On Sampling Lower Bounds for Polynomials

TL;DR

This paper investigates the limitations of sampling distributions defined by low-degree polynomials over finite fields, establishing bounds on their total variation distance from a uniform Bernoulli distribution.

Contribution

It provides new lower bounds on the total variation distance for polynomial-based samplers of constant degree, extending prior results and introducing a novel bound on polynomial bias.

Findings

For degree 1, the TV distance is at least 1 minus an exponential in n.

For degree 2, the TV distance is at least 1 minus an exponential in log(n)/loglog(n).

For degree 3, the TV distance is at least 1 minus an exponential in sqrt(loglog(n)).

Abstract

In this work, we continue the line of research on the complexity of distributions (Viola, Journal of Computing 2012), and study samplers defined by low degree polynomials. An $n$-tuple $P = (P_1,\dots, P_n)$ of functions $P_i \colon \mathbb{F}_2^m \to \mathbb{F}_2$ defines a distribution over $\{0,1\}^n$ in the natural way: draw $X$ uniformly at random from $\mathbb{F}_2^m$ and output $(P_1(X),\dots, P_n(X)) \in \{0,1\}^n$. We show that when $P$ is defined by polynomials of degree $d$, the total variation distance of $P$ from the product distribution $\mathrm{Ber}(1/3)^{\otimes n}$ is $1-o_n(1)$, where $o_n(1)$ is a vanishing function of $n$ for any constant degree $d$. For small values of $d$, we show the following concrete bounds. (i) For $d=1$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-\Omega(n))$. (ii) For $d=2$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-\Omega(\log(n)/\log\log(n)))$. (iii) For $d=3$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-\Omega(\sqrt{\log\log(n)}))$. Our results extend the recent lower bound results for sampling distributions, which have mostly focused on local samplers, small depth decision trees, and small depth circuits. As part of our proof, we establish the following result, that may be of independent interest: for any degree-$d$ polynomial $P\colon\mathbb{F}_2^m \to \mathbb{F}_2$ it holds that $\Pr_X[P(X) = 1]$ is bounded away from $1/3$ by some absolute constant $\delta = \delta_d>0$. Although the statement may seem obvious, we are not aware of an elementary proof of this. The proof techniques rely on the structural results for low degree polynomials, saying that any biased polynomial of degree $d$ can be written as a function of a small number of polynomials of degree $d-1$.