This paper deals with the problem of efficient sampling from a stochastic differential equation, given the drift function and the diffusion matrix. The proposed approach leverages a recent model for probabilities \cite{rudi2021psd} (the positive semi-definite -- PSD model) from which it is possible to obtain independent and identically distributed (i.i.d.) samples at precision $\varepsilon$ with a cost that is $m^2 d \log(1/\varepsilon)$ where $m$ is the dimension of the model, $d$ the dimension of the space. The proposed approach consists in: first, computing the PSD model that satisfies the Fokker-Planck equation (or its fractional variant) associated with the SDE, up to error $\varepsilon$, and then sampling from the resulting PSD model. Assuming some regularity of the Fokker-Planck solution (i.e. $\beta$-times differentiability plus some geometric condition on its zeros) We obtain an algorithm that: (a) in the preparatory phase obtains a PSD model with L2 distance $\varepsilon$ from the solution of the equation, with a model of dimension $m = \varepsilon^{-(d+1)/(\beta-2s)} (\log(1/\varepsilon))^{d+1}$ where $1/2\leq s\leq1$ is the fractional power to the Laplacian, and total computational complexity of $O(m^{3.5} \log(1/\varepsilon))$ and then (b) for Fokker-Planck equation, it is able to produce i.i.d.\ samples with error $\varepsilon$ in Wasserstein-1 distance, with a cost that is $O(d \varepsilon^{-2(d+1)/\beta-2} \log(1/\varepsilon)^{2d+3})$ per sample. This means that, if the probability associated with the SDE is somewhat regular, i.e. $\beta \geq 4d+2$, then the algorithm requires $O(\varepsilon^{-0.88} \log(1/\varepsilon)^{4.5d})$ in the preparatory phase, and $O(\varepsilon^{-1/2}\log(1/\varepsilon)^{2d+2})$ for each sample. Our results suggest that as the true solution gets smoother, we can circumvent the curse of dimensionality without requiring any sort of convexity.

翻译：本文研究给定漂移函数和扩散矩阵条件下，从随机微分方程中高效采样的问题。所提出的方法利用了近期提出的概率正半定模型 \cite{rudi2021psd}（PSD模型），该模型能以精度$\varepsilon$生成独立同分布（i.i.d.）样本，计算成本为$m^2 d \log(1/\varepsilon)$，其中$m$为模型维度，$d$为空间维度。具体方案包括：首先计算满足随机微分方程对应的Fokker-Planck方程（或其分数阶变体）的PSD模型，使其误差不超过$\varepsilon$；然后从所得PSD模型中采样。在假设Fokker-Planck解具备一定正则性（即$\beta$次可微性及零点几何条件）的前提下，我们得到如下算法：（a）预备阶段获得与方程解L2距离为$\varepsilon$的PSD模型，模型维度$m = \varepsilon^{-(d+1)/(\beta-2s)} (\log(1/\varepsilon))^{d+1}$（其中$1/2\leq s\leq1$为拉普拉斯算子的分数阶指数），总计算复杂度为$O(m^{3.5} \log(1/\varepsilon))$；（b）对于Fokker-Planck方程，能以Wasserstein-1距离误差$\varepsilon$生成独立同分布样本，每个样本的计算成本为$O(d \varepsilon^{-2(d+1)/\beta-2} \log(1/\varepsilon)^{2d+3})$。这意味着若随机微分方程对应的概率分布具有足够正则性（即$\beta \geq 4d+2$），则预备阶段算法复杂度为$O(\varepsilon^{-0.88} \log(1/\varepsilon)^{4.5d})$，每个样本的复杂度为$O(\varepsilon^{-1/2}\log(1/\varepsilon)^{2d+2})$。研究结果表明，随着真实解光滑性的提升，我们可在无需任何凸性假设的条件下规避维度灾难问题。