Frequentist parameter estimation with supervised learning

Abstract

Recently, there has been a great deal of interest surrounding the calibration of quantum sensors using machine learning techniques. This work explores the use of regression to infer a machine-learned point estimate of an unknown parameter. Although the analysis is necessarily frequentist - relying on repeated estimates to build up statistics - the authors clarify that this machine-learned estimator converges to the Bayesian maximum a posteriori estimator (subject to some regularity conditions). When the number of training measurements is large, this is identical to the well-known maximum-likelihood estimator (MLE), and using this fact, the authors argue that the Cramér-Rao sensitivity bound applies to the mean-square error cost function and can therefore be used to select optimal model and training parameters. The machine-learned estimator inherits the desirable asymptotic properties of the MLE, up to a limit imposed by the resolution of the training grid. Furthermore, the authors investigate the role of quantum noise in the training process and show that this noise imposes a fundamental limit on the number of grid points. This manuscript paves the way for machine-learning to assist the calibration of quantum sensors, thereby allowing maximum-likelihood inference to play a more prominent role in the design and operation of the next generation of ultra-precise sensors.