Learning-Based Importance Sampling via Stochastic Optimal Control for Stochastic Reaction Networks

arXiv (Cornell University)

Abstract

We explore efficient estimation of statistical quantities, particularly rare event probabilities, for stochastic reaction networks. Consequently, we propose an importance sampling (IS) approach to improve the Monte Carlo (MC) estimator efficiency based on an approximate tau-leap scheme. The crucial step in the IS framework is choosing an appropriate change of probability measure to achieve substantial variance reduction. This task is typically challenging and often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection in the stochastic reaction network context between finding optimal IS parameters within a class of probability measures and a stochastic optimal control formulation. Optimal IS parameters are obtained by solving a variance minimization problem. First, we derive an associated dynamic programming equation. Analytically solving this backward equation is challenging, hence we propose an approximate dynamic programming formulation to find near-optimal control parameters. To mitigate the curse of dimensionality, we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. Our analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators.

Authors 4

1. Chiheb Ben Hammouda Aachen Chair of Mathematics for Uncertainty Quantification

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen

   RWTH Aachen University, Chair of Mathematics for Uncertainty Quantification, Aachen, Germany

2. Nadhir Ben Rached

   University of Leeds

   Affiliation as printed

   University of Leeds, School of Mathematics, Leeds, UK

3. Raul F. Tempone Aachen

   King Abdullah University of Science and Technology · RWTH Aachen University

   Affiliation as printed

   RWTH Aachen

   King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), Thuwal, Saudi Arabia

   RWTH Aachen University, Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, Aachen, Germany

4. Sophia Wiechert corresponding Aachen Chair of Mathematics for Uncertainty Quantification

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen

   RWTH Aachen University, Chair of Mathematics for Uncertainty Quantification, Aachen, Germany