Principled Design of Continuous Stochastic Search: From Theory to Practice

We derive a stochastic search procedure for parameter optimization from two first principles: (1) imposing the least prior assumptions, namely by maximum entropy sampling, unbiasedness and invariance; (2) exploiting all available information under the constraints imposed by (1). We additionally require that two of the most basic functions can be solved reasonably fast. Given these principles, two principal heuristics are used: reinforcing of good solutions and good steps (increasing their likelihood) and rendering successive steps orthogonal. The resulting search algorithm is the covariance matrix adaptation evolution strategy, CMA-ES, that coincides to a great extent to a natural gradient descent. The invariance properties of the CMA-ES are formalized, as are its maximum likelihood and stationarity properties. A small parameter study for a specific heuristic—deduced from the principles of reinforcing good steps and exploiting all information—is presented, namely for the cumulation of an evolution or search path. Experiments on two noisy functions are provided.

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