Stable softmax and cross-entropy

Problem

A classifier gives one row of logits for each example. Softmax turns each row into probabilities. Cross-entropy measures how much probability the model gives to the correct class. Its gradient with respect to the logits is the signal that trains every classifier in the course.

Task

Write three functions.

1. softmax(logits) returns the softmax of each row.
2. cross_entropy(logits, targets) returns the mean cross-entropy loss over the batch.
3. cross_entropy_grad(logits, targets) returns the gradient of that mean loss with respect to logits.

The softmax of a row z is p_i = e^{z_i} / \sum_j e^{z_j}. The loss of one example with the correct class y is -\log p_y. The batch loss is the mean of the example losses.

Large logits overflow np.exp. Your softmax must give correct probabilities for logits such as 1000.0.

Input and output

• softmax returns a float array of shape (N, C).
• cross_entropy returns a Python float.
• cross_entropy_grad returns a float array of shape (N, C).

Constraints

• 1 <= N <= 512 and 2 <= C <= 1000.
• Logits can be any finite float, also values larger than 700.
• Do not change the input arrays.
• Use numpy only.

Examples

pythonCopylogits = np.array([[1.0, 2.0, 3.0]])
softmax(logits)

textCopy[[0.09003057 0.24472847 0.66524096]]

pythonCopylogits = np.array([[2.0, 1.0, 0.1], [0.5, 2.5, 0.3]])
targets = np.array([0, 1])
cross_entropy(logits, targets)

textCopy0.31853977

The gradient of the same batch has one row for each example. Each row sums to 0.