Batch Normalization

As a network trains, every layer's weights shift — so the distribution each layer receives keeps moving under it. Batch norm pins that distribution in place: standardize each feature across the batch, then let the network learn its own scale and shift back.

For each feature j, across the batch:

1normalized = (x - mean[j]) / sqrt(variance[j] + epsilon)
2output     = gamma[j] * normalized + beta[j]

Task: write batch_norm(batch, gamma, beta, epsilon) where batch is a list of samples (each a list of features). Return the same shape, rounded to 4 decimal places.

• Statistics are computed per feature, down the batch — not across the features of one sample.
• Use the population variance (divide by n).
• epsilon goes inside the square root.

gamma and beta exist because forcing every layer to zero mean and unit variance would be a real loss of expressiveness. Learning them back means the network can undo the normalization exactly when that helps — it just no longer has to.