  Natural-gradient descent (NGD) on structured parameter spaces (e.g., low-rank
covariances) is computationally challenging due to difficult Fisher-matrix
computations. We address this issue by using \emph{local-parameter coordinates}
to obtain a flexible and efficient NGD method that works well for a
wide-variety of structured parameterizations. We show four applications where
our method (1) generalizes the exponential natural evolutionary strategy, (2)
recovers existing Newton-like algorithms, (3) yields new structured
second-order algorithms via matrix groups, and (4) gives new algorithms to
learn covariances of Gaussian and Wishart-based distributions. We show results
on a range of problems from deep learning, variational inference, and evolution
strategies. Our work opens a new direction for scalable structured geometric
methods.
