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[Machine Learning] Note of Cross Entropy Loss

Last Updated on 2024-08-18 by Clay

Introduction to Cross Entropy

Cross entropy is a very common loss function in Machine Learning, as it is able to quantify the difference between a model's classification predictions and the actual class labels, particularly in 'classification tasks'.

In classification problems, the formula for cross entropy is as follows:

H(p,q)=-\sum_{i=1}^{N}p_{i}log(q_{i})

  • N is the total number of classes
  • p_i represents the true distribution for the i-th class in the actual class labels (note: this is typically represented using one-hot encoding in programming, marking the correct class)
  • q_{i} is the predicted probability for the i-th class by the model

From this formula, we can conclude the following:

  1. The more accurate the prediction, the smaller the cross-entropy value:
    • If the model's predicted probability for the correct class q_i approaches 1, then log(q_i) will approach 0. This makes the cross-entropy value (the modelโ€™s loss function) smaller, indicating that the model's prediction is very accurate.
    • If the model's predicted probability for the correct class q_i approaches 0, then log(q_i) will tend toward negative infinity, meaning that the cross-entropy value will give a huge penalty, encouraging more improvement in the model.
  2. The role of one-hot encoding:
    • In multi-class classification, the correct answer is usually represented in one-hot encoding format (for example, classifying cats and dogs, where a cat is represented by [1, 0] and a dog by [0, 1]). This means only the correct class is 1, and the rest are 0. The modelโ€™s predicted probability distribution will only calculate the correct position.

Code Implementation

Here, we consider batch input. In addition to implementing the formula with -np.sum(p * np.log(q)), we also average the loss function. This is done to allow a comparison with PyTorch's implementation later.

def my_cross_entropy(p, q):
    return -np.sum(p * np.log(q)) / len(p)


Below is a randomly defined test case. PyTorch accepts logits as input, not probabilities, and internally calculates the probability distribution with softmax. Our implementation, however, defines a softmax function, which converts logits to probabilities before passing them to the custom my_cross_entropy().

import torch
import numpy as np


def my_cross_entropy(p, q):
    return -np.sum(p * np.log(q)) / len(p)


def my_softmax(logits):
    exp_logits = np.exp(logits)
    return exp_logits / np.sum(exp_logits, axis=1, keepdims=True)


# Test case
if __name__ == "__main__":
    # Real data (cat, dog, turtle)
    real_data = [
        [1, 0, 0],
        [1, 0, 0],
        [0, 0, 1],
    ]
    model_predictions = [
        [9.7785, 0.9195, 2.10],
        [3.133, -3.05, 1.02],
        [0.12, 0.432518, 0.470],
    ]

    # Torch
    real_data_indices = np.argmax(real_data, axis=1)
    real_data_torch = torch.tensor(real_data_indices)
    model_predictions_torch = torch.tensor(model_predictions)
    torch_loss = torch.nn.functional.cross_entropy(model_predictions_torch, real_data_torch)
    print("Torch Cross Entropy Loss:", torch_loss.item())

    # My Cross Entropy
    model_predictions_softmax = my_softmax(np.array(model_predictions))  # Apply softmax
    my_loss = my_cross_entropy(np.array(real_data), model_predictions_softmax)
    print("My Cross Entropy Loss:", my_loss)


Output:

Torch Cross Entropy Loss: 0.36594846844673157
My Cross Entropy Loss: 0.36594851568931036


You can see that our implementation is very consistent with PyTorch.

References

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