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

Abstract

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;

Significance-based estimation-of-distribution algorithms

“…So far, such learning-based concepts are rarely used in evolutionary computation. The only theoretical works in this direction propose a history-based choice of the mutation strength [9] and analyze hyperheuristics that stick to a chosen subheuristic until its performance over the last τ iterations, τ a parameter of the algorithms, appears insufficient (see, e.g., [15] and the references therein).…”

Theory of Parameter Control for Discrete Black-Box Optimization: Provable Performance Gains Through Dynamic Parameter Choices

“…The generalized random gradient heuristic was further extended in [DLOW18]. There an operator was defined as successful (which leads to another phase using this operator) if it leads to σ improvements in a phase of at most τ iterations.…”

“…By using a larger value of σ, the algorithm is able to take more robust decisions on what is a success. This was used in [DLOW18] to determine the phase length τ in a self-adjusting manner. While the previous work [LOW17] does not state this explicitly, the choice of τ is crucial.…”

“…Since the choice of τ is that critical, a mechanism successfully adjusting it to the right value is desirable. In [DLOW18] it is shown that by choosing σ ∈ Ω(log 4 n) ∩ o( n/ log n)-note that this is a quite wide range-the value of τ can be easily adjusted on the fly via a multiplicative update rule. This gives again the asymptotically optimal running time of Theorem 14.…”

Maximizing drift is not optimal for solving OneMax

“…All these references consider the optimization of One-Max, the problem of maximizing the counting-ones function f : {0, 1} n → R, x → n i=1 x i . Only few theoretical results analyzing algorithms with adaptive parameters consider different functions, e.g., [LOW17,DLOW18,DDK18] (see [DD18b] for a complete list of references). OneMax also plays a prominent role in empirical research on parameter control.…”