# Unbiased Estimation using a Class of Diffusion Processes

arXiv: https://arxiv.org/abs/2203.03013 · HTML: https://mlsanity.com/paper/2203.03013/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Mar 2022.

## Built on by (most cited first, 4 of 4)

- [Unpaired Image-to-Image Translation via Neural Schr"odinger Bridge](https://mlsanity.com/paper/2305.15086/) (2023)
- [Sampling from multi-modal distributions with polynomial query complexity in fixed dimension via reverse diffusion](https://mlsanity.com/paper/2501.00565/) (2024)
- [Unbiased Estimation using Underdamped Langevin Dynamics](https://mlsanity.com/paper/2206.07202/) (2022)
- [Connections between the F"ollmer process and the denoising diffusion probabilistic model](https://mlsanity.com/paper/2605.18040/) (2026)

## Abstract (arXiv)

We study the problem of unbiased estimation of expectations with respect to (w.r.t.) π a given, general probability measure on (ℝd,B(ℝd)) that is absolutely continuous with respect to a standard Gaussian measure. We focus on simulation associated to a particular class of diffusion processes, sometimes termed the Schr"odinger-F"ollmer Sampler, which is a simulation technique that approximates the law of a particular diffusion bridge process Xtt∈ [0,1] on ℝd, d∈ mathbbN0. This latter process is constructed such that, starting at X0=0, one has X1~ π. Typically, the drift of the diffusion is intractable and, even if it were not, exact sampling of the associated diffusion is not possible. As a result, citesforig,jiao consider a stochastic Euler-Maruyama scheme that allows the development of biased estimators for expectations w.r.t.~π. We show that for this methodology to achieve a mean square error of O(ε2), for arbitrary ε>0, the associated cost is O(ε-5). We then introduce an alternative approach that provides unbiased estimates of expectations w.r.t.~π, that is, it does not suffer from the time discretization bias or the bias related with the approximation of the drift function. We prove that to achieve a mean square error of O(ε2), the associated cost is, with high probability, O(ε-2|log(ε)|2+δ), for any δ>0. We implement our method on several examples including Bayesian inverse problems.