Nesterov-aided stochastic gradient methods using Laplace approximation for Bayesian design optimization

André Gustavo Carlon, Ben Mansour Dia, Luis Espath, Rafael Holdorf Lopez, Raul Tempone

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

Finding the best setup for experiments is the primary concern for Optimal Experimental Design (OED). Here, we focus on the Bayesian experimental design problem of finding the setup that maximizes the Shannon expected information gain. We use the stochastic gradient descent and its accelerated counterpart, which employs Nesterov's method, to solve the optimization problem in OED. We adapt a restart technique, originally proposed for the acceleration in deterministic optimization, to improve stochastic optimization methods. We combine these optimization methods with three estimators of the objective function: the double-loop Monte Carlo estimator (DLMC), the Monte Carlo estimator using the Laplace approximation for the posterior distribution (MCLA) and the double-loop Monte Carlo estimator with Laplace-based importance sampling (DLMCIS). Using stochastic gradient methods and Laplace-based estimators together allows us to use expensive and complex models, such as those that require solving partial differential equations (PDEs). From a theoretical viewpoint, we derive an explicit formula to compute the gradient estimator of the Monte Carlo methods, including MCLA and DLMCIS. From a computational standpoint, we study four examples: three based on analytical functions and one using the finite element method. The last example is an electrical impedance tomography experiment based on the complete electrode model. In these examples, the accelerated stochastic gradient descent method using MCLA converges to local maxima with up to five orders of magnitude fewer model evaluations than gradient descent with DLMC.
Original languageEnglish (US)
Pages (from-to)112909
JournalComputer Methods in Applied Mechanics and Engineering
Volume363
DOIs
StatePublished - Feb 22 2020

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): CRG3 Award Ref: 2281, CRG4 Award Ref: 2584
Acknowledgements: The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST), Saudi Arabia, KAUST CRG3 Award Ref: 2281, and KAUST CRG4 Award Ref: 2584. The authors also gratefully acknowledge the financial support of CNPq (National Counsel of Technological and Scientific Development), Brazil and CAPES (Coordination of Superior Level Staff Improvement), Brazil.

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