Polynomial Threshold Functions for Decision Lists

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 $2^{O(n/d2)}$. This improves the result by Klivans and Servedio [Klivans, Servedio, 2006] by a $\log² d$ 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 $W = 2^{{\Omega(n/d2)}$} for all $d = O(\sqrt{k})$. \end{abstract}