A Rigorous Runtime Analysis of the $(1 + (λ, λ))$ GA on Jump Functions
Denis Antipov, Benjamin Doerr, Vitalii Karavaev
备注：
The full version of the paper "The $(1 + (λ, λ))$ GA Is Even Faster on Multimodal Problems" presented at GECCO 2020 containing all the proofs omitted in the conference paper
The $(1 + (λ,λ))$ genetic algorithm is a younger evolutionary algorithm trying to profit also from inferior solutions. Rigorous runtime analyses on unimodal fitness functions showed that it can indeed be faster than classical evolutionary algorithms, though on these simple problems the gains were only moderate.
In this work, we conduct the first runtime analysis of this algorithm on a multimodal problem class, the jump functions benchmark. We show that with the right parameters, the \ollga optimizes any jump function with jump size $2 \le k \le n/4$ in expected time $O(n^{(k+1)/2} e^{O(k)} k^{-k/2})$, which significantly and already for constant~$k$ outperforms standard mutation-based algorithms with their $Θ(n^k)$ runtime and standard crossover-based algorithms with their $\tilde{O}(n^{k-1})$ runtime guarantee.
For the isolated problem of leaving the local optimum of jump functions, we determine provably optimal parameters that lead to a runtime of $(n/k)^{k/2} e^{Θ(k)}$. This suggests some general advice on how to set the parameters of the \ollga, which might ease the further use of this algorithm.
It was recently observed that the $(1+(λ,λ))$ genetic algorithm can comparably easily escape the local optimum of the jump functions benchmark. Consequently, this algorithm can optimize the jump function with jump size $k$ in an expected runtime of only $n^{(k + 1)/2}k^{-k/2}e^{O(k)}$ fitness evaluations (Antipov, Doerr, Karavaev (GECCO 2020)). To obtain this performance, however, a non-standard parameter setting depending on the jump size $k$ was used. To overcome this difficulty, we propose to choose two parameters of the $(1+(λ,λ))$ genetic algorithm randomly from a power-law distribution. …
In the first runtime analysis of an estimation-of-distribution algorithm (EDA) on the multi-modal jump function class, Hasenöhrl and Sutton (GECCO 2018) proved that the runtime of the compact genetic algorithm with suitable parameter choice on jump functions with high probability is at most polynomial (in the dimension) if the jump size is at most logarithmic (in the dimension), and is at most exponential in the jump size if the jump size is super-logarithmic. The exponential runtime guarantee was achieved with a hypothetical population size that is also exponential in the jump size. Consequen…
In the first and so far only mathematical runtime analysis of an estimation-of-distribution algorithm (EDA) on a multimodal problem, Hasenöhrl and Sutton (GECCO 2018) showed for any $k = o(n)$ that the compact genetic algorithm (cGA) with any hypothetical population size $μ= Ω(ne^{4k} + n^{3.5+\varepsilon})$ with high probability finds the optimum of the $n$-dimensional jump function with jump size $k$ in time $O(μn^{1.5} \log n)$. We significantly improve this result for small jump sizes $k \le \frac 1 {20} \ln n -1$. In this case, already for $μ= Ω(\sqrt n \log n) \cap \text{poly}(n)$ the ru…
Population diversity is crucial in evolutionary algorithms to enable global exploration and to avoid poor performance due to premature convergence. This book chapter reviews runtime analyses that have shown benefits of population diversity, either through explicit diversity mechanisms or through naturally emerging diversity. These works show that the benefits of diversity are manifold: diversity is important for global exploration and the ability to find several global optima. Diversity enhances crossover and enables crossover to be more effective than mutation. Diversity can be crucial in dyn…
Most evolutionary algorithms (EAs) used in practice employ crossover. In contrast, only for few and mostly artificial examples a runtime advantage from crossover could be proven with mathematical means. The most convincing such result shows that the $(μ+1)$ genetic algorithm (GA) with population size $μ=O(n)$ optimizes jump functions with gap size $k \ge 3$ in time $O(n^k / μ+ n^{k-1}\log n)$, beating the $Θ(n^k)$ runtime of many mutation-based EAs. This result builds on a proof that the GA occasionally and then for an expected number of $Ω(μ^2)$ iterations has a population that is not dominat…
