# A Deep Learning-Driven Black-Box Benchmark Generation Method via Exploratory Landscape Analysis
## 5 Experiment Setup
### Landscape Analysis Perspective  
In this study, ELA feature vectors derived from functions in the BBOB benchmark suite are used as target functions to validate the proposed methodology [34]. The initial instance of each function is used in this experiment for dimensions \(D\in\{2,\ 3\}\). A problem instance is defined as \(p_{i}=(FID,D)\), which results in 48 distinct problem instances to be reconstructed. For each problem instance, the CMA-ES implementation from the pycma packages is used with a budget of \(200n\), where \(n\) represents the dimensionality of the topological map point optimization problem rather than the function dimension \(D\) to ensure that neural networks are trained using the optimal topological map point. All reported results correspond to the fits of the best topological map points. Examples of topological map points where the target ELA feature vector is well approximated, along with a counter-example, are shown in Figure 4. Figure 4a,d illustrates the scenario for the BBOB function (FID 3) and function (FID 16) plots for the surface. Figure 4b,e show random samples from the original BBOB function (FID 3) and function (FID 16). SubsequentFigure 4c,f, which present the topological map point optimized for the BBOB function with FID 3 and FID 16. It is observed that the two landscapes are rotated while maintaining structural similarity. However, some function landscapes exhibit marked quadratic patterns and noise, which arise from the low resolution of the topological map point. For accuracy assessment, 200 samples are constructed for each problem instance \(p_{i}\), which employs Latin Hypercube Sampling (LHS) and calculates their ELA feature vectors. All vectors are combined to form a single resultant vector, denoted as \(\overrightarrow{v_{i}}\). The neural network architecture is adopted for all 48 problem instances, but train an individual NN for each instance. Every problem \(p_{i}\) obtains its own NN model. Training involves randomness. NNs are trained for each problem instance, differing only in their initial weights. Interestingly, NNs interpolate topological map point samples similarly [35]. The fitness landscape of the generated function remains unaffected by the initial weights and biases of the neural networks. For the NNs, 200 ELA feature vectors \(z_{i}^{(j)},j\in\{1,2,\cdots,200\}\) are generated. The fitness landscapes of a representative set of functions are thoroughly analyzed, which cover the successful and unsuccessful operational domains of the proposed method [34]. Additionally, examples of both uni-modal and multi-modal problem instances are presented. Figure 5 presents a parallel coordinate plot visualizing the ELA features, in which the target ELA values are plotted in black and the surrogate model outputs are displayed in color. The black lines indicate the eight target ELA vectors. The colored line represents the mean ELA feature vector obtained from the trained NNs, and the shaded regions denote the tenth to ninetieth percentile ranges of the sampled ELA feature values. The top row of Figure 5 displays the Sphere function (FID 1), one of the easiest optimization problems in the BBOB benchmark suite. The approximation exhibits similarity but lacks the artificially smooth characteristics of the BBOB benchmark function. The Ellipsoid separable function (FID 2) is a similar case. Its separability and overall quadratic structure were successfully emulated. Although the generated landscape is rotated, this is not a fundamental issue because it does not alter the characteristics of the problem. The Rosenbrock rotated function (FID 9) poses a significant challenge to optimization  
Figure 4: Comparison of the original BBOB landscapes and generated topological map points. (**a**,**d**) Show 3D surface plots of BBOB functions FID 3 and FID 16. (**b**,**e**) Show random LHS samples from the respective BBOB functions. (**c**,**f**) Topological map points optimized via CMA-ES. Structural similarities and rotation effects are visible.