Dirk Sudholt

Duc-Cuong Dang, Aneta Neumann, Frank Neumann et al.

Quality diversity (QD) algorithms have shown to provide sets of high quality solutions for challenging problems in robotics, games, and combinatorial optimisation. So far, theoretical foundational explaining their good behaviour in practice lack far behind their practical success. We contribute to the theoretical understanding of these algorithms and study the behaviour of QD algorithms for a classical planning problem seeking several solutions. We study the all-pairs-shortest-paths (APSP) problem which gives a natural formulation of the behavioural space based on all pairs of nodes of the given input graph that can be used by Map-Elites QD algorithms. Our results show that Map-Elites QD algorithms are able to compute a shortest path for each pair of nodes efficiently in parallel. Furthermore, we examine parent selection techniques for crossover that exhibit significant speed ups compared to the standard QD approach.

Edgar Covantes Osuna, Dirk Sudholt

Niching methods have been developed to maintain the population diversity, to investigate many peaks in parallel and to reduce the effect of genetic drift. We present the first rigorous runtime analyses of restricted tournament selection (RTS), embedded in a ($μ$+1) EA, and analyse its effectiveness at finding both optima of the bimodal function ${\rm T{\small WO}M{\small AX}}$. In RTS, an offspring competes against the closest individual, with respect to some distance measure, amongst $w$ (window size) population members (chosen uniformly at random with replacement), to encourage competition within the same niche. We prove that RTS finds both optima on ${\rm T{\small WO}M{\small AX}}$ efficiently if the window size $w$ is large enough. However, if $w$ is too small, RTS fails to find both optima even in exponential time, with high probability. We further consider a variant of RTS selecting individuals for the tournament \emph{without} replacement. It yields a more diverse tournament and is more effective at preventing one niche from taking over the other. However, this comes at the expense of a slower progress towards optima when a niche collapses to a single individual. Our theoretical results are accompanied by experimental studies that shed light on parameters not covered by the theoretical results and support a conjectured lower runtime bound.

Jakob Bossek, Frank Neumann, Pan Peng et al.

We contribute to the theoretical understanding of randomized search heuristics for dynamic problems. We consider the classical vertex coloring problem on graphs and investigate the dynamic setting where edges are added to the current graph. We then analyze the expected time for randomized search heuristics to recompute high quality solutions. The (1+1)~Evolutionary Algorithm and RLS operate in a setting where the number of colors is bounded and we are minimizing the number of conflicts. Iterated local search algorithms use an unbounded color palette and aim to use the smallest colors and, consequently, the smallest number of colors. We identify classes of bipartite graphs where reoptimization is as hard as or even harder than optimization from scratch, i.e., starting with a random initialization. Even adding a single edge can lead to hard symmetry problems. However, graph classes that are hard for one algorithm turn out to be easy for others. In most cases our bounds show that reoptimization is faster than optimizing from scratch. We further show that tailoring mutation operators to parts of the graph where changes have occurred can significantly reduce the expected reoptimization time. In most settings the expected reoptimization time for such tailored algorithms is linear in the number of added edges. However, tailored algorithms cannot prevent exponential times in settings where the original algorithm is inefficient.

Mario Alejandro Hevia Fajardo, Dirk Sudholt

Evolutionary algorithms (EAs) are general-purpose optimisers that come with several parameters like the sizes of parent and offspring populations or the mutation rate. It is well known that the performance of EAs may depend drastically on these parameters. Recent theoretical studies have shown that self-adjusting parameter control mechanisms that tune parameters during the algorithm run can provably outperform the best static parameters in EAs on discrete problems. However, the majority of these studies concerned elitist EAs and we do not have a clear answer on whether the same mechanisms can be applied for non-elitist EAs. We study one of the best-known parameter control mechanisms, the one-fifth success rule, to control the offspring population size $λ$ in the non-elitist $(1,λ)$ EA. It is known that the $(1,λ)$ EA has a sharp threshold with respect to the choice of $λ$ where the expected runtime on the benchmark function OneMax changes from polynomial to exponential time. Hence, it is not clear whether parameter control mechanisms are able to find and maintain suitable values of $λ$. For OneMax we show that the answer crucially depends on the success rate $s$ (i.\,e.\ a one-$(s+1)$-th success rule). We prove that, if the success rate is appropriately small, the self-adjusting $(1,λ)$ EA optimises OneMax in $O(n)$ expected generations and $O(n \log n)$ expected evaluations, the best possible runtime for any unary unbiased black-box algorithm. A small success rate is crucial: we also show that if the success rate is too large, the algorithm has an exponential runtime on OneMax and other functions with similar characteristics.

George T. Hall, Pietro Simone Oliveto, Dirk Sudholt

Recent work has shown that the ParamRLS and ParamILS algorithm configurators can tune some simple randomised search heuristics for standard benchmark functions in linear expected time in the size of the parameter space. In this paper we prove a linear lower bound on the expected time to optimise any parameter tuning problem for ParamRLS, ParamILS as well as for larger classes of algorithm configurators. We propose a harmonic mutation operator for perturbative algorithm configurators that provably tunes single-parameter algorithms in polylogarithmic time for unimodal and approximately unimodal (i.e., non-smooth, rugged with an underlying gradient towards the optimum) parameter spaces. It is suitable as a general-purpose operator since even on worst-case (e.g., deceptive) landscapes it is only by at most a logarithmic factor slower than the default ones used by ParamRLS and ParamILS. An experimental analysis confirms the superiority of the approach in practice for a number of configuration scenarios, including ones involving more than one parameter.

Jakob Bossek, Frank Neumann, Pan Peng et al.

Dynamic optimization problems have gained significant attention in evolutionary computation as evolutionary algorithms (EAs) can easily adapt to changing environments. We show that EAs can solve the graph coloring problem for bipartite graphs more efficiently by using dynamic optimization. In our approach the graph instance is given incrementally such that the EA can reoptimize its coloring when a new edge introduces a conflict. We show that, when edges are inserted in a way that preserves graph connectivity, Randomized Local Search (RLS) efficiently finds a proper 2-coloring for all bipartite graphs. This includes graphs for which RLS and other EAs need exponential expected time in a static optimization scenario. We investigate different ways of building up the graph by popular graph traversals such as breadth-first-search and depth-first-search and analyse the resulting runtime behavior. We further show that offspring populations (e. g. a (1+$λ$) RLS) lead to an exponential speedup in $λ$. Finally, an island model using 3 islands succeeds in an optimal time of $Θ(m)$ on every $m$-edge bipartite graph, outperforming offspring populations. This is the first example where an island model guarantees a speedup that is not bounded in the number of islands.

George T. Hall, Pietro Simone Oliveto, Dirk Sudholt

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.

George T. Hall, Pietro S. Oliveto, Dirk Sudholt

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.

Per Kristian Lehre, Dirk Sudholt

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.

Dirk Sudholt

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.

Edgar Covantes Osuna, Wanru Gao, Frank Neumann et al.

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.

Phan Trung Hai Nguyen, Dirk Sudholt

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.

Edgar Covantes Osuna, Dirk Sudholt

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.

Edgar Covantes Osuna, Dirk Sudholt

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.

Dirk Sudholt

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.

Duc-Cuong Dang, Tobias Friedrich, Timo Kötzing et al.

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 $\text{Jump}_k$.

Dirk Sudholt, Carsten Witt

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.

Tiago Paixão, Jorge Pérez Heredia, Dirk Sudholt et al.

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.

Dirk Sudholt

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.

Jörg Lässig, Dirk Sudholt

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.