用多级方法加速扩散模型采样，理论可实现多项式提速。
Polynomial Speedup in Diffusion Models with the Multilevel Euler-Maruyama Method
- 通过分级近似漂移项，减少高精度模型调用次数。
- 在γ≈2.5时，图像生成速度提升最高达4倍。
- 适合大模型扩散采样，尤其对超大规模网络效果更显著。
我们提出多级欧拉-马鲁雅米（ML-EM）方法，通过一系列精度递增、成本递增的漂移函数近似器 $f^1,\dots,f^k$ 求解随机微分方程（SDE）与常微分方程（ODE）。当漂移项处于“比蒙特卡洛更难”（HTMC）情形，即需 $ε^{-γ}$ 计算量才能达到 $ε$ 精度（$γ>2$）时，ML-EM 仅需 $ε^{-γ}$ 计算即可实现 $ε$ 精度的解，优于传统欧拉-马鲁雅米方法的 $ε^{-γ-1}$ 复杂度。这意味着可在等价于单次高精度漂移评估的成本下求解整个 SDE。在扩散模型中，各层级 $f^1,\dots,f^k$ 对应逐步增大尺寸的 UNet，ML-EM 实现了等效于一次最大尺寸 UNet 的采样效率。数值实验验证了理论：在 64x64 缩放的 CelebA 数据集上，速度提升最高达 4 倍，此时 $γ\approx2.5$。由于这是多项式加速，实际应用中使用更大规模网络时预期提速更强。
原文摘要 · Abstract (English)
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.
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