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There has been recently a lot of interest in the analysis of the Steingradient descent method, a deterministic sampling algorithm. It is based on aparticle system moving along the gradient flow of the Kullback-Leiblerdivergence towards the asymptotic state corresponding to the desireddistribution. Mathematically, the method can be formulated as a joint limit oftime $t$ and number of particles $N$ going to infinity. We first observe thatthe recent work of Lu, Lu and Nolen (2019) implies that if $t \approx \log \logN$, then the joint limit can be rigorously justified in the Wassersteindistance. Not satisfied with this time scale, we explore what happens forlarger times by investigating the stability of the method: if the particles areinitially close to the asymptotic state (with distance $\approx 1/N$), how longwill 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 findinga functional invariant for the linearized equation. This allows to eliminatelinear terms and arrive at an improved Gronwall-type estimate.

Convergence and stability results for the particle system in the Stein gradient descent method