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

Black-Box Complexity of the Binary Value Function

The binary value function, or BinVal, has appeared in several studies in theory of evolutionary computation as one of the extreme examples of linear pseudo-Boolean functions. Its unbiased black-box complexity was previously shown to be at most $\lceil \log_2 n \rceil + 2$, where $n$ is the problem size. We augment it with an upper bound of $\log_2 n + 2.42141558 - o(1)$, which is more precise for many values of $n$. We also present a lower bound of $\log_2 n + 1.1186406 - o(1)$. Additionally, we prove that BinVal is an easiest function among all unimodal pseudo-Boolean functions at least for unbiased algorithms.Comment: 24 pages, one figure. An extended two-page abstract of this work will appear in proceedings of the Genetic and Evolutionary Computation Conference, GECCO'1

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

Improving Time and Memory Efficiency of Genetic Algorithms by Storing Populations as Minimum Spanning Trees of Patches

In many applications of evolutionary algorithms the computational cost of applying operators and storing populations is comparable to the cost of fitness evaluation. Furthermore, by knowing what exactly has changed in an individual by an operator, it is possible to recompute fitness value much more efficiently than from scratch. The associated time and memory improvements have been available for simple evolutionary algorithms, few specific genetic algorithms and in the context of gray-box optimization, but not for all algorithms, and the main reason is that it is difficult to achieve in algorithms using large arbitrarily structured populations. This paper makes a first step towards improving this situation. We show that storing the population as a minimum spanning tree, where vertices correspond to individuals but only contain meta-information about them, and edges store structural differences, or patches, between the individuals, is a viable alternative to the straightforward implementation. Our experiments suggest that significant, even asymptotic, improvements -- including execution of crossover operators! -- can be achieved in terms of both memory usage and computational costs.Comment: Accepted to the GECCO'23 conference, EvoSoft worksho

An Asynchronous Implementation of the Limited Memory CMA-ES

We present our asynchronous implementation of the LM-CMA-ES algorithm, which is a modern evolution strategy for solving complex large-scale continuous optimization problems. Our implementation brings the best results when the number of cores is relatively high and the computational complexity of the fitness function is also high. The experiments with benchmark functions show that it is able to overcome its origin on the Sphere function, reaches certain thresholds faster on the Rosenbrock and Ellipsoid function, and surprisingly performs much better than the original version on the Rastrigin function.Comment: 9 pages, 4 figures, 4 tables; this is a full version of a paper which has been accepted as a poster to IEEE ICMLA conference 201

Better Fixed-Arity Unbiased Black-Box Algorithms

In their GECCO'12 paper, Doerr and Doerr proved that the $k$-ary unbiased black-box complexity of OneMax on $n$ bits is $O(n/k)$ for $2\le k\le O(\log n)$. We propose an alternative strategy for achieving this unbiased black-box complexity when $3\le k\le\log_2 n$. While it is based on the same idea of block-wise optimization, it uses $k$-ary unbiased operators in a different way. For each block of size $2^{k-1}-1$ we set up, in $O(k)$ queries, a virtual coordinate system, which enables us to use an arbitrary unrestricted algorithm to optimize this block. This is possible because this coordinate system introduces a bijection between unrestricted queries and a subset of $k$-ary unbiased operators. We note that this technique does not depend on OneMax being solved and can be used in more general contexts. This together constitutes an algorithm which is conceptually simpler than the one by Doerr and Doerr, and at the same time achieves better constant factors in the asymptotic notation. Our algorithm works in $(2+o(1))\cdot n/(k-1)$, where $o(1)$ relates to $k$. Our experimental evaluation of this algorithm shows its efficiency already for $3\le k\le6$.Comment: An extended abstract will appear at GECCO'1

Better Fixed-Arity Unbiased Black-Box Algorithms

In their GECCO'12 paper, Doerr and Doerr proved that the $k$-ary unbiased black-box complexity of OneMax on $n$ bits is $O(n/k)$ for $2\le k\le O(\log n)$. We propose an alternative strategy for achieving this unbiased black-box complexity when $3\le k\le\log_2 n$. While it is based on the same idea of block-wise optimization, it uses $k$-ary unbiased operators in a different way. For each block of size $2^{k-1}-1$ we set up, in $O(k)$ queries, a virtual coordinate system, which enables us to use an arbitrary unrestricted algorithm to optimize this block. This is possible because this coordinate system introduces a bijection between unrestricted queries and a subset of $k$-ary unbiased operators. We note that this technique does not depend on OneMax being solved and can be used in more general contexts. This together constitutes an algorithm which is conceptually simpler than the one by Doerr and Doerr, and at the same time achieves better constant factors in the asymptotic notation. Our algorithm works in $(2+o(1))\cdot n/(k-1)$, where $o(1)$ relates to $k$. Our experimental evaluation of this algorithm shows its efficiency already for $3\le k\le6$.Comment: An extended abstract will appear at GECCO'1