PaperlayerView exact revision on arXiv

Source-linked AI summary

A Flexible and Generic Approach for Explainable Landscape Analysis and the pyXla Toolbox

Tony Ombaso, Anna S. Bosman, Arnaud Liefooghe, Katherine M. Malan, Sébastien Verel

arXiv:2610.06359v1cs.NEcs.MS

TL;DR

Landscape analysis has many techniques and tools, but choosing and implementing them is difficult, and many features are hard to interpret. The paper introduces XLA and the Python package pyXla to provide interpretable analysis across varied optimisation problems. It demonstrates pyXla’s visualisations and metrics on a selection of continuous and discrete problems, including constrained and multi-objective cases.

  • Problem

    Choosing and implementing landscape-analysis techniques is difficult, existing tools have domain or measurement limits, and many features are not easily interpretable.

    Source: §1 · pp. 2–3

  • Method

    XLA combines intuitively interpretable landscape features with visualisations and metrics, separating data loading and sampling from analysis so it can support varied inputs and problem domains.

    Source: §1 · p. 3

  • Results

    pyXla demonstrates interpretable visualisations and metrics on continuous and discrete optimisation problems with and without constraints or multiple objectives.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The framework’s analysis can begin with objective-value samples, while additional inputs such as violation values, neighbourhood information, or solution samples make more features applicable.

    Source: §1 · p. 3

Abstract

from arXiv · show

Landscape analysis has been successfully applied to understand complex optimisation problems, gain insights into algorithm behaviour, and automate algorithm selection and configuration. Although many landscape analysis techniques have been developed over the last decades, it remains difficult for researchers and practitioners to decide which approaches are appropriate and to implement them in practice. Some tools are available, but these are either restricted to particular problem domains (e.g., unconstrained black-box continuous optimisation), or are limited in what they model and measure. In addition, output from landscape analysis is often not easily interpretable, especially when computed landscape features do not correspond with aspects of problems that practitioners are familiar with. In this paper, we introduce a principled approach for explainable landscape analysis (XLA) with an associated Python package called pyXla. The approach is generic in that it applies to problems with different representations (continuous or combinatorial), with single or multiple objectives, with or without constraints. The extent of analysis provided by the XLA framework depends on the data available, with richer analysis offered as additional information is provided by the user. We demonstrate the explainable output produced by pyXla on a selection of hand-crafted problems with diverse landscape characteristics.

1 Introduction

Landscape analysis can help practitioners understand optimisation problems, but the field’s growing emphasis on high-dimensional features and predictive automation has made concrete interpretation difficult. The paper proposes XLA and pyXla to provide interpretable analysis across varied optimisation domains.

1 Introduction

  • High-dimensional ELA features and machine-learning pipelines make it difficult to state which precise features distinguish problem classes, despite efforts to relate low-level features to properties such as separability and multimodality.The authors give neutrality as an example of a property whose high and low levels are hard to distinguish concretely.Source: §1 · p. 2
  • The authors argue that landscape features should provide actionable meaning for practitioners, rather than being valued only for predictive accuracy.They note that predictive performance depends on context, such as the algorithm or portfolio being considered within a problem class.Source: §1 · pp. 2–3
  • LA tools and techniques remain difficult to choose and implement, while existing libraries are limited by domain or feature focus and no unified library is known to combine interpretable features across problem domains.The authors describe flacco and pflacco as specialised to continuous problems and ELA, and Paradiseo as focused mostly on unconstrained combinatorial single-objective optimisation.Source: §1 · p. 3
  • The paper proposes XLA, which combines intuitively interpretable features with visualisations and metrics, and separates data loading and sampling from analysis to support different inputs and problem domains.The framework can produce output from objective-value samples alone, with more features becoming applicable when violation values, neighbourhood information, or solution samples are supplied.Source: §1 · p. 3
  • The authors implement XLA as the open-source Python package pyXla and demonstrate its interpretable visualisations and metrics on continuous and discrete problems, with and without constraints or multiple objectives.Source: §1 · p. 3

2 Overview of the Proposed XLA Framework

The XLA framework separates data representation from analysis. It applies visual and numerical landscape-analysis techniques to supplied or generated data, with available analysis depending on the inputs.

