Improving Monte Carlo randomized approximation schemes

TL;DR

This paper presents an improved Monte Carlo approximation scheme that reduces the number of samples needed for estimating a mean within a specified error and confidence level, enhancing efficiency in various computational problems.

Contribution

It introduces a new estimation method that significantly lowers the sample complexity from the previous best, making Monte Carlo approximations more efficient.

Findings

Reduces sample complexity from 19.35 to 6.96 times (c/ε)^2 ln(1/δ)

Maintains accuracy and confidence guarantees with fewer samples

Applicable to a wide range of randomized approximation problems

Abstract

Consider a central problem in randomized approximation schemes that use a Monte Carlo approach. Given a sequence of independent, identically distributed random variables $X_1,X_2,\ldots$ with mean $\mu$ and standard deviation at most $c \mu$, where $c$ is a known constant, and $\epsilon,\delta > 0$, create an estimate $\hat \mu$ for $\mu$ such that $\text{P}(|\hat \mu - \mu| > \epsilon \mu) \leq \delta$. This technique has been used for building randomized approximation schemes for the volume of a convex body, the permanent of a nonnegative matrix, the number of linear extensions of a poset, the partition function of the Ising model and many other problems. Existing methods use (to the leading order) $19.35 (c/\epsilon)^2 \ln(\delta^{-1})$ samples. This is the best possible number up to the constant factor, and it is an open question as to what is the best constant possible. This work gives an easy to apply estimate that only uses $6.96 (c/\epsilon)^2 \ln(\delta^{-1})$ samples in the leading order.