Finite-Sample Guarantees for Fully Discrete Conditional Diffusion Models

Abstract: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 $\beta>0$, a suitable grid yields averaged conditional Kullback--Leibler (KL) divergence $\widetilde O_p(n^{-2\beta/(d_x+d_y+2\beta)})$ relative to the original target with or without early stopping. Both the full and early-stopped samplers attain average target total-variation (TV) error $\widetilde O_p(n^{-\beta/(d_x+d_y+2\beta)})$. Here $n$ is the sample size, $d_x,d_y$ 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.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2610.13851 [stat.ML]

(or arXiv:2610.13851v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2610.13851 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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