arXiv subjects

Dirk Sudholt

Publications and source records attributed to Dirk Sudholt.

At least 37 records · Page 2Linked to original sources

arXiv2020

Analysis of the Performance of Algorithm Configurators for Search Heuristics with Global Mutation Operators

Recently it has been proved that a simple algorithm configurator called ParamRLS can efficiently identify the optimal neighbourhood size to be used by stochastic local search to optimise two standard benchmark problem classes. In this paper we analyse the performance of algorithm configurators for tuning the more sophisticated global mutation operator used in standard evolutionary algorithms, which flips each of the $n$ bits independently with probability $χ/n$ and the best value for $χ$ has to be identified. We compare the performance of configurators when the best-found fitness values within the cutoff time $κ$ are used to compare configurations against the actual optimisation time for two standard benchmark problem classes, Ridge and LeadingOnes. We rigorously prove that all algorithm configurators that use optimisation time as performance metric require cutoff times that are at least as large as the expected optimisation time to identify the optimal configuration. Matters are considerably different if the fitness metric is used. To show this we prove that the simple ParamRLS-F configurator can identify the optimal mutation rates even when using cutoff times that are considerably smaller than the expected optimisation time of the best parameter value for both problem classes.

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arXiv2019

On the Impact of the Cutoff Time on the Performance of Algorithm Configurators

Algorithm configurators are automated methods to optimise the parameters of an algorithm for a class of problems. We evaluate the performance of a simple random local search configurator (ParamRLS) for tuning the neighbourhood size $k$ of the RLS$_k$ algorithm. We measure performance as the expected number of configuration evaluations required to identify the optimal value for the parameter. We analyse the impact of the cutoff time $κ$ (the time spent evaluating a configuration for a problem instance) on the expected number of configuration evaluations required to find the optimal parameter value, where we compare configurations using either best found fitness values (ParamRLS-F) or optimisation times (ParamRLS-T). We consider tuning RLS$_k$ for a variant of the Ridge function class (Ridge*), where the performance of each parameter value does not change during the run, and for the OneMax function class, where longer runs favour smaller $k$. We rigorously prove that ParamRLS-F efficiently tunes RLS$_k$ for Ridge* for any $κ$ while ParamRLS-T requires at least quadratic $κ$. For OneMax ParamRLS-F identifies $k=1$ as optimal with linear $κ$ while ParamRLS-T requires a $κ$ of at least $Ω(n\log n)$. For smaller $κ$ ParamRLS-F identifies that $k>1$ performs better while ParamRLS-T returns $k$ chosen uniformly at random.

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arXiv2019

Parallel Black-Box Complexity with Tail Bounds

We propose a new black-box complexity model for search algorithms evaluating $λ$ search points in parallel. The parallel unary unbiased black-box complexity gives lower bounds on the number of function evaluations every parallel unary unbiased black-box algorithm needs to optimise a given problem. It captures the inertia caused by offspring populations in evolutionary algorithms and the total computational effort in parallel metaheuristics. We present complexity results for LeadingOnes and OneMax. Our main result is a general performance limit: we prove that on every function every $λ$-parallel unary unbiased algorithm needs at least $Ω(\frac{λn}{\ln λ} + n \log n)$ evaluations to find any desired target set of up to exponential size, with an overwhelming probability. This yields lower bounds for the typical optimisation time on unimodal and multimodal problems, for the time to find any local optimum, and for the time to even get close to any optimum. The power and versatility of this approach is shown for a wide range of illustrative problems from combinatorial optimisation. Our performance limits can guide parameter choice and algorithm design; we demonstrate the latter by presenting an optimal $λ$-parallel algorithm for OneMax that uses parallelism most effectively.

