Finite-Particle Convergence Rates for Conservative and Non-Conservative Drifting Models
AuthorsKrishnakumar Balasubramanian
Resources
This paper gives a mathematical guarantee for a new one-step generative modeling method by showing how particle-based drifting can converge and how its errors can be explicitly bounded.
Key results
For the conservative drifting method, the optimized root residual-velocity rate holds under h-uniform quadrature regularity.
For conservative drift with quadrature growth BA,N(h)+BV,N(h)=O(h^-β), the optimized root residual-velocity rate becomes this expression for 0≤β<2.
For Gaussian kernels, the paper states -ΔKh(0)=h^-d-2 a1 with a1=d(2π)^-d/2.
What the paper found
This paper studies finite-particle convergence for one-step generative “drifting” models and shows that the geometry of the drift field fundamentally changes the error rate. For the conservative variant, the velocity is defined as the difference of KDE scores, b_{ν,µ,h}(z)=∇log ρ_{ν,h}(z)−∇log ρ_{µ,h}(z), making the field a true gradient, unlike the displacement-based drift of Deng et al. (2026). Using a joint-entropy identity adapted from recent finite-particle SVGD analysis, the paper proves that the time-averaged squared residual velocity V_N contracts at rate H_N(0)/(NT) plus a reciprocal-KDE self-interaction term and an explicit quadrature error term. Under h-uniform quadrature regularity, the optimized root residual-velocity rate is N^{-1/(d+4)}; more generally, if the quadrature constants grow like h^{-β} with 0≤β<2, the rate becomes N^{-(2−β)/(2(d+4−β))}. For Gaussian kernels, the conservative drift is exactly 1/h^2 times the original displacement field, so the two dynamics coincide after time rescaling. The paper then analyzes the original non-conservative Laplace drift using a sharp companion kernel, decomposing it into a positive scale factor times a sharp-score mismatch plus an unavoidable Laplace scale-mismatch residual ∆_h. The resulting finite-particle bound has the form κ_0/(γ_h N)+β_h∆_h^2/(γ_h h^2)+approximation errors, showing that convergence is only to a residual neighborhood unless local Laplace radii align. The main novelty is the explicit separation of conservative score-driven convergence from non-conservative scale-mismatch error, with all rates kept bandwidth-explicit.
Original 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 $η$.
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