The Effect of Stochasticity in Score-Based Diffusion Sampling: a KL Divergence Analysis

Abstract: Sampling in score-based diffusion models can be performed by solving either a probability flow ODE or a reverse-time stochastic differential equation (SDE) parameterized by an arbitrary stochasticity parameter. In this work, we study the effect of stochasticity on the generation process through bounds on the Kullback-Leibler (KL) divergence and complement the analysis with numerical and analytical examples. Our results apply to general forward SDEs with additive noise and Lipschitz-continuous score functions, and quantify how errors from the prior distribution and score approximation propagate under different choices of the stochasticity parameter. The theoretical bounds are derived using log-Sobolev inequalities for the marginals of the forward process, which enable a more effective control of the KL divergence decay along sampling. For exact score functions, we find that stochasticity acts as an error-correcting mechanism, decreasing KL divergence along the sampling trajectory. For an approximate score function, there is a trade-off between error correction and score error amplification, so that stochasticity can either improve or worsen the performance, depending on the structure of the score error. Numerical experiments on simple datasets and a fully analytical example are included to illustrate and enlighten the theoretical results.