Natural gradient descent with momentum

2026-04-16 · Source: Artificial Intelligence · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

This work introduces a natural version of classical inertial dynamic methods, such as Heavy-Ball or Nesterov, to improve the learning process for nonlinear model classes. Natural Gradient Descent (NGD) approximates a function using a nonlinear manifold, like neural networks with differentiable activation functions or tensor networks. NGD operates as a preconditioned gradient descent, where parameter updates are guided by a functional perspective, utilizing the Gram matrix of the tangent space's generating system instead of the Hessian. While NGD aims for locally optimal updates in function space, both NGD and standard gradient descent can become trapped in local minima. Additionally, for nonlinear manifolds or poorly conditioned loss functions (e.g., KL-divergence, PDE residuals), even NGD may produce suboptimal directions. The proposed natural inertial methods aim to address these limitations.

Key takeaway

For research scientists optimizing nonlinear models, you should investigate integrating natural versions of inertial methods like Heavy-Ball or Nesterov into your NGD implementations. This approach offers a path to mitigate issues with local minima and suboptimal update directions, potentially leading to faster convergence and better performance in complex landscapes, especially with challenging loss functions or model architectures.

Key insights

Natural gradient descent with momentum improves optimization for nonlinear models by overcoming local minima and suboptimal directions.

Principles

• NGD uses the Gram matrix, not the Hessian.
• NGD aims for locally optimal functional updates.
• Inertial dynamics can enhance NGD performance.

Method

The work introduces natural versions of Heavy-Ball and Nesterov inertial dynamic methods, applying them to Natural Gradient Descent to improve learning in nonlinear model classes.

In practice

• Apply to neural networks with differentiable activations.
• Consider for tensor networks optimization.
• Useful for KL-divergence loss functions.

Topics

• Natural Gradient Descent
• Nonlinear Manifolds
• Neural Networks
• Tensor Networks
• Inertial Dynamic Methods

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Artificial Intelligence.