The optimization of functions to find the best solution according to one or several objectives has a central role in many engineering and research fields. Recently, a new family of optimization algorithms, named Quality-Diversity optimization, has been introduced, and contrasts with classic algorithms. Instead of searching for a single solution, Quality-Diversity algorithms are searching for a large collection of both diverse and high-performing solutions. The role of this collection is to cover the range of possible solution types as much as possible, and to contain the best solution for each type. The contribution of this paper is threefold. Firstly, we present a unifying framework of Quality-Diversity optimization algorithms that covers the two main algorithms of this family (Multi-dimensional Archive of Phenotypic Elites and the Novelty Search with Local Competition), and that highlights the large variety of variants that can be investigated within this family. Secondly, we propose algorithms with a new selection mechanism for Quality-Diversity algorithms that outperforms all the algorithms tested in this paper. Lastly, we present a new collection management that overcomes the erosion issues observed when using unstructured collections. These three contributions are supported by extensive experimental comparisons of Quality-Diversity algorithms on three different experimental scenarios.
Following different inspirations from the works presented above, the Multi-dimensional Archive of Phenotypic Elites () algorithm @cite_44 has been recently introduced. While this algorithm was first designed to illuminate'' the landscape of objective functions @cite_9 , it showed itself to be an effective algorithm to generate a collection of solutions that are both diverse and high-performing. The main difference with NSLC and BR-Evolution is that, in MAP-Elites, the population of the algorithms is the collection itself, and the selection, mutations and preservation mechanisms directly consider the solutions that are stored in the collection.
@cite_44 @cite_9
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The recently introduced Multi-dimensional Archive of Phenotypic Elites (MAP-Elites) is an evolutionary algorithm capable of producing a large archive of diverse, high-performing solutions in a single run. It works by discretizing a continuous feature space into unique regions according to the desired discretization per dimension. While simple, this algorithm has a main drawback: it cannot scale to high-dimensional feature spaces since the number of regions increase exponentially with the number of dimensions. In this paper, we address this limitation by introducing a simple extension of MAP-Elites that has a constant, pre-defined number of regions irrespective of the dimensionality of the feature space. Our main insight is that methods from computational geometry could partition a high-dimensional space into well-spread geometric regions. In particular, our algorithm uses a centroidal Voronoi tessellation (CVT) to divide the feature space into a desired number of regions; it then places every generated individual in its closest region, replacing a less fit one if the region is already occupied. We demonstrate the effectiveness of the new "CVT-MAP-Elites" algorithm in high-dimensional feature spaces through comparisons against MAP-Elites in a hexapod locomotion task. Exploration of the search space through the optimisation of phenotypic diversity is of increasing interest within the field of evolutionary robotics. Novelty search and the more recent MAP-Elites are two state of the art evolutionary algorithms which diversify low dimensional phenotypic traits for divergent exploration. In this paper we introduce a novel alternative for rapid divergent search of the feature space. Unlike previous phenotypic search procedures, our proposed Spatial, Hierarchical, Illuminated Neuro-Evolution (SHINE) algorithm utilises a tree structure for the maintenance and selection of potential candidates. SHINE penalises previous solutions in more crowded areas of the landscape. Our experimental results show that SHINE significantly outperforms novelty search and MAP-Elites in both performance and exploration. We conclude that the SHINE algorithm is a viable method for rapid divergent search of low dimensional, phenotypic landscapes. Many fields use search algorithms, which automatically explore a search space to find high-performing solutions: chemists search through the space of molecules to discover new drugs; engineers search for stronger, cheaper, safer designs, scientists search for models that best explain data, etc. The goal of search algorithms has traditionally been to return the single highest-performing solution in a search space. Here we describe a new, fundamentally different type of algorithm that is more useful because it provides a holistic view of how high-performing solutions are distributed throughout a search space. It creates a map of high-performing solutions at each point in a space defined by dimensions of variation that a user gets to choose. This Multi-dimensional Archive of Phenotypic Elites (MAP-Elites) algorithm