The Information-Geometric Optimization (IGO) has been introduced as a unified
framework for stochastic search algorithms. Given a parametrized family of
probability distributions on the search space, the IGO turns an arbitrary
optimization problem on the search space into an optimization problem on the
parameter space of the probability distribution family and defines a natural
gradient ascent on this space. From the natural gradients defined over the
entire parameter space we obtain continuous time trajectories which are the
solutions of an ordinary differential equation (ODE). Via discretization, the
IGO naturally defines an iterated gradient ascent algorithm. Depending on the
chosen distribution family, the IGO recovers several known algorithms such as
the pure rank-\mu update CMA-ES. Consequently, the continuous time
IGO-trajectory can be viewed as an idealization of the original algorithm. In
this paper we study the continuous time trajectories of the IGO given the
family of isotropic Gaussian distributions. These trajectories are a
deterministic continuous time model of the underlying evolution strategy in the
limit for population size to infinity and change rates to zero. On functions
that are the composite of a monotone and a convex-quadratic function, we prove
the global convergence of the solution of the ODE towards the global optimum.
We extend this result to composites of monotone and twice continuously
differentiable functions and prove local convergence towards local optima.Comment: PPSN - 12th International Conference on Parallel Problem Solving from
Nature - 2012 (2012