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

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 $η$.

• The paper establishes explicit finite-particle convergence rates for drifting-based generative methods, contrasting conservative and non-conservative dynamics.
• It applies an entropy-dissipation framework and quadrature error analysis to derive rates such as O(N^{-1/(d+4)}) and reveal irreducible residuals in non-conservative settings.
• The study also provides precise conditions on kernel bandwidth and local occupancy to ensure stability and guide practical algorithmic design.

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

Overview and Motivation

This paper develops a rigorous finite-particle convergence theory for drifting-based one-step generative modeling, with a focus on contrasting conservative and non-conservative drift dynamics. The work addresses the gap in understanding finite-sample effects in generative drifting, establishing explicit convergence rates that depend on kernel bandwidth, sample size, particle configuration, and the structure of the underlying drift field. The analysis reveals how conservative and non-conservative forms of drift exhibit fundamentally distinct mathematical properties—most notably in their convergence behavior and the nature of their residual error terms.

Drifting Methods: Conservative and Non-Conservative Formulations

The paper considers two classes of drift-based generative models:

1. Conservative Drifting: In this formulation, the drift field is constructed as the difference between the scores (gradients of log-densities) of kernel density estimators (KDEs) formed from the data and from particles representing the model distribution. This yields a gradient field and ensures the dynamics are conservative by construction. For Gaussian kernels, this conservative score-difference field coincides (up to a scaling) with the original mean-shift displacement field, but for non-Gaussian kernels the two can differ substantially.
2. Non-Conservative Drifting: This is the original mean-shift (or displacement) based approach as in [deng2026drifting], where the drift at each location is the difference between the local mean-shifts of the data and model, using a smoothing kernel. Except in special cases (notably Gaussian kernels), this drift is not a gradient field, which has significant implications for stability, convergence, and theoretical interpretability.

The work provides detailed formalization of both approaches, including their particle-based ODE systems, empirical fields, and finite-particle loss measures.

Main Theoretical Results

Entropy-Based Analysis and Rates for Conservative Drifting

The analysis employs a joint-entropy dissipation framework inspired by SVGD theory [balasubramanian2024improved]. For the conservative dynamics (with kernel $K_h$ and $N$ particles), the main results establish that the empirical squared drift magnitude, the smoothed Fisher divergence, and other residuals satisfy time-averaged convergence bounds of the form: $$\frac{1}{N}\int_0^T \mathbb{E} [\mathsf{V}_N(X(t))] \, dt \leq O\left( \frac{1}{N} + \frac{1}{N h^{d+2}} + h^2 C_{\mathrm{quad}}(h) \right)$$ where:

• $h$ is the KDE bandwidth,
• $C_{\mathrm{quad}}(h)$ is an explicit quadrature error constant with bandwidth dependence,
• $\mathsf{V}_N(x)$ is the empirical squared norm of the conservative velocity,
• The $1/(Nh^{d+2})$ term arises from finite-particle self-interaction.

After optimizing over $h$ (given the scaling of $C_{\mathrm{quad}}$), the asymptotic root mean squared residual rate is

$O\left( N^{-1/(d+4)} \right)$

under uniform KDE regularity, and more generally

$N$0

for quadrature growth exponent $N$1.

Non-Conservative Drifting with Laplace Kernel

For the mean-shift-based (non-conservative) Laplace drift, the analysis reveals a sharp structural decomposition: the drift is not a pure score difference, but rather decomposes into a positive scaler multiple of a sharp kernel score-mismatch and an irreducible residual term (Laplace scale-mismatch). Formally,

$N$2

where $N$3 is the sharp-score mismatch, $N$4 is a local scaling factor, and $N$5 is the residual. The main convergence rate result is: $N$6 where $N$7 is an irreducible residual depending on the scale-mismatch between data and model KDEs—a term not present in the conservative case and which does not vanish asymptotically unless the radii of data and model KDEs align.

Bandwidth and Local Occupancy

A notable contribution is the detailed non-asymptotic analysis of the effect of kernel bandwidth $N$8, particle count $N$9, and local KDE denominator control (occupancy). Explicit sufficient conditions for reciprocal KDE denominator lower bounds are derived—both deterministic and in high probability for i.i.d. data. These conditions are essential for avoiding singularities in the velocity field and for the validity of convergence rates.

Numerical Strength and Contrasting Claims

• Explicit convergence rates: The theory yields non-asymptotic rates for finite $$\frac{1}{N}\int_0^T \mathbb{E} [\mathsf{V}_N(X(t))] \, dt \leq O\left( \frac{1}{N} + \frac{1}{N h^{d+2}} + h^2 C_{\mathrm{quad}}(h) \right)$$0, with explicit dependence on $$\frac{1}{N}\int_0^T \mathbb{E} [\mathsf{V}_N(X(t))] \, dt \leq O\left( \frac{1}{N} + \frac{1}{N h^{d+2}} + h^2 C_{\mathrm{quad}}(h) \right)$$1, $$\frac{1}{N}\int_0^T \mathbb{E} [\mathsf{V}_N(X(t))] \, dt \leq O\left( \frac{1}{N} + \frac{1}{N h^{d+2}} + h^2 C_{\mathrm{quad}}(h) \right)$$2, and quadrature constants, allowing for principled bandwidth optimization.
• Sharp residual term in non-conservative case: The paper highlights an irreducible residual error for non-conservative Laplace drift that is absent in conservative methods, establishing a precise limitation on using non-conservative drift fields outside the Gaussian case.
• Occupancy-dependent stability: Explicit, checkable high-probability occupancy conditions guarantee denominator stability both at initialization and along the trajectory, providing practical criteria for algorithmic reliability.

Implications and Theoretical Significance

The results rigorously clarify when and how one-step generative drifting methods can approximate population smoothed scores or effect substantial distributional transport, and when they are fundamentally limited by finite-particle and field structure effects. The distinction between conservative and non-conservative fields is demonstrated to be nontrivial, especially for non-Gaussian kernels, impacting both theoretical guarantees and algorithmic design.

The explicit tracking of bandwidth dependence in all terms exposes the delicate interplay between statistical accuracy, quadrature approximation, and particle system regularity. This clarifies both statistical and numerical aspects of kernel-based generative models.

Future Directions

Potential avenues for further study include:

• Extending the analysis to minibatch stochastic versions as used in practice, where leave-one-out and empirical errors interact with batch selection and stochastic drift.
• Closing the gap between finite-particle convergence rates and vanishing optimization errors in practical neural generator training.
• Generalizing drift constructions to other classes of kernels or to adaptive, data-dependent kernels, and developing regularization strategies for improved quadrature behavior in high dimensions.
• Exploring the implications for the expressive power and stability of one-step generative models in settings beyond translation-invariant kernels or Euclidean distances.

Conclusion

This work delivers a highly detailed finite-sample convergence analysis for both conservative and non-conservative drifting methods in one-step generative modeling. It provides both foundational theoretical tools—entropy-dissipation identities, quadrature analysis, occupancy arguments—and practical rate formulas, and it draws precise conceptual boundaries between structurally different classes of drift fields. These results clarify the conditions under which drifting-based generative models can be expected to provide principled finite-sample behavior and inform further developments in the theory and practical performance of generative modeling frameworks (2605.22795).