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arXiv2018

Analysing the Robustness of Evolutionary Algorithms to Noise: Refined Runtime Bounds and an Example Where Noise is Beneficial

We analyse the performance of well-known evolutionary algorithms (1+1)EA and (1+$λ$)EA in the prior noise model, where in each fitness evaluation the search point is altered before evaluation with probability $p$. We present refined results for the expected optimisation time of the (1+1)EA and the (1+$λ$)EA on the function LeadingOnes, where bits have to be optimised in sequence. Previous work showed that the (1+1)EA on LeadingOnes runs in polynomial expected time if $p = O((\log n)/n^2)$ and needs superpolynomial expected time if $p = ω((\log n)/n)$, leaving a huge gap for which no results were known. We close this gap by showing that the expected optimisation time is $Θ(n^2) \cdot \exp(Θ(\min\{pn^2, n\}))$ for all $p \le 1/2$, allowing for the first time to locate the threshold between polynomial and superpolynomial expected times at $p = Θ((\log n)/n^2)$. Hence the (1+1)EA on LeadingOnes is much more sensitive to noise than previously thought. We also show that offspring populations of size $λ\ge 3.42\log n$ can effectively deal with much higher noise than known before. Finally, we present an example of a rugged landscape where prior noise can help to escape from local optima by blurring the landscape and allowing a hill climber to see the underlying gradient. We prove that in this particular setting noise can have a highly beneficial effect on performance.

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arXiv2018

Design and Analysis of Diversity-Based Parent Selection Schemes for Speeding Up Evolutionary Multi-objective Optimisation

Parent selection in evolutionary algorithms for multi-objective optimisation is usually performed by dominance mechanisms or indicator functions that prefer non-dominated points. We propose to refine the parent selection on evolutionary multi-objective optimisation with diversity-based metrics. The aim is to focus on individuals with a high diversity contribution located in poorly explored areas of the search space, so the chances of creating new non-dominated individuals are better than in highly populated areas. We show by means of rigorous runtime analysis that the use of diversity-based parent selection mechanisms in the Simple Evolutionary Multi-objective Optimiser (SEMO) and Global SEMO for the well known bi-objective functions ${\rm O{\small NE}M{\small IN}M{\small AX}}$ and ${\rm LOTZ}$ can significantly improve their performance. Our theoretical results are accompanied by experimental studies that show a correspondence between theory and empirical results and motivate further theoretical investigations in terms of stagnation. We show that stagnation might occur when favouring individuals with a high diversity contribution in the parent selection step and provide a discussion on which scheme to use for more complex problems based on our theoretical and experimental results.

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arXiv2018

Memetic Algorithms Beat Evolutionary Algorithms on the Class of Hurdle Problems

Memetic algorithms are popular hybrid search heuristics that integrate local search into the search process of an evolutionary algorithm in order to combine the advantages of rapid exploitation and global optimisation. However, these algorithms are not well understood and the field is lacking a solid theoretical foundation that explains when and why memetic algorithms are effective. We provide a rigorous runtime analysis of a simple memetic algorithm, the $(1+1)$ MA, on the Hurdle problem class, a landscape class of tuneable difficulty that shows a "big valley structure", a characteristic feature of many hard problems from combinatorial optimisation. The only parameter of this class is the hurdle width w, which describes the length of fitness valleys that have to be overcome. We show that the $(1+1)$ EA requires $Θ(n^w)$ expected function evaluations to find the optimum, whereas the $(1+1)$ MA with best-improvement and first-improvement local search can find the optimum in $Θ(n^2+n^3/w^2)$ and $Θ(n^3/w^2)$ function evaluations, respectively. Surprisingly, while increasing the hurdle width makes the problem harder for evolutionary algorithms, the problem becomes easier for memetic algorithms. We discuss how these findings can explain and illustrate the success of memetic algorithms for problems with big valley structures.

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arXiv2018

Runtime Analysis of Probabilistic Crowding and Restricted Tournament Selection for Bimodal Optimisation

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 crowding, the offspring compete with their parents and the survivor is chosen proportionally to its fitness. On Twomax probabilistic crowding fails to find any reasonable solution quality even in exponential time. In RTS the offspring compete against the closest individual amongst $w$ (window size) individuals. We prove that RTS fails if $w$ is too small, leading to exponential times with high probability. However, if w is chosen large enough, it finds both optima for Twomax in time $O(μn \log{n})$ with high probability. Our theoretical results are accompanied by experimental studies that match the theoretical results and also shed light on parameters not covered by the theoretical results.

