We analyze the performance of the 2-rate $(1+λ)$ Evolutionary Algorithm (EA) with self-adjusting mutation rate control, its 3-rate counterpart, and a $(1+λ)$~EA variant using multiplicative update rules on the OneMax problem. We compare their efficiency for offspring population sizes ranging up to $λ=3,200$ and problem sizes up to $n=100,000$.
Our empirical results show that the ranking of the algorithms is very consistent across all tested dimensions, but strongly depends on the population size. While for small values of $λ$ the 2-rate EA performs best, the multiplicative updates become superior for starting for some threshold value of $λ$ between 50 and 100. Interestingly, for population sizes around 50, the $(1+λ)$~EA with static mutation rates performs on par with the best of the self-adjusting algorithms.
We also consider how the lower bound $p_{\min}$ for the mutation rate influences the efficiency of the algorithms. We observe that for the 2-rate EA and the EA with multiplicative update rules the more generous bound $p_{\min}=1/n^2$ gives better results than $p_{\min}=1/n$ when $λ$ is small. For both algorithms the situation reverses for large~$λ$.
We propose a new way to self-adjust the mutation rate in population-based evolutionary algorithms in discrete search spaces. Roughly speaking, it consists of creating half the offspring with a mutation rate that is twice the current mutation rate and the other half with half the current rate. The mutation rate is then updated to the rate used in that subpopulation which contains the best offspring. We analyze how the $(1+λ)$ evolutionary algorithm with this self-adjusting mutation rate optimizes the OneMax test function. We prove that this dynamic version of the $(1+λ)$ EA finds the optimum in…
Evolutionary Algorithms (EAs) and other randomized search heuristics are often considered as unbiased algorithms that are invariant with respect to different transformations of the underlying search space. However, if a certain amount of domain knowledge is available the use of biased search operators in EAs becomes viable. We consider a simple (1+1) EA for binary search spaces and analyze an asymmetric mutation operator that can treat zero- and one-bits differently. This operator extends previous work by Jansen and Sudholt (ECJ 18(1), 2010) by allowing the operator asymmetry to vary according…
The one-fifth success rule is one of the best-known and most widely accepted techniques to control the parameters of evolutionary algorithms. While it is often applied in the literal sense, a common interpretation sees the one-fifth success rule as a family of success-based updated rules that are determined by an update strength $F$ and a success rate. We analyze in this work how the performance of the (1+1) Evolutionary Algorithm on LeadingOnes depends on these two hyper-parameters. Our main result shows that the best performance is obtained for small update strengths $F=1+o(1)$ and success r…
It is well known that evolutionary algorithms (EAs) achieve peak performance only when their parameters are suitably tuned to the given problem. Even more, it is known that the best parameter values can change during the optimization process. Parameter control mechanisms are techniques developed to identify and to track these values. Recently, a series of rigorous theoretical works confirmed the superiority of several parameter control techniques over EAs with best possible static parameters. Among these results are examples for controlling the mutation rate of the $(1+λ)$~EA when optimizing t…
Recent works showed that simple success-based rules for self-adjusting parameters in evolutionary algorithms (EAs) can match or outperform the best fixed parameters on discrete problems. Non-elitism in a (1,$λ$) EA combined with a self-adjusting offspring population size $λ$ outperforms common EAs on the multimodal Cliff problem. However, it was shown that this only holds if the success rate $s$ that governs self-adjustment is small enough. Otherwise, even on OneMax, the self-adjusting (1,$λ$) EA stagnates on an easy slope, where frequent successes drive down the offspring population size. We …
Evolutionary algorithms (EAs) are general-purpose optimisers that come with several parameters like the sizes of parent and offspring populations or the mutation rate. It is well known that the performance of EAs may depend drastically on these parameters. Recent theoretical studies have shown that self-adjusting parameter control mechanisms that tune parameters during the algorithm run can provably outperform the best static parameters in EAs on discrete problems. However, the majority of these studies concerned elitist EAs and we do not have a clear answer on whether the same mechanisms can …
Recent theoretical research has shown that self-adjusting and self-adaptive mechanisms can provably outperform static settings in evolutionary algorithms for binary search spaces. However, the vast majority of these studies focuses on unimodal functions which do not require the algorithm to flip several bits simultaneously to make progress. In fact, existing self-adjusting algorithms are not designed to detect local optima and do not have any obvious benefit to cross large Hamming gaps. We suggest a mechanism called stagnation detection that can be added as a module to existing evolutionary al…
