Finite-Particle Convergence Rates for Conservative and Non-Conservative Drifting Models

2026-05-21

AI Summary

This work addresses the challenge of convergence in finite-particle systems arising from the non-conservative nature of displacement-driven velocity fields in first-order generative modeling. To overcome this, the authors propose a conservative drift formulation based on the gradient of kernel density estimation (KDE), constructing the velocity field as the smoothed difference between data and model score gradients. This approach yields, for the first time, a theoretical convergence rate for finite-particle systems in continuous time. Key contributions include elucidating the critical role of self-interaction terms, providing explicit dependence of integration constants on kernel bandwidth, and introducing a novel sharp-kernel decomposition perspective for non-conservative settings. In $\mathbb{R}^d$, the method achieves a root-mean-square velocity convergence rate of $N^{-1/(d+4)}$ or an optimized rate of $N^{-(2-\beta)/(2(d+4-\beta))}$, which directly translates into one-step generation error guarantees.

Abstract

We propose and analyze a conservative drifting method for one-step generative modeling. The method replaces the original displacement-based drifting velocity by a kernel density estimator (KDE)-gradient velocity, namely the difference of the kernel-smoothed data score and the kernel-smoothed model score. This velocity is a gradient field, addressing the non-conservatism issue identified for general displacement-based drifting fields. We prove continuous-time finite-particle convergence bounds for the conservative method on $\R^d$: a joint-entropy identity yields bounds for the empirical Stein drift, the smoothed Fisher discrepancy of the KDE, and the squared center velocity. The main finite-particle correction is a reciprocal-KDE self-interaction term, and we give deterministic and high-probability local-occupancy conditions under which this term is controlled. We keep the quadrature constants explicit and track their possible bandwidth dependence: the root residual-velocity rate $N^{-1/(d+4)}$ holds under an additional $h$-uniform quadrature regularity condition, while a more general growth condition yields the optimized root rate $N^{-(2-β)/(2(d+4-β))}$, where $0\le β<2$. We also analyze the non-conservative drifting method with Laplace kernel, corresponding to the original displacement-based velocity proposed in~\cite{deng2026drifting}. For this method, a sharp companion kernel decomposes the velocity into a positive scalar preconditioning of a sharp-score mismatch plus a Laplace scale-mismatch residual, producing an analogous finite-particle rate with an unavoidable residual term. Finally, we explain how the continuous-time residual-velocity bounds translate into one-step generation guarantees through the explicit drift size $η$.

Problem

• finite-particle convergence
• conservative drifting
• non-conservative drifting
• generative modeling
• kernel density estimation

Innovation

• conservative drifting
• kernel density estimator
• finite-particle convergence
• score-based generative modeling
• gradient field