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arXiv2018

On the Runtime Analysis of the Clearing Diversity-Preserving Mechanism

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 optimising all functions of unitation for small niches in polynomial time, while a genotypic distance function requires exponential time. Finally, we prove that with phenotypic and genotypic distances clearing is able to find both optima for Twomax and several general classes of bimodal functions in polynomial expected time. We use empirical analysis to highlight some of the characteristics that makes it a useful mechanism and to support the theoretical results.

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arXiv2018

The Benefits of Population Diversity in Evolutionary Algorithms: A Survey of Rigorous Runtime Analyses

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 dynamic optimization, when the problem landscape changes over time. And, finally, it facilitates search for the whole Pareto front in evolutionary multiobjective optimization. The presented analyses rigorously quantify the performance of evolutionary algorithms in the light of population diversity, laying the foundation for a rigorous understanding of how search dynamics are affected by the presence or absence of population diversity and the introduction of diversity mechanisms.

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arXiv2016

Escaping Local Optima using Crossover with Emergent or Reinforced Diversity

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 algorithms like (1+1) EA. Moreover, increasing the mutation rate by an arbitrarily small constant factor can facilitate the generation of diversity, leading to speedups of order $Ω(n)$. We also compare seven commonly used diversity mechanisms and evaluate their impact on runtime bounds for the ($μ$+1) GA. All previous results in this context only hold for unrealistically low crossover probability $p_c=O(k/n)$, while we give analyses for the setting of constant $p_c < 1$ in all but one case. For the typical case of constant $k > 2$ and constant $p_c$, we can compare the resulting expected runtimes for different diversity mechanisms assuming an optimal choice of $μ$: $O(n^{k-1})$ for duplicate elimination/minim., $O(n^2\log n)$ for maximising the convex hull, $O(n\log n)$ for deterministic crowding (assuming $p_c = k/n$), $O(n\log n)$ for maximising Hamming distance, $O(n\log n)$ for fitness sharing, $O(n\log n)$ for single-receiver island model. This proves a sizeable advantage of all variants of the ($μ$+1) GA compared to (1+1) EA, which requires time $Θ(n^k)$. Experiments complement our theoretical findings and further highlight the benefits of crossover and diversity on $\text{Jump}_k$.

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arXiv2016

Update Strength in EDAs and ACO: How to Avoid Genetic Drift

We provide a rigorous runtime analysis concerning the update strength, a vital parameter in probabilistic model-building GAs such as the step size $1/K$ in the compact Genetic Algorithm (cGA) and the evaporation factor $ρ$ in ACO. While a large update strength is desirable for exploitation, there is a general trade-off: too strong updates can lead to genetic drift and poor performance. We demonstrate this trade-off for the cGA and a simple MMAS ACO algorithm on the OneMax function. More precisely, we obtain lower bounds on the expected runtime of $Ω(K\sqrt{n} + n \log n)$ and $Ω(\sqrt{n}/ρ+ n \log n)$, respectively, showing that the update strength should be limited to $1/K, ρ= O(1/(\sqrt{n} \log n))$. In fact, choosing $1/K, ρ\sim 1/(\sqrt{n}\log n)$ both algorithms efficiently optimize OneMax in expected time $O(n \log n)$. Our analyses provide new insights into the stochastic behavior of probabilistic model-building GAs and propose new guidelines for setting the update strength in global optimization.

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arXiv2015

First Steps Towards a Runtime Comparison of Natural and Artificial Evolution

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 evolution can be modelled as a (1+1)-type process where the probability of accepting a new genotype (improvements or worsenings) depends on the change in fitness. We present an initial runtime analysis of SSWM, quantifying its performance for various parameters and investigating differences to the (1+1)EA. We show that SSWM can have a moderate advantage over the (1+1)EA at crossing fitness valleys and study an example where SSWM outperforms the (1+1)EA by taking advantage of information on the fitness gradient.

