Polynomial Threshold Functions for Decision Lists

TL;DR

This paper investigates polynomial threshold function representations of decision lists, providing tight bounds on degree and weight over Boolean cubes and Hamming balls, improving previous results and establishing new lower bounds.

Contribution

It introduces improved upper bounds for representing decision lists as PTFs with specific degree and weight constraints, and proves matching lower bounds, advancing understanding of PTF complexity.

Findings

Decision lists over 00^n can be represented by PTFs with degree O((n/log n)^{1/3}) and weight 2^{O(n/d^2)}.

The bounds are tight due to matching lower bounds by Beigel.

For decision lists over Hamming balls, the weight bound improves to n^{O(sqrt{k})} for degree (sqrt{k}).

Abstract

For $S \subseteq \{0,1\}^n$ a Boolean function $f \colon S \to \{-1,1\}$ is a polynomial threshold function (PTF) of degree $d$ and weight $W$ if there is a polynomial $p$ with integer coefficients of degree $d$ and with sum of absolute coefficients $W$ such that $f(x) = \text{sign}(p(x))$ for all $x \in S$. We study a representation of decision lists as PTFs over Boolean cubes $\{0,1\}^n$ and over Hamming balls $\{0,1\}^{n}_{\leq k}$. As our first result, we show that for all $d = O\left( \left( \frac{n}{\log n}\right)^{1/3}\right)$ any decision list over $\{0,1\}^n$ can be represented by a PTF of degree $d$ and weight $2^{O(n/d^2)}$. This improves the result by Klivans and Servedio [Klivans, Servedio, 2006] by a $\log^2 d$ factor in the exponent of the weight. Our bound is tight for all $d = O\left( \left( \frac{n}{\log n}\right)^{1/3}\right)$ due to the matching lower bound by Beigel [Beigel, 1994]. For decision lists over a Hamming ball $\{0,1\}^n_{\leq k}$ we show that the upper bound on weight above can be drastically improved to $n^{O(\sqrt{k})}$ for $d = \Theta(\sqrt{k})$. We also show that similar improvement is not possible for smaller degrees by proving the lower bound $W = 2^{\Omega(n/d^2)}$ for all $d = O(\sqrt{k})$. \end{abstract}