Hilbert curves for efficient exploratory landscape analysis neighbourhood sampling

Abstract: Landscape analysis aims to characterise optimisation problems based on their objective (or fitness) function landscape properties. The problem search space is typically sampled, and various landscape features are estimated based on the samples. One particularly salient set of features is information content, which requires the samples to be sequences of neighbouring solutions, such that the local relationships between consecutive sample points are preserved. Generating such spatially correlated samples that also provide good search space coverage is challenging. It is therefore common to first obtain an unordered sample with good search space coverage, and then apply an ordering algorithm such as the nearest neighbour to minimise the distance between consecutive points in the sample. However, the nearest neighbour algorithm becomes computationally prohibitive in higher dimensions, thus there is a need for more efficient alternatives. In this study, Hilbert space-filling curves are proposed as a method to efficiently obtain high-quality ordered samples. Hilbert curves are a special case of fractal curves, and guarantee uniform coverage of a bounded search space while providing a spatially correlated sample. We study the effectiveness of Hilbert curves as samplers, and discover that they are capable of extracting salient features at a fraction of the computational cost compared to Latin hypercube sampling with post-factum ordering. Further, we investigate the use of Hilbert curves as an ordering strategy, and find that they order the sample significantly faster than the nearest neighbour ordering, without sacrificing the saliency of the extracted features.

• The paper demonstrates that Hilbert curves deliver efficient, uniform sampling of high-dimensional search spaces for effective landscape analysis.
• It shows that Hilbert curves outperform traditional methods like Latin Hypercube Sampling and random walks in computation cost and sampling quality.
• Results confirm that landscape features derived from Hilbert curve samples are highly predictive of function classes in optimization tasks.

Summary of "Hilbert curves for efficient exploratory landscape analysis neighbourhood sampling"

The paper "Hilbert curves for efficient exploratory landscape analysis neighbourhood sampling" by Johannes J. Pienaar, Anna S. Bosman, and Katherine M. Malan investigates the employment of Hilbert space-filling curves as a method for generating spatially correlated samples from continuous search spaces in the context of optimization problem landscape analysis. The key contributions of the paper are twofold: using Hilbert curves for sampling and ordering other sampling methodologies, particularly in Exploratory Landscape Analysis (ELA).

Motivation and Background

Landscape analysis aids in understanding optimization problems by characterizing them based on the properties of their objective or fitness function landscapes. The process typically involves sampling the problem's search space and estimating various landscape features from these samples. A salient set of features in this context is the information content, which necessitates sequences of neighboring solutions to preserve the local relationships between consecutive sample points. The generation of spatially correlated samples that ensure good coverage of the search space is challenging, especially in high-dimensional spaces.

Common approaches for obtaining such samples often use nearest-neighbor algorithms to order an unordered sample, but these become computationally prohibitive as dimensions increase. The paper proposes Hilbert curves, a type of fractal curve, as an efficient alternative providing uniform search space coverage and spatial correlation at a reduced computational cost.

Hilbert Curves as Samplers

Hilbert curves are desirable due to their ability to ensure uniform coverage of a bounded search space and generate spatially correlated samples. By leveraging these properties, the authors assess the efficacy of Hilbert curves against traditional methods like Latin Hypercube Sampling (LHS) and random walks. The study finds that Hilbert curves provide comparable or superior search space coverage to LHS, as measured by Hausdorff distance, and significantly outperform random walks.

Moreover, Hilbert curve sampling proves to be more computationally efficient when calculating information content metrics over other strategies, especially as sample sizes grow. Simulation results indicate that the computational cost of generating a Hilbert curve sample scales acceptably even for large dimensions, making it a practical alternative for landscape analysis tasks.

Predictive Performance and Step Size Consistency

The paper evaluates landscape features extracted from Hilbert-curve-sampled solutions by classifying the landscape of BBOB functions into their respective groups. The evaluation reveals that features derived from Hilbert curve samples are highly predictive of the function class and exhibit similar or better performance compared to those generated using other sampling techniques.

Additionally, the authors analyze the consistency of step sizes in Hilbert curve sampling, noting that the step sizes are more regular compared to nearest-neighbor-ordered LHS samples, which usually tend to display an increasing trend in step sizes as the number of unvisited points decreases.

Hilbert Curves as an Ordering Tool

In addition to its sampling utility, the paper explores the efficiency of using Hilbert curves to order samples generated by other methodologies like LHS. Since a Hilbert curve provides a mapping between a high-dimensional space and a one-dimensional curve, it can order points spatially with minimal computational overhead. The study confirms that Hilbert curve ordering substantially reduces computation time compared to the nearest-neighbor ordering method while maintaining saliency of landscape features.

Conclusion and Implications

The findings suggest that Hilbert curves offer a robust and computationally efficient method for both sampling and ordering in landscape analysis. This approach is particularly beneficial for calculating landscape features that require spatially correlated samples, like the information content metrics.

Future work may explore refining Hilbert curve techniques to further reduce computation time or exploring their applicability in other domains requiring spatial correlation and uniform coverage. The implications extend to automated algorithm design and configuration, where efficient and salient landscape features are crucial.

Hilbert curves present as a viable alternative to traditional methods, balancing computational efficiency with the need for high-quality sampling and ordering in high-dimensional spaces.

For a thorough understanding of the theoretical underpinnings of the discussed methodologies and their computational implementations, the reader is directed to the references cited in the original paper, particularly those related to Hilbert curves, Latin hypercube sampling, and landscape analysis techniques.