An asymptotically optimal Bernoulli factory for certain functions that can be expressed as power series

TL;DR

This paper introduces an asymptotically optimal Bernoulli factory algorithm for functions expressed as power series with non-negative coefficients, including fractional powers, achieving optimal efficiency for certain functions.

Contribution

It presents a new Bernoulli factory algorithm for functions as power series, with proven asymptotic optimality and a non-randomized variant, extending previous methods.

Findings

Algorithm efficiently simulates f(p) for power series functions.

Achieves asymptotic optimality in input usage for specific functions.

Includes a non-randomized version and discusses extensions.

Abstract

Given a sequence of independent Bernoulli variables with unknown parameter $p$, and a function $f$ expressed as a power series with non-negative coefficients that sum to at most $1$, an algorithm is presented that produces a Bernoulli variable with parameter $f(p)$. In particular, the algorithm can simulate $f(p)=p^a$, $a\in(0,1)$. For functions with a derivative growing at least as $f(p)/p$ for $p\rightarrow 0$, the average number of inputs required by the algorithm is asymptotically optimal among all simulations that are fast in the sense of Nacu and Peres. A non-randomized version of the algorithm is also given. Some extensions are discussed.