A comparison of illumination algorithms in unbounded spaces

International audienceIllumination algorithms are a new class of evolutionary algorithms capable of producing large archives of diverse and high-performing solutions. Examples of such algorithms include Novelty Search with Local Competition (NSLC), the Multi-dimensional Archive of Phenotypic Elites (MAP-Elites) and the newly introduced Cen-troidal Voronoi Tessellation (CVT) MAP-Elites. While NSLC can be used in unbounded behavioral spaces, MAP-Elites and CVT-MAP-Elites require the user to manually specify the bounds. In this study, we introduce variants of these algorithms that expand their bounds based on the discovered solutions. In addition, we introduce a novel algorithm called "Cluster-Elites" that can adapt its bounds to non-convex spaces. We compare all algorithms in a maze navigation problem and illustrate that Cluster-Elites and the expansive variants of MAP-Elites and CVT-MAP-Elites have comparable or better performance than NSLC, MAP-Elites and CVT-MAP-Elites

Using Centroidal Voronoi Tessellations to Scale Up the Multi-dimensional Archive of Phenotypic Elites Algorithm

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 maze navigation and hexapod locomotion tasks

Comparing and Combining Lexicase Selection and Novelty Search

Lexicase selection and novelty search, two parent selection methods used in evolutionary computation, emphasize exploring widely in the search space more than traditional methods such as tournament selection. However, lexicase selection is not explicitly driven to select for novelty in the population, and novelty search suffers from lack of direction toward a goal, especially in unconstrained, highly-dimensional spaces. We combine the strengths of lexicase selection and novelty search by creating a novelty score for each test case, and adding those novelty scores to the normal error values used in lexicase selection. We use this new novelty-lexicase selection to solve automatic program synthesis problems, and find it significantly outperforms both novelty search and lexicase selection. Additionally, we find that novelty search has very little success in the problem domain of program synthesis. We explore the effects of each of these methods on population diversity and long-term problem solving performance, and give evidence to support the hypothesis that novelty-lexicase selection resists converging to local optima better than lexicase selection

On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.Comment: 23 pages, 19 figure

Bounded-Distortion Metric Learning

Metric learning aims to embed one metric space into another to benefit tasks like classification and clustering. Although a greatly distorted metric space has a high degree of freedom to fit training data, it is prone to overfitting and numerical inaccuracy. This paper presents {\it bounded-distortion metric learning} (BDML), a new metric learning framework which amounts to finding an optimal Mahalanobis metric space with a bounded-distortion constraint. An efficient solver based on the multiplicative weights update method is proposed. Moreover, we generalize BDML to pseudo-metric learning and devise the semidefinite relaxation and a randomized algorithm to approximately solve it. We further provide theoretical analysis to show that distortion is a key ingredient for stability and generalization ability of our BDML algorithm. Extensive experiments on several benchmark datasets yield promising results

Efficient Evaluation of the Number of False Alarm Criterion

This paper proposes a method for computing efficiently the significance of a parametric pattern inside a binary image. On the one hand, a-contrario strategies avoid the user involvement for tuning detection thresholds, and allow one to account fairly for different pattern sizes. On the other hand, a-contrario criteria become intractable when the pattern complexity in terms of parametrization increases. In this work, we introduce a strategy which relies on the use of a cumulative space of reduced dimensionality, derived from the coupling of a classic (Hough) cumulative space with an integral histogram trick. This space allows us to store partial computations which are required by the a-contrario criterion, and to evaluate the significance with a lower computational cost than by following a straightforward approach. The method is illustrated on synthetic examples on patterns with various parametrizations up to five dimensions. In order to demonstrate how to apply this generic concept in a real scenario, we consider a difficult crack detection task in still images, which has been addressed in the literature with various local and global detection strategies. We model cracks as bounded segments, detected by the proposed a-contrario criterion, which allow us to introduce additional spatial constraints based on their relative alignment. On this application, the proposed strategy yields state-of the-art results, and underlines its potential for handling complex pattern detection tasks

Windowed Green Function method for layered-media scattering

This paper introduces a new Windowed Green Function (WGF) method for the numerical integral-equation solution of problems of electromagnetic scattering by obstacles in presence of dielectric or conducting half-planes. The WGF method, which is based on use of smooth windowing functions and integral kernels that can be expressed directly in terms of the free-space Green function, does not require evaluation of expensive Sommerfeld integrals. The proposed approach is fast, accurate, flexible and easy to implement. In particular, straightforward modifications of existing (accelerated or unaccelerated) solvers suffice to incorporate the WGF capability. The mathematical basis of the method is simple: the method relies on a certain integral equation posed on the union of the boundary of the obstacle and a small flat section of the interface between the penetrable media. Numerical experiments demonstrate that both the near- and far-field errors resulting from the proposed approach decrease faster than any negative power of the window size. In the examples considered in this paper the proposed method is up to thousands of times faster, for a given accuracy, than a corresponding method based on the layer-Green-function.Comment: 17 page