illuminates search spaces, allowing researchers to understand how interesting attributes of solutions combine to affect performance, either positively or, equally of interest, negatively. For example, a drug company may wish to understand how performance changes as the size of molecules and their cost-to-produce vary. MAP-Elites produces a large diversity of high-performing, yet qualitatively different solutions, which can be more helpful than a single, high-performing solution. Interestingly, because MAP-Elites explores more of the search space, it also tends to find a better overall solution than state-of-the-art search algorithms. We demonstrate the benefits of this new algorithm in three different problem domains ranging from producing modular neural networks to designing simulated and real soft robots. Because MAP- Elites (1) illuminates the relationship between performance and dimensions of interest in solutions, (2) returns a set of high-performing, yet diverse solutions, and (3) improves finding a single, best solution, it will advance science and engineering. Abstract In the last years, multi-objective evolutionary algorithms (MOEAs) have been extensively used to generate sets of fuzzy rule-based classifiers (FRBCs) with different trade-offs between accuracy and interpretability. Since the computation of the accuracy for each chromosome evaluation requires the scan of the overall training set, these approaches have proved to be very expensive in terms of execution time and memory occupation. For this reason, they have not been applied to very large datasets yet. On the other hand, just for these datasets, interpretability of classifiers would be very desirable. In the last years the advent of a number of open source cluster computing frameworks has however opened new interesting perspectives. In this paper, we exploit one of these frameworks, namely Apache Spark, and propose the first distributed multi-objective evolutionary approach to learn concurrently the rule and data bases of FRBCs by maximizing accuracy and minimizing complexity. During the evolutionary process, the computation of the fitness is divided among the cluster nodes, thus allowing the designer to distribute both the computational complexity and the dataset storing. We have performed a number of experiments on ten real-world big datasets, evaluating our distributed approach in terms of both classification rate and scalability, and comparing it with two well-known state-of-art distributed classifiers. Finally, we have evaluated the achievable speedup on a small computer cluster. We present that the distributed version can efficiently extract compact rule bases with high accuracy, preserving the interpretability of the rule base, and can manage big datasets even with modest hardware support. Evolutionary algorithms integrating various niching techniques have been widely used to find multiple optima of an optimization problem. In recent years, an increasing amount of research has been focused on the design and application of speciation-based niching techniques. These techniques rely on speciation to partition a population into subpopulations (species) such that each occupies a different region of attraction (niche) on the fitness landscape. Existing speciation methods are either distance-based or topology-based. Topology-based methods are more flexible and have fewer assumptions than distance-based methods. However, existing topology-based methods all require sampling and evaluating new individuals in order to capture the landscape topography. This incurs additional fitness evaluations (FEs), which is a drawback, especially when the FE budget is limited. In this paper, a new topology-based speciation method named history-based topological speciation (HTS) is proposed. It relies exclusively on search history to capture the landscape topography and, therefore, does not require any additional FEs to be performed. To the best of our knowledge, HTS is the only parameter-free speciation method at the moment. Both theoretical and empirical analyses have been conducted. Theoretical analysis shows that HTS incurs acceptable computational overhead. In the experimental study, HTS outperformed existing topology-based methods on benchmark functions in up to 32-D space and with as many as 50 optima, and the time overhead was practically negligible if a single FE took seconds. This paper shows how multiobjective evolutionary algorithms can be used to procedurally generate complete and playable maps for real-time strategy (RTS) games. We devise heuristic objective functions that measure properties of maps that impact important aspects of gameplay experience. To show the generality of our approach, we design two different evolvable map representations, one for an imaginary generic strategy game based on heightmaps, and one for the classic RTS game StarCraft. The effect of combining tuples or triples of the objective functions are investigated in systematic experiments, in particular which of the objectives are partially conflicting. A selection of generated maps are visually evaluated by a population of skilled StarCraft players, confirming that most of our objectives