  In optimization, the natural gradient method is well-known for likelihood
maximization. The method uses the Kullback-Leibler divergence, corresponding
infinitesimally to the Fisher-Rao metric, which is pulled back to the parameter
space of a family of probability distributions. This way, gradients with
respect to the parameters respect the Fisher-Rao geometry of the space of
distributions, which might differ vastly from the standard Euclidean geometry
of the parameter space, often leading to faster convergence. However, when
minimizing an arbitrary similarity measure between distributions, it is
generally unclear which metric to use. We provide a general framework that,
given a similarity measure, derives a metric for the natural gradient. We then
discuss connections between the natural gradient method and multiple other
optimization techniques in the literature. Finally, we provide computations of
the formal natural gradient to show overlap with well-known cases and to
compute natural gradients in novel frameworks.

Machine Learning;Machine Learning