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A Multi-level procedure for enhancing accuracy of machine learning algorithms

by K. Lye and S. Mishra and R. Molinaro

(Report number 2019-54)

Abstract
We propose a multi-level method to increase the accuracy of machine learning algorithms for approximating observables in scientific computing, particularly those that arise in systems modeled by differential equations. The algorithm relies on judiciously combining a large number of computationally cheap training data on coarse resolutions with a few expensive training samples on fine grid resolutions. Theoretical arguments for lowering the generalization error, based on reducing the variance of the underlying maps, are provided and numerical evidence, indicating significant gains over underlying single-level machine learning algorithms, are presented. Moreover, we also apply the multi-level algorithm in the context of forward uncertainty quantification and observe a considerable speed-up over competing algorithms.

Keywords: machine learning, multi-level Monte Carlo, multi-level, deep learning, uncertainty quantifications

@Techreport{LMM19_858,
  author = {K. Lye and S. Mishra and R. Molinaro},
  title = {A Multi-level procedure for enhancing accuracy of machine learning algorithms},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-54},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-54.pdf },
  year = {2019}
}

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