Natural Gradient Descent is an approximate secondorder optimisation method. It has an interpretation as optimizing over a Riemannian manifold using an intrinsic distance metric, which implies the updates are invariant to transformations such as whitening. By using the positive semidefinite (PSD) GaussNewton matrix to approximate the (possibly negative definite) Hessian, NGD can often work better than exact secondorder methods.
Given the gradient of $z$, $g = \frac{\delta{f}\left(z\right)}{\delta{z}}$, NGD computes the update as:
$$\Delta{z} = \alpha{F}^{−1}g$$
where the Fisher information matrix $F$ is defined as:
$$ F = \mathbb{E}_{p\left(t\mid{z}\right)}\left[\nabla\ln{p}\left(t\mid{z}\right)\nabla\ln{p}\left(t\mid{z}\right)^{T}\right] $$
The loglikelihood function $\ln{p}\left(t\mid{z}\right)$ typically corresponds to commonly used error functions such as the cross entropy loss.
Source: LOGAN
Image: Fast Convergence of Natural Gradient Descent for Overparameterized Neural Networks
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