Precision, Local Search and Unimodal Functions

Abstract

We investigate the effects of precision on the efficiency of various local search algorithms on 1-D unimodal functions. We present a (1 + 1)-EA with adaptive step size which finds the optimum in O(log n) steps, where n is the number of points used. We then consider binary and Gray representations with single bit mutations. The standard binary method does not guarantee locating the optimum, whereas using Gray code does so in O((log n)2 ) steps. A (1 + 1)-EA with a fixed mutation probability distribution is then presented which also runs in O((log n)2 ). Moreover, a recent result shows that this is optimal (up to some constant scaling factor), in that there exist unimodal functions for which a lower bound of Ω((log n)2 ) holds regardless of the choice of mutation distribution. Finally, we show that it is not possible for a black box algorithms to efficiently optimise unimodal functions for two or more dimensions (in terms of the precision used).

First Steps Towards a Runtime Analysis of Neuroevolution

“…, 𝑟 }, where 𝐻 𝑟 = 𝑟 𝑖=1 1/𝑖. This operator was used before in [2,3,13] for similar search spaces. Table 1 summarizes the parameters and settings of the (1+1) NA.…”

“…Moreover, we will present problems with local optima which are hard to overcome. While analyzing the runtimes, we compare different mutation operators, more precisely a local one and the harmonic mutation operator introduced in [2], and prove exponentially (in the desired resolution of the discretized search space) smaller bounds for the harmonic mutation in several cases. Our proposed classification problems may serve as examples of typical optimization (sub)scenarios in neuroevolution and as a starting point for the runtime analysis of more advanced scenarios.…”