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arXiv2014

How Crossover Speeds Up Building-Block Assembly in Genetic Algorithms

We re-investigate a fundamental question: how effective is crossover in Genetic Algorithms in combining building blocks of good solutions? Although this has been discussed controversially for decades, we are still lacking a rigorous and intuitive answer. We provide such answers for royal road functions and OneMax, where every bit is a building block. For the latter we show that using crossover makes every ($μ$+$λ$) Genetic Algorithm at least twice as fast as the fastest evolutionary algorithm using only standard bit mutation, up to small-order terms and for moderate $μ$ and $λ$. Crossover is beneficial because it effectively turns fitness-neutral mutations into improvements by combining the right building blocks at a later stage. Compared to mutation-based evolutionary algorithms, this makes multi-bit mutations more useful. Introducing crossover changes the optimal mutation rate on OneMax from $1/n$ to $(1+\sqrt{5})/2 \cdot 1/n \approx 1.618/n$. This holds both for uniform crossover and $k$-point crossover. Experiments and statistical tests confirm that our findings apply to a broad class of building-block functions.

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arXiv2012

General Upper Bounds on the Running Time of Parallel Evolutionary Algorithms

We present a new method for analyzing the running time of parallel evolutionary algorithms with spatially structured populations. Based on the fitness-level method, it yields upper bounds on the expected parallel running time. This allows to rigorously estimate the speedup gained by parallelization. Tailored results are given for common migration topologies: ring graphs, torus graphs, hypercubes, and the complete graph. Example applications for pseudo-Boolean optimization show that our method is easy to apply and that it gives powerful results. In our examples the possible speedup increases with the density of the topology. Surprisingly, even sparse topologies like ring graphs lead to a significant speedup for many functions while not increasing the total number of function evaluations by more than a constant factor. We also identify which number of processors yield asymptotically optimal speedups, thus giving hints on how to parametrize parallel evolutionary algorithms.

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arXiv2011

A New Method for Lower Bounds on the Running Time of Evolutionary Algorithms

We present a new method for proving lower bounds on the expected running time of evolutionary algorithms. It is based on fitness-level partitions and an additional condition on transition probabilities between fitness levels. The method is versatile, intuitive, elegant, and very powerful. It yields exact or near-exact lower bounds for LO, OneMax, long k-paths, and all functions with a unique optimum. Most lower bounds are very general: they hold for all evolutionary algorithms that only use bit-flip mutation as variation operator---i.e. for all selection operators and population models. The lower bounds are stated with their dependence on the mutation rate. These results have very strong implications. They allow to determine the optimal mutation-based algorithm for LO and OneMax, i.e., which algorithm minimizes the expected number of fitness evaluations. This includes the choice of the optimal mutation rate.

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arXiv2011

Memetic Algorithms: Parametrization and Balancing Local and Global Search

This is a preprint of a book chapter from the Handbook of Memetic Algorithms, Studies in Computational Intelligence, Vol. 379, ISBN 978-3-642-23246-6, Springer, edited by F. Neri, C. Cotta, and P. Moscato. It is devoted to the parametrization of memetic algorithms and how to find a good balance between global and local search.

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arXiv2011

Analysis of Speedups in Parallel Evolutionary Algorithms for Combinatorial Optimization

Evolutionary algorithms are popular heuristics for solving various combinatorial problems as they are easy to apply and often produce good results. Island models parallelize evolution by using different populations, called islands, which are connected by a graph structure as communication topology. Each island periodically communicates copies of good solutions to neighboring islands in a process called migration. We consider the speedup gained by island models in terms of the parallel running time for problems from combinatorial optimization: sorting (as maximization of sortedness), shortest paths, and Eulerian cycles. Different search operators are considered. The results show in which settings and up to what degree evolutionary algorithms can be parallelized efficiently. Along the way, we also investigate how island models deal with plateaus. In particular, we show that natural settings lead to exponential vs. logarithmic speedups, depending on the frequency of migration.

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arXiv2011

Adaptive Population Models for Offspring Populations and Parallel Evolutionary Algorithms

We present two adaptive schemes for dynamically choosing the number of parallel instances in parallel evolutionary algorithms. This includes the choice of the offspring population size in a (1+$λ$) EA as a special case. Our schemes are parameterless and they work in a black-box setting where no knowledge on the problem is available. Both schemes double the number of instances in case a generation ends without finding an improvement. In a successful generation, the first scheme resets the system to one instance, while the second scheme halves the number of instances. Both schemes provide near-optimal speed-ups in terms of the parallel time. We give upper bounds for the asymptotic sequential time (i.e., the total number of function evaluations) that are not larger than upper bounds for a corresponding non-parallel algorithm derived by the fitness-level method.

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