Adversarial Network for Gaussian Distribution Synthesis

In this challenge, you will implement and train a Generative Adversarial Network (GAN) to synthesize samples from a one-dimensional Gaussian distribution. GANs, introduced by Ian Goodfellow in 2014, represent one of the most influential breakthroughs in generative modeling, enabling neural networks to learn complex data distributions through an elegant adversarial training paradigm.

The Adversarial Learning Framework

A GAN consists of two competing neural networks engaged in a minimax game:

1. Generator Network (G): This network transforms random noise from a simple latent distribution (typically standard normal) into samples that should resemble the target data distribution. The generator's goal is to produce samples so realistic that the discriminator cannot distinguish them from real data.

2. Discriminator Network (D): This network acts as a binary classifier, estimating the probability that a given sample originated from the real data distribution rather than being synthesized by the generator. The discriminator's goal is to correctly identify real samples from fake ones.

Network Architecture Requirements

Both networks should be implemented as simple feedforward neural networks with the following specifications:

Generator Architecture:

• Input Layer: Accepts latent noise vector z (sampled from standard normal distribution)
• Hidden Layer: One fully-connected layer with ReLU activation
• Output Layer: Linear activation (no non-linearity) to produce generated samples

Discriminator Architecture:

• Input Layer: Accepts real or generated data samples
• Hidden Layer: One fully-connected layer with ReLU activation
• Output Layer: Sigmoid activation to output probability ∈ [0, 1]

Training Objectives

The training follows the non-saturating GAN loss formulation:

Discriminator Loss (Binary Cross-Entropy): $$\mathcal{L}D = -\mathbb{E}{x \sim p_{data}}[\log D(x)] - \mathbb{E}_{z \sim p_z}[\log(1 - D(G(z)))]$$

Generator Loss (Non-Saturating): $$\mathcal{L}G = -\mathbb{E}{z \sim p_z}[\log D(G(z))]$$

The non-saturating formulation for the generator provides stronger gradients early in training compared to the original minimax objective, helping to avoid the vanishing gradient problem when the discriminator is too strong.

Training Procedure

During training, you should alternate between:

1. Update Discriminator: Sample a batch of real data and a batch of generated data, compute the discriminator loss, and update discriminator weights using gradient descent.
2. Update Generator: Generate a batch of fake samples, compute the generator loss based on discriminator's response, and update generator weights using gradient descent.

Parameters should be updated using vanilla gradient descent with an appropriate learning rate.

Your Task

Implement a function that:

1. Initializes the generator and discriminator networks with appropriate weights
2. Trains the GAN for the specified number of epochs
3. Returns a trained generator forward function that, given latent noise, produces generated samples along with their mean and standard deviation

The goal is for the trained generator to produce samples that match the target Gaussian distribution's statistical properties (mean and standard deviation).