2 Overview of the Proposed XLA Framework

  • XLA separates data representation, which includes loading and sampling, from analysis, which applies landscape-analysis techniques to the represented data.Source: §2 · p. 4
  • The input format identifies sampled solutions as X, objective values as F, violations as V, distances as D, and neighbourhood information as N.Source: §2 · p. 4
  • XLA features are relative to the supplied sample or full enumeration, and the user controls the sample budget because the framework performs no additional evaluations.Supported sampling methods include random uniform, Latin hypercube, random walk, adaptive walk, Hilbert curve, and full enumeration for small combinatorial spaces.Source: §2 · p. 4
  • Analysis techniques are grouped as statistical, rank-based, distance-based, and neighbourhood-based, and produce both visual and numerical outputs; where possible, they also analyse constraint-violation values.Extending fitness distance correlation to violation values yields violation distance correlation for analysing violation landscapes.Source: §2 · p. 4

3 XLA Features and the pyXla Package

pyXla implements the XLA framework in Python and provides its analysis techniques. The input description identifies the data used for analysis and how records are linked across files.

3 XLA Features and the pyXla Package

  • pyXla is the Python implementation of XLA; its package description covers input handling, data-generation support, and the landscape-analysis techniques it provides.Source: §3 · p. 4

3.1 Input Data

  • Only objective-function values (F) are required for basic pyXla analysis.Source: §3.1 · p. 4
  • Table 1 describes pyXla’s input data, with identifiers in X, F, V, and I and the id1/id2 fields in D and N linking data between files.Source: Table 1 · p. 5

D Pairwise distance between solutions

The D input represents pairwise distances between solutions. If D is unavailable, pyXla can generate it from solution data and a specified distance metric.

D Pairwise distance between solutions

N Neighbouring pairs of solutions

If neighbourhood relationships N are not supplied, pyXla can derive them from a neighbourhood function or from pairwise distances using a specified neighbourhood size rule.

N Neighbouring pairs of solutions

I Additional information

pyXla separates data representation from landscape analysis: users can supply or generate inputs, and additional input components support richer analysis. Its features span statistical, rank-based, distance-based, and neighbourhood-based categories in a unified toolbox.

I Additional information

  • All input components must refer to the same solution sample, with a unique identifier for each solution across input files.Source: I Additional information · p. 6

Auxiliary Functions for Input Data Generation

  • pyXla supports supplied or generated inputs and five samplers; adaptive walks accept improving neighbours until local optima, while Hilbert-curve sampling spatially correlates samples.Hilbert-curve ordering avoids reordering to determine neighbourhood and is more efficient than LHS when some metrics require post-factum ordering.Source: Auxiliary Functions for Input Data… · pp. 6–7

3.2 XLA Features

  • The four feature classes are statistical, rank-based, distance-based, and neighbourhood-based, with Figure 3 mapping the inputs or input combinations required by each feature.Source: §3.2 · p. 7, Figure 3 · p. 8

3.2.1 Statistical Features

  • Statistical features include objective and violation distributions, value and rank correlations, and variable importance, with distribution outputs summarized by nine descriptive statistics.Source: §3.2.1 · pp. 8–9

3.2.2 Rank-based Features

  • Pareto-rank and Deb-rule distributions summarize rankings; the Deb rule ranks by violations before objectives, so feasible solutions always outrank infeasible ones.Pareto ranks apply to objectives, violations, or their combination; for a single objective they correspond to objective-value ordering, with lower ranks better.Source: §3.2.2 · pp. 9–10

3.2.3 Distance-based Features

  • Distance-based features include fitness, violation, rank, and pairwise distance correlations, plus dispersion of the best solutions.Source: §3.2.3 · p. 10

3.2.4 Neighbourhood-based Features

  • Neighbourhood-based features measure correlations in neighbouring solutions’ objective values, violations, or ranks, and changes in feasibility.Source: §3.2.4 · p. 10

3.3 The pyXla Package

  • pyXla implements the four feature categories in an open-source toolbox for continuous and combinatorial, single- and multi-objective, constrained and unconstrained problems.The package deliberately limits its feature set to support explainability and problem understanding rather than downstream automation.Source: §3.3 · p. 10

4 Illustrative Use Cases of Explainable Landscape Analysis

The use-case section illustrates the framework’s flexibility, generality, and explainability through selected problem instances, then discusses feature outputs across categories.

4 Illustrative Use Cases of Explainable Landscape Analysis

  • Selected problem instances illustrate framework flexibility, generality, and explainability, with feature outputs discussed by category across use cases.Source: §4 · p. 13

4.1 Selected Problem Instances

The selected instances span continuous and combinatorial, single- and multi-objective, constrained and unconstrained problems, with varied complexity. Together, they illustrate pyXla across diverse problem settings.

