How can explainable landscape analysis work across diverse optimization problems?

XLA reframes landscape features as explanations, not predictors

Exploratory landscape analysis (ELA) extracts large numbers of low-level features cheaply and feeds them into machine learning pipelines for automated algorithm selection. The paper argues this shift has cost explainability: many low-level features are not directly interpretable, and it has become difficult to state concretely which features distinguish problem classes, such as problems with high versus low neutrality.

The proposed alternative, explainable landscape analysis (XLA), keeps the goal of intuitive understanding. Its stated premise is that a landscape feature is most valuable when it provides actionable meaning for the practitioner, not necessarily when it achieves high predictive accuracy, because predictive performance is context-dependent. Explainability here is delivered by features meant for intuitive interpretation and by pairing visualisations with numerical metrics.

A unified input format is what makes the approach generic

The framework splits into two components that are separate from each other: data representation (loading and sampling) and analysis (applying landscape techniques). Inputs are named by letter: X for sampled solutions, F for objective values, V for constraint violation values, D for pairwise distances, N for neighbourhood relationships, and I for any additional information. All inputs must refer to the same sample of solutions, and each solution carries a unique identifier across files.

Only the objective values F are required for basic analysis. As additional inputs are provided, more features become applicable, so the extent of analysis depends on the data available. This separation is the mechanism behind the paper's claim of generality across continuous and combinatorial representations, single and multiple objectives, and constrained and unconstrained problems.

Users can either supply the input as CSV files in the specified format or provide alternative information, in which case pyXla computes the required input. Auxiliary functions generate each input type: sample X, compute F, compute V, compute D, and compute N. All auxiliary functions except sample X require an X file, which can itself be produced by sampling the problem space.

Sampling choice changes which landscape you actually see

Because landscape features depend on the sample, different sampling algorithms can yield different perspectives of the same problem instance. pyXla implements five: random uniform sampling, Latin hypercube sampling, random walk sampling, adaptive walk sampling, and Hilbert curve sampling. In the current version, random uniform sampling applies to both combinatorial and continuous domains, while the other four are for continuous domains.

The algorithms serve different purposes. Random uniform and Latin hypercube sampling are included for simplicity and ubiquity. A walk is an ordered sequence in which each sample neighbours the previous one; a random walk picks a neighbour at random at each step and is typically used to characterise ruggedness. An adaptive walk selects an improving neighbour at each step until it is trapped in a local optimum, targeting features suited to measuring hill-climbing traces and capturing multimodality. Hilbert curve sampling provides coverage with spatial correlation when samples are mapped to one dimension, removing the need to re-order samples to determine neighbourhood.

The paper illustrates the difference with a 2D comparison of 100 samples within the bounding box [-5, 5], run with the same seed. Random walk and adaptive walk sampling use a step size of 0.5, and the adaptive walk is guided by the sphere function. This is a demonstration of sampling behaviour, not a benchmark of which sampler is best.

Sixteen features are grouped by the inputs they need

The toolbox composes 16 explainable landscape analysis features in four categories. Statistical features are distribution of objective values (distr F), distribution of violation values (distr V), correlation of values (corr), correlation of ranks (corr ranks), and variable importance (X imp). Rank-based features are distribution of Pareto ranks (distr Par) and distribution of Deb's ranks (distr Deb). Distance-based features are fitness distance correlation (FDC), violation distance correlation (VDC), rank distance correlation (RDC), pairwise distance correlation (PDC), and dispersion of the best solutions (disp best). Neighbourhood features are neighbouring solutions' objective values correlation (NFC), neighbouring solutions' violation value correlation (NVC), neighbouring solutions' ranks correlation (NRC), and neighbouring change in feasibility (NCF).

The features are implemented generically where possible, so the same technique can analyse not only fitness values but also violation values for constrained problems. For example, the fitness distance correlation technique is extended to violation values, yielding violation distance correlation and enabling analysis of violation landscapes. The numerical output of distr F and distr V uses the same descriptive statistics throughout pyXla: mean, median, minimum, maximum, first quartile, third quartile, standard deviation, skewness, and kurtosis.

Searchability separates the constrained sphere from the Easom function

The paper demonstrates the output on hand-crafted problems with diverse landscape characteristics. For the neighbouring solutions' ranks correlation (NRC) feature, the constrained sphere problem yields 0.9970 for objective ranks, 0.9949 for the ranks of Constraint (2), -0.2022 for the ranks of Constraint (3), 0.8840 for paretoV ranks, 0.8898 for Deb's ranks, and 0.9961 for paretoFV ranks. The Easom function is unconstrained with a single objective, so it has only one rank and an NRC coefficient of 0.3308.

The paper reads these numbers as a contrast in searchability: the constrained sphere objective has high searchability given its almost perfect NRC coefficient, while the Easom function is much less searchable because of its neutrality. Within the sphere problem, all ranks have high NRC except the ranks of Constraint (3), which is not highly searchable because of the cyclic nature of the sine function. These are feature values on specific demonstration problems, not general rankings of problem classes.

Feasibility transitions expose how constraints shape neighbourhoods

The neighbouring change in feasibility (NCF) feature uses a neighbourhood file to compute the proportion of transitions between solutions that change feasibility and the proportion that do not. For Constraint (2) of the constrained sphere problem, contiguous feasibility is 0.9798 versus 0.0202 discontiguous, which the paper expects because the feasible region is contiguous and sandwiched between two contiguous infeasible regions. For Constraint (3), discontiguous feasibility is 0.596 versus 0.404 contiguous, which the paper attributes to the cyclic sine function making transitions to dissimilar feasibility more likely.

When the two constraints are considered together, contiguous feasibility is 0.8383 versus 0.1616 discontiguous. The contrast between the per-constraint and combined proportions matters: a single constraint can look highly discontiguous while the combined feasibility structure still looks largely contiguous. Readers comparing constraint difficulty should therefore check whether a reported proportion is per constraint or aggregated across constraints.

The toolbox is a starting point, not an exhaustive feature set

The paper states that the developed toolbox is not exhaustive in its collection of XLA features. Planned additions include local optima networks, an entropy-based ruggedness measure, the neutrality measures M1 and M2, multi-objective measures based on dominance and indicators, a non-linear variable importance measure based on the influence of mutating a variable on fitness, and features capturing variable interactions.

The demonstration is also bounded: pyXla is validated on a selection of continuous and discrete optimisation problems with and without constraints or multiple objectives, and the explainable output is shown on hand-crafted problems with diverse landscape characteristics. Future work named by the authors is to use pyXla to analyse an entire problem suite and real-world problems. The reported feature values therefore illustrate what the toolbox produces on designed examples rather than establishing how the features behave across a full benchmark suite.

Match the feature to the inputs and sample behind it

The paper's practical priority follows from its design: before interpreting any pyXla feature value, check which input components were available, because the applicable feature set is determined by the combination of X, F, V, D, N, and I. A feature that requires neighbourhood information cannot be read from an objective-value sample alone, and the framework performs no additional evaluations beyond the user's sample budget.

The second check is the sample itself. Landscape features are relative to the provided sample, and the paper states that different sampling algorithms can yield different perspectives of the same problem instance. The third check is the denominator or aggregation level of the metric: NRC coefficients are reported per rank type, and NCF proportions are reported per constraint and combined. The most useful reading of this paper is therefore not a single headline number but the pairing of a feature with the inputs, sampler, and aggregation that produced it.

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