There has been recently a lot of interest in the analysis of the Stein gradient descent method, a deterministic sampling algorithm. It is based on a particle system moving along the gradient flow of the Kullback-Leibler divergence towards the asymptotic state corresponding to the desired distribution. Mathematically, the method can be formulated as a joint limit of time $t$ and number of particles $N$ going to infinity. We first observe that the recent work of Lu, Lu and Nolen (2019) implies that if $t \approx \log \log N$, then the joint limit can be rigorously justified in the Wasserstein distance. Not satisfied with this time scale, we explore what happens for larger times by investigating the stability of the method: if the particles are initially close to the asymptotic state (with distance $\approx 1/N$), how long will they remain close? We prove that this happens in algebraic time scales $t \approx \sqrt{N}$ which is significantly better. The exploited method, developed by Caglioti and Rousset for the Vlasov equation, is based on finding a functional invariant for the linearized equation. This allows to eliminate linear terms and arrive at an improved Gronwall-type estimate.

翻译：近年来，学界对Stein梯度下降法（一种确定性采样算法）的分析产生了浓厚兴趣。该方法基于沿Kullback-Leibler散度梯度流运动的粒子系统，其渐近状态对应目标分布。从数学角度，该方法可被表述为时间$t$与粒子数$N$趋于无穷的联合极限。我们首先注意到，Lu、Lu与Nolen（2019）的近期工作表明：若$t \approx \log \log N$，则该联合极限可在Wasserstein距离下被严格证明。鉴于该时间尺度不尽理想，我们通过探究方法稳定性考察更长时间范围内的行为：当粒子初始状态接近渐近态（距离$\approx 1/N$）时，这种接近性能维持多久？我们证明该状态可在代数时间尺度$t \approx \sqrt{N}$上保持，这一结果显著优于前者。本文所采用的方法由Caglioti与Rousset针对Vlasov方程提出，其核心在于寻找线性化方程的函数不变性，从而消除线性项并推导出改进的Gronwall型估计。