Deriving and improving CMA-ES with Information geometric trust regions

CMA-ES is one of the most popular stochastic search algorithms. It performs favourably in many tasks without the need of extensive parameter tuning. The algorithm has many beneficial properties, including automatic step-size adaptation, efficient covariance updates that incorporates the current samples as well as the evolution path and its invariance properties. Its update rules are composed of well established heuristics where the theoretical foundations of some of these rules are also well understood. In this paper we will fully derive all CMA-ES update rules within the framework of expectation-maximisation-based stochastic search algorithms using information-geometric trust regions. We show that the use of the trust region results in similar updates to CMA-ES for the mean and the covariance matrix while it allows for the derivation of an improved update rule for the step-size. Our new algorithm, Trust-Region Covariance Matrix Adaptation Evolution Strategy (TR-CMA-ES) is fully derived from first order optimization principles and performs favourably in compare to standard CMA-ES algorithm

Convergence of the Continuous Time Trajectories of Isotropic Evolution Strategies on Monotonic C^2-composite Functions

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

Cooperative Coevolution for Non-Separable Large-Scale Black-Box Optimization: Convergence Analyses and Distributed Accelerations

Given the ubiquity of non-separable optimization problems in real worlds, in this paper we analyze and extend the large-scale version of the well-known cooperative coevolution (CC), a divide-and-conquer optimization framework, on non-separable functions. First, we reveal empirical reasons of why decomposition-based methods are preferred or not in practice on some non-separable large-scale problems, which have not been clearly pointed out in many previous CC papers. Then, we formalize CC to a continuous game model via simplification, but without losing its essential property. Different from previous evolutionary game theory for CC, our new model provides a much simpler but useful viewpoint to analyze its convergence, since only the pure Nash equilibrium concept is needed and more general fitness landscapes can be explicitly considered. Based on convergence analyses, we propose a hierarchical decomposition strategy for better generalization, as for any decomposition there is a risk of getting trapped into a suboptimal Nash equilibrium. Finally, we use powerful distributed computing to accelerate it under the multi-level learning framework, which combines the fine-tuning ability from decomposition with the invariance property of CMA-ES. Experiments on a set of high-dimensional functions validate both its search performance and scalability (w.r.t. CPU cores) on a clustering computing platform with 400 CPU cores