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

arXiv:2605.22795v1 Announce Type: cross 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-eta)/(2(d+4-eta))}, where 0le eta<2. We also analyze the non-conservative drifting method with Laplace kernel, corresponding to the original displacement-based velocity proposed in~ite{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 eta.

Source: arXiv cs.LG | 2026-05-23