A Regularity Lemma, and Low-Weight Approximators, for Low-Degree Polynomial Threshold Functions

We give a “regularity lemma” for degree-d polynomial threshold functions (PTFs) over the Boolean cube {−1, 1}n . Roughly speaking, this result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a “regular” PTF is a PTF sign(p(x)) where the influence of each variable on the polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for any constants d ≥ 1, > 0, every degree-d PTF over n variables can be approximated to accuracy by a constant-degree PTF that has integer weights of total magnitude O(nd ). This weight bound is shown to be optimal up to logarithmic factors. ∗ Supported by NSF grant CCF-0728736, and by an Alexander S. Onassis Foundation Fellowship. Part of this work was done while visiting IBM Almaden. † Supported by NSF grants CCF-0347282, CCF-0523664 and CNS-0716245, and by D...

• Type: Conference
• Year: 2010
• Where: COCO
• Authors: Ilias Diakonikolas, Rocco A. Servedio, Li-Yang Tan, Andrew Wan
• Keywords: Algorithms, COCO 2010, Degree-d Ptf, NSF Grants, Regularity Lemma