Categorical Cross-Entropy Loss

A multi-class classifier outputs one probability per class (for example from softmax). Categorical cross-entropy punishes it for giving the correct class a low probability:

loss = -(ln(p₁) + ln(p₂) + … + ln(pₙ)) / n

where pᵢ is the probability the model gave to the correct class of sample i. To avoid ln(0), first raise any probability below 0.0000001 to 0.0000001.

Implement categorical_cross_entropy(probabilities, labels).

Input

probabilities: One row per sample, with one probability per class.

labels: The index of the correct class for each sample (0-based).

Output

Return the average loss as a decimal number.

Constraints

1 ≤ samples ≤ 1,000, 2 ≤ classes ≤ 100

0 ≤ each probability ≤ 1

0 ≤ labels[i] < number of classes

Examples

Example 1

Input:
probabilities = [
    [0.7, 0.2, 0.1],
    [0.1, 0.8, 0.1]
]
labels = [0, 1]

Expected output:
0.289909

Explanation

The correct classes got 0.7 and 0.8: -(ln(0.7) + ln(0.8)) / 2 = -(-0.356675 - 0.223144) / 2 ≈ 0.289909.

Example 2

Input:
probabilities = [
    [1.0, 0.0]
]
labels = [1]

Expected output:
16.118096

Explanation

The correct class got probability 0, which is raised to 0.0000001, so the loss is -ln(0.0000001) ≈ 16.118096: a confident wrong answer is punished hard.

References

Run the sample tests to check your code without submitting.

Example 1 Not run

Input:

probabilities = [
    [0.7, 0.2, 0.1],
    [0.1, 0.8, 0.1]
]
labels = [0, 1]

Expected output:

0.289909

Your output

Example 2 Not run

Input:

probabilities = [
    [1.0, 0.0]
]
labels = [1]

Expected output:

16.118096

Your output

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