4.1 Selected Problem Instances

  • The suite spans continuous and combinatorial, single- and multi-objective, constrained and unconstrained problems, and includes instances of varying complexity.Table 7 summarizes their representations and sampling strategies.Source: §4.1 · p. 13

4.1.1 Constrained Sphere Problem

  • One hundred 1-dimensional points sampled within [−5, 5] define a constrained sphere minimization problem whose objective and two constraint violations are visualized in Figure 4.The points are obtained using Hilbert curve sampling.Source: §4.1.1 · pp. 13–14, Figure 4 · p. 14

4.1.2 Constrained Deceptive Problem

  • The constrained deceptive problem uses 100 Hilbert-sampled 1-dimensional solutions in [−5, 5] and has two local minima, with the global minimum in a narrow basin.Figure 5 visualizes the objective and constraint values.Source: §4.1.2 · p. 14, Figure 5 · p. 15

4.1.3 (Unconstrained) Easom Function

  • The unconstrained Easom example uses 100 Hilbert-sampled 2-dimensional points in [−5, 5]; its surface is unimodal and has high neutrality.The surface is shown in Figure 6.Source: §4.1.3 · p. 14, Figure 6 · p. 15

4.1.4 Knapsack Instances

  • The easy knapsack instance combines high feasibility with uncorrelated profits and weights, while the hard instance has low feasibility and strongly correlated profits and weights.The paper describes uncorrelated cases as generally easier and strongly correlated cases as generally harder to solve.Source: §4.1.4 · p. 15

4.1.5 NK Landscape Instances

  • The NK suite has three unconstrained single-objective instances with n = 14 and k = 1, 2, and 4, plus one constrained three-objective instance with k = 2.Higher k makes the landscape more rugged and harder to solve; each bit's contribution depends on its interacting variables, with lookup-table sub-functions assigning objective values.Source: §4.1.5 · p. 16

4.1.6 Simulating XOR Gate using a Neural Network

  • The XOR study optimizes a 9-parameter network using SGD for 1000 epochs or randomly samples 1000 configurations, comparing binary cross-entropy loss across the two scenarios.The network uses tanh in its hidden layer and a sigmoid output converted to Boolean logic using a 0.5 threshold; both scenarios use the same seed and corresponding parameter-sampling distributions.Source: §4.1.6 · pp. 16–17

4.2 pyXla in Action

pyXla demonstrates how statistical, rank-based, distance-based, and neighborhood-based features provide interpretable analyses of landscape structure, constraint behavior, and searchability across diverse problem instances. Its neighborhood-based outputs also distinguish search support and feasibility transitions, including the low searchability and more discontiguous feasibility transitions associated with constraint (3)’s cyclic sine behavior.

4.2.1 Statistical Features

  • The objective-value distributions distinguish the parabolic sphere from neutral Easom, while XOR loss distributions show lower losses along the SGD trajectory than under random sampling.Sphere skewness is 0.6142; Easom skewness is -4.6114 and kurtosis is 19.8784. The Easom distribution’s many near-equal values reflect neutrality.Source: §4.2.1 · p. 18, Figure 8 · p. 18, Table 9 · p. 19
  • Easy and hard knapsack instances have feasibility rates of 88.18% and 0.0977%, respectively, with mostly zero violations in the easy instance and normally distributed violations in the hard one.The easy instance’s violation values and ranks are left-skewed; the hard instance’s values and ranks are normally distributed.Source: §4.2.1 · pp. 18–19, Figure 10 · p. 20
  • For the three-objective NK instance, objectives are weakly correlated; on the constrained sphere, sinusoidal Constraint v1 is weakly correlated with other ranks, while Constraint v0 correlates strongly with them.A strong positive correlation with the objective makes low-objective solutions more likely to have lower violations and be feasible.Source: §4.2.1 · pp. 19–20, Figure 11 · p. 20, Figure 12 · p. 21
  • Variable importance identifies different influential neural-network parameters for trained and randomly sampled XOR scenarios, while NK variable importance generally decreases as k increases.For SGD, w3 and w1 are most important; for random sampling, b2, w5, and w4 are most important. NK importance scores agree with Spearman correlations, and higher k increases variable dependencies.Source: §4.2.1 · pp. 20–22, Figure 12 · p. 22, Figure 13 · p. 23

