Optimal linear Bernoulli factories for small mean problems

Suppose a coin with unknown probability p of heads can be flipped as often as desired. A Bernoulli factory for a function f is an algorithm that uses flips of the coin together with auxiliary randomness to flip a single coin with probability f(p) of heads. Applications include perfect sampling from the stationary distribution of certain regenerative processes. When f is analytic, the problem can be reduced to a Bernoulli factory of the form f(p) = Cp for constant C. Presented here is a new algorithm that for small values of Cp, requires roughly only C coin flips. From information theoretic considerations, this is also conjectured to be (to first order) the minimum number of flips needed by any such algorithm. For large values of Cp, the new algorithm can also be used to build a new Bernoulli factory that uses only 80% of the expected coin flips of the older method. In addition, the new method also applies to the more general problem of a linear multivariate Bernoulli factory, where there are k coins, the kth coin has unknown probability pk of heads, and the goal is to simulate a coin flip with probability C1p1+· · ·+Ckpk of heads.