the set of increments in a Hilbert space, we show that under a $C^2$ network with locally non-degenerate Jacobian this increment set is a boundaryless smooth embedded submanifold, on which a natural preconditioned (Gauss--Newton-type) gradient flow in parameter space induces exactly the Riemannian gradient flow. Under a strict interior-localization condition and an explicit data condition, the reached sublevel set is geodesically convex and the subproblem objective is geodesically strongly convex on it; both the continuous Riemannian gradient flow and its discrete companion via the exponential map converge linearly to the unique subproblem minimizer. Propagating finite-time inner-solver inexactness and neural-approximation error through the MMS iterations yields a uniform function-space tracking bound and an explicit trajectory budget, so the inexact neural iterates converge to an $O(δ)$-neighborhood of the global minimum. Numerical experiments on nonlinear regression and a small-scale latent-diffusion testbed indicate that the Gauss--Newton-type inner solver achieves smaller trajectory errors with substantially fewer inner iterations than first-order baselines. |
| title | Global Convergence and Error Propagation in Neural Gradient Flows: A Riemannian Optimization Framework |
| topic | Optimization and Control Numerical Analysis |
| url | https://arxiv.org/abs/2605.27779 |