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

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

From Understanding Genetic Drift to a Smart-Restart Parameter-less Compact Genetic Algorithm

One of the key difficulties in using estimation-of-distribution algorithms is choosing the population size(s) appropriately: Too small values lead to genetic drift, which can cause enormous difficulties. In the regime with no genetic drift, however, often the runtime is roughly proportional to the population size, which renders large population sizes inefficient. Based on a recent quantitative analysis which population sizes lead to genetic drift, we propose a parameter-less version of the compact genetic algorithm that automatically finds a suitable population size without spending too much time in situations unfavorable due to genetic drift. We prove a mathematical runtime guarantee for this algorithm and conduct an extensive experimental analysis on four classic benchmark problems both without and with additive centered Gaussian posterior noise. The former shows that under a natural assumption, our algorithm has a performance very similar to the one obtainable from the best problem-specific population size. The latter confirms that missing the right population size in the original cGA can be detrimental and that previous theory-based suggestions for the population size can be far away from the right values; it also shows that our algorithm as well as a previously proposed parameter-less variant of the cGA based on parallel runs avoid such pitfalls. Comparing the two parameter-less approaches, ours profits from its ability to abort runs which are likely to be stuck in a genetic drift situation.Comment: 4 figures. Extended version of a paper appearing at GECCO 202

On the runtime analysis of generalised selection hyper-heuristics for pseudo-boolean optimisation

© 2017 ACM. Selection hyper-heuristics are randomised search methodologies which choose and execute heuristics from a set of low-level heuristics. Recent time complexity analyses for the LeadingOnes benchmark function have shown that the standard simple random, permutation, random gradient, greedy and reinforcement learning selection mechanisms show no effects of learning. The idea behind the learning mechanisms is to continue to exploit the currently selected heuristic as long as it is successful. However, the probability that a promising heuristic is successful in the next step is relatively low when perturbing a reasonable solution to a combinatorial optimisation problem. In this paper we generalise the classical selection-perturbation mechanisms so success can be measured over some fixed period of length t, rather than in a single iteration. We present a benchmark function where it is necessary to learn to exploit a particular low-level heuristic, rigorously proving that it makes the difference between an efficient and an inefficient algorithm. For LeadingOnes we prove that the generalised random gradient mechanism approaches optimal performance while generalised greedy, although not as fast, still outperforms random local search. An experimental analysis shows that combining the two generalised mechanisms leads to even better performance

Fast re-optimization via structural diversity

International audienceWhen a problem instance is perturbed by a small modification, one would hope to find a good solution for the new instance by building on a known good solution for the previous one. Via a rigorous mathematical analysis, we show that evolutionary algorithms, despite usually being robust problem solvers, can have unexpected difficulties to solve such re-optimization problems. When started with a random Hamming neighbor of the optimum, the (1+1) evolutionary algorithm takes $\Omega(n^2)$ time to optimize the LeadingOnes benchmark function, which is the same asymptotic optimization time when started in a randomly chosen solution. There is hence no significant advantage from re-optimizing a structurally good solution. We then propose a way to overcome such difficulties. As our mathematical analysis reveals, the reason for this undesired behavior is that during the optimization structurally good solutions can easily be replaced by structurally worse solutions of equal or better fitness. We propose a simple diversity mechanism that prevents this behavior, thereby reducing the re-optimization time for LeadingOnes to $O(\gamma\delta n)$, where $\gamma$ is the population size used by the diversity mechanism and $\delta \le \gamma$ the Hamming distance of the new optimum from the previous solution. We show similarly fast re-optimization times for the optimization of linear functions with changing constraints and for the minimum spanning tree problem

Self-adjusting mutation rates with provably optimal success rules

International audienceThe 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 s. We analyze in this work how the performance of the (1+1) Evolutionary Algorithm (EA) 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 rate 1/e. We also prove that the runtime obtained by this parameter setting is asymptotically optimal among all dynamic choices of the mutation rate for the (1+1) EA, up to lower order error terms. We show similar results for the resampling variant of the (1+1) EA, which enforces to flip at least one bit per iteration

Does Comma Selection Help To Cope With Local Optima?

One hope when using non-elitism in evolutionary computation is that the ability to abandon the current-best solution aids leaving local optima. To improve our understanding of this mechanism, we perform a rigorous runtime analysis of a basic non-elitist evolutionary algorithm (EA), the (µ, λ) EA, on the most basic benchmark function with a local optimum, the jump function. We prove that for all reasonable values of the parameters and the problem, the expected runtime of the (µ, λ) EA is, apart from lower order terms, at least as large as the expected runtime of its elitist counterpart, the (µ + λ) EA (for which we conduct the first runtime analysis on jump functions to allow this comparison). Consequently, the ability of the (µ, λ) EA to leave local optima to inferior solutions does not lead to a runtime advantage. We complement this lower bound with an upper bound that, for broad ranges of the parameters, is identical to our lower bound apart from lower order terms. This is the first runtime result for a non-elitist algorithm on a multi-modal problem that is tight apart from lower order terms

Dynamic parameter choices in evolutionary computation (tutorial at GECCO 2019)

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