A Statistical Analysis for Per-Instance Evaluation of Stochastic Optimizers: How Many Repeats Are Enough?

A key trait of stochastic optimizers is that multiple runs of the same optimizer in attempting to solve the same problem can produce different results. As a result, their performance is evaluated over several repeats, or runs, on the problem. However, the accuracy of the estimated performance metrics depends on the number of runs and should be studied using statistical tools. We present a statistical analysis of the common metrics, and develop guidelines for experiment design to measure the optimizer's performance using these metrics to a high level of confidence and accuracy. To this end, we first discuss the confidence interval of the metrics and how they are related to the number of runs of an experiment. We then derive a lower bound on the number of repeats in order to guarantee achieving a given accuracy in the metrics. Using this bound, we propose an algorithm to adaptively adjust the number of repeats needed to ensure the accuracy of the evaluated metric. Our simulation results demonstrate the utility of our analysis and how it allows us to conduct reliable benchmarking as well as hyperparameter tuning and prevent us from drawing premature conclusions regarding the performance of stochastic optimizers.

A Fast MR Fingerprinting Simulator for Direct Error Estimation and Sequence Optimization

MR Fingerprinting is a novel quantitative MR technique that could simultaneously provide multiple tissue property maps. When optimizing MRF scans, modeling undersampling errors and field imperfections in cost functions will make the optimization results more practical and robust. However, this process is computationally expensive and impractical for sequence optimization algorithms when MRF signal evolutions need to be generated for each optimization iteration. Here, we introduce a fast MRF simulator to simulate aliased images from actual scan scenarios including undersampling and system imperfections, which substantially reduces computational time and allows for direct error estimation and efficient sequence optimization. By constraining the total number of tissues present in a brain phantom, MRF signals from highly undersampled scans can be simulated as the product of the spatial response functions based on sampling patterns and sequence-dependent temporal functions. During optimization, the spatial response function is independent of sequence design and does not need to be recalculated. We evaluate the performance and computational speed of the proposed approach by simulations and in vivo experiments. We also demonstrate the power of applying the simulator in MRF sequence optimization. The simulation results from the proposed method closely approximate the signals and MRF maps from in vivo scans, with 158 times shorter processing time than the conventional simulation method using Non-uniform Fourier transform. Incorporating the proposed simulator in the MRF optimization framework makes direct estimation of undersampling errors during the optimization process feasible, and provide optimized MRF sequences that are robust against undersampling factors and system inhomogeneity.