Rocco Servedio, Li-Yang Tan, Justin Thaler

We study the challenging problem of learning decision lists attribute-efficiently, giving both positive and negative results.

Our main positive result is a new tradeoff between the running time and mistake bound for learning length-$k$ decision lists over $n$ Boolean variables. When the allowed running time is relatively high, our new ... more >>>

Kristoffer Arnsfelt Hansen, Vladimir Podolskii

We study the complexity of computing Boolean functions on general

Boolean domains by polynomial threshold functions (PTFs). A typical

example of a general Boolean domain is $\{1,2\}^n$. We are mainly

interested in the length (the number of monomials) of PTFs, with

their degree and weight being of secondary interest. We ...
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Anindya De, Ilias Diakonikolas, Rocco Servedio

Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$ be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic} algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ and an accuracy parameter $\eps>0$, approximates \[

\Pr_{x \sim \{-1,1\}^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1] \]

to within an additive $\pm \eps$. For any constant ...
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Anindya De, Ilias Diakonikolas, Rocco Servedio

We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-$2$ polynomial threshold function. Given a degree-2 input polynomial $p(x_1,\dots,x_n)$ and a parameter $\eps > 0$, the algorithm approximates

\[

\Pr_{x \sim \{-1,1\}^n}[p(x) \geq 0]

\]

to within an additive $\pm \eps$ in ...
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Anindya De, Rocco Servedio

We give a deterministic algorithm for

approximately counting satisfying assignments of a degree-$d$ polynomial threshold function

(PTF).

Given a degree-$d$ input polynomial $p(x_1,\dots,x_n)$ over $\mathbb{R}^n$

and a parameter $\epsilon > 0$, our algorithm approximates

$

\mathbf{P}_{x \sim \{-1,1\}^n}[p(x) \geq 0]

$

to within an additive $\pm \epsilon$ in time $O_{d,\epsilon}(1)\cdot ...
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Alexander A. Sherstov

The threshold degree of a Boolean function $f$ is the minimum degree of

a real polynomial $p$ that represents $f$ in sign: $f(x)\equiv\mathrm{sgn}\; p(x)$. In a seminal 1969

monograph, Minsky and Papert constructed a polynomial-size constant-depth

$\{\wedge,\vee\}$-circuit in $n$ variables with threshold degree $\Omega(n^{1/3}).$ This bound underlies ...
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Alexander A. Sherstov

The threshold degree of a Boolean function $f$ is the minimum degree of

a real polynomial $p$ that represents $f$ in sign: $f(x)\equiv\mathrm{sgn}\; p(x)$. Introduced

in the seminal work of Minsky and Papert (1969), this notion is central to

some of the strongest algorithmic and complexity-theoretic results for

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Valentine Kabanets, Daniel Kane, Zhenjian Lu

A polynomial threshold function (PTF) of degree $d$ is a boolean function of the form $f=\mathrm{sgn}(p)$, where $p$ is a degree-$d$ polynomial, and $\mathrm{sgn}$ is the sign function. The main result of the paper is an almost optimal bound on the probability that a random restriction of a PTF is ... more >>>