On the runtime analysis of selection hyper-heuristics with adaptive learning periods

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 LeadingOnes 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, 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 during the run. To achieve this we equip the algorithm with a simple self-adjusting 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 LeadingOnes in optimal runtime by automatically adapting and achieving a 1 − o(1) ratio of the desired behaviour. Complementary experiments for realistic problem sizes show the value of adapting as desired and that the hyper-heuristic with adaptive learning period outperforms the hyper-heuristic with xed learning periods.