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

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

Expected output:
expected = 0.289909

Input:
std::vector<long long> labels = {0, 1};
std::vector<std::vector<double>> probabilities = {
    {0.7, 0.2, 0.1},
    {0.1, 0.8, 0.1}
};

Expected output:
double expected = 0.289909;

Input:
long[] labels = new long[] {0, 1};
double[][] probabilities = new double[][] {
    new double[] {0.7, 0.2, 0.1},
    new double[] {0.1, 0.8, 0.1}
};

Expected output:
double expected = 0.289909;

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

Expected output:
expected = 16.118096

Input:
std::vector<long long> labels = {1};
std::vector<std::vector<double>> probabilities = {
    {1.0, 0.0}
};

Expected output:
double expected = 16.118096;

Input:
long[] labels = new long[] {1};
double[][] probabilities = new double[][] {
    new double[] {1.0, 0.0}
};

Expected output:
double expected = 16.118096;

References

This challenge is adapted for CodeUtility. Follow the statement and constraints above.

• scikit-learn, log_loss
• Goodfellow, Bengio and Courville, Deep Learning