Bi-population CMA-ES agorithms with surrogate models and line searches

Abstract

In this paper, three extensions of the BI-population Covariance Matrix Adaptation Evolution Strategy with weighted active covariance matrix update (BIPOP-aCMA-ES) are investigated. First, to address expensive optimization, we benchmark a recently proposed extension of the self-adaptive surrogate-assisted CMA-ES which benefits from more intensive surrogate model exploitation (BIPOP-saACM-k). Second, to address separable optimization, we propose a hybrid of BIPOP-aCMA-ES and STEP algorithm with coordinatewise line search (BIPOP-aCMA-STEP). Third, we propose HCMA, a hybrid of BIPOP-saACM-k, STEP and NEWUOA to benefit both from surrogate models and line searches. All algorithms were tested on the noiseless BBOB testbed using restarts till a total number of function evaluations of 10 6 n was reached, where n is the dimension of the function search space.The comparison shows that BIPOP-saACM-k outperforms its predecessor BIPOP-saACM up to a factor of 2 on illconditioned problems, while BIPOP-aCMA-STEP outperforms the original BIPOP-based algorithms on separable functions. The hybrid HCMA algorithm demonstrates the best overall performance compared to the best algorithms of the BBOB-2009, BBOB-2010 and BBOB-2012 when running for more than 100n function evaluations.

Global Line Search Algorithm Hybridized with Quadratic Interpolation and Its Extension to Separable Functions

“…• HCMA [8] which is a competitive hybrid algorithm combining NEWUOA, STEP and a CMA-ES variant (and uses a variant of interleaved STEP on separable functions until non-separability is detected).…”

“…Such an approach is quite usual in local search methods, e.g. Rosenbrock's [12], but for global line search methods in general, and for the STEP algorithm in particular, it was first proposed recently as part of the HCMA algorithm [8]. In our implementation, a single dimension is chosen in a round-robin fashion, and the algorithm associated with that dimension performs a single iteration.…”

“…In HCMA [8], the context was updated only after a complete pass through all dimensions, which allowed the hybrid algorithm to test the separability of the function.…”

“…Although these algorithms rely on the function separability and despite the above disadvantages, both of these methods proved to be useful in algorithmic portfolios [1] and hybrid algorithms [8] that strive to be successful on a broad range of functions, including the separable ones. This paper builds on the above mentioned methods, and provides the following contributions:…”

“…We suggest a better way of making a multidimensional variant of this method. As opposed to solving the 1D problem in all dimensions sequentially, we use the general idea [8] to interleave the steps in individual dimensions, but modify it by updating the full coordinates of sampled points based on results obtained in other dimensions so far.…”