Rigorous Runtime Analyses Research Articles

The Influence of Noise on Multi-Parent Crossover for an Island Model Genetic Algorithm

Many optimization problems tackled by evolutionary algorithms are not only computationally expensive, but also complicated with one or more sources of noise. One technique to deal with high computational overhead is parallelization. However, though the existing literature gives good insights about the expected behavior of parallelized evolutionary algorithms, we still lack an understanding of their performance in the presence of noise. This paper considers how parallelization might be leveraged together with multi-parent crossover in order to handle noisy problems. We present a rigorous running time analysis of an island model with weakly connected topology tasked with hill climbing in the presence of general additive noise (i.e., noisy OneMax ). Our proofs yield insights into the relationship between the noise intensity and number of required parents. We translate this into positive and negative results for two kinds of multi-parent crossover operators. We then empirically analyze and extend this framework to investigate the trade-offs between noise impact, optimization time, and limits of computation power to deal with noise.

Runtime Analysis of Restricted Tournament Selection for Bimodal Optimisation.

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 TwoMax. 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 TwoMax 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 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.

A Rigorous Runtime Analysis of the $$(1 + (\lambda , \lambda ))$$ GA on Jump Functions

The $(1 + (\lambda,\lambda))$ 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 $\Theta(n^k)$ 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 $(n/k)^{k/2} e^{\Theta(k)}$. This suggests some general advice on how to set the parameters of the \ollga, which might ease the further use of this algorithm.

Runtime Analyses of the Population-Based Univariate Estimation of Distribution Algorithms on LeadingOnes

We perform rigorous runtime analyses for the univariate marginal distribution algorithm (UMDA) and the population-based incremental learning (PBIL) Algorithm on LeadingOnes. For the UMDA, the currently known expected runtime on the function is {mathcal {O}}left( nlambda log lambda +n^2right) under an offspring population size lambda =Omega (log n) and a parent population size mu le lambda /(e(1+delta )) for any constant delta >0 (Dang and Lehre, GECCO 2015). There is no lower bound on the expected runtime under the same parameter settings. It also remains unknown whether the algorithm can still optimise the LeadingOnes function within a polynomial runtime when mu ge lambda /(e(1+delta )). In case of the PBIL, an expected runtime of {mathcal {O}}(n^{2+c}) holds for some constant c in (0,1) (Wu, Kolonko and Möhring, IEEE TEVC 2017). Despite being a generalisation of the UMDA, this upper bound is significantly asymptotically looser than the upper bound of {mathcal {O}}left( n^2right) of the UMDA for lambda =Omega (log n)cap {mathcal {O}}left( n/log nright). Furthermore, the required population size is very large, i.e., lambda =Omega (n^{1+c}). Our contributions are then threefold: (1) we show that the UMDA with mu =Omega (log n) and lambda le mu e^{1-varepsilon }/(1+delta ) for any constants varepsilon in (0,1) and 0<delta le e^{1-varepsilon }-1 requires an expected runtime of e^{Omega (mu )} on LeadingOnes, (2) an upper bound of {mathcal {O}}left( nlambda log lambda +n^2right) is shown for the PBIL, which improves the current bound {mathcal {O}}left( n^{2+c}right) by a significant factor of Theta (n^{c}), and (3) we for the first time consider the two algorithms on the LeadingOnes function in a noisy environment and obtain an expected runtime of {mathcal {O}}left( n^2right) for appropriate parameter settings. Our results emphasise that despite the independence assumption in the probabilistic models, the UMDA and the PBIL with fine-tuned parameter choices can still cope very well with variable interactions.

