THE FRAUD MODEL''S FOCAL LOSS

Difficulty: Hard Tags: Deep Learning, Imbalanced Classification, Loss Functions

Story

A fraud model sees very few positive cases. The team adopts binary focal loss to emphasize hard examples, but an implementation mixes up the true-class probability and the predicted positive probability for negative examples. Compute the per-example focal loss exactly as defined.

The Math

For logit z, label y in {0, 1}, define p = sigmoid(z) and:

p_t = p       if y == 1
p_t = 1 - p   if y == 0

alpha_t = alpha      if y == 1
alpha_t = 1 - alpha  if y == 0

The focal loss for one example is:

L = -alpha_t * (1 - p_t)^gamma * log(p_t)

Return the mean loss over all examples.

Input Format

logits (length n), labels (length n, binary), alpha, gamma.

Output Format

Mean focal loss, 6 decimal places.

Constraints

• 1 <= n <= 10^5, -20 <= z_i <= 20, 0 <= alpha <= 1, 0 <= gamma <= 5.
• Time limit: 1.0 second.

Example

Input

logits = [0, 2], labels = [1, 0], alpha = 0.25, gamma = 2

Output

0.640440

Explanation: For the positive example, p_t = 0.5. For the negative example with logit 2, p_t = 1 - sigmoid(2) = 0.119203. Applying the class weights and focal factors gives a mean loss of about 0.640440.