# A Deep Learning-Driven Black-Box Benchmark Generation Method via Exploratory Landscape Analysis
## 3 Exploratory Landscape Analysis and BBOB Benchmark
### Landscape Analysis  
Landscape analysis involves mathematical and statistical methods for quantifying the features of an optimization problem, thereby describing the feature space [19]. Generally, Landscape analysis is formulated as a function: \(\zeta:(X,y)\longrightarrow v\), which maps the sampled \(X\) and its corresponding objective values \(y\) to a feature vector \(v\), each dimension of which indicates some problem-specific properties of the target optimization problem. Subsequently, the feature vector \(v\) fed into machine learning algorithms to solve some specific algorithmic tasks. While the difficulty and characteristics of an optimization problem is visually characterized for up to two-dimensional problems given enough samples, a visualization of higher-dimensional problems is generally infeasible [20]. Therefore, other methods are required to quantify critical landscape characteristics, which include modality density and structural funnel configurations, and ultimately determine the difficulty of a problem. These methods generate a set of numerical features. In single-objective continuous optimization domains, these features are grouped under the term ELA [21]. Landscape analysis encompasses both ELA and fitness landscape analysis. An individual ELA measure is typically linked to multiple high-level properties, whereas an individual FLA measure usually represents a single high-level characteristic of an optimization problem. ELA employs stochastic sampling techniques to compute numerical landscape descriptors from objective function evaluations across the search domain, which enable quantitative characterization of black-box optimization problems [22]. Over the years, various sampling procedures have been studied. The sampling size depends on the analysis scope. In automated algorithm selection, the size needs to be small to maintain efficiency, while theoretical studies use larger samples. Regardless of scope, the sample size usually increases with the dimensionality of the problem [23]. In this work, a sampling size of 200 \(\times\)\(D\) (\(D\) denoting the decision space dimensionality) is adopted to balance search space coverage with computational efficiency. This size is also recommender to classify all BBOB problems. Latin hypercube sampling [24] is used as a sampling strategy. Additionally, the objective values of the fitness function are normalized to the \([0,1]\) range using sample minimum and maximum. This has two advantages: It corrects for the fact that some ELA features are not invariant to linear transformations of the objective space and it helps with neural networks, which use a sigmoid activation function and mapping all values to the \([0,1]\) interval. Many of the features of ELA are quite similar, and in practice, when computed for the BBOB functions, they exhibit a high degree of correlation. The correlation between the selected ELA feature on three dimensions is shown in Figure 1. These features are selected since they do not require further objective function evaluations other than the initial sample [25]. However, the complexity of optimization challenges inherently relates to fitness landscape characteristics, thus making strategic ELA feature selection a critical design component. This work implements an iterative refinement framework grounded in prior research [26], while dynamically adjusting feature subsets to enhance topological diversity across synthesized benchmark functions. The final chosen ELA features to balance descriptive power and computational tractability, and are grounded in prior studies that evaluate their discriminative capability, computed using the pflacco [27] package, are listed below:  
* Meta model (ela_meta): Defined by [PERSON] et al. [28]. These measures determine how well the sampled fitness values align with linear and quadratic models. The article selects four ela_meta features: \begin{tabular}{}
* ela_meta.lin_simple.adj_r2, ela_meta.lin_w_interact.adj_r2, \\ ela_meta.quad_simple.adj_r2, ela_meta.quad_w_interact.adj_r2. \\ \end{tabular}