  We provide the first convergence guarantees for the Consistency Models (CMs),
a newly emerging type of one-step generative models that can generate
comparable samples to those generated by Diffusion Models. Our main result is
that, under the basic assumptions on score-matching errors, consistency errors
and smoothness of the data distribution, CMs can efficiently sample from any
realistic data distribution in one step with small $W_2$ error. Our results (1)
hold for $L^2$-accurate score and consistency assumption (rather than
$L^\infty$-accurate); (2) do note require strong assumptions on the data
distribution such as log-Sobelev inequality; (3) scale polynomially in all
parameters; and (4) match the state-of-the-art convergence guarantee for
score-based generative models (SGMs). We also provide the result that the
Multistep Consistency Sampling procedure can further reduce the error comparing
to one step sampling, which support the original statement of "Consistency
Models, Yang Song 2023". Our result further imply a TV error guarantee when
take some Langevin-based modifications to the output distributions.

Numerical Analysis;Artificial Intelligence;Numerical Analysis;Analysis of PDEs;Probability