Non-Asymptotic Error Bounds for SMC with Biased Proposals: Application to Conditional Diffusion Sampling
Provides theoretical guarantees for SMC-based conditional sampling with diffusion models, addressing a known bottleneck in post-hoc conditioning of pretrained generative models.
The paper develops non-asymptotic error bounds for sequential Monte Carlo methods with biased proposals, decomposing the total error into kernel bias and finite-particle Monte Carlo error. Applied to conditional diffusion sampling, it provides the first joint error bound controlling initialization, time discretization, score approximation, and particle error.
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.