We introduce the Multilevel Euler-Maruyama (ML-EM) method compute solutions of SDEs and ODEs using a range of approximators $f^1,\dots,f^k$ to the drift $f$ with increasing accuracy and computational cost, only requiring a few evaluations of the most accurate $f^k$ and many evaluations of the less costly $f^1,\dots,f^{k-1}$. 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 $f^{1},\dots,f^{k}$ 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 $γ\approx2.5$. Given that this is a polynomial speedup, we expect even stronger speedups in practical applications which involve orders of magnitude larger networks.

翻译：我们引入多水平欧拉-丸山（ML-EM）方法，通过一系列精度与计算成本递增的近似函数 $f^1,\dots,f^k$ 逼近漂移项 $f$，仅需对最高精度 $f^k$ 进行少量评估，而对低成本的 $f^1,\dots,f^{k-1}$ 进行大量评估，从而求解随机微分方程（SDE）和常微分方程（ODE）。若漂移项处于所谓"比蒙特卡罗更难"（HTMC）区域（即需 $ε^{-γ}$ 计算量才能实现 $ε$ 逼近，其中 $γ>2$），则ML-EM方法能以 $ε^{-γ}$ 的计算量实现SDE解的 $ε$ 逼近，优于传统欧拉-丸山方法的 $ε^{-γ-1}$ 阶复杂度。换言之，该方法使求解SDE的计算成本与单次漂移项评估相当。在扩散模型场景中，不同层级 $f^{1},\dots,f^{k}$ 通过训练不同规模的UNet获得，而ML-EM方法仅需等效于单次最大UNet评估即可完成采样。数值实验证实了理论结果：在降采样至64x64的CelebA数据集上，我们实现了最高四倍的图像生成加速（测得 $γ\approx2.5$）。鉴于这种多项式加速特性，我们预期在采用数量级更大网络的实际应用中可获得更显著的加速效果。