We prove that the compact genetic algorithm (cGA) with hypothetical population size $μ= Ω(\sqrt n \log n) \cap \text{poly}(n)$ with high probability finds the optimum of any $n$-dimensional jump function with jump size $k < \frac 1 {20} \ln n$ in $O(μ\sqrt n)$ iterations. Since it is known that the cGA with high probability needs at least $Ω(μ\sqrt n + n \log n)$ iterations to optimize the unimodal OneMax function, our result shows that the cGA in contrast to most classic evolutionary algorithms here is able to cross moderate-sized valleys of low fitness at no extra cost. Our runtime guarantee…
We perform a rigorous runtime analysis for the Univariate Marginal Distribution Algorithm on the LeadingOnes function, a well-known benchmark function in the theory community of evolutionary computation with a high correlation between decision variables. For a problem instance of size $n$, the currently best known upper bound on the expected runtime is $\mathcal{O}(nλ\logλ+n^2)$ (Dang and Lehre, GECCO 2015), while a lower bound necessary to understand how the algorithm copes with variable dependencies is still missing. Motivated by this, we show that the algorithm requires a $e^{Ω(μ)}$ runtime…
For genetic algorithms using a bit-string representation of length~$n$, the general recommendation is to take $1/n$ as mutation rate. In this work, we discuss whether this is really justified for multimodal functions. Taking jump functions and the $(1+1)$ evolutionary algorithm as the simplest example, we observe that larger mutation rates give significantly better runtimes. For the $\jump_{m,n}$ function, any mutation rate between $2/n$ and $m/n$ leads to a speed-up at least exponential in $m$ compared to the standard choice. The asymptotically best runtime, obtained from using the mutation r…
Laboratoire d’Informatique (LIX)（计算机科学实验室（LIX））；CNRS（法国国家科学研究中心）；École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）
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 $…
Despite significant progress in the theory of evolutionary algorithms, the theoretical understanding of evolutionary algorithms which use non-trivial populations remains challenging and only few rigorous results exist. Already for the most basic problem, the determination of the asymptotic runtime of the $(μ+λ)$ evolutionary algorithm on the simple OneMax benchmark function, only the special cases $μ=1$ and $λ=1$ have been solved. In this work, we analyze this long-standing problem and show the asymptotically tight result that the runtime $T$, the number of iterations until the optimum is foun…
Evolutionary algorithms (EAs) form a popular optimisation paradigm inspired by natural evolution. In recent years the field of evolutionary computation has developed a rigorous analytical theory to analyse their runtime on many illustrative problems. Here we apply this theory to a simple model of natural evolution. In the Strong Selection Weak Mutation (SSWM) evolutionary regime the time between occurrence of new mutations is much longer than the time it takes for a new beneficial mutation to take over the population. In this situation, the population only contains copies of one genotype and e…
Laboratoire d’Informatique (LIX)（计算机科学实验室（LIX））；CNRS（法国国家科学研究中心）；École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）
While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems. To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the \textsc{LeadingO…
Clearing is a niching method inspired by the principle of assigning the available resources among a niche to a single individual. The clearing procedure supplies these resources only to the best individual of each niche: the winner. So far, its analysis has been focused on experimental approaches that have shown that clearing is a powerful diversity-preserving mechanism. Using rigorous runtime analysis to explain how and why it is a powerful method, we prove that a mutation-based evolutionary algorithm with a large enough population size, and a phenotypic distance function always succeeds in o…
Parameterized runtime analysis seeks to understand the influence of problem structure on algorithmic runtime. In this paper, we contribute to the theoretical understanding of evolutionary algorithms and carry out a parameterized analysis of evolutionary algorithms for the Euclidean traveling salesperson problem (Euclidean TSP). We investigate the structural properties in TSP instances that influence the optimization process of evolutionary algorithms and use this information to bound the runtime of simple evolutionary algorithms. Our analysis studies the runtime in dependence of the number of …
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 show…
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…
