# SBNNR: Small-Size Bat-Optimized KNN Regression
## 4 Proposed Method
### Generative Adversarial Networks (GANs) for Synthetic Data Generation  
Generative adversarial networks (GANs) represent a powerful framework for generating synthetic data that resemble real data distributions. In our methodology, GANs are utilized to address data sparsity by creating realistic synthetic samples that augment our existing dataset. This is particularly important in scenarios where acquiring additional labeled data is challenging or costly. By employing a dual network structure consisting of a generator and a discriminator, GANs iteratively improve their performance through adversarial training. The generator creates synthetic samples, while the discriminator evaluates their authenticity. This dynamic process not only enriches our dataset but also helps in reducing overfitting and enhancing the generalization of our models.  
GANs play a crucial role in enhancing model performance, especially when handling small datasets. GANs address data scarcity by generating synthetic data that replicate the underlying distribution of the original data. In this process, GANs use a generator network to create synthetic samples and a discriminator network to evaluate their authenticity. The generator aims to produce samples (\(G(z)\)) that closely resemble real data, where \(z\) is a random noise vector sampled from a prior distribution (e.g., Gaussian). The discriminator (\(D(x)\)) distinguishes between real data points (\(x\)) and generated samples (\(G(z)\)). The networks are trained in an adversarial setup, iterating until the generator produces realistic data that the discriminator cannot reliably distinguish from the real data.  
This adversarial setup effectively allows GANs to generate synthetic data that preserve complex, non-linear relationships within the original data distribution. The optimization objective for GAN training can be expressed as [35]  
\[\min_{G}\max_{D}\mathbb{E}_{x\sim p_{data}(x)}[\log D(x)]+\mathbb{E}_{z\sim p_{ z}(z)}[\log(1-D(G(z)))] \tag{7}\]  
where \(p_{data}(x)\) is the data distribution and \(p_{z}(z)\) is the prior distribution of the noise vector. Through this minimax game, the generator learns to produce data that follow the intricate patterns and variations within the original dataset, resulting in high-quality synthetic samples that enhance model robustness and generalization.  
GANs are also beneficial for conditional generation, where the generation process is guided by specific data attributes or class labels. Conditional GANs (cGANs) introduce acondition (\(y\)) to both the generator and discriminator, enabling the generation of samples for under-represented categories within the dataset. This is achieved by modifying the objective function to incorporate the conditional variable (\(y\)):  
\[\min_{G}\max_{D}\mathbb{E}_{x\sim p_{data}(x)}[\log D(x|y)]+\mathbb{E}_{z\sim p _{z}(z)}[\log(1-D(G(z|y)))] \tag{8}\]  
By generating targeted synthetic data, cGANs ensure that even rare classes are adequately represented, improving model performance on imbalanced data and enhancing predictive power.  
In summary, GANs facilitate model training on limited datasets by expanding the effective data size and preserving complex data relationships. This approach reduces overfitting and increases model reliability, making GANs a powerful tool for improving machine learning outcomes in data-scarce domains.