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Selection HHs are randomised search methodologies which choose and executeheuristics during the optimisation process from a set of low-level heuristics.A machine learning mechanism is generally used to decide which low-levelheuristic should be applied in each decision step. In this paper we analysewhether sophisticated learning mechanisms are always necessary for HHs toperform well. To this end we consider the most simple HHs from the literatureand rigorously analyse their performance for the LeadingOnes function. Ouranalysis shows that the standard Simple Random, Permutation, Greedy and RandomGradient HHs show no signs of learning. While the former HHs do not attempt tolearn from the past performance of low-level heuristics, the idea behind theRandom Gradient HH is to continue to exploit the currently selected heuristicas long as it is successful. Hence, it is embedded with a reinforcementlearning mechanism with the shortest possible memory. However, the probabilitythat a promising heuristic is successful in the next step is relatively lowwhen perturbing a reasonable solution to a combinatorial optimisation problem.We generalise the simple Random Gradient HH so success can be measured over afixed period of time tau, instead of a single iteration. For LO we prove thatthe Generalised Random Gradient HH can learn to adapt the neighbourhood size ofRLS to optimality during the run. We prove it has the best possible performanceachievable with the low-level heuristics. We also prove that the performance ofthe HH improves as the number of low-level local search heuristics to choosefrom increases. Finally, we show that the advantages of GRG over RLS and EAsusing standard bit mutation increase if the anytime performance is considered.Experimental analyses confirm these results for different problem sizes.

Simple Hyper-heuristics Control the Neighbourhood Size of Randomised Local Search Optimally for LeadingOnes