Contour-Assisted Evolutionary Computation: Estimating Peaks From the Contour Lines of Evaluated Solutions

원문 초록

Evolutionary algorithms (EAs) evaluate many candidate solutions but use them mainly to rank individuals. This paper treats the evaluated solutions as the spot heights of a topographic survey and draws the contour lines they imply. The contour map decomposes the evaluated region into peaks: in two dimensions a contour tree with persistence separates significant peaks from interpolation ripples, and in more dimensions hill climbing on the interpolated surface locates the peaks and ray-cast level sets describe them. A memory of the explored peaks directs the search to new ones. The data-supported contours of each peak are extrapolated to zero size by an area law, exact for quadratic peaks and second-order accurate for smooth ones, that estimates a peak's location, height and shape before it is sampled. A contour-assisted EA (CA-EA) inserts contour steps with local search and peak memory into a genetic algorithm (GA), differential evolution or particle swarm optimization. A contour-assisted surrogate-assisted EA (CA-SAEA), for expensive problems, spends each evaluation on surrogate-screened offspring or on sampling around a peak, entered at the maximum of the interpolated surface. On 2-D benchmarks, CA-EA reduces the evaluations a plain GA needs by factors of 12 to 112; controls attribute its gain over classical memetic search to restarting local search on distinct peaks. On the CEC 2015 expensive suite in 10 and 30 dimensions, CA-SAEA has the second-best average rank, behind DYCORS and ahead of lq-CMA-ES and the competition winner MVMO, although in 30 dimensions MVMO wins significantly on 9 functions and loses on 6; there the map beats sampling around the best point on 5 of 15 functions and loses on 3. Entering the peaks at their extrapolated locations brought no gain at these budgets, although, with enough samples, it improves the estimated heights on most two-dimensional functions.