On the choice of the parameter control mechanism in the (1+(λ, λ)) genetic algorithm

The self-adjusting (1 + (λ, λ)) GA is the best known genetic algorithm for problems with a good fitness-distance correlation as in OneMax. It uses a parameter control mechanism for the parameter λ that governs the mutation strength and the number of offspring. However, on multimodal problems, the parameter control mechanism tends to increase λ uncontrollably. We study this problem and possible solutions to it using rigorous runtime analysis for the standard Jumpk benchmark problem class. The original algorithm behaves like a (1+n) EA whenever the maximum value λ = n is reached. This is ineffective for problems where large jumps are required. Capping λ at smaller values is beneficial for such problems. Finally, resetting λ to 1 allows the parameter to cycle through the parameter space. We show that this strategy is effective for all Jumpk problems: the (1 + (λ, λ)) GA performs as well as the (1 + 1) EA with the optimal mutation rate and fast evolutionary algorithms, apart from a small polynomial overhead. Along the way, we present new general methods for bounding the runtime of the (1 + (λ, λ)) GA that allows to translate existing runtime bounds from the (1 + 1) EA to the self-adjusting (1 + (λ, λ)) GA. Our methods are easy to use and give upper bounds for novel classes of functions

Self-adjusting Population Sizes for Non-elitist Evolutionary Algorithms:Why Success Rates Matter

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 be applied for non-elitist EAs. We study one of the best-known parameter control mechanisms, the one-fifth success rule, to control the offspring population size λ in the non-elitist (1, λ) EA. It is known that the (1, λ) EA has a sharp threshold with respect to the choice of λ where the expected runtime on the benchmark function OneMax changes from polynomial to exponential time. Hence, it is not clear whether parameter control mechanisms are able to find and maintain suitable values of λ. For OneMax we show that the answer crucially depends on the success rate s (i. e. a one-(s + 1)-th success rule). We prove that, if the success rate is appropriately small, the self-adjusting (1, λ) EA optimises OneMax in O(n) expected generations and O(n log n) expected evaluations, the best possible runtime for any unary unbiased black-box algorithm. A small success rate is crucial: we also show that if the success rate is too large, the algorithm has an exponential runtime on OneMax and other functions with similar characteristics

Runtime Analysis for Self-adaptive Mutation Rates

We propose and analyze a self-adaptive version of the $(1,\lambda)$ 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 $O(n\lambda/\log\lambda+n\log n)$ when $\lambda$ is at least $C \ln n$ for some constant $C > 0$. For all values of $\lambda \ge C \ln n$, this performance is asymptotically best possible among all $\lambda$-parallel mutation-based unbiased black-box algorithms. Our result shows that self-adaptation in evolutionary computation can find complex optimal parameter settings on the fly. At the same time, it proves that a relatively complicated self-adjusting scheme for the mutation rate proposed by Doerr, Gie{\ss}en, Witt, and Yang~(GECCO~2017) can be replaced by our simple endogenous scheme. On the technical side, the paper contributes new tools for the analysis of two-dimensional drift processes arising in the analysis of dynamic parameter choices in EAs, including bounds on occupation probabilities in processes with non-constant drift

Offspring Population Size Matters when Comparing Evolutionary Algorithms with Self-Adjusting Mutation Rates

We analyze the performance of the 2-rate $(1+\lambda)$ Evolutionary Algorithm (EA) with self-adjusting mutation rate control, its 3-rate counterpart, and a $(1+\lambda)$~EA variant using multiplicative update rules on the OneMax problem. We compare their efficiency for offspring population sizes ranging up to $\lambda=3,200$ 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 $\lambda$ the 2-rate EA performs best, the multiplicative updates become superior for starting for some threshold value of $\lambda$ between 50 and 100. Interestingly, for population sizes around 50, the $(1+\lambda)$~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 $\lambda$ is small. For both algorithms the situation reverses for large~$\lambda$.Comment: To appear at Genetic and Evolutionary Computation Conference (GECCO'19). v2: minor language revisio

Optimal Parameter Choices Through Self-Adjustment: Applying the 1/5-th Rule in Discrete Settings

