{"@context":"https://schema.org","@type":"CreativeWork","@id":"https://forgecascade.org/public/capsules/a0107e93-332c-4490-807f-d9def8565c66","identifier":"a0107e93-332c-4490-807f-d9def8565c66","url":"https://forgecascade.org/public/capsules/a0107e93-332c-4490-807f-d9def8565c66","name":"Finite-Particle Convergence Rates for Conservative and Non-Conservative Drifting Models","text":"# Finite-Particle Convergence Rates for Conservative and Non-Conservative Drifting Models\n\n**Authors:** Krishnakumar Balasubramanian\n**arXiv:** https://arxiv.org/abs/2605.22795v1\n**Published:** 2026-05-21T17:49:09Z\n\n## Abstract\nWe 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 $η$.","keywords":["stat.ML","cs.AI","cs.LG","math.ST"],"about":[],"citation":[],"isPartOf":{"@type":"Dataset","name":"Forge Cascade Knowledge Graph","url":"https://forgecascade.org"},"publisher":{"@type":"Organization","name":"Forge Cascade","url":"https://forgecascade.org"},"dateCreated":"2026-05-22T06:00:05.984000Z","dateModified":"2026-05-22T06:00:05.984000Z"}