An Optimal Algorithm for Monte Carlo Estimation

Reviewer: Panamalai R. Parthasarathy

One of the simplest design problems is to decide when to stop sampling. For example, suppose $Z_1,Z_2,...$ are independently and identically distributed according to $Z$ in the interval [0,1] with mean $\mu_Z$ and variance ${\sigma_Z}^2$. From Bernstein's inequality,we know that if N is fixed proportional to $\frac{ln(\frac{1}{\delta})}{\epsilon^2}$ and $S_N=Z_1+..+Z_N$, then with probability of atleast $1-\delta$, $\frac{S}{N}$ approsimates $\mu_Z$ with absolute error $\epsilon$. Often, however, $\mu_Z$ is small and a good absolute error estimate of $\mu_Z$ is typically a poor relative error approximation of $\mu_Z$.If $Pr[\mu_Z(1-\epsilon) \leq \bar{\mu}_Z \leq \mu_Z(1+\epsilon)] \geq 1-\delta$ then $\bar{\mu}_Z$ is an $(\epsilon,\delta)$-approximation of $\mu_Z$. The authors describe an approximation algorithm based on a simple stopping rule. Using the stopping rule, the approximation algorithm gives an $(\epsilon,\delta)$ approximation of $\mu_Z$ after an expected number of experiments proportionsl to $\frac{\gamma}{\mu_Z}$ where $\gamma=\frac{4\lambda ln(\frac{2}{\delta})}{\epsilon^2}$. The authors present a powerful algorithm, the $\cal{AA}$ algorithm that, given $\epsilon, \delta$ and independently and identically distributed outcomes $Z_1, Z_2,...$ generated from any random variable Z distributed in [0,1], obtains an $(\epsilon,\delta)$ approximation of $\mu_Z$ after an expected number of experiments proportional to $\frac{\gamma \rho_Z}{\mu_Z^2}$. They prove that for all Z, $\cal{AA}$ runs the optimal number of experiments to within a constant factor. Also $\cal{AA}$ provides substantial computational savings in applications that employ a poor upper bound on $\frac{\rho_Z}{\mu_Z^2}$.