correspond to perceived gameplay qualities. Our method could be used to completely automate in-game controlled map generation, enabling player-adaptive games, or as a design support tool for human designers. Evolutionary algorithms (EAs) are often well-suited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in a single run. However, the few comparative studies of different methods presented up to now remain mostly qualitative and are often restricted to a few approaches. In this paper, four multiobjective EAs are compared quantitatively where an extended 0 1 knapsack problem is taken as a basis. Furthermore, we introduce a new evolutionary approach to multicriteria optimization, the strength Pareto EA (SPEA), that combines several features of previous multiobjective EAs in a unique manner. It is characterized by (a) storing nondominated solutions externally in a second, continuously updated population, (b) evaluating an individual's fitness dependent on the number of external nondominated points that dominate it, (c) preserving population diversity using the Pareto dominance relationship, and (d) incorporating a clustering procedure in order to reduce the nondominated set without destroying its characteristics. The proof-of-principle results obtained on two artificial problems as well as a larger problem, the synthesis of a digital hardware-software multiprocessor system, suggest that SPEA can be very effective in sampling from along the entire Pareto-optimal front and distributing the generated solutions over the tradeoff surface. Moreover, SPEA clearly outperforms the other four multiobjective EAs on the 0 1 knapsack problem. We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES). We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm, in its simplest form, is a (1 + 1) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors. (1 + 1)-PAES is intended to be a baseline approach against which more involved methods may be compared. It may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods. We introduce (1 + λ) and (μ | λ) variants of PAES as extensions to the basic algorithm. Six variants of PAES are compared to variants of the Niched Pareto Genetic Algorithm and the Nondominated Sorting Genetic Algorithm over a diverse suite of six test functions. Results are analyzed and presented using techniques that reduce the attainment surfaces generated from several optimization runs into a set of univariate distributions. This allows standard statistical analysis to be carried out for comparative purposes. Our results provide strong evidence that PAES performs consistently well on a range of multiobjective optimization tasks. Evolutionary computation techniques have had limited capabilities in solving large-scale problems due to the large search space demanding large memory and much longer training times. In the work presented here, a genetic programming like rich encoding scheme has been constructed to identify building blocks of knowledge in a learning classifier system. The fitter building blocks from the learning system trained against smaller problems have been utilized in a higher complexity problem in the domain to achieve scalable learning. The proposed system has been examined and evaluated on four different Boolean problem domains: 1) multiplexer, 2) majority-on, 3) carry, and 4) even-parity problems. The major contribution of this paper is to successfully extract useful building blocks from smaller problems and reuse them to learn more complex large-scale problems in the domain, e.g., 135-bit multiplexer problem, where the number of possible instances is 2 135 ≈ 4 × 10 40 , is solved by reusing the extracted knowledge from the learned lower level solutions in the domain. Autonomous scaling is, for the first time, shown to be possible in learning classifier systems. It improves effectiveness and reduces the number of training instances required in large problems, but requires more time due to its sequential build-up of knowledge. We propose a novel method called deep convolutional decision jungle (CDJ) and its learning algorithm for image classification. The CDJ maintains the structure of standard convolutional neural networks (CNNs), i.e. multiple layers of multiple response maps fully connected. Each response map-or node-in both the convolutional and fully-connected layers selectively respond to class labels s.t. each data sample travels via a specific soft route of those activated nodes. The proposed method CDJ automatically learns features, whereas decision forests and jungles require pre-defined feature sets. Compared to CNNs, the method embeds the benefits of using data-dependent discriminative functions, which better handles multi-modal heterogeneous data; further,the method offers more diverse sparse network responses, which in turn can be used for cost-effective learning classification. The network is learnt by combining conventional softmax and proposed entropy losses in each layer. The entropy loss,as used in decision tree growing, measures the purity of data activation according to the class label distribution. The back-propagation rule for the proposed loss function is derived from stochastic