Error Bounds for a Diffusion Model-Based Drift Estimator
2026-06-01 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study a method introduced by Tapia Costa et al. for estimating the drift in stochastic differential equations using noisy data from many paths. They focus on understanding how well this method works by providing a clear mathematical bound on its average error over time. Their analysis breaks down the total error into different parts, like the way the data is approximated, noise handling, and randomness in sampling. This helps clarify how different settings affect the accuracy of the drift estimate.
stochastic differential equationsparameter estimationdrift estimationdiffusion parameterEuler-Maruyama discretizationscore-matchingdenoisingrisk boundmean-squared errorsampling variance
Authors
Ioar Casado-Telletxea, Omar Rivasplata
Abstract
Parameter estimation in stochastic differential equations is a classical statistical problem of much importance in many scientific fields. Recent work of Tapia Costa et al. (2026) introduced a novel technique for estimating the drift when the diffusion parameter is known, using discrete samples from multiple trajectories. Their method treats drift estimation as a denoising problem, and leverages tools from (conditional) score-matching diffusion models. Although their experiments showed promising results across different drift classes, the question of theoretical guarantees for their estimator was left unanswered. In this note, we address this gap by exploiting techniques from diffusion model theory. More concretely, we derive an explicit risk bound for the time-averaged mean-squared error of said drift estimator. Our bound decomposes the risk into the (i) Euler-Maruyama discretization, (ii) score/denoiser approximation, (iii) noise initialization, and (iv) sampling variance, revealing the trade-offs between the different hyperparameters and sources of error in the estimator.