# A Deep Learning-Driven Black-Box Benchmark Generation Method via Exploratory Landscape Analysis
## 4 New Benchmark Function Generation
### Surrogate Models  
In order to create a new function based on an optimized topological map point \((W,x^{*})\), the work uses pre-trained NNs as surrogate approximation models. This surrogate model \(s:\mathbb{R}^{m\times d}\rightarrow\mathbb{R},s(w)=\hat{a}\). It is fitted on the dataset \(\mathcal{T}=\left\{w^{(i)},x^{*(i)}\right\}\ (i=1,\cdots,n)\), where \(w^{(i)}\) is the \(i\)th row of \(W\) and \(x^{*(i)}\) the \(i\)th element of \(x^{*}\). A feed-forward network is used for this purpose. The network has two hidden layers, each with 256 units. The tanh activation function is adopted for the final hidden layer. A sigmoid activation function is usedfor the final output layer. This choice is made because the objective values are scaled via min-max normalization. This study ensures that the surrogate model predictions maintain scale consistency with the normalized objective values across all experimental conditions. The selection of a surrogate model is driven by its theoretical properties. Feed-forward neural networks are known as universal approximators, which means that each hidden-layer network approximates the fitness function to arbitrary levels of accuracy when the network contains a sufficient number of hidden units. Additionally, neural networks provide practical benefits due to their flexibility, which enables them to model intricate data relationships and efficiency in processing large-scale datasets. When properly trained, neural networks generalize well to new datasets. Their seamless deployment and integration into benchmark suites, along with minimal latency overhead during evaluation, further reinforces their suitability as surrogate models. Neural networks are particularly advantageous in this scenario, as they outperform alternative models. However, it is important to acknowledge their inherent tendency to learn smooth functions, particularly when employing the tanh and sigmoid activation functions. This inductive bias can only be adjusted to a limited extent by modifying the network architecture. Deliberate overfitting is acceptable and even necessary because the generated neural surrogate is not intended for predictive deployment but rather for reconstructing a function landscape with a prescribed topological signature. In other words, the training set is the specification of the desired behavior, not a sample from a broader data distribution. The training procedure, as detailed in Algorithm 1, employs gradient descent optimization with the mean squared error serving as the loss function to minimize discrepancies between predicted and target values. The core equations are necessary to understand the optimization and surrogate learning processes. For clarity, all variables and operators are introduced in Algorithm 1. Additional mathematical details (e.g., regularization or batch strategies) are omitted for readability but are available upon request. The study utilizes the [PERSON] optimization algorithm with default hyperparameters and a learning rate of 0.01. The training was conducted over 5000 epochs to achieve an exact fit to the training data, thereby intentionally overfitting. This is required to ensure that the surrogate model \(s\) reproduce the target ELA feature vector \(v\), or a close approximation, when assessed at the anchor \(W\). In other words, the work applies \(z(x)=\parallel f(W,x)-v\parallel_{2}\) to be sufficiently small. A single batch containing all available data is adopted. This is due to the training objective being deliberately set to strongly overfit the training data. Neural networks are implemented using PyTorch (2.3 version) packages [31]. ```
Input: optimized point \(\{x_{i}\}_{i=1}^{n}\), fitness value \(\{w(x_{i})\}_{i=1}^{n}\), initial parameter \(\mu\), size of gradient descent steps \(J\), neural networks width \(m\), regularization parameter \(\tau\), learning rate \(\gamma\).