Diffusion Models are Kelly Gamblers

Abstract

We draw a connection between diffusion models and the Kelly criterion for maximizing returns in betting games. A signal that is correlated with the outcome of such a game can be used to focus the bets on a narrow range of high probability predictions. Diffusion models share the same paradigm in that they gradually concentrate the probability mass to fit the training data. We show that the information stored in an unconditional diffusion model captures, in part, the joint correlation between the components of the data variable $X$. Conditional diffusion models store additional information to bind the signal $X$ with the conditioning information $Y$, equal to the mutual information between them. The latter is only a small fraction of the total information in the neural network if the data is low-dimensional. We examine why this does not hinder conditional generation.