gradient descent (SGD) optimization of CNNs. We show that our proposed method outperforms state-of-the-art methods on three public image classification benchmarks and one face verification dataset. We also demonstrate the use of auxiliary data labels, when available, which helps our method to learn more discriminative routing and representations and leads to improved classification. Evolutionary algorithms (EAs) are increasingly popular approaches to multi-objective optimization. One of their significant advantages is that they can directly optimize the Pareto front by evolving a population of solutions, where the recombination (also called crossover) operators are usually employed to reproduce new and potentially better solutions by mixing up solutions in the population. Recombination in multi-objective evolutionary algorithms is, however, mostly applied heuristically. In this paper, we investigate how from a theoretical viewpoint a recombination operator will affect a multi-objective EA. First, we employ artificial benchmark problems: the Weighted LPTNO problem (a generalization of the well-studied LOTZ problem), and the well-studied COCZ problem, for studying the effect of recombination. Our analysis discloses that recombination may accelerate the filling of the Pareto front by recombining diverse solutions and thus help solve multi-objective optimization. Because of this, for these two problems, we find that a multi-objective EA with recombination enabled achieves a better expected running time than any known EAs with recombination disabled. We further examine the effect of recombination on solving the multi-objective minimum spanning tree problem, which is an NP-hard problem. Following our finding on the artificial problems, our analysis shows that recombination also helps accelerate filling the Pareto front and thus helps find approximate solutions faster. By synthesizing a growing body ofwork in search processes that are not driven by explicit objectives, this paper advances the hypothesis that there is a fundamental problem with the dominant paradigm of objective-based search in evolutionary computation and genetic programming: Most ambitious objectives do not illuminate a path to themselves. That is, the gradient of improvement induced by ambitious objectives tends to lead not to the objective itself but instead to deadend local optima. Indirectly supporting this hypothesis, great discoveries often are not the result of objective-driven search. For example, the major inspiration for both evolutionary computation and genetic programming, natural evolution, innovates through an open-ended process that lacks a final objective. Similarly, large-scale cultural evolutionary processes, such as the evolution of technology, mathematics, and art, lack a unified fixed goal. In addition, direct evidence for this hypothesis is presented from a recently-introduced search algorithm called novelty search. Though ignorant of the ultimate objective of search, in many instances novelty search has counter-intuitively outperformed searching directly for the objective, including a wide variety of randomly-generated problems introduced in an experiment in this chapter. Thus a new understanding is beginning to emerge that suggests that searching for a fixed objective, which is the reigning paradigm in evolutionary computation and even machine learning as a whole, may ultimately limit what can be achieved. Yet the liberating implication of this hypothesis argued in this paper is that by embracing search processes that are not driven by explicit objectives, the breadth and depth of what is reachable through evolutionary methods such as genetic programming may be greatly expanded. While evolutionary computation and evolutionary robotics take inspiration from nature, they have long focused mainly on problems of performance optimization. Yet evolution in nature can be interpreted as more nuanced than a process of simple optimization. In particular, natural evolution is a divergent search that optimizes locally within each niche as it simultaneously diversifies. This tendency to discover both quality and diversity at the same time differs from many of the conventional algorithms of machine learning, and also thereby suggests a different foundation for inferring the approach of greatest potential for evolutionary algorithms. In fact, several recent evolutionary algorithms called quality diversity (QD) algorithms(e.g. novelty search with local competition and MAP-Elites) have drawn inspiration from this more nuanced view, aiming to fill a space of possibilities with the best possible example of each type of achievable behavior. The result is a new class of algorithms that return an archive of diverse, high-quality behaviors in a single run. The aim in this paper is to study the application of QD algorithms in challenging environments (in particular complex mazes) to establish their best practices for ambitious domains in the future. In addition to providing insight into cases when QD succeeds and fails, a new approach is investigated that hybridizes multiple views of behaviors (called behavior characterizations) in the same run, which succeeds in