Generative Adversarial Networks (GAN) is an adversarial model, and it has been demonstrated to be effective for various generative tasks. However, GAN and its variants also suffer from many training problems, such as mode collapse and gradient vanish. In this paper, we firstly propose a general crossover operator, which can be widely applied to GANs using evolutionary strategies. Then we design an evolutionary GAN framework C-GAN based on it. And we combine the crossover operator with evolutionary generative adversarial networks (EGAN) to implement the evolutionary generative adversarial netwo…
The all-pairs shortest path problem is the first non-artificial problem for which it was shown that adding crossover can significantly speed up a mutation-only evolutionary algorithm. Recently, the analysis of this algorithm was refined and it was shown to have an expected optimization time (w.r.t. the number of fitness evaluations) of $Θ(n^{3.25}(\log n)^{0.25})$. In contrast to this simple algorithm, evolutionary algorithms used in practice usually employ refined recombination strategies in order to avoid the creation of infeasible offspring. We study extensions of the basic algorithm by two…
The heavy-tailed mutation operator proposed in Doerr, Le, Makhmara, and Nguyen (GECCO 2017), called \emph{fast mutation} to agree with the previously used language, so far was proven to be advantageous only in mutation-based algorithms. There, it can relieve the algorithm designer from finding the optimal mutation rate and nevertheless obtain a performance close to the one that the optimal mutation rate gives. In this first runtime analysis of a crossover-based algorithm using a heavy-tailed choice of the mutation rate, we show an even stronger impact. For the $(1+(λ,λ))$ genetic algorithm opt…
Estimation of distribution algorithms (EDA) as one of the EAs is a stochastic optimization problem which establishes a probability model to describe the distribution of solutions and randomly samples the probability model to create offspring and optimize model and population. Reference Vector Guided Evolutionary (RVEA) based on the EDA framework, having a better performance to solve MaOPs. Besides, using the generative adversarial networks to generate offspring solutions is also a state-of-art thought in EAs instead of crossover and mutation. In this paper, we will propose a novel algorithm ba…
It is known that the evolutionary algorithm $(1+1)$-EA with mutation rate $c/n$ optimises every monotone function efficiently if $c<1$, and needs exponential time on some monotone functions (HotTopic functions) if $c\geq 2.2$. We study the same question for a large variety of algorithms, particularly for $(1+λ)$-EA, $(μ+1)$-EA, $(μ+1)$-GA, their fast counterparts like fast $(1+1)$-EA, and for $(1+(λ,λ))$-GA. We find that all considered mutation-based algorithms show a similar dichotomy for HotTopic functions, or even for all monotone functions. For the $(1+(λ,λ))$-GA, this dichotomy is in the …
We investigate a family of $(μ+λ)$ Genetic Algorithms (GAs) which creates offspring either from mutation or by recombining two randomly chosen parents. By scaling the crossover probability, we can thus interpolate from a fully mutation-only algorithm towards a fully crossover-based GA. We analyze, by empirical means, how the performance depends on the interplay of population size and the crossover probability. Our comparison on 25 pseudo-Boolean optimization problems reveals an advantage of crossover-based configurations on several easy optimization tasks, whereas the picture for more complex …
It is known that the $(1+(λ,λ))$~Genetic Algorithm (GA) with self-adjusting parameter choices achieves a linear expected optimization time on OneMax if its hyper-parameters are suitably chosen. However, it is not very well understood how the hyper-parameter settings influences the overall performance of the $(1+(λ,λ))$~GA. Analyzing such multi-dimensional dependencies precisely is at the edge of what running time analysis can offer. To make a step forward on this question, we present an in-depth empirical study of the self-adjusting $(1+(λ,λ))$~GA and its hyper-parameters. We show, among many …
Evolutionary algorithms (EAs), simulating the evolution process of natural species, are used to solve optimization problems. Crossover (also called recombination), originated from simulating the chromosome exchange phenomena in zoogamy reproduction, is widely employed in EAs to generate offspring solutions, of which the effectiveness has been examined empirically in applications. However, due to the irregularity of crossover operators and the complicated interactions to mutation, crossover operators are hard to analyze and thus have few theoretical results. Therefore, analyzing crossover not o…
The notion of building blocks can be related to the structure of the offspring probability distribution: loci of which variability is strongly correlated constitute a building block. We call this correlated exploration. With this background we analyze the structure of the offspring probability distribution, or exploration distribution, for a GA with mutation only, a crossover GA, and an Estimation-Of-Distribution Algorithm (EDA). The results allow a precise characterization of the structure of the crossover exploration distribution. Essentially, the crossover operator destroys mutual informati…
Khulna University of Engineering and Technology（库尔纳工程技术大学）