Runtime Analysis of Crowding Mechanisms for Multimodal Optimization

Many real-world optimization problems lead to multimodal domains and require the identification of multiple optima. Crowding methods have been developed to maintain population diversity, to investigate many peaks in parallel and to reduce genetic drift. We present the first rigorous runtime analyses of probabilistic crowding and generalized crowding, embedded in a (μ +1) EA. In probabilistic crowding the offspring compete with their parent in a fitness-proportional selection. Generalized crowding decreases the fitness of the inferior solution by a scaling factor during selection. We consider the bimodal function TwoMax and introduce a novel and natural notion for functions with bounded gradients. For a broad range of such functions we prove that probabilistic crowding needs exponential time with overwhelming probability to find solutions significantly closer to any global optimum than those found by random search. Even when the fitness function is scaled exponentially, probabilistic crowding still fails badly. Only if the exponential's base is linear in the problem size, probabilistic crowding becomes efficient on TwoMax. A similar threshold behavior holds for generalized crowding on TwoMax with respect to the scaling factor. Our theoretical results are accompanied by experiments for TwoMax showing that the threshold behaviors also apply to the best fitness found.

Rigorous Running Time Analysis of a Simple Immune-Based Multi-Objective Optimizer for Bi-Objective Pseudo-Boolean Functions

A simple immune-based multi-objective optimizer (IBMO) is proposed, and a rigorous running time analysis of IBMO on three proposed bi-objective pseudo-Boolean functions (Bi-Trap, Bi-Plateau and Bi-Jump) is presented. The running time of a global simple evolutionary multi-objective optimizer (GSEMO) using standard bit mutation operator with IBMO using somatic contiguous hypermutation (CHM) operator is compared with these three functions. The results show that the immune-based hypermutation can significantly beat standard bit mutation on some well-known multi-objective pseudo-Boolean functions. The proofs allow us to understand the relationship between the characteristics of the problems and the features of the algorithms more deeply. These analysis results also give us a good inspiration to analyze and design a bio-inspired search heuristics.

Island Models Meet Rumor Spreading

Island models in evolutionary computation solve problems by a careful interplay of independently running evolutionary algorithms on the island and an exchange of good solutions between the islands. In this work, we conduct rigorous run time analyses for such island models trying to simultaneously obtain good run times and low communication effort. We improve the existing upper bounds for both measures (i) by improving the run time bounds via a careful analysis, (ii) by balancing individual computation and communication in a more appropriate manner, and (iii) by replacing the usual communicate-with-all approach with randomized rumor spreading. In the latter, each island contacts a randomly chosen neighbor. This epidemic communication paradigm is known to lead to very fast and robust information dissemination in many applications. Our results concern island models running simple $$(1+1)$$ evolutionary algorithms to optimize the classic test functions OneMax and LeadingOnes. We investigate binary trees, d-dimensional tori, and complete graphs as communication topologies.

Running time analysis of the Pareto archived evolution strategy on pseudo-Boolean functions

Evolutionary algorithms have long been quite successfully applied to solve multi-objective optimization problems. However, theoretical analysis of multi-objective evolutionary algorithms (MOEAs) is mainly restricted to the simple evolutionary multi-objective optimizer (SEMO). The Pareto archived evolution strategy (PAES) is a simple but important multi-objective evolutionary algorithm which is come from the study of telecommunication problems, and it has been successfully applied to many optimization problems, such as image processing and signal processing. In this paper, we make a first step toward studying the rigorous running time analysis for PAES. We show that the PAES outperforms the SEMO on function PATH when the PAES uses a simple mutation operator. However, it can not find the whole Pareto front with overwhelming probability on the well-studied function LOTZ. Additional experiments show that the experimental results are in agreement with the theoretical results.

On the Effectiveness of Sampling for Evolutionary Optimization in Noisy Environments.

In real-world optimization tasks, the objective (i.e., fitness) function evaluation is often disturbed by noise due to a wide range of uncertainties. Evolutionary algorithms are often employed in noisy optimization, where reducing the negative effect of noise is a crucial issue. Sampling is a popular strategy for dealing with noise: to estimate the fitness of a solution, it evaluates the fitness multiple ([Formula: see text]) times independently and then uses the sample average to approximate the true fitness. Obviously, sampling can make the fitness estimation closer to the true value, but also increases the estimation cost. Previous studies mainly focused on empirical analysis and design of efficient sampling strategies, while the impact of sampling is unclear from a theoretical viewpoint. In this article, we show that sampling can speed up noisy evolutionary optimization exponentially via rigorous running time analysis. For the (1[Formula: see text]1)-EA solving the OneMax and the LeadingOnes problems under prior (e.g., one-bit) or posterior (e.g., additive Gaussian) noise, we prove that, under a high noise level, the running time can be reduced from exponential to polynomial by sampling. The analysis also shows that a gap of one on the value of [Formula: see text] for sampling can lead to an exponential difference on the expected running time, cautioning for a careful selection of [Formula: see text]. We further prove by using two illustrative examples that sampling can be more effective for noise handling than parent populations and threshold selection, two strategies that have shown to be robust to noise. Finally, we also show that sampling can be ineffective when noise does not bring a negative impact.

