Why Cross-Entropy Is the Key Loss Function for Classification Models

In machine learning prediction tasks we use loss functions to measure model performance. For regression problems the mean squared error is common, while classification problems require a loss that reflects the difference between predicted and true categories.

One‑Hot Encoding and Probability Distribution

In multi‑class classification we represent the true label with a one‑hot vector, where the element corresponding to the correct class is 1 and all others are 0. The model’s raw outputs are passed through a Softmax function, converting them into a probability distribution that sums to 1.

Definition of the Cross‑Entropy Loss

Cross‑entropy measures the difference between the true one‑hot distribution y and the predicted probability distribution p . For a single sample the loss is

-∑_i y_i log(p_i)

Because y is one‑hot, only the term for the correct class contributes, simplifying to -log(p_{true}) . Thus, the higher the predicted probability for the correct class, the lower the loss.

Example: for a three‑class problem with true class A, if the model predicts [0.7, 0.2, 0.1] , the loss is -log(0.7) . If the prediction changes to [0.1, 0.2, 0.7] , the loss becomes -log(0.1) , which is much larger.

Consequently, cross‑entropy heavily penalizes predictions that assign low probability to the true class.

Cross‑entropy is widely used in classification tasks such as image classification with convolutional neural networks, language modeling and machine translation, and speech recognition, where it quantifies the discrepancy between predicted and actual word or phoneme distributions.

The mathematical foundation of cross‑entropy comes from information theory and maximum likelihood estimation. However, because it involves logarithms, the loss can become very large when predicted probabilities are close to zero, potentially affecting training stability.