{
  "id": "batch-norm",
  "title": "Batch normalization forward and backward",
  "kind": "implement",
  "summary": "Training and inference forward passes with running statistics, and the full backward pass for x, gamma and beta.",
  "difficulty": "hard",
  "tags": ["batch-norm", "normalization", "gradients", "backpropagation"],
  "lesson": "phases/03-deep-learning-core/07-regularization",
  "cheatsheet": "deep-learning-training",
  "minutes": 45,
  "entry": ["batchnorm_forward", "batchnorm_backward"],
  "collections": ["deep-learning-core"],
  "time_limit_s": 10,
  "mutants": {
    "backward_ignores_batch_stats": "Returns grad_output * gamma / std for dx and forgets that x also moves the batch mean and variance.",
    "eps_outside_sqrt": "Divides by sqrt(var) + eps, not by sqrt(var + eps).",
    "in_place_running_mean": "Updates running_mean in place, so the caller's array changes.",
    "inference_uses_batch_stats": "Uses the batch mean and variance in inference mode, not the running statistics.",
    "momentum_swapped": "Gives the old running value the weight momentum and the batch value the weight 1 - momentum.",
    "unbiased_variance": "Uses the unbiased variance with N - 1, not the biased variance with N."
  }
}