New Drifting Method Guarantees Finite-Particle Convergence for Generative Models

A new theoretical paper by Krishnakumar Balasubramanian addresses a fundamental challenge in one-step generative modeling: convergence guarantees when using finitely many particles. The proposed method replaces the original displacement-based drifting velocity with a kernel density estimator (KDE)-gradient velocity, which is the difference between kernel-smoothed data score and kernel-smoothed model score. This gradient field formulation resolves the non-conservatism issue that plagued earlier displacement-based drifting approaches. The paper proves continuous-time finite-particle convergence bounds on R^d, deriving a joint-entropy identity that yields bounds for empirical Stein drift, smoothed Fisher discrepancy, and squared center velocity.

The key finite-particle correction turns out to be a reciprocal-KDE self-interaction term. Balasubramanian provides deterministic and high-probability local-occupancy conditions to control this term. The convergence rates depend on bandwidth and quadrature regularity: under uniform quadrature regularity, the optimal root rate is N^{-1/(d+4)}, while a more general growth condition yields N^{-(2-β)/(2(d+4-β))} for β in [0,2). The paper also analyzes the original non-conservative method with Laplace kernel, decomposing the velocity into a preconditioned score mismatch plus a scale-mismatch residual. These theoretical bounds translate into explicit one-step generation guarantees through the drift size η, offering a rigorous foundation for deploying these methods in high-dimensional generative AI.