THE CLASS-WEIGHTED LOSS

Difficulty: Medium Tags: Deep Learning, Multiclass Classification, Cross-Entropy

Story

A product classifier trains on imbalanced categories. Each example has a class-specific weight. The implementation must compute mean weighted negative log likelihood with the reduction convention explicitly defined by the training team.

The Math

For each example i, logits z_i (length C), true class y_i, and class weight w_{y_i}:

p_ic = softmax(z_i)_c
L = -(1/n) * sum_i w_{y_i} * log(p_i,y_i)

Do not divide by the sum of class weights; divide by n.

Input Format

logits (n x C), labels (length n, 0-based), weights (length C).

Output Format

The scalar loss, 6 decimal places.

Constraints

• 1 <= n <= 10^4, 2 <= C <= 100, |z_ic| <= 10^3, 0 <= w_c <= 100.
• Time limit: 1.0 second.

Example

Input

logits = [[2, 0], [0, 2]], labels = [0, 1], weights = [1, 3]

Output

0.253856

Explanation: Each row's unweighted negative log-likelihood is log(1 + e^-2) ~= 0.126928. The class weights are 1 and 3, so the mean weighted loss is (0.126928 + 3*0.126928) / 2 ~= 0.253856.