Automated Importance Sampling via Optimal Control for Stochastic Reaction Networks: A Markovian Projection-based Approach
arxiv(2023)
摘要
We propose a novel alternative approach to our previous work (Ben Hammouda et
al., 2023) to improve the efficiency of Monte Carlo (MC) estimators for rare
event probabilities for stochastic reaction networks (SRNs). In the same spirit
of (Ben Hammouda et al., 2023), an efficient path-dependent measure change is
derived based on a connection between determining optimal importance sampling
(IS) parameters within a class of probability measures and a stochastic optimal
control formulation, corresponding to solving a variance minimization problem.
In this work, we propose a novel approach to address the encountered curse of
dimensionality by mapping the problem to a significantly lower-dimensional
space via a Markovian projection (MP) idea. The output of this model reduction
technique is a low-dimensional SRN (potentially even one dimensional) that
preserves the marginal distribution of the original high-dimensional SRN
system. The dynamics of the projected process are obtained by solving a related
optimization problem via a discrete L^2 regression. By solving the resulting
projected Hamilton-Jacobi-Bellman (HJB) equations for the reduced-dimensional
SRN, we obtain projected IS parameters, which are then mapped back to the
original full-dimensional SRN system, resulting in an efficient IS-MC estimator
for rare events probabilities of the full-dimensional SRN. Our analysis and
numerical experiments reveal that the proposed MP-HJB-IS approach substantially
reduces the MC estimator variance, resulting in a lower computational
complexity in the rare event regime than standard MC estimators.
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