Finite-Time Convergence Analysis of ODE-based Generative Models for Stochastic Interpolants
Yuhao Liu, Rui Hu, Yu Chen, Longbo Huang
TL;DR
This paper provides finite-time convergence guarantees and error bounds for numerical schemes used in ODE-based stochastic interpolants, advancing the theoretical understanding of their efficiency and accuracy in generative modeling.
Contribution
It establishes novel finite-time error bounds for Euler and Heun's methods applied to stochastic interpolant ODEs, and analyzes iteration complexity for improved efficiency.
Findings
Finite-time error bounds in total variation distance for Euler and Heun's methods.
Validated theoretical bounds through numerical experiments.
Provided optimized iteration schedules for computational efficiency.
Abstract
Stochastic interpolants offer a robust framework for continuously transforming samples between arbitrary data distributions, holding significant promise for generative modeling. Despite their potential, rigorous finite-time convergence guarantees for practical numerical schemes remain largely unexplored. In this work, we address the finite-time convergence analysis of numerical implementations for ordinary differential equations (ODEs) derived from stochastic interpolants. Specifically, we establish novel finite-time error bounds in total variation distance for two widely used numerical integrators: the first-order forward Euler method and the second-order Heun's method. Furthermore, our analysis on the iteration complexity of specific stochastic interpolant constructions provides optimized schedules to enhance computational efficiency. Our theoretical findings are corroborated by…
Peer Reviews
Decision·ICLR 2026 Poster
- The paper addresses a significant theoretical gap for SI-based ODEs by providing finite-time TV bounds and iteration complexities for both Euler and Heun methods within the SI framework. - Mathematical proofs are solid and clear. Discrete-to-continuous interpolation yields a piecewise ODE, enabling drift/divergence comparisons to bound TV. - 2D tasks and Gaussian-mixture tests verify $O(h)$ (Euler) and $O(h^2)$ (Heun) discretization orders.
- The requirement of uniform Lipschitz on $\hat{b}$ and its divergence (Assumption 4.4) seems kind of idealized. Is it possible to provide an empirical verification? - Experiment demos are limited to three 2D transformations and d-dim Gaussian mixtures; Is it possible to provide results on real-data benchmarks? - The theory in this paper establishes error bounds that scale with $d^2$ and $d^3$. However, the empirical findings indicate a roughly linear correlation, suggesting room for further ref
The paper is the first to give an analysis of the ODE discretization for stochastic interpolants under general conditions. This works for any interpolations with path $I$ and noise magnitude $\gamma(t)$ satisfying some smoothness conditions, and estimates satisfying the assumptions. Although the analysis techniques do not seem particularly novel (in being similar to previous ODE flow analyses), it is nevertheless valuable to work out the precise bounds in new setting under carefully detailed gen
There is an additional factor of $d$ in Euler's method compared to diffusion models, and it is unknown whether this is a limitation of the analysis, or necessary in this general setting. (The experiments suggest the true scaling is linear.)
**(S1) Important research theme.** Investigating the iteration complexity of ODE solvers to ensure accurate generation with stochastic interpolants is an important research direction. Although diffusion models (in discretization form of neural ODE) are a major focus in machine learning, most evidence remains empirical. This work may provide the missing theoretical analyses underpinning diffusion models. **(S2) Theoretical contributions.** For general stochastic-interpolant ODEs, we derive the i
**(W1) Positioning relative to prior work.** In the Introduction (lines 048–049), you state that for ODE-based transformations, “the analysis has been limited to the continuous-time setting.” Meanwhile, Sec. 2.2 (lines 123–127) and Appendix B (Table 1) summarize prior results on the time needed for an ODE to reach $\epsilon$-TV error. Am I correct that all of these results are derived in continuous-time setting and do not account for discretization error? If YES, please revise Sec. 2.2 to make t
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Taxonomy
TopicsNumerical methods for differential equations · Model Reduction and Neural Networks · Probabilistic and Robust Engineering Design