A backward SDE method for uncertainty quantification in deep learning

Richard Archibald; Feng Bao; Yanzhao Cao; He Zhang

doi:10.3934/dcdss.2022062

Article Contents

2022, Volume 15, Issue 10: 2807-2835. Doi: 10.3934/dcdss.2022062

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A backward SDE method for uncertainty quantification in deep learning

1.
Computational Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA
2.
Department of Mathematics, Florida State University, Tallahassee, Florida, USA
3.
Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, USA
4.
School of Mathematics, Jilin University, Changchun, China

^* Corresponding author: He Zhang
^* Corresponding author: He Zhang

Received: August 31, 2021

Revised: December 31, 2021

Early access: March 2022

Published: September 2022

The second and third authors are partially supported by U.S. Department of Energy under grant numbers DE-SC0022297 and DE-SC0022253, the last author is supported by NSFC12071175 and Science and Technology Development of Jilin Province, China no. 201902013020

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Abstract

We develop a backward stochastic differential equation based probabilistic machine learning method, which formulates a class of stochastic neural networks as a stochastic optimal control problem. An efficient stochastic gradient descent algorithm is introduced with the gradient computed through a backward stochastic differential equation. Convergence analysis for stochastic gradient descent optimization and numerical experiments for applications of stochastic neural networks are carried out to validate our methodology in both theory and performance.

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Mathematics Subject Classification: 660H35, 68T07, 93E20.

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Figure 1. Classified samples of the deterministic and random classification functions

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Figure 2. Comparison between BNN output and SNN output

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Figure 3. Weight surface for classification results obtained by the SNN

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Figure 4. Example of data samples

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Figure 5. Comparison of the BNN approach and the backward SDE approach for SNNs

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Figure 6. Example of data samples

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Figure 7. Comparison of the BNN approach and the backward SDE approach for SNNs

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Figure 8. 4D Function approximation – surface views

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Figure 9. 4D Function approximation – section views

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Figure 10. Behavior of model curve with different parameters

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Figure 11. Prediction performance of SNNs for parameter estimation

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Figure 12. Distributions for SNN output

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Figure 13. Predicted function values corresponding to different parameters

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Table 1. Numerical implementation of the backward SDE method for SNNs

Algorithm 1.

Formulate the SNN model (1) as the stochastic optimal control problem (3) - (5) and give a partition $ \Pi^N $ to the control problem.
Choose the number of SGD iteration steps $ K \in \mathbb{N} $ corresponding to a stopping criteria, the learning rate $ \{\eta_k\}_k $ and the initial guess for the optimal control $ \{u_n^0\}_n $
for SGD iteration steps $ k =0, 1, 2, \cdots, K-1 $,
  : Simulate one realization of the state process $ \{X^{k}_{n}\}_{n} $ through the scheme (28).
  : Simulate one pair of solution paths $ \{\big( \hat{Y}^{k}_{n}, \ \hat{Z}^k_n \big)\}_{n} $ of the adjoint BSDEs system (7) corresponding to $ \{X^{k}_{n}\}_{n} $ through the schemes (29);
  : Calculate the gradient process and update the estimated optimal control $ \{u^{k+1}_n\}_n $ through the SGD iteration scheme (30);
end for
- The estimated optimal control is given by $ \{ \hat{u} _n\}_n := \{ u_{n}^{K} \}_n $;

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