Relationship Between Cross-Entropy and KL Divergence

Cross-entropy and KL divergence are intimately related through the equation:

H(P,Q) = H(P) + D_KL(P||Q)

where H(P) is the entropy of distribution P. This relationship reveals why cross-entropy is so effective for training models: minimizing cross-entropy H(P,Q) is equivalent to minimizing KL divergence D_KL(P||Q) when the true entropy H(P) is fixed (which is the case when training on a fixed dataset).

Intuitive Analogy: Cross-entropy is like the total fuel cost of a journey, while KL divergence represents the extra fuel burned compared to the optimal route. If the shortest path length (entropy) is fixed, minimizing total fuel consumption (cross-entropy) is the same as minimizing wasted fuel (KL divergence).

This connection explains why many machine learning objectives that appear different on the surface (maximum likelihood, cross-entropy minimization, KL divergence reduction) are mathematically equivalent under certain conditions, providing a unified theoretical foundation for diverse learning approaches.