Finite-Time Analysis of Discrete-Time Stochastic Interpolants

TL;DR

This paper provides the first finite-time analysis of discrete-time stochastic interpolants, offering new insights into convergence behavior and practical design of efficient sampling schedules for generative models.

Contribution

It introduces a novel discrete-time sampler and derives a finite-time error bound, advancing understanding of convergence in stochastic interpolant frameworks.

Findings

Finite-time upper bound on distribution estimation error.

Factors affecting convergence rate are quantitatively characterized.

Numerical experiments validate theoretical predictions.

Abstract

The stochastic interpolant framework offers a powerful approach for constructing generative models based on ordinary differential equations (ODEs) or stochastic differential equations (SDEs) to transform arbitrary data distributions. However, prior analyses of this framework have primarily focused on the continuous-time setting, assuming a perfect solution of the underlying equations. In this work, we present the first discrete-time analysis of the stochastic interpolant framework, where we introduce an innovative discrete-time sampler and derive a finite-time upper bound on its distribution estimation error. Our result provides a novel quantification of how different factors, including the distance between source and target distributions and estimation accuracy, affect the convergence rate and also offers a new principled way to design efficient schedules for convergence acceleration. Finally, numerical experiments are conducted on the discrete-time sampler to corroborate our theoretical findings.