Consistency models, which were proposed to mitigate the high computational overhead during the sampling phase of diffusion models, facilitate single-step sampling while attaining state-of-the-art empirical performance. When integrated into the training phase, consistency models attempt to train a sequence of consistency functions capable of mapping any point at any time step of the diffusion process to its starting point. Despite the empirical success, a comprehensive theoretical understanding of consistency training remains elusive. This paper takes a first step towards establishing theoretical underpinnings for consistency models. We demonstrate that, in order to generate samples within $\varepsilon$ proximity to the target in distribution (measured by some Wasserstein metric), it suffices for the number of steps in consistency learning to exceed the order of $d^{5/2}/\varepsilon$, with $d$ the data dimension. Our theory offers rigorous insights into the validity and efficacy of consistency models, illuminating their utility in downstream inference tasks.

一步采样的连贯性模型在训练阶段中具有良好的实证效果，通过训练一系列连贯性函数，能够将扩散过程的任何时间步骤的任意点映射到其起始点，其理论基础表明，为了生成与目标分布在ε接近程度内的样本，连贯性学习的步骤数应超过d^5/2/ε，其中d为数据维度，这一理论为连贯性模型的有效性和实用性提供了深入洞察，从而启示其在下游推理任务中的实用性。

扩散模型中一种一致性训练的数学理论