Understanding Approximate Fisher Information for Fast Convergence of Natural Gradient Descent in Wide Neural Networks

by   Ryo Karakida, et al.

Natural Gradient Descent (NGD) helps to accelerate the convergence of gradient descent dynamics, but it requires approximations in large-scale deep neural networks because of its high computational cost. Empirical studies have confirmed that some NGD methods with approximate Fisher information converge sufficiently fast in practice. Nevertheless, it remains unclear from the theoretical perspective why and under what conditions such heuristic approximations work well. In this work, we reveal that, under specific conditions, NGD with approximate Fisher information achieves the same fast convergence to global minima as exact NGD. We consider deep neural networks in the infinite-width limit, and analyze the asymptotic training dynamics of NGD in function space via the neural tangent kernel. In the function space, the training dynamics with the approximate Fisher information are identical to those with the exact Fisher information, and they converge quickly. The fast convergence holds in layer-wise approximations; for instance, in block diagonal approximation where each block corresponds to a layer as well as in block tri-diagonal and K-FAC approximations. We also find that a unit-wise approximation achieves the same fast convergence under some assumptions. All of these different approximations have an isotropic gradient in the function space, and this plays a fundamental role in achieving the same convergence properties in training. Thus, the current study gives a novel and unified theoretical foundation with which to understand NGD methods in deep learning.


page 1

page 2

page 3

page 4


Analysis and Comparison of Two-Level KFAC Methods for Training Deep Neural Networks

As a second-order method, the Natural Gradient Descent (NGD) has the abi...

Efficient Approximations of the Fisher Matrix in Neural Networks using Kronecker Product Singular Value Decomposition

Several studies have shown the ability of natural gradient descent to mi...

Component-Wise Natural Gradient Descent – An Efficient Neural Network Optimization

Natural Gradient Descent (NGD) is a second-order neural network training...

Fast Approximate Natural Gradient Descent in a Kronecker-factored Eigenbasis

Optimization algorithms that leverage gradient covariance information, s...

Limitations of the Empirical Fisher Approximation

Natural gradient descent, which preconditions a gradient descent update ...

Are Saddles Good Enough for Deep Learning?

Recent years have seen a growing interest in understanding deep neural n...

Natural Wake-Sleep Algorithm

The benefits of using the natural gradient are well known in a wide rang...

Please sign up or login with your details

Forgot password? Click here to reset