Quasi-Monte Carlo for unbounded integrands with importance sampling

TL;DR

This paper analyzes the convergence rates of quasi-Monte Carlo methods for unbounded integrands, showing how importance sampling with a t distribution can significantly improve error rates.

Contribution

It introduces a novel approach using a smoothed projection operator and importance sampling to enhance QMC convergence for unbounded functions.

Findings

Error rate of O(n^{-1+ε}) for functions with slow growth.

Error rate of O(n^{-1+2M+ε}) for exponential growth functions.

Importance sampling with t distribution improves RQMC error to O(n^{-3/2+ε}).

Abstract

We consider the problem of estimating an expectation $ \mathbb{E}\left[ h(W)\right]$ by quasi-Monte Carlo (QMC) methods, where $ h $ is an unbounded smooth function on $ \mathbb{R}^d $ and $ W$ is a standard normal distributed random variable. To study rates of convergence for QMC on unbounded integrands, we use a smoothed projection operator to project the output of $W$ to a bounded region, which differs from the strategy of avoiding the singularities along the boundary of the unit cube $ [0,1]^d $ in 10.1137/S0036144504441573. The error is then bounded by the quadrature error of the transformed integrand and the projection error. If the function $h(\boldsymbol{x})$ and its mixed partial derivatives do not grow too fast as the Euclidean norm $|\boldsymbol{x}|$ goes to infinity, we obtain an error rate of $O(n^{-1+\epsilon})$ for QMC and randomized QMC (RQMC) with a sample size $n$ and an arbitrarily small $\epsilon>0$. However, the rate turns out to be $O(n^{-1+2M+\epsilon})$ if the functions grow exponentially with a rate of $O(\exp\{M|\boldsymbol{x}|^2\})$ for a constant $M\in(0,1/2)$. Superisingly, we find that using importance sampling with t distribution as the proposal can improve the root mean squared error of RQMC from $O(n^{-1+2M+\epsilon})$ to $O( n^{-3/2+\epsilon})$ for any $M\in(0,1/2)$.