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Higher-order Quasi-Monte Carlo Training of Deep Neural Networks

by M. Longo and S. Mishra and T. K. Rusch and Ch. Schwab

(Report number 2020-57)

Abstract
We present a novel algorithmic approach and an error analysis leveraging Quasi-Monte Carlo (QMC) points for training deep neural network (DNN) surrogates of holomorphic Data- to-Observable (DtO) maps in engineering design. Our analysis reveals higher-order consistent, deterministic choices of training points in the input parameter space for both deep and shallow Neural Networks with holomorphic activation functions such as tanh. We prove that higher order QMC training points facilitate higher-order decay (in terms of the number of training samples) of the underlying generalization error, with consistency error bounds that are free from the curse of dimensionality in terms of the number of input param- eters, provided that DNN weights in hidden layers satisfy certain summability conditions. We present numerical experiments for DtO maps from elliptic and parabolic PDEs with uncertain inputs that confirm the theoretical analysis.

Keywords: deep learning, higher-order QMC, generalization error, deep neural networks, scientific computing

@Techreport{LMRS20_930,
  author = {M. Longo and S. Mishra and T. K. Rusch and Ch. Schwab},
  title = {Higher-order Quasi-Monte Carlo Training of Deep Neural Networks},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-57},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-57.pdf },
  year = {2020}
}

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