Hardness of reconstructing multivariate polynomials over finite fields

Parikshit Gopalan, Subhash Khot, Rishi Saket

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We study the polynomial reconstruction problem for low-degree multivariate polynomials over double-struck F sign[2]. In this problem, we are given a set of points x ∈{0, 1}n and target values f(x) ∈ {0, 1} for each of these points, with the promise that there is a polynomial over double-struck F sign[2] of degree at most d that agrees with f at 1 - ε 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 + δ fraction of the points for any ε, δ > 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 [16]. In the setting of Computational Learning, our result shows the hardness of (non-proper)agnostic learning of parities, where the learner is allowed a low-degree polynomial over double-struck F sign[2] as a hypothesis. This is the first nonproper hardness result for this central problem in computational learning. Our results extend to multivariate polynomial reconstruction over any finite field.

Original languageEnglish (US)
Title of host publicationProceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007
Pages349-359
Number of pages11
DOIs
StatePublished - 2007
Event48th Annual Symposium on Foundations of Computer Science, FOCS 2007 - Providence, RI, United States
Duration: Oct 20 2007Oct 23 2007

Other

Other48th Annual Symposium on Foundations of Computer Science, FOCS 2007
CountryUnited States
CityProvidence, RI
Period10/20/0710/23/07

Fingerprint

Hardness
Polynomials

ASJC Scopus subject areas

  • Engineering(all)

Cite this

Gopalan, P., Khot, S., & Saket, R. (2007). Hardness of reconstructing multivariate polynomials over finite fields. In Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007 (pp. 349-359). [4389506] https://doi.org/10.1109/FOCS.2007.4389506

Hardness of reconstructing multivariate polynomials over finite fields. / Gopalan, Parikshit; Khot, Subhash; Saket, Rishi.

Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007. 2007. p. 349-359 4389506.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Gopalan, P, Khot, S & Saket, R 2007, Hardness of reconstructing multivariate polynomials over finite fields. in Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007., 4389506, pp. 349-359, 48th Annual Symposium on Foundations of Computer Science, FOCS 2007, Providence, RI, United States, 10/20/07. https://doi.org/10.1109/FOCS.2007.4389506
Gopalan P, Khot S, Saket R. Hardness of reconstructing multivariate polynomials over finite fields. In Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007. 2007. p. 349-359. 4389506 https://doi.org/10.1109/FOCS.2007.4389506
Gopalan, Parikshit ; Khot, Subhash ; Saket, Rishi. / Hardness of reconstructing multivariate polynomials over finite fields. Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007. 2007. pp. 349-359
@inproceedings{e1f793c0892640fd9029b6b7a6d10aca,
title = "Hardness of reconstructing multivariate polynomials over finite fields",
abstract = "We study the polynomial reconstruction problem for low-degree multivariate polynomials over double-struck F sign[2]. In this problem, we are given a set of points x ∈{0, 1}n and target values f(x) ∈ {0, 1} for each of these points, with the promise that there is a polynomial over double-struck F sign[2] of degree at most d that agrees with f at 1 - ε 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 + δ fraction of the points for any ε, δ > 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 H{\aa}stad [16]. In the setting of Computational Learning, our result shows the hardness of (non-proper)agnostic learning of parities, where the learner is allowed a low-degree polynomial over double-struck F sign[2] as a hypothesis. This is the first nonproper hardness result for this central problem in computational learning. Our results extend to multivariate polynomial reconstruction over any finite field.",
author = "Parikshit Gopalan and Subhash Khot and Rishi Saket",
year = "2007",
doi = "10.1109/FOCS.2007.4389506",
language = "English (US)",
isbn = "0769530109",
pages = "349--359",
booktitle = "Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007",

}

TY - GEN

T1 - Hardness of reconstructing multivariate polynomials over finite fields

AU - Gopalan, Parikshit

AU - Khot, Subhash

AU - Saket, Rishi

PY - 2007

Y1 - 2007

N2 - We study the polynomial reconstruction problem for low-degree multivariate polynomials over double-struck F sign[2]. In this problem, we are given a set of points x ∈{0, 1}n and target values f(x) ∈ {0, 1} for each of these points, with the promise that there is a polynomial over double-struck F sign[2] of degree at most d that agrees with f at 1 - ε 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 + δ fraction of the points for any ε, δ > 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 [16]. In the setting of Computational Learning, our result shows the hardness of (non-proper)agnostic learning of parities, where the learner is allowed a low-degree polynomial over double-struck F sign[2] as a hypothesis. This is the first nonproper hardness result for this central problem in computational learning. Our results extend to multivariate polynomial reconstruction over any finite field.

AB - We study the polynomial reconstruction problem for low-degree multivariate polynomials over double-struck F sign[2]. In this problem, we are given a set of points x ∈{0, 1}n and target values f(x) ∈ {0, 1} for each of these points, with the promise that there is a polynomial over double-struck F sign[2] of degree at most d that agrees with f at 1 - ε 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 + δ fraction of the points for any ε, δ > 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 [16]. In the setting of Computational Learning, our result shows the hardness of (non-proper)agnostic learning of parities, where the learner is allowed a low-degree polynomial over double-struck F sign[2] as a hypothesis. This is the first nonproper hardness result for this central problem in computational learning. Our results extend to multivariate polynomial reconstruction over any finite field.

UR - http://www.scopus.com/inward/record.url?scp=46749152630&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=46749152630&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2007.4389506

DO - 10.1109/FOCS.2007.4389506

M3 - Conference contribution

SN - 0769530109

SN - 9780769530109

SP - 349

EP - 359

BT - Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007

ER -