Randomized Dimension Reduction for Monte Carlo Simulations

Abstract

We present a new unbiased algorithm that estimates the expected value of f (U ) via Monte Carlo simulation, where U is a vector of d independent random variables, and f is a function of d variables. We assume that f does not depend equally on all its arguments. Under certain conditions we prove that, for the same computational cost, the variance of our estimator is lower than the variance of the standard Monte Carlo estimator by a factor of order d. Our method can be used to obtain a low-variance unbiased estimator for the expectation of a function of the state of a Markov chain at a given time-step. We study applications to volatility forecasting and time-varying queues. Numerical experiments show that our algorithm dramatically improves upon the standard Monte Carlo method for large values of d, and is highly resilient to discontinuities.

On the effective dimension and multilevel Monte Carlo

“…This section considers a class of MLMC unbiased estimators of E(f (U )) based on successive approximations of f by deterministic functions of its first components. In (Kahalé 2020b), a lower bound on the work-normalized variance of such estimators is given in terms of that of the randomized dimension reduction estimator. This section provides a lower bound on the work-normalized variance of these estimators in terms of the truncation dimension.…”

“…Owen (2019) gives a recent survey on the effective dimension. Kahalé (2020b) studies the relationship between the truncation dimension and the randomized dimension reduction method, a recent variance reduction technique applicable to high-dimensional problems.…”

“…The proof of Proposition 5.2 is essentially the same as that of(Kahalé 2020b, Proposition 10), and is therefore omitted. A justification of (20) for timevarying queues and numerical examples showing the efficiency of MLMC for estimating Markov chain functionals are given in(Kahalé 2020b).…”

Optimal unbiased estimators via convex hulls

“…Vihola (2018) describes stratified RMLMC methods that, under certain conditions, are shown to be asymptotically as efficient as MLMC. The randomized dimension reduction method, recently introduced in (Kahalé 2016, Kahalé 2019, is another technique that can provably achieve substantial variance reduction in high-dimensional settings, such as the estimation of the expectation of a functional of a timevarying Markov chain at a long horizon.…”

“…In (Rhee and Glynn 2015, Section 3), an algorithm that finds in O(m 3 ) time an m-truncated distribution that optimizes the efficiency of these estimators is given. On the other hand, the asymptotic efficiency of the randomized dimension reduction method is maximized in (Kahalé 2019) via a new geometric algorithm that solves an m-dimensional optimization problem in O(m) time. Kahalé (2019) points out that the same geometric algorithm solves the optimization problem in (Rhee and Glynn 2015, Section 3) in O(m) time.…”