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Center for Nonlinear Studies |
Many problems or sub-problems in modeling, analysis, and optimization of stochastic systems require repeated estimation of a quantity of interest at different decision or model parameters. There are also parametric estimation problems that need to be solved in an “on-line" fashion where once the parameter value becomes known, the problem has to be solved within a limited time and computational budget. With these settings in mind, we present a learning framework for efficient MC simulation that uses sampling at some parameter values to construct effective variance reducing algorithms for estimation at other parameters. We illustrate the strategy in the context of the variance reduction technique of Control Variates (CV). Our approach deviates from other implementations of the CV technique in two important ways: (i) in the type of control variates we consider, and (ii) in how we seek computational efficiency and account for it.
Using a sample-wise approximation approach, we show that a number of standard deterministic function approximation methods imply very effective control variates. To evaluate the controlled estimators, we implement a two stage estimation procedure called DataBase Monte Carlo (DBMC). In the first stage a probability measure with finite support is defined that approximates the original probability measure. Next we define an approximate parametric estimation problem over a finite set called the DataBase, representing the support of the approximating measure. We apply the classical CV procedures for optimally controlled estimation to the approximate problem. There is a cost associated with constructing the database and evaluating the control means. We justify this cost in two settings: (a) solving many instances of the estimation problem where we gain efficiency in each instance; (b) solving an “online" estimation problem where we can obtain high quality estimators within the given time and computational budget constraints.
Host: Frank Alexander