On the Fractal Nature of Local Optima Networks

Thomson, Sarah L.; Vérel, Sebástien; Ochoa, Gabriela; Veerapen, Nadarajen; McMenemy, Paul

doi:10.1007/978-3-319-77449-7_2

Cited by 7 publications

(12 citation statements)

References 10 publications

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“…Another engaging observation from the sampling validation attempt (Section 5.1) was that generally, sampled LON features explained more search variance than those of the enumerated LONs. This is important because the vast majority of LON research Daolio et al, 2011;Verel et al, 2011;Herrmann et al, 2016) dissected fully enumerated LONs, even using their features for algorithm performance prediction (Daolio et al, 2012;Herrmann et al, 2018;Thomson et al, 2018). Our results suggest that LON construction algorithms may approximate or infer yet unseen fitness landscapes better than best-improvement exhaustive enumeration of the local optima level.…”

Section: Discussionmentioning

confidence: 89%

Inferring Future Landscapes: Sampling the Local Optima Level

Thomson

Ochoa

Vérel

et al. 2020

Evolutionary Computation

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Connection patterns among Local Optima Networks (LONs) can inform heuristic design for optimisation. LON research has predominantly required complete enumeration of a fitness landscape, thereby restricting analysis to problems diminutive in size compared to real-life situations. LON sampling algorithms are therefore important. In this article, we study LON construction algorithms for the Quadratic Assignment Problem (QAP). Using machine learning, we use estimated LON features to predict search performance for competitive heuristics used in the QAP domain. The results show that by using random forest regression, LON construction algorithms produce fitness landscape features which can explain almost all search variance. We find that LON samples better relate to search than enumerated LONs do. The importance of fitness levels of sampled LONs in search predictions is crystallised. Features from LONs produced by different algorithms are combined in predictions for the first time, with promising results for this “super-sampling”: a model to predict tabu search success explained 99% of variance. Arguments are made for the use-case of each LON algorithm and for combining the exploitative process of one with the exploratory optimisation of the other.

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Section: Discussionmentioning

confidence: 89%

Inferring Future Landscapes: Sampling the Local Optima Level

Thomson

Ochoa

Vérel

et al. 2020

Evolutionary Computation

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“…While in continuous optimization a large set of features has been defined and can be computed with the flacco package [Kerschke and Trautmann(2016)], the research community currently lacks a meaningful analog for discrete optimization problems. We note though, that several advances in this direction have been made, including the above-introduced features covered by the W-model (size of the effective dimension, neutrality, epistasis, ruggedness) and the local optima networks (see [Thomson et al(2018a) Thomson, Vérel, Ochoa, Veerapen, and McMenemy, Thomson et al(2018b) Thomson, Vérel, Ochoa, Veerapen, and Cairns] and references mentioned therein). We suggest to start a first prototype using these existing features, while at the same time intensifying research efforts to find additional landscape features that can be used to characterize pseudo-Boolean optimization problems.…”

Section: Discussionmentioning

confidence: 99%

Benchmarking discrete optimization heuristics with IOHprofiler

Doerr

Ye²,

Horesh³

et al. 2019

Proceedings of the Genetic and Evolutionary Computation Conference Companion

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Automated benchmarking environments aim to support researchers in understanding how different algorithms perform on different types of optimization problems. Such comparisons provide insights into the strengths and weaknesses of different approaches, which can be leveraged into designing new algorithms and into the automation of algorithm selection and configuration. With the ultimate goal to create a meaningful benchmark set for iterative optimization heuristics, we have recently released IOHprofiler, a software built to create detailed performance comparisons between iterative optimization heuristics. With this present work we demonstrate that IOHprofiler provides a suitable environment for automated benchmarking. We compile and assess a selection of 23 discrete optimization problems that subscribe to different types of fitness landscapes. For each selected problem we compare performances of twelve different heuristics, which are as of now available as baseline algorithms in IOHprofiler. We also provide a new module for IOHprofiler which extents the fixed-target and fixed-budget results for the individual problems by ECDF results, which allows one to derive aggregated performance statistics for groups of problems.

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“…A previous study modified box-counting [14] for local optima networks such that fitness distance was considered as well as link distance: two nodes can be boxed together iff d(lo i , lo j ) < s and…”

Section: A Fractional Dimensionmentioning

confidence: 99%

“…In a previous study, fractal dimensions were calculated on the LONs of a set of NK Landscape instances [14]. The box-counting algorithm used considered edge distance between nodes as a scaling factor for dimension calculation.…”

Section: Probabilistic Dimensionsmentioning

confidence: 99%

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Multifractality and dimensional determinism in local optima networks

Thomson

Vérel

Ochoa

et al. 2018

Proceedings of the Genetic and Evolutionary Computation Conference

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We conduct a study of local optima networks (LONs) in a search space using fractal dimensions. The fractal dimension (FD) of these networks is a complexity index which assigns a non-integer dimension to an object. We propose a fine-grained approach to obtaining the FD of LONs, using the probabilistic search transitions encoded in LON edge weights. We then apply multi-fractal calculations to LONs for the first time, comparing with mono-fractal analysis. For complex systems such as LONs, the dimensionality may be different between two sub-systems and multi-fractal analysis is needed. Here we focus on the Quadratic Assignment Problem (QAP), conducting fractal analyses on sampled LONs of reasonable size for the first time. We also include fully enumerated LONs of smaller size. Our results show that local optima spaces can be multi-fractal and that valuable information regarding probabilistic self-similarity is encoded in the edge weights of local optima networks. Links are drawn between these phenomena and the performance of two competitive metaheuristic algorithms.

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On the Fractal Nature of Local Optima Networks

Cited by 7 publications

References 10 publications

Inferring Future Landscapes: Sampling the Local Optima Level

Inferring Future Landscapes: Sampling the Local Optima Level

Benchmarking discrete optimization heuristics with IOHprofiler

Multifractality and dimensional determinism in local optima networks

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