Summary of Convergence Analysis Of Natural Gradient Descent For Over-parameterized Physics-informed Neural Networks, by Xianliang Xu et al.
Convergence Analysis of Natural Gradient Descent for Over-parameterized Physics-Informed Neural Networks
by Xianliang Xu, Ting Du, Wang Kong, Ye Li, Zhongyi Huang
First submitted to arxiv on: 1 Aug 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: None
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary The paper explores the effectiveness of first-order methods like gradient descent (GD) and stochastic gradient descent (SGD) in training neural networks. While previous research has shown that randomly initialized GD converges to a globally optimal solution at a linear rate for quadratic loss, the learning rate of GD for two-layer neural networks exhibits poor dependence on sample size and Gram matrix, leading to slow training. The authors demonstrate that for L2 regression problems, the learning rate can be improved from O(λ0/n2) to O(1/‖H∞‖2), implying a faster convergence rate. They also generalize this method to GD in Physics-Informed Neural Networks (PINNs). Although the improved learning rate has mild dependence on Gram matrix, it still needs to be set small enough due to unknown eigenvalues. The authors provide convergence analysis of natural gradient descent (NGD) in training PINNs, showing that the learning rate can be O(1), with a convergence rate independent of Gram matrix. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary The paper is about improving how neural networks learn from data. Right now, there’s a problem with how quickly these networks learn new things. The authors want to fix this by finding a better way for the networks to learn. They tested different methods and found that one method, called natural gradient descent, can learn much faster than before. This is important because it means we can train neural networks more quickly and make them work better. |
Keywords
* Artificial intelligence * Gradient descent * Regression * Stochastic gradient descent
