This paper extends the runtime analysis of non-elitist evolutionary algorithms (EAs) with fitness-proportionate selection from the simple OneMax function to the linear functions. Not only does our analysis cover a larger class of fitness functions, it also holds for a wider range of mutation rates. We show that with overwhelmingly high probability, no linear function can be optimised in less than exponential time, assuming bitwise mutation rate $Θ(1/n)$ and population size $λ=n^k$ for any constant $k>2$. In contrast to this negative result, we also show that for any linear function with polynomially bounded weights, the EA achieves a polynomial expected runtime if the mutation rate is reduced to $Θ(1/n^2)$ and the population size is sufficiently large. Furthermore, the EA with mutation rate $χ/n=Θ(1/n)$ and modest population size $λ=Ω(\ln n)$ optimises the scaled fitness function $e^{(χ+\varepsilon)f(x)}$ for any linear function $f$ and any $\varepsilon>0$ in expected time $O(nλ\lnλ+n^2)$. These upper bounds also extend to some additively decomposed fitness functions, such as the Royal Road functions. We expect that the obtained results may be useful not only for the development of the theory of evolutionary algorithms, but also for biological applications, such as the directed evolution.
We consider the expected runtime of non-elitist evolutionary algorithms (EAs), when they are applied to a family of fitness functions with a plateau of second-best fitness in a Hamming ball of radius r around a unique global optimum. On one hand, using the level-based theorems, we obtain polynomial upper bounds on the expected runtime for some modes of non-elitist EA based on unbiased mutation and the bitwise mutation in particular. On the other hand, we show that the EA with fitness proportionate selection is inefficient if the bitwise mutation is used with the standard settings of mutation p…
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…
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…
We study evolutionary algorithms in a dynamic setting, where for each generation a different fitness function is chosen, and selection is performed with respect to the current fitness function. Specifically, we consider Dynamic BinVal, in which the fitness functions for each generation is given by the linear function BinVal, but in each generation the order of bits is randomly permuted. For the (1+1)-EA it was known that there is an efficiency threshold $c_0$ for the mutation parameter, at which the runtime switches from quasilinear to exponential. A previous empirical evidence suggested that …
The lexicase parent selection method selects parents by considering performance on individual data points in random order instead of using a fitness function based on an aggregated data accuracy. While the method has demonstrated promise in genetic programming and more recently in genetic algorithms, its applications in other forms of evolutionary machine learning have not been explored. In this paper, we investigate the use of lexicase parent selection in Learning Classifier Systems (LCS) and study its effect on classification problems in a supervised setting. We further introduce a new varia…
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…
To gain a better theoretical understanding of how evolutionary algorithms (EAs) cope with plateaus of constant fitness, we propose the $n$-dimensional Plateau$_k$ function as natural benchmark and analyze how different variants of the $(1 + 1)$ EA optimize it. The Plateau$_k$ function has a plateau of second-best fitness in a ball of radius $k$ around the optimum. As evolutionary algorithm, we regard the $(1 + 1)$ EA using an arbitrary unbiased mutation operator. Denoting by $α$ the random number of bits flipped in an application of this operator and assuming that $\Pr[α= 1]$ has at least some…
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…
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…
In this paper, we consider a fitness-level model of a non-elitist mutation-only evolutionary algorithm (EA) with tournament selection. The model provides upper and lower bounds for the expected proportion of the individuals with fitness above given thresholds. In the case of so-called monotone mutation, the obtained bounds imply that increasing the tournament size improves the EA performance. As corollaries, we obtain an exponentially vanishing tail bound for the Randomized Local Search on unimodal functions and polynomial upper bounds on the runtime of EAs on 2-SAT problem and on a family of …
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. …
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^{…
A fitness assignment process transforms the features (such as the objective value) of a candidate solution to a scalar fitness, which then is the basis for selection. Under Frequency Fitness Assignment (FFA), the fitness corresponding to an objective value is its encounter frequency in selection steps and is subject to minimization. FFA creates algorithms that are not biased towards better solutions and are invariant under all injective transformations of the objective function value. We investigate the impact of FFA on the performance of two theory-inspired, state-of-the-art EAs, the Greedy (…
Many real optimisation problems lead to multimodal domains and so require the identification of multiple optima. Niching methods have been developed to maintain the population diversity, to investigate many peaks in parallel and to reduce the effect of genetic drift. Using rigorous runtime analysis, we analyse for the first time two well known niching methods: probabilistic crowding and restricted tournament selection (RTS). We incorporate both methods into a $(μ+1)~EA$ on the bimodal function Twomax where the goal is to find two optima at opposite ends of the search space. In probabilistic cr…
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…
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…
We present a time complexity analysis of the Opt-IA artificial immune system (AIS). We first highlight the power and limitations of its distinguishing operators (i.e., hypermutations with mutation potential and ageing) by analysing them in isolation. Recent work has shown that ageing combined with local mutations can help escape local optima on a dynamic optimisation benchmark function. We generalise this result by rigorously proving that, compared to evolutionary algorithms (EAs), ageing leads to impressive speed-ups on the standard Cliff benchmark function both when using local and global mu…
Various studies have shown that immune system inspired hypermutation operators can allow artificial immune systems (AIS) to be very efficient at escaping local optima of multimodal optimisation problems. However, this efficiency comes at the expense of considerably slower runtimes during the exploitation phase compared to standard evolutionary algorithms. We propose modifications to the traditional `hypermutations with mutation potential' (HMP) that allow them to be efficient at exploitation as well as maintaining their effective explorative characteristics. Rather than deterministically evalu…
Self-adjustment of parameters can significantly improve the performance of evolutionary algorithms. A notable example is the $(1+(λ,λ))$ 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 …
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…
Signed Sensitivity of Expected Hitting Time to Mutation Rate in the (1+1) EA: Per-State Sign Theorems and Verifiable Certificates for Non-Lumpable Families
Institute of Automation, Chinese Academy of Sciences (CASIA)（中国科学院自动化研究所）；University of Chinese Academy of Sciences (UCAS)（中国科学院大学）
