On the runtime analysis of generalised selection hyper-heuristics for pseudo-boolean optimisation

Abstract

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.

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

“…Consequently, this new hyper-heuristic outperforms the previously investigated ones when c is large enough. For c tending to infinity, its performance approaches the best-possible performance that can be obtained from the two mutation operators, which is, as also shown in [LOW17], ln(2)+1 4 n 2 + o(n 2 ). The following variant of this result appeared in the preprint [LOW18].…”

“…Given that the probabilities to find a true improvement are very low in this discrete optimization problem, one would expect that the four selection mechanisms all use the two operators in a very balanced manner and thus lead to very similar running times. This is indeed the first set of results in the remarkable work of Lissovoi, Oliveto, and Warwicker [LOW17]. Building on the precise analysis method of [BDN10] instead of the fitness level method, they show that the expected running time for all four selection mechanisms is 1 2 ln(3) n 2 + o(n 2 ) ≈ 0.549n 2 .…”

“…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. A τ -value of smaller asymptotic order than Θ(n) leads to typically no improvement within a phase and thus reverts the algorithms to the simple random heuristic.…”

“…Theorem 13 (Theorem 4.2 and Corollary 4.3 in [LOW17]). The (1 + 1) EA using one of the selection mechanisms simple random, random gradient, greedy, or permutation to choose between the 1-bit or the 2-bit flip operator optimizes the LeadingOnes function in an expected number of 1 2 ln(3) n 2 + o(n 2 ) ≈ 0.549n 2 iterations.…”

Simple on-the-fly parameter selection mechanisms for two classical discrete black-box optimization benchmark problems

“…Before describing this algorithm in detail, we note that already allowing 1-and 2-bitflips (i.e., mut with = 1 and = 2) decreases the optimal 1 + n 2 /2 expected optimization time of static unary unbiased algorithms to about 0.4233n 2 [LOW17] (the fact that mut is defined slightly different in [LOW17] has a negligible impact on this result). This running time can be further reduced by allowing larger step sizes.…”