Globally convergent evolution strategies

Abstract

In this paper we show how to modify a large class of evolution strategies (ES's) for unconstrained optimization to rigorously achieve a form of global convergence, meaning convergence to stationary points independently of the starting point. The type of ES under consideration recombines the parent points by means of a weighted sum, around which the offspring points are computed by random generation. One relevant instance of such an ES is CMA-ES (covariance matrix adaptation ES).The modifications consist essentially of the reduction of the size of the steps whenever a sufficient decrease condition on the function values is not verified. When such a condition is satisfied, the step size can be reset to the step size maintained by the ES's themselves, as long as this latter one is sufficiently large. We suggest a number of ways of imposing sufficient decrease for which global convergence holds under reasonable assumptions (in particular density of certain limit directions in the unit sphere).Given a limited budget of function evaluations, our numerical experiments have shown that the modified CMA-ES is capable of further progress in function values. Moreover, we have observed that such an improvement in efficiency comes without weakening significantly the performance of the underlying method in the presence of several local minimizers.

A parallel evolution strategy for an earth imaging problem in geophysics

“…ES's are a class of evolutionary algorithms designed for searching the global minimum of a function in a continuous space. We are motivated by the modifications in ES's recently proposed in [16] to ensure some form of rigorous convergence and a better computational performance under moderate budgets of function evaluations. Thirdly, based on one of the modified ES's given in [16], we propose a highly parallel ES adapted to the full-waveform inversion setting.…”

“…We are motivated by the modifications in ES's recently proposed in [16] to ensure some form of rigorous convergence and a better computational performance under moderate budgets of function evaluations. Thirdly, based on one of the modified ES's given in [16], we propose a highly parallel ES adapted to the full-waveform inversion setting. By combining model reduction and ES's in a parallel environment, we aim at solving realistic instances of the problem.…”

“…Its main drawback is, however, that a large budget is required to provide outstanding results. Recently, the authors in [16] have proposed modifications to the class of algorithms in (µ/µ W , λ)-ES to make them enjoying a favorable convergence property and performing better for smaller budgets. The modifications have been essentially the imposition of a sufficient decrease on the objective function values to accept new iterates and the reduction of the step size when such a condition is not satisfied.…”

“…Under such modifications, these ES's can converge globally (meaning independently of the starting point) for a first-order stationary point. Algorithm 1 shows an adaptation of the globally convergent ES's proposed in [16] to the context of full-waveform inversion.…”

“…The authors in [16] have proposed three different globally convergent ES versions named mean/mean, max/max, and max/mean. On a large data set of problems, the mean/mean version has performed numerically the best.…”