4.2.2 Rank-based Features

  • Pareto ranks are evenly spread for the constrained deceptive problem (maximum 100) but shared more often for constrained sphere (maximum 52); Deb ranks are normal for easy knapsack and uniform for hard knapsack.The deceptive problem’s single constraint is correlated with its objective, yielding unit-size Pareto fronts and uniformly distributed ranks; combining objective and violation ranks produces more distinct Deb ranks for hard knapsack.Source: §4.2.2 · pp. 22–24, Figure 14 · p. 24, Figure 15 · p. 25

4.2.3 Distance-based Features

  • Fitness-distance correlation distinguishes nondeceptive sphere from deceptive constrained deceptive, and decreases across NK instances as k rises from 1 to 4.The NK values are 0.42, 0.28, and 0.10 for k = 1, 2, and 4. XOR’s SGD trajectory has near-perfect correlation, whereas random sampling yields 0.09, indicating that gradient information makes good solutions easier to find.Source: §4.2.3 · pp. 24–27
  • VDC indicates Constraint (2) is easier to cross into feasibility than Constraint (3), while RDC shows local search readily finds improving objective and violation ranks, less easily for Constraint (3).Constraint (3)’s VDC Spearman coefficient is 0.24; RDC coefficients range from 0.94 to 1.00 except Constraint (3), which is 0.91.Source: §4.2.3 · pp. 27–29, Figure 21 · p. 28, Figure 22 · p. 29
  • Pairwise-distance analysis gives Easom the weakest objective and objective-rank correlations, while dispersion distinguishes unimodal sphere structure from deceptive-problem funnels.Easom’s objective-space and objective-rank PDC values are 0.23 and 0.19; dispersion is -2.331 for sphere and 0.88 for constrained deceptive.Source: §4.2.3 · pp. 29–31, Figure 23 · p. 30, Figure 24 · p. 31

4.2.4 Neighbourhood-based Features

  • Constraint (3) has low searchability because neighboring solutions are less likely to traverse continuously toward improving feasibility.The passage attributes this behavior to the constraint’s neighboring solution values.Source: §4.2.4 · p. 34, Figure 29 · p. 34
  • NRC indicates high searchability for the constrained sphere objective (0.9970), but lower searchability for neutral Easom (0.3308).The constrained sphere’s NRC values are 0.9949 for Constraint (2), -0.2022 for Constraint (3), 0.8840 for paretoV, 0.8898 for Deb’s ranks, and 0.9961 for paretoFV; Constraint (3)’s cyclic sine behavior limits searchability.Source: §4.2.4 · p. 34, Figure 30 · p. 35, Figure 31 · p. 35
  • 0.8383 contiguous-feasibility transitions versus 0.1616 discontiguous-feasibility transitions occur when both constraints are considered.Constraint (2) has proportions of 0.9798 contiguous and 0.0202 discontiguous feasibility; Constraint (3) has 0.404 contiguous and 0.596 discontiguous feasibility, consistent with its cyclic sine function.Source: §4.2.4 · pp. 35–36, Figure 32 · p. 36

5 Conclusion

XLA and pyXla separate input representation and sampling from analysis, enabling explainable analysis across continuous or combinatorial, constrained or unconstrained, single- or multi-objective problems. pyXla provides interpretable visual and numerical outputs, and its feature coverage remains incomplete.

5 Conclusion

  • The framework supports continuous or combinatorial, constrained or unconstrained, single- or multi-objective problems by separating input representation and sampling from analysis.Its techniques produce visual and numerical outputs intended to be intuitively interpretable.Source: §5 · p. 36
  • pyXla composes 16 explainable landscape features and five sampling algorithms, with inputs supplied as data files or generated by specified functions.Inputs can include objective values, solutions, violation values, pairwise distances, neighbourhood relationships, and additional information.Source: §5 · pp. 36–37
  • The 16 features span statistical, rank-based, distance-based, and neighbourhood categories, while uniform random sampling applies to both combinatorial and continuous domains.The other four sampling techniques are for continuous domains in the current version.Source: §5 · p. 37
  • The toolbox is not exhaustive; future work includes analyses of entire problem suites and real-world problems, plus additional features and sampling techniques.Planned features include local optima networks, ruggedness and neutrality measures, multi-objective measures, and features for variable interactions.Source: §5 · p. 37
Loading 2610.06359v1…