# An optimal $(\epsilon,\delta)$-approximation scheme for the mean of random variables with bounded relative variance

## TL;DR

This paper introduces an efficient $(\epsilon,\delta)$-approximation scheme for estimating the mean of random variables with bounded relative variance, matching optimal sample complexity without requiring higher moment bounds.

## Contribution

It presents a simple, easy-to-implement approximation method that achieves optimal sample complexity for mean estimation under bounded relative variance.

## Key findings

- Achieves optimal sample complexity of approximately $2c^2\epsilon^{-2}\ln(1/\delta)$ samples.
- Does not require bounds on third or fourth moments.
- Provides a practical scheme for randomized approximation in complex counting problems.

## Abstract

Randomized approximation algorithms for many #P-complete problems (such as the partition function of a Gibbs distribution, the volume of a convex body, the permanent of a $\{0,1\}$-matrix, and many others) reduce to creating random variables $X_1,X_2,\ldots$ with finite mean $\mu$ and standard deviation$\sigma$ such that $\mu$ is the solution for the problem input, and the relative standard deviation $|\sigma/\mu| \leq c$ for known $c$. Under these circumstances, it is known that the number of samples from the $\{X_i\}$ needed to form an $(\epsilon,\delta)$-approximation $\hat \mu$ that satisfies $\mathbb{P}(|\hat \mu - \mu| > \epsilon \mu) \leq \delta$ is at least $(2-o(1))\epsilon^{-2} c^2\ln(1/\delta)$. We present here an easy to implement $(\epsilon,\delta)$-approximation $\hat \mu$ that uses $(2+o(1))c^2\epsilon^{-2}\ln(1/\delta)$ samples. This achieves the same optimal running time as other estimators, but without the need for extra conditions such as bounds on third or fourth moments.