Early Stopping for Ensemble Kalman-Bucy Inversion

Abstract: Bayesian linear inverse problems aim to recover an unknown signal from noisy observations, incorporating prior knowledge. This paper analyses a data dependent method to choose the scale parameter of a Gaussian prior. The method we study arises from early stopping methods, which have been successfully applied to a range of problems, such as statistical inverse problems, in the frequentist setting. These results are extended to the Bayesian setting. We study the use of a discrepancy-based stopping rule in the setting of random noise, which allows for adaptation. Our proposed stopping rule results in optimal rates under certain conditions on the prior covariance operator. We furthermore derive for which class of signals this method is adaptive. It is also shown that the associated posterior contracts at the optimal rate and provides a conservative measure of uncertainty. We implement the proposed stopping rule using the continuous-time ensemble Kalman--Bucy filter (EnKBF). The fictitious time parameter replaces the scale parameter, and the ensemble size is appropriately adjusted in order to not lose the statistical optimality of the computed estimator. With this Monte Carlo algorithm, we extend our results numerically to a non-linear problem.