Solving Problems with Unknown Solution Length at (Almost) No Extra Cost

Abstract

Most research in the theory of evolutionary computation assumes that the problem at hand has a fixed problem size. This assumption does not always apply to real-world optimization challenges, where the length of an optimal solution may be unknown a priori.Following up on previous work of Cathabard, Lehre, and Yao [FOGA 2011] we analyze variants of the (1+1) evolutionary algorithm for problems with unknown solution length. For their setting, in which the solution length is sampled from a geometric distribution, we provide mutation rates that yield an expected optimization time that is of the same order as that of the (1+1) EA knowing the solution length.We then show that almost the same run times can be achieved even if no a priori information on the solution length is available.Finally, we provide mutation rates suitable for settings in which neither the solution length nor the positions of the relevant bits are known. Again we obtain almost optimal run times for the OneMax and LeadingOnes test functions, thus solving an open problem from Cathabard et al.

Self-Adaptation in Nonelitist Evolutionary Algorithms on Discrete Problems With Unknown Structure

“…As discussed in more detail in Section 1.2, this is a significant speedup compared to an EA using a static choice of mutation rate, which can only achieve Θ(nk) on LeadingOnes k . This is also an asymptotic speedup from the best-known runtime shown in [16], and indeed is asymptotically optimal among all unary unbiased black-box algorithms [3].…”

“…Rather than adjusting the mutation rate in each generation during the actual search process, the algorithm instead spends O(k) generations approximating the hidden value k, and then O(k log k) generations actually optimising f k now that k is approximately known. This algorithm not only improves the bound from O(k(log k) 2+ε ) in [16] to Θ(k log k) for OneMax k under the hidden subset model, but the implicit constants of (1 ± o(1))en ln n are found as well, matching the performance of a (1 + 1) EA which knows k in advance. However, it remained to be demonstrated whether an EA could similarly solve the LeadingOnes k problem at no extra cost when k was unknown.…”

“…The setting where k corresponds to an unknown initial number of bits which impact fitness has become known as the initial segment uncertainty model. The closely related hidden subset problem, which is analogous to the initial segment model except the k meaningful bits can be anywhere in the bitstring, has also been studied for LeadingOnes k and OneMax k [15,16,26]. Since our algorithm always flips all bits with equal probability during mutation, our results immediately extend to this class of problems.…”

“…However, several extensions of the (1 + 1) EA have been proposed which are more effective for optimisation in this uncertain environment. In [16], Doerr et al consider two different variants of the (1 + 1) EA, one which assigns different flip probabilities to each bit, and one which samples a new bitflip probability from a distribution Q in each generation, both of which they show to have an expected runtime of O(k 2 (log k) 1+ε ) on LeadingOnes k . They also show the log 1+ε k term can be further reduced by more carefully choosing the positional bitflip probabilities or the distribution Q; however, in a follow-up work, it is shown that the upper bound for both of these algorithms is nearly tight, that is, the expected runtime is ω(k 2 log k) [15].…”

“…However, progress in the empirical and theoretical study of EAs has shown many exceptions to these statements. It is now known that even small changes to the basic parameters of an EA can drastically increase the runtime on some problems [20,31], and more recently [34], and that hiding some aspects of the problem structure from an EA can decrease performance [5,15,16,26].…”