Runtime analysis of ant colony optimization on dynamic shortest path problems

A simple ACO algorithm called λ-MMAS for dynamic variants of the single-destination shortest paths problem is studied by rigorous runtime analyses. Building upon previous results for the special case of 1-MMAS, it is studied to what extent an enlarged colony using λ ants per vertex helps in tracking an oscillating optimum. It is shown that easy cases of oscillations can be tracked by a constant number of ants. However, the paper also identifies more involved oscillations that with overwhelming probability cannot be tracked with any polynomial-size colony. Finally, parameters of dynamic shortest-path problems which make the optimum difficult to track are discussed. Experiments illustrate theoretical findings and conjectures.

Population size matters: Rigorous runtime results for maximizing the hypervolume indicator

Evolutionary multi-objective optimization is one of the most successful areas in the field of evolutionary computation. Using the hypervolume indicator to guide the search of evolutionary multi-objective algorithms has become very popular in recent years. We contribute to the theoretical understanding of these algorithms by carrying out rigorous runtime analyses. We consider multi-objective variants of the problems OneMax and LeadingOnes called OneMinMax and LOTZ, respectively, and investigate hypervolume-based algorithms with population sizes that do not allow coverage of the entire Pareto front. Our results show that LOTZ is easier to optimize than OneMinMax for hypervolume-based evolutionary multi-objective algorithms, which is contrary to the results on their single-objective variants and the well-studied (1+1) EA. Furthermore, we study multi-objective genetic programming using the hypervolume indicator. We show that the classical ORDER problem is easy to optimize if the population size is large enough to cover the whole Pareto front and point out situations where a small population size leads to an exponential optimization time.

Analysis of Diversity-Preserving Mechanisms for Global Exploration

Maintaining diversity is important for the performance of evolutionary algorithms. Diversity-preserving mechanisms can enhance global exploration of the search space and enable crossover to find dissimilar individuals for recombination. We focus on the global exploration capabilities of mutation-based algorithms. Using a simple bimodal test function and rigorous runtime analyses, we compare well-known diversity-preserving mechanisms like deterministic crowding, fitness sharing, and others with a plain algorithm without diversification. We show that diversification is necessary for global exploration, but not all mechanisms succeed in finding both optima efficiently. Our theoretical results are accompanied by additional experiments for different population sizes.

Plateaus can be harder in multi-objective optimization

In recent years a lot of progress has been made in understanding the behavior of evolutionary computation methods for single- and multi-objective problems. Our aim is to analyze the diversity mechanisms that are implicitly used in evolutionary algorithms for multi-objective problems by rigorous runtime analyses. We show that, even if the population size is small, the runtime can be exponential where corresponding single-objective problems are optimized within polynomial time. To illustrate this behavior we analyze a simple plateau function in a first step and extend our result to a class of instances of the well-known SetCover problem.

