Polynomial Speedup in Diffusion Models with the Multilevel Euler-Maruyama Method

We introduce the Multilevel Euler-Maruyama (ML-EM) method compute solutions of SDEs and ODEs using a range of approximators f1,dots,fk to the drift f with increasing accuracy and computational cost, only requiring a few evaluations of the most accurate fk and many evaluations of the less costly f1,dots,fk-1. If the drift lies in the so-called Harder than Monte Carlo (HTMC) regime, i.e. it requires ε-γ compute to be ε-approximated for some γ>2, then ML-EM ε-approximates the solution of the SDE with ε-γ compute, improving over the traditional EM rate of ε-γ-1. In other terms it allows us to solve the SDE at the same cost as a single evaluation of the drift. In the context of diffusion models, the different levels f1,dots,fk are obtained by training UNets of increasing sizes, and ML-EM allows us to perform sampling with the equivalent of a single evaluation of the largest UNet. Our numerical experiments confirm our theory: we obtain up to fourfold speedups for image generation on the CelebA dataset downscaled to 64x64, where we measure a γ≈2.5. Given that this is a polynomial speedup, we expect even stronger speedups in practical applications which involve orders of magnitude larger networks.