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Combinatorial Bernoulli Factories

Rad Niazadeh, Renato Paes Leme, Jon Schneider

TL;DR

This work characterizes exactly when a polytope $P\subseteq[0,1]^n$ admits a Bernoulli factory and provides a constructive, algebraic-geometric framework based on Bernstein polynomials and zonotope tilings. It proves a factory exists if and only if $P$ is the intersection of $[0,1]^n$ with an affine subspace, and then builds strong factories for generic subspaces, extending to non-generic cases via limits. The results yield explicit factories for the perfect matching polytope (via arborescences and the matrix-tree theorem) and recover Sampford sampling for $k$-subsets, linking exact Bernoulli simulation to classical combinatorial constructs. Overall, the paper bridges polyhedral geometry, Bernstein polynomial identities, and exact combinatorial sampling with potential implications for precise sampling in combinatorial and statistical contexts.

Abstract

A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter $p \in [0,1]$ means the algorithm does not know $p$, but has sample access to independent draws of a Bernoulli random variable with mean equal to $p$. In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector $x\in P$ for a given polytope $P\subset [0,1]^n$, output a randomized vertex such that the expected value of the $i$-th coordinate is \emph{exactly} equal to $x_i$. For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix $[x_{ij}]$ and asked to sample a matching such that the probability of each edge $(i,j)$ be present in the matching is exactly equal to $x_{ij}$. We show that a polytope $P$ admits a Bernoulli factory if and and only if $P$ is the intersection of $[0,1]^n$ with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on $P$. The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings. We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the $k$-uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.

Combinatorial Bernoulli Factories

TL;DR

This work characterizes exactly when a polytope admits a Bernoulli factory and provides a constructive, algebraic-geometric framework based on Bernstein polynomials and zonotope tilings. It proves a factory exists if and only if is the intersection of with an affine subspace, and then builds strong factories for generic subspaces, extending to non-generic cases via limits. The results yield explicit factories for the perfect matching polytope (via arborescences and the matrix-tree theorem) and recover Sampford sampling for -subsets, linking exact Bernoulli simulation to classical combinatorial constructs. Overall, the paper bridges polyhedral geometry, Bernstein polynomial identities, and exact combinatorial sampling with potential implications for precise sampling in combinatorial and statistical contexts.

Abstract

A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter means the algorithm does not know , but has sample access to independent draws of a Bernoulli random variable with mean equal to . In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector for a given polytope , output a randomized vertex such that the expected value of the -th coordinate is \emph{exactly} equal to . For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix and asked to sample a matching such that the probability of each edge be present in the matching is exactly equal to . We show that a polytope admits a Bernoulli factory if and and only if is the intersection of with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on . The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings. We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the -uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.

Paper Structure

This paper contains 26 sections, 34 theorems, 99 equations, 9 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Bernoulli factories
  4. Bernstein polynomials
  5. Combinatorial factories
  6. A Factory for Matching
  7. Proof of Theorem \ref{['prop:matching-factory']}
  8. Necessary Conditions for Factories for Polytopes
  9. Bernoulli Factories for Generic Polytopes
  10. Proof of Theorem \ref{['thm:valid_factory']}
  11. Checking Condition \ref{['eq:not_vanish']}
  12. Rewriting Condition \ref{['eq:factory']}
  13. Partitions and Zonotopes
  14. Geometric illustration of Lemma \ref{['lemma:identity']} for $k=2$
  15. Proof of Lemma \ref{['lemma:identity']} for general $k$.
  16. ...and 11 more sections

Key Result

Lemma 2.4

Let $P(x) = \sum_{j=1}^{k}c_{j}M_j(x)$ be a Bernstein polynomial in $n$ variables, and let $C = \sum_{j=1}^{k} c_j$. Then there exists a finite one-bit Bernoulli factory for $P(x)/C$.

Figures (9)

  • Figure 1: Arborescences in $\T_1(3)$ corresponding to the monomials in $P_\eps$.
  • Figure 2: An example of the bijection between pairs $(\pi,T)\in S_n\times \mathcal{T}_r(n)$ with $\pi(r)=c$(left hand side) and $r$-bi-trees $G'\in \mathcal{G}_r$(right hand side): $n=5$, root $r=1$, $\pi=(3,1,2,5,4)$, $c=\pi(r)=3$, and $T=\{2\rightarrow 1, 3\rightarrow 1, 5\rightarrow 3,4 \rightarrow 3\}$; solid black edges belong to $T$; dashed purple edges are matching edges corresponding to $\pi$ excluding the green dashed edge $\left(r,\pi(r)\right)$, i.e., $\{(u,\pi(u))\}_{u\in\{2,3,4,5\}}$;
  • Figure 3: The factory should output $v$ at $x_2$ but not at $x_1$.
  • Figure 4: $\sigma_1 = \sigma_2 = +1$
  • Figure 5: $\sigma_1 = +1, \sigma_2 = -1$
  • ...and 4 more figures

Theorems & Definitions (73)

  • Definition 2.1: Bernoulli factory
  • Definition 2.2: One-bit Bernoulli factory
  • Definition 2.3: Bernstein polynomial
  • Lemma 2.4
  • proof
  • Theorem 2.5: Bernoulli race
  • proof
  • Corollary 2.6: Bernoulli race over Bernstein polynomials
  • proof
  • Definition 2.7: Bernoulli factory for a polytope $\P$
  • ...and 63 more