Abstract

In this note we compare two measures of the complexity of a class $\mathcal F$ of Boolean functions studied in (unconditional) pseudorandomness: $\mathcal F$'s ability to distinguish between biased and uniform coins (the coin problem), and the norms of the different levels of the Fourier expansion of functions in $\mathcal F$ (the Fourier growth). We show that for coins with low bias $\varepsilon = o(1/n)$, a function's distinguishing advantage in the coin problem is essentially equivalent to $\varepsilon$ times the sum of its level $1$ Fourier coefficients, which in particular shows that known level $1$ and total influence bounds for some classes of interest (such as constant-width read-once branching programs) in fact follow as a black-box from the corresponding coin theorems, thereby simplifying the proofs of some known results in the literature. For higher levels, it is well-known that Fourier growth bounds on all levels of the Fourier spectrum imply coin theorems, even for large $\varepsilon$, and we discuss here the possibility of a converse.

Highlights

  • A natural question one can ask when studying a limited model of computation is how well it can solve some basic computational task, such as distinguishing between an unfair and a fair coin: Definition 1.1 (Coin problem)

  • We study implications between bounds on the advantage of a class of Boolean functions in the coin problem and bounds on the Fourier spectrum

  • This allows us to give simple “black-box” proofs of existing level 1 bounds in the literature, such as the bound for constant-width read-once branching programs due to Steinke, Vadhan, and Wan [23, §6] in 2014, which was conjectured by [20]

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Summary

Introduction

A natural question one can ask when studying a limited model of computation is how well it can solve some basic computational task, such as distinguishing between an unfair and a fair coin: Definition 1.1 (Coin problem). Perhaps surprisingly, even coin theorems for only a small range of ε are sufficient to bound the level 1 spectrum of the class F, even for F not closed under restriction This allows us to give simple “black-box” proofs of existing level 1 bounds in the literature, such as the bound for constant-width read-once branching programs due to Steinke, Vadhan, and Wan [23, §6] in 2014, which was conjectured by [20]. These results are summarized, showing the implications between bounds on coin problems and Fourier growth for a class F, arranged from top to bottom in order of strength (under the assumption that F is monotone or closed under negation of input variables, but need not be closed under restriction). We hope that this result may help point the way to giving some additional constraints under which one could hope for a L11 or coin problem bound to imply a general Lk1 bound

Low bias and level 1
Larger bias and beyond Level 1
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