Non-Asymptotic Error Bounds for SMC with Biased Proposals: Application to Conditional Diffusion Sampling
2026-07-06 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study a method called Sequential Monte Carlo (SMC), which is used to fine-tune pretrained generative models by adjusting samples after they are created. They analyze the errors that come up when the steps in this method aren’t perfectly accurate, breaking down the total error into two parts: one due to using approximate steps instead of ideal ones, and the other due to having a limited number of samples. Their work extends mathematical tools to better understand and control these errors, particularly for models that use score-based diffusion processes. This leads to the first clear error bounds that consider all major sources of error in these sampling methods.
Sequential Monte Carlo (SMC)Feynman-Kac flowmutation kernelskernel biasMonte Carlo errorDoeblin conditionLyapunov driftscore-based diffusion modelstime discretizationreverse diffusion dynamics
Authors
Stanislas Strasman, Gabriel Victorino Cardoso, Sylvain Le Corff, Vincent Lemaire, Antonio Ocello
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.