Analysis of different MMAS ACO algorithms on unimodal functions and plateaus

Recently, the first rigorous runtime analyses of ACO algorithms appeared, cover- ing variants of the MAX-MIN ant system and their runtime on pseudo-Boolean functions. Interestingly, a variant called 1-ANT is very sensitive to the evaporation factor while Gutjahr and Sebastiani proved partly opposite results for their variant MMASbs. These algorithms differ in their pheromone update mechanisms and, moreover, 1-ANT accepts equally fit so- lutions in contrast to MMASbs. By analyzing variants of MMASbs, we prove that the different behavior of 1-ANT and MMASbs results from the different pheromone update mechanisms. Building upon results by Gutjahr and Sebastiani, we extend their analyses of MMASbs to the class of unimodal functions and show improved results for test functions using new and specialized techniques; in particular, we present new lower bounds. Finally, we compare MMASbs with a variant that also accepts equally fit solutions as this enables the exploration of plateaus. For well-known plateau functions we prove that this drastically reduces the optimization time. Our findings are complemented by experiments that support our asymptotic analyses and yield additional insights.

When to use bit-wise neutrality

Representation techniques are important issues when designing successful evolutionary algorithms. Within this field the use of neutrality plays an important role. We examine the use of bit-wise neutrality introduced by Poli and López (2007) from a theoretical point of view and show that this mechanism only enhances mutation-based evolutionary algorithms if not the same number of genotypic bits for each phenotypic bit is used. Using different numbers of genotypic bits for the bits in the phenome we point out by rigorous runtime analyses that it may reduce the optimization time significantly.

A rigorous analysis of the compact genetic algorithm for linear functions

Estimation of distribution algorithms (EDAs) solve an optimization problem heuristically by finding a probability distribution focused around its optima. Starting with the uniform distribution, points are sampled with respect to this distribution and the distribution is changed according to the function values of the sampled points. Although there are many successful experiments suggesting the usefulness of EDAs, there are only few rigorous theoretical results apart from convergence results without time bounds. Here we present first rigorous runtime analyses of a simple EDA, the compact genetic algorithm (cGA), for linear pseudo-Boolean functions on n variables. We prove a general lower bound for all functions and a general upper bound for all linear functions. Simple test functions show that not all linear functions are optimized in the same runtime by the cGA.

Analysis of a Multiobjective Evolutionary Algorithm on the 0–1 knapsack problem

Multiobjective Evolutionary Algorithms (MOEAs) are increasingly being used for effectively solving many real-world problems, and many empirical results are available. However, theoretical analysis is limited to a few simple toy functions. In this work, we select the well-known knapsack problem for the analysis. The multiobjective knapsack problem in its general form is NP-complete. Moreover, the size of the set of Pareto-optimal solutions can grow exponentially with the number of items in the knapsack. Thus, we formalize a ( 1 + ε ) -approximate set of the knapsack problem and attempt to present a rigorous running time analysis of a MOEA to obtain the formalized set. The algorithm used in the paper is based on a restricted mating pool with a separate archive to store the remaining population; we call the algorithm a Restricted Evolutionary Multiobjective Optimizer (REMO). We also analyze the running time of REMO on a special bi-objective linear function, known as LOTZ (Leading Ones : Trailing Zeros), whose Pareto set is shown to be a subset of the knapsack. An extension of the analysis to the Simple Evolutionary Multiobjective Optimizer (SEMO) is also presented. A strategy based on partitioning of the decision space into fitness layers is used for the analysis.

Running Time Analysis of Multiobjective Evolutionary Algorithms on Pseudo-Boolean Functions

This paper presents a rigorous running time analysis of evolutionary algorithms on pseudo-Boolean multiobjective optimization problems. We propose and analyze different population-based algorithms, the simple evolutionary multiobjective optimizer (SEMO), and two improved versions, fair evolutionary multiobjective optimizer (FEMO) and greedy evolutionary multiobjective optimizer (GEMO). The analysis is carried out on two biobjective model problems, leading ones trailing zeroes (LOTZ) and count ones count zeroes (COCZ), as well as on the scalable m-objective versions mLOTZ and mCOCZ. Results on the running time of the different population-based algorithms and for an alternative approach, a multistart (1+1)-EA based on the /spl epsi/-constraint method, are derived. The comparison reveals that for many problems, the simple algorithm SEMO is as efficient as this (1+1)-EA. For some problems, the improved variants FEMO and GEMO are provably better. For the analysis, we propose and apply two general tools, an upper bound technique based on a decision space partition and a randomized graph search algorithm, which facilitate the analysis considerably.