The interplay between mutation and selection plays a fundamental role in the behaviour of evolutionary algorithms (EAs). However, this interplay is still not completely understood. This paper presents a rigorous runtime analysis of a non-elitist population-based EA that uses the linear ranking selection mechanism. The analysis focuses on how the balance between parameter $η$, controlling the selection pressure in linear ranking, and parameter $χ$ controlling the bit-wise mutation rate, impacts the runtime of the algorithm. The results point out situations where a correct balance between select…
We introduce a new benchmark problem called Deceptive Leading Blocks (DLB) to rigorously study the runtime of the Univariate Marginal Distribution Algorithm (UMDA) in the presence of epistasis and deception. We show that simple Evolutionary Algorithms (EAs) outperform the UMDA unless the selective pressure $μ/λ$ is extremely high, where $μ$ and $λ$ are the parent and offspring population sizes, respectively. More precisely, we show that the UMDA with a parent population size of $μ=Ω(\log n)$ has an expected runtime of $e^{Ω(μ)}$ on the DLB problem assuming any selective pressure $\fracμλ \geq …
Under Frequency Fitness Assignment (FFA), the fitness corresponding to an objective value is its encounter frequency in fitness assignment steps and is subject to minimization. FFA renders optimization processes invariant under bijective transformations of the objective function value. On TwoMax, Jump, and Trap functions of dimension s, the classical (1+1)-EA with standard mutation at rate 1/s can have expected runtimes exponential in s. In our experiments, a (1+1)-FEA, the same algorithm but using FFA, exhibits mean runtimes that seem to scale as $s^2\ln{s}$. Since Jump and Trap are bijective…
Theoretical Analyses of Multiobjective Evolutionary Algorithms on Multimodal Objectives
School of Computer Science and Technology（计算机科学与技术学院）；International Research Institute for Artificial Intelligence（国际人工智能研究院）；Harbin Institute of Technology（哈尔滨工业大学）；Southern University of Science and Technology（南方科技大学）；Laboratoire d’Informatique (LIX)（信息学实验室）；École Polytechnique（巴黎综合理工学院）；CNRS（法国国家科学研究中心）；Institut Polytechnique de Paris（巴黎理工学院）
Due to the more complicated population dynamics of the NSGA-II, none of the existing runtime guarantees for this algorithm is accompanied by a non-trivial lower bound. Via a first mathematical understanding of the population dynamics of the NSGA-II, that is, by estimating the expected number of individuals having a certain objective value, we prove that the NSGA-II with suitable population size needs $Ω(Nn\log n)$ function evaluations to find the Pareto front of the OneMinMax problem and $Ω(Nn^k)$ evaluations on the OneJumpZeroJump problem with jump size $k$. These bounds are asymptotically ti…
Structural plasticity of the brain describes the creation of new and the deletion of old synapses over time. Rinke et al. (JPDC 2018) introduced a scalable algorithm that simulates structural plasticity for up to one billion neurons on current hardware using a variant of the Barnes-Hut algorithm. They demonstrate good scalability and prove a runtime complexity of $O(n \log^2 n)$. In this comment paper, we show that with careful consideration of the algorithm and a rigorous proof, the theoretical runtime can even be classified as $O(n \log n)$.
Runtime Analysis for the NSGA-II: Proving, Quantifying, and Explaining the Inefficiency For Many Objectives
School of Computer Science and Technology（计算机科学与技术学院）；International Research Institute for Artificial Intelligence（国际人工智能研究院）；Harbin Institute of Technology（哈尔滨工业大学）；Laboratoire d’Informatique (LIX)（信息学实验室（LIX））；École Polytechnique（巴黎综合理工学院）；CNRS（法国国家科学研究中心）；Institut Polytechnique de Paris（巴黎理工学院）
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…
Population diversity is essential for avoiding premature convergence in Genetic Algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous runtime analysis to gain insight into population dynamics and GA performance for the ($μ$+1) GA and the $\text{Jump}_k$ test function. We show that the interplay of crossover and mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to improvements of the expected optimisation time of order $Ω(n/\log n)$ compared to mutation-only alg…
The runtime of evolutionary algorithms (EAs) depends critically on their parameter settings, which are often problem-specific. Automated schemes for parameter tuning have been developed to alleviate the high costs of manual parameter tuning. Experimental results indicate that self-adaptation, where parameter settings are encoded in the genomes of individuals, can be effective in continuous optimisation. However, results in discrete optimisation have been less conclusive. Furthermore, a rigorous runtime analysis that explains how self-adaptation can lead to asymptotic speedups has been missing.…