overcoming some of the challenges associated with searching for QD with respect to a behavior characterization that is not necessarily sufficient for generating both quality and diversity at the same time. Many different algorithms have been developed in the last few decades for solving complex real-world search and optimization problems. The main focus in this research has been on the development of a single universal genetic operator for population evolution that is always efficient for a diverse set of optimization problems. In this paper, we argue that significant advances to the field of evolutionary computation can be made if we embrace a concept of self-adaptive multimethod optimization in which multiple different search algorithms are run concurrently, and learn from each other through information exchange using a common population of points. We present an evolutionary algorithm, entitled A Multialgorithm Genetically Adaptive Method for Single Objective Optimization (AMALGAM-SO), that implements this concept of self adaptive multimethod search. This method simultaneously merges the strengths of the covariance matrix adaptation (CMA) evolution strategy, genetic algorithm (GA), and particle swarm optimizer (PSO) for population evolution and implements a self-adaptive learning strategy to automatically tune the number of offspring these three individual algorithms are allowed to contribute during each generation. Benchmark results in 10, 30, and 50 dimensions using synthetic functions from the special session on real-parameter optimization of CEC 2005 show that AMALGAM-SO obtains similar efficiencies as existing algorithms on relatively simple unimodal problems, but is superior for more complex higher dimensional multimodal optimization problems. The new search method scales well with increasing number of dimensions, converges in the close proximity of the global minimum for functions with noise induced multimodality, and is designed to take full advantage of the power of distributed computer networks. In this paper, we describe a novel deep convolutional neural network (CNN) that is deeper and wider than other existing deep networks for hyperspectral image classification. Unlike current state-of-the-art approaches in CNN-based hyperspectral image classification, the proposed network, called contextual deep CNN, can optimally explore local contextual interactions by jointly exploiting local spatio-spectral relationships of neighboring individual pixel vectors. The joint exploitation of the spatio-spectral information is achieved by a multi-scale convolutional filter bank used as an initial component of the proposed CNN pipeline. The initial spatial and spectral feature maps obtained from the multi-scale filter bank are then combined together to form a joint spatio-spectral feature map. The joint feature map representing rich spectral and spatial properties of the hyperspectral image is then fed through a fully convolutional network that eventually predicts the corresponding label of each pixel vector. The proposed approach is tested on three benchmark data sets: the Indian Pines data set, the Salinas data set, and the University of Pavia data set. Performance comparison shows enhanced classification performance of the proposed approach over the current state-of-the-art on the three data sets. To extend multiobjective evolutionary algorithm based on decomposition (MOEA D) in higher dimensional objective spaces, this paper proposes a new version of MOEA D with uniform design, named the uniform design multiobjective evolutionary algorithm based on decomposition (UMOEA D), and compares the proposed algorithm with MOEA D and NSGA-II on some scalable test problems with three to five objectives. UMOEA D adopts the uniform design method to set the aggregation coefficient vectors of the subproblems. Compared with MOEA D, distribution of the coefficient vectors is more uniform over the design space, and the population size neither increases nonlinearly with the number of objectives nor considers a formulaic setting. The experimental results indicate that UMOEA D outperforms MOEA D and NSGA-II on almost all these many-objective test instances, especially on problems with higher dimensional objectives and complicated Pareto set shapes. Experimental results also show that UMOEA D runs faster than NSGA-II for the problems used in this paper. In additional, the results obtained are very competitive when comparing UMOEA D with some other algorithm on the multiobjective knapsack problems. We present a new method for clustering based on compression. The method does not use subject-specific features or background knowledge, and works as follows: First, we determine a parameter-free, universal, similarity distance, the normalized compression distance or NCD, computed from the lengths of compressed data files (singly and in pairwise concatenation). Second, we apply a hierarchical clustering method. The NCD is not restricted to a specific application area, and works across application area boundaries. A theoretical precursor, the normalized information distance, co-developed by one of the authors, is provably optimal. However, the optimality