Abstract

$ \newcommand{\R}{{\mathbb R}} \newcommand{\zoon}{\{0,1\}^n} \newcommand{\tildeg}{\,\deg\hspace{-18pt}\widetilde{\rule{0mm}{6pt}\hspace{18pt}}} \newcommand{\robust}{{\operatorname{robust}}} $ A basic question in any model of computation is how to reliably compute a given function when its inputs are subject to noise. Buhrman, Newman, Röhrig, and de Wolf (2003) posed the noisy computation problem for real polynomials. We give a complete solution to this problem. For any $\delta>0$ and any polynomial $p\colon\zoon\to[-1,1],$ we construct a corresponding polynomial $p_\robust\colon\R^n\to\R$ of degree $O(\deg p+\log1/\delta)$ that is robust to noise in the inputs: $|p(x)-p_\robust(x+\epsilon)|<\delta$ for all $x\in\zoon$ and all $\epsilon\in[-1/3,1/3]^n$. This result is optimal with respect to all parameters. We construct $p_\robust$ explicitly for each $p$. Previously, it was open to give such a construction even for $p=x_1\oplus x_2\oplus \cdots\oplus x_n$ (Buhrman et al., 2003). The proof contributes a technique of independent interest, which allows one to force partial cancellation of error terms in a polynomial. An extended abstract of this article appeared in the Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing (STOC'12), pages 747--758, 2012.

Highlights

  • Noise is a well-studied phenomenon in the computing literature

  • Research has shown that the answer depends crucially on the computational model in question. Models studied in this context include decision trees [21, 43, 19, 17, 38], circuits [40, 22, 18, 30, 52, 53], broadcast networks [23, 36, 20, 38, 25, 14, 15], and communication protocols [45, 46, 7, 24]

  • Combinatorial argument, the authors of [10] obtained an upper bound of O( deg ( f ) log deg ( f )) on the degree of a robust polynomial for any given Boolean function f

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Summary

Introduction

Noise is a well-studied phenomenon in the computing literature. It arises naturally in several ways. The contribution of this paper is to answer a question posed ten years ago by Buhrman et al [10] These authors asked whether real polynomials, as a computational model, are robust to noise. Buhrman et al [10] asked whether a polynomial p can be made robust with only a constant-factor increase in degree, provided that p approximates a Boolean function. Combinatorial argument, the authors of [10] obtained an upper bound of O( deg ( f ) log deg ( f )) on the degree of a robust polynomial for any given Boolean function f This combinatorial argument seems to be of no use in proving Theorem 1.1. Even though the error in the approximation of an individual monomial is relatively large, we show that the errors across the monomials behave in a coordinated way and essentially cancel each other out

Notation and preliminaries
A robust polynomial for parity
Reduction to homogeneous polynomials
Error cancellation in homogeneous polynomials
Main result

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