Selection hyper-heuristics are randomised optimisation techniques that select from a set of low-level heuristics which one should be applied in the next step of the optimisation process. Recently it has been proven that a Random Gradient hyper-heuristic optimises the L O benchmark function in the best runtime achievable with any combination of its low-level heuristics, up to lower order terms. To achieve this runtime, the learning period τ , used to evaluate the performance of the currently chosen heuristic, should be set appropriately, i.e., super-linear in the problem size but not excessively larger. In this paper we automate the hyper-heuristic further by allowing it to self-adjust the learning period τ during the run. To achieve this we equip the algorithm with a simple selfadjusting mechanism, called 1 − o(1) rule, inspired by the 1/5 rule traditionally used in continuous optimisation. We rigorously prove that the resulting hyper-heuristic solves L O in optimal runtime by automatically adapting τ and achieving a 1 − o(1) ratio of the desired behaviour. Complementary experiments for realistic problem sizes show the value of τ adapting as desired and that the hyper-heuristic with adaptive learning period outperforms the hyper-heuristic with xed learning periods. CCS CONCEPTS• eory of computation → eory of randomized search heuristics;

Selection hyper-heuristics are randomised search methodologies which choose and execute heuristics from a set of low-level heuristics. Recent time complexity analyses for the L O benchmark function have shown that the standard simple random, permutation, random gradient, greedy and reinforcement learning selection mechanisms show no e ects of learning. e idea behind the learning mechanisms is to continue to exploit the currently selected heuristic as long as it is successful. However, the probability that a promising heuristic is successful in the next step is relatively low when perturbing a reasonable solution to a combinatorial optimisation problem. In this paper we generalise the classical selection-perturbation mechanisms so success can be measured over some xed period of length τ , rather than in a single iteration. We present a benchmark function where it is necessary to learn to exploit a particular low-level heuristic, rigorously proving that it makes the di erence between an e cient and an ine cient algorithm. For L O we prove that the generalised random gradient mechanism approaches optimal performance while generalised greedy, although not as fast, still outperforms random local search. An experimental analysis shows that combining the two generalised mechanisms leads to even be er performance.

Selection hyper-heuristics are automated algorithm selection methodologies that choose between different heuristics during the optimisation process. Recently selection hyperheuristics choosing between a collection of elitist randomised local search heuristics with different neighbourhood sizes have been shown to optimise a standard unimodal benchmark function from evolutionary computation in the optimal expected runtime achievable with the available low-level heuristics. In this paper we extend our understanding to the domain of multimodal optimisation by considering a hyper-heuristic from the literature that can switch between elitist and nonelitist heuristics during the run. We first identify the range of parameters that allow the hyper-heuristic to hillclimb efficiently and prove that it can optimise a standard hillclimbing benchmark function in the best expected asymptotic time achievable by unbiased mutation-based randomised search heuristics. Afterwards, we use standard multimodal benchmark functions to highlight function characteristics where the hyper-heuristic is efficient by swiftly escaping local optima and ones where it is not. For a function class called CLIFFd where a new gradient of increasing fitness can be identified after escaping local optima, the hyper-heuristic is extremely efficient while a wide range of established elitist and non-elitist algorithms are not, including the well-studied Metropolis algorithm. We complete the picture with an analysis of another standard benchmark function called JUMPd as an example to highlight problem characteristics where the hyper-heuristic is inefficient. Yet, it still outperforms the wellestablished non-elitist Metropolis algorithm.

Selection hyper-heuristics (HHs) are randomised search methodologies which choose and execute heuristics during the optimisation process from a set of low-level heuristics. A machine learning mechanism is generally used to decide which low-level heuristic should be applied in each decision step. In this paper we analyse whether sophisticated learning mechanisms are always necessary for HHs to perform well. To this end we consider the most simple HHs from the literature and rigorously analyse their performance for the LEADINGONES benchmark function. Our analysis shows that the standard Simple Random, Permutation, Greedy and Random Gradient HHs show no signs of learning. While the former HHs do not attempt to learn from the past performance of low-level heuristics, the idea behind the Random Gradient HH is to continue to exploit the currently selected heuristic as long as it is successful. Hence, it is embedded with a reinforcement learning mechanism with the shortest possible memory. However, the probability that a promising heuristic is successful in the next step is relatively low when perturbing a reasonable solution to a combinatorial optimisation problem. We generalise the 'simple' Random Gradient HH so success can be measured over a fixed period of time τ , instead of a single iteration. For LEADINGONES we prove that the Generalised Random Gradient (GRG) HH can learn to adapt the neighbourhood size of Randomised Local Search to optimality during the run. As a result, we prove it has the best possible performance achievable with the low-level heuristics (Randomised Local Search with different neighbourhood sizes), up to lower order terms. We also prove that the performance of the HH improves as the number of low-level local search heuristics to choose from increases. In particular, with access to k low-level local search heuristics, it outperforms the best-possible algorithm using any subset of the k heuristics. Finally, we show that the advantages of GRG over Randomised Local Search and Evolutionary Algorithms using standard bit mutation increase if the anytime performance is considered (i.e., the performance gap is larger if approximate solutions are sought rather than exact ones). Experimental analyses confirm these results for different problem sizes (up to n = 10 8 ) and shed some light on the best choices for the parameter τ in various situations. * An extended abstract of this manuscript has appeared at the 2017 Genetic and Evolutionary Computation Conference (GECCO 2017) (Lissovoi et al., 2017).

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