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

Problem
This paper addresses the gap in the literature regarding the convergence rates of finite-particle methods in one-step generative modeling, particularly focusing on the non-conservatism issues associated with displacement-based drifting fields. The authors propose a conservative drifting method that utilizes a kernel density estimator (KDE) to derive a gradient velocity, which is essential for ensuring the conservation of probability mass during the generative process. This work is presented as a preprint and has not yet undergone peer review.

Method
The core technical contribution is the introduction of a conservative drifting method that replaces the traditional displacement-based drifting velocity with a KDE-gradient velocity. This velocity is defined as the difference between the kernel-smoothed data score and the kernel-smoothed model score, effectively creating a gradient field that mitigates non-conservatism. The authors derive continuous-time finite-particle convergence bounds on (\mathbb{R}^d) using a joint-entropy identity, which leads to bounds for the empirical Stein drift, the smoothed Fisher discrepancy of the KDE, and the squared center velocity. The main finite-particle correction involves a reciprocal-KDE self-interaction term, with deterministic and high-probability local-occupancy conditions provided to control this term. The paper also discusses the non-conservative drifting method using a Laplace kernel, which corresponds to the original displacement-based velocity. The convergence rates are characterized by the root residual-velocity rate (N^{-1/(d+4)}) under a specific (h)-uniform quadrature regularity condition, while a more general growth condition yields an optimized root rate of (N^{-(2-β)/(2(d+4-β))}), where (0 \leq β < 2).

Results
The proposed conservative drifting method demonstrates improved convergence rates compared to traditional methods. Specifically, the root residual-velocity rate of (N^{-1/(d+4)}) and the optimized rate (N^{-(2-β)/(2(d+4-β))}) indicate significant improvements in convergence behavior. The authors provide empirical evidence supporting these rates, although specific numerical results against named baselines on standard benchmarks are not detailed in the abstract.

Limitations
The authors acknowledge that their method’s performance is contingent on the control of the reciprocal-KDE self-interaction term, which may not be trivial in practice. Additionally, the reliance on specific quadrature regularity conditions may limit the applicability of the results in more general settings. The paper does not address potential computational overhead associated with the KDE calculations, which could impact scalability in high-dimensional spaces.

Why it matters
This work has significant implications for the field of generative modeling, particularly in enhancing the reliability and efficiency of one-step generative processes. By addressing the non-conservatism of traditional methods, the proposed approach could lead to more robust generative models that maintain probability mass conservation, which is crucial for applications in areas such as image synthesis, natural language processing, and reinforcement learning. The findings may also inspire further research into alternative drifting methods and their convergence properties.

Authors: Krishnakumar Balasubramanian
Source: arXiv:2605.22795
URL: https://arxiv.org/abs/2605.22795v1