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 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;

Evolutionary algorithms (EAs) perform well in settings involving uncertainty, including settings with stochastic or dynamic fitness functions. In this paper, we analyze the (1+1) EA on dynamically changing OneMax, as introduced by Droste (2003). We re-prove the known results on first hitting times using the modern tool of drift analysis. We extend these results to search spaces which allow for more than two values per dimension.Furthermore, we make an anytime analysis as suggested by Jansen and Zarges (2014), analyzing how closely the (1+1) EA can track the dynamically moving optimum over time. We get tight bounds both for the case of bit strings, as well as for the case of more than two values per position. Surprisingly, in the latter setting, the expected quality of the search point maintained by the (1+1) EA does not depend on the number of values per dimension.

Genetic programming (GP) is a general purpose bio-inspired meta-heuristic for the evolution of computer programs. In contrast to the several successful applications, there is little understanding of the working principles behind GP. In this paper we present a performance analysis that sheds light on the behaviour of simple GP systems for evolving conjunctions of n variables (ANDn). The analysis of a random local search GP system with minimal terminal and function sets reveals the relationship between the number of iterations and the progress the GP makes toward finding the target function. Afterwards we consider a more realistic GP system equipped with a global mutation operator and prove that it can efficiently solve ANDn by producing programs of linear size that fit a training set to optimality and with high probability generalise well. Additionally, we consider more general problems which extend the terminal set with undesired variables or negated variables. In the presence of undesired variables, we prove that, if non-strict selection is used, then the algorithm fits the complete training set efficiently while the strict selection algorithm may fail with high probability unless the substitution operator is switched off. If negations are allowed, we show that while the algorithms fail to fit the complete training set, the constructed solutions generalise well. Finally, from a problem hardness perspective, we reveal the existence of small training sets that allow the evolution of the exact conjunctions even with access to negations or undesired variables.

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.