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This paper provides the first finite-sample guarantees for fully discrete conditional diffusion models trained from data and executed with a finite number of denoising steps. By aligning the score-training objective with the sampling grid, it enables joint analysis of statistical estimation and numerical discretization errors, yielding high-probability bounds on generated distribution error and quantifying necessary score evaluations. The derived bounds show that for any smoothness $\beta>0$, an appropriate grid achieves averaged conditional KL divergence of $\widetilde O_p(n^{-2\beta/(d_x+d_y+2\beta)})$ and total-variation error of $\widetilde O_p(n^{-\beta/(d_x+d_y+2\beta)})$, matching or improving upon existing distribution estimation rates.
Finite-sample guarantees for discrete conditional diffusion models reveal that optimal error rates are achievable with a finite number of denoising steps, matching theoretical bounds for distribution estimation.
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.