comes at the price of using the noncomputable notion of Kolmogorov complexity. We propose axioms to capture the real-world setting, and show that the NCD approximates optimality. To extract a hierarchy of clusters from the distance matrix, we determine a dendrogram (ternary tree) by a new quartet method and a fast heuristic to implement it. The method is implemented and available as public software, and is robust under choice of different compressors. To substantiate our claims of universality and robustness, we report evidence of successful application in areas as diverse as genomics, virology, languages, literature, music, handwritten digits, astronomy, and combinations of objects from completely different domains, using statistical, dictionary, and block sorting compressors. In genomics, we presented new evidence for major questions in Mammalian evolution, based on whole-mitochondrial genomic analysis: the Eutherian orders and the Marsupionta hypothesis against the Theria hypothesis. Top-k dominating queries combine the natural idea of selecting the k best items with a comprehensive “goodness” criterion based on dominance. A point p1 dominates p2 if p1 is as good as p2 in all attributes and is strictly better in at least one. Existing works address the problem in settings where data objects are multidimensional points. However, there are domains where we only have access to the distance between two objects. In cases like these, attributes reflect distances from a set of input objects and are dynamically generated as the input objects change. Consequently, prior works from the literature cannot be applied, despite the fact that the dominance relation is still meaningful and valid. For this reason, in this work, we present the first study for processing top-k dominating queries over distance-based dynamic attribute vectors, defined over a metric space. We propose four progressive algorithms that utilize the properties of the underlying metric space to efficiently solve the problem and present an extensive, comparative evaluation on both synthetic and real-world datasets. Over the last two decades, many sophisticated evolutionary algorithms have been introduced for solving constrained optimization problems. Due to the variability of characteristics in different COPs, no single algorithm performs consistently over a range of problems. In this paper, for a better coverage of the problem characteristics, we introduce an algorithm framework that uses multiple search operators in each generation. The appropriate mix of the search operators, for any given problem, is determined adaptively. The framework is tested by implementing two different algorithms. The performance of the algorithms is judged by solving 60 test instances taken from two constrained optimization benchmark sets from specialized literature. The first algorithm, which is a multi-operator based genetic algorithm (GA), shows a significant improvement over different versions of GA (each with a single one of these operators). The second algorithm, using differential evolution (DE), also confirms the benefit of the multi-operator algorithm by providing better and consistent solutions. The overall results demonstrated that both GA and DE based algorithms show competitive, if not better, performance as compared to the state of the art algorithms. This paper describes a fast and scalable strategy for constructing a radiation hybrid (RH) map from data on different RH panels. The maps on each panel are then integrated to produce a single RH map for the genome. Recurring problems in using maps from several sources are that the maps use different markers, the maps do not place the overlapping markers in same order, and the objective functions for map quality are incomparable. We use methods from combinatorial optimization to develop a strategy that addresses these issues. We show that by the standard objective functions of obligate chromosome breaks and maximum likelihood, software for the traveling salesman problem produces RH maps with better quality much more quickly than using software specifically tailored for RH mapping. We use known algorithms for the longest common subsequence problem as part of our map integration strategy. We demonstrate our methods by reconstructing and integrating maps for markers typed on the Genebridge 4 (GB4) and the Stanford G3 panels publicly available from the RH database. We compare map quality of our integrated map with published maps for GB4 panel and G3 panel by considering whether markers occur in the same order on a map and in DNA sequence contigs submitted to GenBank. We find that all of the maps are inconsistent with the sequence data for at least 50 of the contigs, but our integrated maps are more consistent. The map integration strategy not only scales to multiple RH maps but also to any maps that have comparable criteria for measuring map quality. Our software improves on current technology for doing RH mapping in areas of computation time and algorithms for considering a large number of markers for mapping. The essential impediments to producing dense high-quality RH maps are data quality and panel size, not computation.