  Information geometry applies concepts in differential geometry to probability
and statistics and is especially useful for parameter estimation in exponential
families where parameters are known to lie on a Riemannian manifold.
Connections between the geometric properties of the induced manifold and
statistical properties of the estimation problem are well-established. However
developing first-order methods that scale to larger problems has been less of a
focus in the information geometry community. The best known algorithm that
incorporates manifold structure is the second-order natural gradient descent
algorithm introduced by Amari. On the other hand, stochastic approximation
methods have led to the development of first-order methods for optimizing noisy
objective functions. A recent generalization of the Robbins-Monro algorithm
known as mirror descent, developed by Nemirovski and Yudin is a first order
method that induces non-Euclidean geometries. However current analysis of
mirror descent does not precisely characterize the induced non-Euclidean
geometry nor does it consider performance in terms of statistical relative
efficiency. In this paper, we prove that mirror descent induced by Bregman
divergences is equivalent to the natural gradient descent algorithm on the dual
Riemannian manifold. Using this equivalence, it follows that (1) mirror descent
is the steepest descent direction along the Riemannian manifold of the
exponential family; (2) mirror descent with log-likelihood loss applied to
parameter estimation in exponential families asymptotically achieves the
classical Cram\'er-Rao lower bound and (3) natural gradient descent for
manifolds corresponding to exponential families can be implemented as a
first-order method through mirror descent.
