Adam Klivans, Alexander A. Sherstov

We give the first representation-independent hardness results for

PAC learning intersections of halfspaces, a central concept class

in computational learning theory. Our hardness results are derived

from two public-key cryptosystems due to Regev, which are based on the

worst-case hardness of well-studied lattice problems. Specifically, we

prove that a polynomial-time ...
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Alexander A. Sherstov

The threshold degree of a Boolean function

$f\colon\{0,1\}\to\{-1,+1\}$ is the least degree of a real

polynomial $p$ such $f(x)\equiv\mathrm{sgn}\; p(x).$ We

construct two halfspaces on $\{0,1\}^n$ whose intersection has

threshold degree $\Theta(\sqrt n),$ an exponential improvement on

previous lower bounds. This solves an open problem due to Klivans

(2002) and ...
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Prahladh Harsha, Adam Klivans, Raghu Meka

Let $X$ be randomly chosen from $\{-1,1\}^n$, and let $Y$ be randomly

chosen from the standard spherical Gaussian on $\R^n$. For any (possibly unbounded) polytope $P$

formed by the intersection of $k$ halfspaces, we prove that

$$\left|\Pr\left[X \in P\right] - \Pr\left[Y \in P\right]\right| \leq \log^{8/5}k ...
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Alexander A. Sherstov

The threshold degree of a function

$f\colon\{0,1\}^n\to\{-1,+1\}$ is the least degree of a

real polynomial $p$ with $f(x)\equiv\mathrm{sgn}\; p(x).$ We

prove that the intersection of two halfspaces on

$\{0,1\}^n$ has threshold degree $\Omega(n),$ which

matches the trivial upper bound and completely answers

a question due to Klivans (2002). The best ...
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Ryan O'Donnell, Rocco Servedio, Li-Yang Tan

We give a pseudorandom generator that fools $m$-facet polytopes over $\{0,1\}^n$ with seed length $\mathrm{polylog}(m) \cdot \log n$. The previous best seed length had superlinear dependence on $m$. An immediate consequence is a deterministic quasipolynomial time algorithm for approximating the number of solutions to any $\{0,1\}$-integer program.

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