Non-Asymptotic Error Bounds for SMC with Biased Proposals: Application to Conditional Diffusion Sampling

2026-07-06

AI Summary

This work addresses the absence of non-asymptotic error bounds for sequential Monte Carlo (SMC) methods employing biased proposal kernels in conditional sampling with pretrained generative models. It introduces the first non-asymptotic analysis framework that jointly controls kernel bias and particle approximation error. By extending local Doeblin conditions and Lyapunov drift arguments to conditional distributions, and integrating Feynman–Kac flow approximations with score-based diffusion models, the total error is decomposed under a forward-smoothing forgetfulness assumption into contributions from initialization, time discretization, score approximation, and finite-particle effects. This study establishes the first comprehensive non-asymptotic error bounds for conditional diffusion sampling, thereby providing theoretical guarantees for the reliability of SMC in generative modeling.

Abstract

Sequential Monte Carlo (SMC) methods are a natural tool for post-hoc conditioning of pretrained generative models, but in many applications the mutation kernels used by the particle system are biased approximations of an ideal Feynman--Kac flow. This paper develops a non-asymptotic error analysis for such SMC samplers. Under forward-smoothing forgetting conditions, we decompose the total error into a kernel bias, measuring the effect of replacing the ideal transition kernels by approximate ones, and a finite-particle Monte Carlo error. Our approach relies on extending local Doeblin-type conditions and Lyapunov drift arguments for Markov kernels to conditional distributions, thereby enabling a principled control of the bias. We then instantiate this general framework for conditional sampling with score-based diffusion models, and derive the first non-asymptotic error bound that jointly controls initialization error, time discretization, and score approximation in the reverse diffusion dynamics as well as finite-particle Monte Carlo error.

Problem

• Sequential Monte Carlo
• biased proposals
• non-asymptotic error bounds
• conditional diffusion sampling
• score-based diffusion models

Innovation

• Sequential Monte Carlo
• non-asymptotic error bounds
• biased proposals
• conditional diffusion sampling
• score-based generative models