条件扩散模型有限样本误差上界研究：训练与离散化误差可联合分析
Finite-Sample Guarantees for Fully Discrete Conditional Diffusion Models
扩散模型的理论保证很少做到条件版本，这篇把训练误差和离散化步数一起算清了，做生成模型理论的可以看看。
一篇 arXiv 论文为从数据学习、并用有限步去噪实现的条件扩散模型建立了有限样本保证。方法上将 score 训练目标与采样网格对齐，使统计估计误差和数值离散化误差可以放在一起分析。在平滑条件密度假设下，论文给出了生成分布误差的高概率上界，并量化了所需 score 评估次数。对每个平滑度指数 β>0，合适的网格可让平均条件 KL 散度达到 O_p(n^{-2β/(d_x+d_y+2β)})，全采样和早停采样器的平均 TV 误差达到 O_p(n^{-β/(d_x+d_y+2β)})，后者在忽略对数因子后匹配已有条件分布估计理论的最优速率。
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 $\widetilde O_p(n^{-2β/(d_x+d_y+2β)})$ relative to the original target with or without early stopping. Both the full and early-stopped samplers attain average target total-variation (TV) error $\widetilde O_p(n^{-β/(d_x+d_y+2β)})$. 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.
