Finite-Sample Guarantees for Fully Discrete Conditional Diffusion Models

We establish finite-sample guarantees for conditional diffusion models learned from data and implemented with finitely many denoising steps. Our approach aligns the score-training objective with the sampling grid, allowing statistical estimation and numerical discretization errors to be analyzed together. Under a smooth conditional density model, we derive high-probability bounds on the generated distribution's error and quantify the required number of score evaluations. For every smoothness index β>0, a suitable grid yields averaged conditional Kullback--Leibler (KL) divergence widetilde Op(n-2β/(dx+dy+2β)) relative to the original target with or without early stopping. Both the full and early-stopped samplers attain average target total-variation (TV) error widetilde Op(n-β/(dx+dy+2β)). Here n is the sample size, dx,dy are the data and conditioning dimensions, and errors are averaged over the conditioning variable. The total-variation bound matches the rate established in existing conditional distribution-estimation theory, up to logarithmic factors.