  This paper explores the theoretical basis of the covariance matrix adaptation
evolution strategy (CMA-ES) from the information geometry viewpoint.
  To establish a theoretical foundation for the CMA-ES, we focus on a geometric
structure of a Riemannian manifold of probability distributions equipped with
the Fisher metric. We define a function on the manifold which is the
expectation of fitness over the sampling distribution, and regard the goal of
update of the parameters of sampling distribution in the CMA-ES as maximization
of the expected fitness. We investigate the steepest ascent learning for the
expected fitness maximization, where the steepest ascent direction is given by
the natural gradient, which is the product of the inverse of the Fisher
information matrix and the conventional gradient of the function.
  Our first result is that we can obtain under some types of parameterization
of multivariate normal distribution the natural gradient of the expected
fitness without the need for inversion of the Fisher information matrix. We
find that the update of the distribution parameters in the CMA-ES is the same
as natural gradient learning for expected fitness maximization. Our second
result is that we derive the range of learning rates such that a step in the
direction of the exact natural gradient improves the parameters in the expected
fitness. We see from the close relation between the CMA-ES and natural gradient
learning that the default setting of learning rates in the CMA-ES seems
suitable in terms of monotone improvement in expected fitness. Then, we discuss
the relation to the expectation-maximization framework and provide an
information geometric interpretation of the CMA-ES.
