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- W2017008844 abstract "We study the polynomial reconstruction problem for low-degree multivariate polynomials over finite field $mathbb{F}[2]$. In this problem, we are given a set of points $mathbf{x}in{0,1}^n$ and target values $f(mathbf{x})in{0,1}$ for each of these points, with the promise that there is a polynomial over $mathbb{F}[2]$ of degree at most d that agrees with f at $1-varepsilon$ fraction of the points. Our goal is to find a degree d polynomial that has good agreement with f. We show that it is NP-hard to find a polynomial that agrees with f on more than $1-2^{-d}+delta$ fraction of the points for any $epsilon,delta>0$. This holds even with the stronger promise that the polynomial that fits the data is in fact linear, whereas the algorithm is allowed to find a polynomial of degree d. Previously the only known hardness of approximation (or even NP-completeness) was for the case when $d =1$, which follows from a celebrated result of Håstad [J. ACM, 48 (2001), pp. 798–859]. In the setting of Computational Learning, our result shows the hardness of nonproper agnostic learning of parities, where the learner is allowed a low-degree polynomial over $mathbb{F}[2]$ as a hypothesis. This is the first nonproper hardness result for this central problem in computational learning. Our results can be extended to multivariate polynomial reconstruction over any finite field." @default.
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- W2017008844 date "2010-01-01" @default.
- W2017008844 modified "2023-09-25" @default.
- W2017008844 title "Hardness of Reconstructing Multivariate Polynomials over Finite Fields" @default.
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- W2017008844 doi "https://doi.org/10.1137/070705258" @default.
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