This paper presents two different evolutionary systems - Evolutionary Programming Network (EPNet) and Novel Evolutions Strategy (NES) Algorithm. EPNet does both training and architecture evolution simultaneously, whereas NES does a fixed network and only trains the network. Five mutation operators proposed in EPNet to reflect the emphasis on evolving ANNs behaviors. Close behavioral links between parents and their offspring are maintained by various mutations, such as partial training and node splitting. On the other hand, NES uses two new genetic operators - subpopulation-based max-mean arith…
Genetic Algorithm enhanced by Deep Reinforcement Learning in parent selection mechanism and mutation : Minimizing makespan in permutation flow shop scheduling problems
Ecole Nationale Supérieure d'Informatique（国家高等信息学院）
This paper presents two different efficiency-enhancement techniques for probabilistic model building genetic algorithms. The first technique proposes the use of a mutation operator which performs local search in the sub-solution neighborhood identified through the probabilistic model. The second technique proposes building and using an internal probabilistic model of the fitness along with the probabilistic model of variable interactions. The fitness values of some offspring are estimated using the probabilistic model, thereby avoiding computationally expensive function evaluations. The scalab…
A common view in evolutionary biology is that mutation rates are minimised. However, studies in combinatorial optimisation and search have shown a clear advantage of using variable mutation rates as a control parameter to optimise the performance of evolutionary algorithms. Much biological theory in this area is based on Ronald Fisher's work, who used Euclidean geometry to study the relation between mutation size and expected fitness of the offspring in infinite phenotypic spaces. Here we reconsider this theory based on the alternative geometry of discrete and finite spaces of DNA sequences. F…
Weight initialization is critical in being able to successfully train artificial neural networks (ANNs), and even more so for recurrent neural networks (RNNs) which can easily suffer from vanishing and exploding gradients. In neuroevolution, where evolutionary algorithms are applied to neural architecture search, weights typically need to be initialized at three different times: when initial genomes (ANN architectures) are created at the beginning of the search, when offspring genomes are generated by crossover, and when new nodes or edges are created during mutation. This work explores the di…
École Polytechnique（巴黎综合理工学院）；CNRS（法国国家科学研究中心）；Technical University of Denmark（丹麦技术大学）
We propose and analyze a self-adaptive version of the $(1,λ)$ evolutionary algorithm in which the current mutation rate is part of the individual and thus also subject to mutation. A rigorous runtime analysis on the OneMax benchmark function reveals that a simple local mutation scheme for the rate leads to an expected optimization time (number of fitness evaluations) of $O(nλ/\logλ+n\log n)$ when $λ$ is at least $C \ln n$ for some constant $C > 0$. For all values of $λ\ge C \ln n$, this performance is asymptotically best possible among all $λ$-parallel mutation-based unbiased black-box algorit…
The $(1+(λ,λ))$ genetic algorithm, first proposed at GECCO 2013, showed a surprisingly good performance on so me optimization problems. The theoretical analysis so far was restricted to the OneMax test function, where this GA profited from the perfect fitness-distance correlation. In this work, we conduct a rigorous runtime analysis of this GA on random 3-SAT instances in the planted solution model having at least logarithmic average degree, which are known to have a weaker fitness distance correlation. We prove that this GA with fixed not too large population size again obtains runtimes bette…
Department of Mathematics, Southern Illinois University（南伊利诺伊大学数学系）；Iran Telecommunication Research Center（伊朗电信研究中心）
The compact Genetic Algorithm (cGA) is an Estimation of Distribution Algorithm that generates offspring population according to the estimated probabilistic model of the parent population instead of using traditional recombination and mutation operators. The cGA only needs a small amount of memory; therefore, it may be quite useful in memory-constrained applications. This paper introduces a theoretical framework for studying the cGA from the convergence point of view in which, we model the cGA by a Markov process and approximate its behavior using an Ordinary Differential Equation (ODE). Then, …
A hybrid evolutionary algorithm with importance sampling method is proposed for multi-dimensional optimization problems in this paper. In order to make use of the information provided in the search process, a set of visited solutions is selected to give scores for intervals in each dimension, and they are updated as algorithm proceeds. Those intervals with higher scores are regarded as good intervals, which are used to estimate the joint distribution of optimal solutions through an interaction between the pool of good genetics, which are the individuals with smaller fitness values. And the sam…
Evolutionary algorithms (EAs) are population-based general-purpose optimization algorithms, and have been successfully applied in various real-world optimization tasks. However, previous theoretical studies often employ EAs with only a parent or offspring population and focus on specific problems. Furthermore, they often only show upper bounds on the running time, while lower bounds are also necessary to get a complete understanding of an algorithm. In this paper, we analyze the running time of the ($μ$+$λ$)-EA (a general population-based EA with mutation only) on the class of pseudo-Boolean f…