While evolutionary algorithms are known to be very successful for a broad range of applications, the algorithm designer is often left with many algorithmic choices, for example, the size of the population, the mutation rates, and the crossover rates of the algorithm. These parameters are known to have a crucial influence on the optimization time, and thus need to be chosen carefully, a task that often requires substantial efforts. Moreover, the optimal parameters can change during the optimization process. It is therefore of great interest to design mechanisms that dynamically choose best-possible parameters. An example for such an update mechanism is the one-fifth success rule for step-size adaption in evolutionary strategies. While in continuous domains this principle is well understood also from a mathematical point of view, no comparable theory is available for problems in discrete domains. In this work we show that the one-fifth success rule can be effective also in discrete settings. We regard the $(1+(\lambda,\lambda))$~GA proposed in [Doerr/Doerr/Ebel: From black-box complexity to designing new genetic algorithms, TCS 2015]. We prove that if its population size is chosen according to the one-fifth success rule then the expected optimization time on \textsc{OneMax} is linear. This is better than what \emph{any} static population size $\lambda$ can achieve and is asymptotically optimal also among all adaptive parameter choices.Comment: This is the full version of a paper that is to appear at GECCO 201

Runtime Analysis of the (1+(λ,λ))(1+(\lambda,\lambda)) Genetic Algorithm on Random Satisfiable 3-CNF Formulas

The $(1+(\lambda,\lambda))$ 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 better than $\Theta(n \log n)$, which is a lower bound for most evolutionary algorithms on pseudo-Boolean problems with unique optimum. However, the self-adjusting version of the GA risks reaching population sizes at which the intermediate selection of the GA, due to the weaker fitness-distance correlation, is not able to distinguish a profitable offspring from others. We show that this problem can be overcome by equipping the self-adjusting GA with an upper limit for the population size. Apart from sparse instances, this limit can be chosen in a way that the asymptotic performance does not worsen compared to the idealistic OneMax case. Overall, this work shows that the $(1+(\lambda,\lambda))$ GA can provably have a good performance on combinatorial search and optimization problems also in the presence of a weaker fitness-distance correlation.Comment: An extended abstract of this report will appear in the proceedings of the 2017 Genetic and Evolutionary Computation Conference (GECCO 2017

Towards a Theory-Guided Benchmarking Suite for Discrete Black-Box Optimization Heuristics: Profiling (1+λ)(1+\lambda) EA Variants on OneMax and LeadingOnes

Theoretical and empirical research on evolutionary computation methods complement each other by providing two fundamentally different approaches towards a better understanding of black-box optimization heuristics. In discrete optimization, both streams developed rather independently of each other, but we observe today an increasing interest in reconciling these two sub-branches. In continuous optimization, the COCO (COmparing Continuous Optimisers) benchmarking suite has established itself as an important platform that theoreticians and practitioners use to exchange research ideas and questions. No widely accepted equivalent exists in the research domain of discrete black-box optimization. Marking an important step towards filling this gap, we adjust the COCO software to pseudo-Boolean optimization problems, and obtain from this a benchmarking environment that allows a fine-grained empirical analysis of discrete black-box heuristics. In this documentation we demonstrate how this test bed can be used to profile the performance of evolutionary algorithms. More concretely, we study the optimization behavior of several $(1+\lambda)$ EA variants on the two benchmark problems OneMax and LeadingOnes. This comparison motivates a refined analysis for the optimization time of the $(1+\lambda)$ EA on LeadingOnes

The 1/5-th Rule with Rollbacks: On Self-Adjustment of the Population Size in the (1+(λ,λ))(1+(\lambda,\lambda)) GA

Self-adjustment of parameters can significantly improve the performance of evolutionary algorithms. A notable example is the $(1+(\lambda,\lambda))$ genetic algorithm, where the adaptation of the population size helps to achieve the linear runtime on the OneMax problem. However, on problems which interfere with the assumptions behind the self-adjustment procedure, its usage can lead to performance degradation compared to static parameter choices. In particular, the one fifth rule, which guides the adaptation in the example above, is able to raise the population size too fast on problems which are too far away from the perfect fitness-distance correlation. We propose a modification of the one fifth rule in order to have less negative impact on the performance in scenarios when the original rule reduces the performance. Our modification, while still having a good performance on OneMax, both theoretically and in practice, also shows better results on linear functions with random weights and on random satisfiable MAX-SAT instances.Comment: 17 pages, 2 figures, 1 table. An extended two-page abstract of this work will appear in proceedings of the Genetic and Evolutionary Computation Conference, GECCO'1

Inheritance-Based Diversity Measures for Explicit Convergence Control in Evolutionary Algorithms

Diversity is an important factor in evolutionary algorithms to prevent premature convergence towards a single local optimum. In order to maintain diversity throughout the process of evolution, various means exist in literature. We analyze approaches to diversity that (a) have an explicit and quantifiable influence on fitness at the individual level and (b) require no (or very little) additional domain knowledge such as domain-specific distance functions. We also introduce the concept of genealogical diversity in a broader study. We show that employing these approaches can help evolutionary algorithms for global optimization in many cases.Comment: GECCO '18: Genetic and Evolutionary Computation Conference, 2018, Kyoto, Japa