Cross-Entropy and Log Loss: Mathematical Foundations and Their Use in Classification

Introduction: Cross-Entropy and Log Loss in Classification

In the realm of machine learning, especially in classification problems, choosing the right loss function is pivotal to building accurate and reliable models. Among various loss functions, one that frequently emerges at the heart of both theory and practice is the Cross-Entropy Loss, also known in some contexts as Log Loss. But what makes this function so special? Why is it the default choice for so many classification tasks — especially in deep learning?

Before diving into the math and applications, it’s important to understand the intuition and history behind Cross-Entropy. Originally rooted in information theory, the concept of cross-entropy was introduced by Claude Shannon in the 1940s as a way to measure the difference between two probability distributions. The name “entropy” itself comes from thermodynamics and reflects the uncertainty or randomness in a system. In the context of machine learning, we use this idea to quantify how far our model’s predicted probability distribution is from the actual distribution (usually a one-hot encoded label).

In simpler terms:

Cross-Entropy answers the question: “How surprised am I by the…