On Proving Linear Convergence of Comparison-based Step-size Adaptive Randomized Search on Scaling-Invariant Functions via Stability of Markov Chains
Anne Auger, Nikolaus Hansen · arXiv (Cornell University) · 2013
Abstract. In the context of numerical optimization, this paper develops a methodology to ana-lyze the linear convergence of comparison-based step-size adaptive randomized search (CB-SARS), a class of probabilistic derivative-free optimization algorithms where the function is solely used through comparisons of candidate solutions. Various algorithms are included in the class of CB-SARS algo-rithms. On the one hand, a few methods introduced already in the 60’s: the step-size adaptive random search by Schumer and Steiglitz, the compound random search by Devroye and simplified versions of Matyas ’ random optimization algorithm or Kjellstrom and Taxen Gaussian adaptation. On the other hand, it includes simplified versions of several recent algorithms: the covariance-matrix-adaptation evolution strategy algorithm (CMA-ES), the exponential natural evolution strategy (xNES), or the cross entropy method. CB-SARS algorithms typically exhibit several invariances. First of all, invariance to composing the objective function with a strictly monotonic transformation which is a direct consequence of the fact that the algorithms only use comparisons. Second, scale invariance that translates the fact that the algorithm has no intrinsic absolute notion of scale. The algorithms are investigated on scaling-invariant functions defined as functions that preserve