Hardness of Learning Noisy Halfspaces using Polynomial Thresholds

Bhattacharyya, A and Ghoshal, S and Saket, R (2018) Hardness of Learning Noisy Halfspaces using Polynomial Thresholds. In: 31st Annual Conference on Learning Theory, COLT 2018, 6 - 9 July 2018, Stockholm, pp. 876-917.

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Abstract

We prove the hardness of weakly learning halfspaces in the presence of adversarial noise using polynomial threshold functions (PTFs). In particular, we prove that for any constants d ∈ Z+ and ε > 0, it is NP-hard to decide: given a set of {−1, 1}-labeled points in Rn whether (YES Case) there exists a halfspace that classifies (1 − ε)-fraction of the points correctly, or (NO Case) any degree-d PTF classifies at most (1/2 + ε)-fraction of the points correctly. This strengthens to all constant degrees the previous NP-hardness of learning using degree-2 PTFs shown by Diakonikolas et al. (2011). The latter result had remained the only progress over the works of Feldman et al. (2006) and Guruswami et al. (2006) ruling out weakly proper learning adversarially noisy halfspaces.

Item Type:	Conference Paper
Publication:	Proceedings of Machine Learning Research
Publisher:	ML Research Press
Additional Information:	The copyright for this article belongs to the ML Research Press.
Keywords:	Halfspaces; Hardness; Learning; PTFs
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	17 Jul 2023 10:07
Last Modified:	17 Jul 2023 10:07
URI:	https://eprints.iisc.ac.in/id/eprint/82438

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