Natural gradient descent with momentum
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915941760106496 |
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| author | Nouy, Anthony Somacal, Agustín |
| author_facet | Nouy, Anthony Somacal, Agustín |
| contents | We consider the problem of approximating a function by an element of a nonlinear manifold which admits a differentiable parametrization, typical examples being neural networks with differentiable activation functions or tensor networks. Natural gradient descent (NGD) for the optimization of a loss function can be seen as a preconditioned gradient descent where updates in the parameter space are driven by a functional perspective. In a spirit similar to Newton's method, a NGD step uses, instead of the Hessian, the Gram matrix of the generating system of the tangent space to the approximation manifold at the current iterate, with respect to a suitable metric. This corresponds to a locally optimal update in function space, following a projected gradient onto the tangent space to the manifold. Still, both gradient and natural gradient descent methods get stuck in local minima. Furthermore, when the model class is a nonlinear manifold or the loss function is not ideally conditioned (e.g., the KL-divergence for density estimation, or a norm of the residual of a partial differential equation in physics informed learning), even the natural gradient might yield non-optimal directions at each step. This work introduces a natural version of classical inertial dynamic methods like Heavy-Ball or Nesterov and show how it can improve the learning process when working with nonlinear model classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15554 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Natural gradient descent with momentum Nouy, Anthony Somacal, Agustín Machine Learning Artificial Intelligence Numerical Analysis Optimization and Control 68T05 G.1.6; G.1.2; G.1.3 We consider the problem of approximating a function by an element of a nonlinear manifold which admits a differentiable parametrization, typical examples being neural networks with differentiable activation functions or tensor networks. Natural gradient descent (NGD) for the optimization of a loss function can be seen as a preconditioned gradient descent where updates in the parameter space are driven by a functional perspective. In a spirit similar to Newton's method, a NGD step uses, instead of the Hessian, the Gram matrix of the generating system of the tangent space to the approximation manifold at the current iterate, with respect to a suitable metric. This corresponds to a locally optimal update in function space, following a projected gradient onto the tangent space to the manifold. Still, both gradient and natural gradient descent methods get stuck in local minima. Furthermore, when the model class is a nonlinear manifold or the loss function is not ideally conditioned (e.g., the KL-divergence for density estimation, or a norm of the residual of a partial differential equation in physics informed learning), even the natural gradient might yield non-optimal directions at each step. This work introduces a natural version of classical inertial dynamic methods like Heavy-Ball or Nesterov and show how it can improve the learning process when working with nonlinear model classes. |
| title | Natural gradient descent with momentum |
| topic | Machine Learning Artificial Intelligence Numerical Analysis Optimization and Control 68T05 G.1.6; G.1.2; G.1.3 |
| url | https://arxiv.org/abs/2604.15554 |