On the Expected Runtime and the Success Probability of Evolutionary Algorithms

Evolutionary algorithms are randomized search heuristics whose general variants have been successfully applied in black box optimization. In this scenario the function f to be optimized is not known in advance and knowledge on f can be obtained only by sampling search points a revealing the value of f a . In order to analyze the behavior of different variants of evolutionary algorithms on certain functions f, the expected runtime until some optimal search point is sampled and the success probability i.e., the probability that an optimal search point is among the first sampled points , are of particular interest. Here a simple method for the analysis is discussed and applied to several functions. For specific situations more involved techniques are necessary. Two such results are presented. First, it is shown that the most simple evolutionary algorithm optimizeseach pseudo-boolean linear function in an expected time of O n log n . Second, an example is shown where crossover decreases the expected runtime from superpolynomial to polynomial

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

Level-Based Analysis of the Population-Based Incremental Learning Algorithm

The Population-Based Incremental Learning (PBIL) algorithm uses a convex combination of the current model and the empirical model to construct the next model, which is then sampled to generate offspring. The Univariate Marginal Distribution Algorithm (UMDA) is a special case of the PBIL, where the current model is ignored. Dang and Lehre (GECCO 2015) showed that UMDA can optimise LeadingOnes efficiently. The question still remained open if the PBIL performs equally well. Here, by applying the level-based theorem in addition to Dvoretzky--Kiefer--Wolfowitz inequality, we show that the PBIL optimises function LeadingOnes in expected time $\mathcal{O}(n\lambda \log \lambda + n^2)$ for a population size $\lambda = \Omega(\log n)$, which matches the bound of the UMDA. Finally, we show that the result carries over to BinVal, giving the fist runtime result for the PBIL on the BinVal problem.Comment: To appea

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

Improved Runtime Bounds for the Univariate Marginal Distribution Algorithm via Anti-Concentration

Unlike traditional evolutionary algorithms which produce offspring via genetic operators, Estimation of Distribution Algorithms (EDAs) sample solutions from probabilistic models which are learned from selected individuals. It is hoped that EDAs may improve optimisation performance on epistatic fitness landscapes by learning variable interactions. However, hardly any rigorous results are available to support claims about the performance of EDAs, even for fitness functions without epistasis. The expected runtime of the Univariate Marginal Distribution Algorithm (UMDA) on OneMax was recently shown to be in $\mathcal{O}\left(n\lambda\log \lambda\right)$ by Dang and Lehre (GECCO 2015). Later, Krejca and Witt (FOGA 2017) proved the lower bound $\Omega\left(\lambda\sqrt{n}+n\log n\right)$ via an involved drift analysis. We prove a $\mathcal{O}\left(n\lambda\right)$ bound, given some restrictions on the population size. This implies the tight bound $\Theta\left(n\log n\right)$ when $\lambda=\mathcal{O}\left(\log n\right)$, matching the runtime of classical EAs. Our analysis uses the level-based theorem and anti-concentration properties of the Poisson-Binomial distribution. We expect that these generic methods will facilitate further analysis of EDAs.Comment: 19 pages, 1 figur

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