Batch normalization forward and backward

Problem

Batch normalization normalizes each feature over the examples of a mini-batch. Then it scales and shifts each feature with two learned vectors. In training, it uses the statistics of the batch and updates running averages. In inference, it uses only the running averages. You write the forward pass for both modes and the backward pass for training mode.

Task

Write two functions.

1. batchnorm_forward(x, gamma, beta, running_mean, running_var, momentum=0.1, eps=1e-5, training=True) returns (out, cache, new_running_mean, new_running_var).
2. batchnorm_backward(grad_output, cache) returns (dx, dgamma, dbeta).

In training mode, compute for each feature column j over the batch of N rows:

The variance is the biased variance. It divides by N, not by N - 1. The running statistics also use this biased variance:

Here m is momentum.

In inference mode (training=False), use running_mean and running_var in place of \mu and \sigma^2. Return copies of the running statistics without a change. The cache in inference mode can be None.

batchnorm_backward gets the gradient of the loss with respect to out and the cache from a training mode call. It returns the gradients of the loss with respect to x, gamma and beta. The gradient dx must include the effect of x on \mu and on \sigma^2.

The cache is your own choice. The checks pass the cache from your forward pass to your backward pass and do not read it.

Input and output

• batchnorm_forward returns a tuple (out, cache, new_running_mean, new_running_var). out has shape (N, D). The two running arrays have shape (D,).
• batchnorm_backward returns a tuple (dx, dgamma, dbeta) with shapes (N, D), (D,) and (D,).

Constraints

• 1 <= N <= 64 and 1 <= D <= 32.
• A column can be constant. Its variance is then 0.
• Do not change any input array. Return new arrays for the running statistics.
• Use numpy only.

Examples

pythonCopyx = np.array([[1.0, 2.0], [3.0, 6.0], [5.0, 10.0]])
gamma = np.array([1.0, 0.5])
beta = np.array([0.0, 1.0])
out, cache, mean, var = batchnorm_forward(x, gamma, beta, np.zeros(2), np.ones(2))
dx, dgamma, dbeta = batchnorm_backward(np.array([[1.0, 0.0], [0.0, 1.0], [1.0, 1.0]]), cache)

textCopyout
[[-1.22474258  0.38762785]
 [ 0.          1.        ]
 [ 1.22474258  1.61237215]]
mean
[0.3 0.6]
var
[1.16666667 1.96666667]
dx
[[ 0.20412376 -0.02551558]
 [-0.40824753  0.05103101]
 [ 0.20412376 -0.02551543]]
dgamma
[0.        1.2247443]
dbeta
[2. 2.]

pythonCopyout, cache, mean, var = batchnorm_forward(np.array([[2.0, 4.0]]), gamma, beta, np.array([0.3, 0.6]), np.array([1.16666667, 1.96666667]), training=False)

textCopyout
[[1.57388742 2.21222277]]
cache
None