  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.
