Implement and Debug a Minimal Neural Network Training Loop (PyTorch)

Problem: Implement and Debug a Minimal Trainable Neural Network (PyTorch) Given a toy binary classification dataset (feature matrix X and labels y), implement a 2-layer MLP (Linear -> ReLU -> Linear) in PyTorch and write a working training loop. You must: Implement the forward pass and output logits. Compute the loss using BCEWithLogitsLoss. Update parameters with a gradient-based optimizer (e.g., torch.optim.SGD or Adam). Print training loss and ensure it decreases substantially. If provided with buggy code, identify and fix common issues (e.g., missing zero_grad(), shape mismatches, incorrect sigmoid + BCELoss usage, improper detach(), wrong train()/eval() mode). Input X: float tensor of shape (N, D) y: label tensor of shape (N,) or (N, 1) with values in {0,1} Hyperparameters: learning rate, batch size, epochs (your choice) Output Final loss (or a loss curve) Final training accuracy (or AUC) Constraints Must use PyTorch; autograd is allowed. Must run on CPU. Example test setup You may validate with synthetic data: N=256, D=20, X ~ N(0,1) Sample random true weights w, set p = sigmoid(X @ w), y = Bernoulli(p) Training should reduce loss significantly (e.g., >30% from the initial loss). Example Input (synthetic) N=256 D=20 seed=0 Output loss decreases significantly; prints loss/acc every 10 epochs