Communications of the ACM
Perceptrons: expanded edition
Surveys in combinatorics, 1993
Learning decision trees using the Fourier spectrum
SIAM Journal on Computing
Efficient noise-tolerant learning from statistical queries
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Weakly learning DNF and characterizing statistical query learning using Fourier analysis
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
PP is closed under intersection
Selected papers of the 23rd annual ACM symposium on Theory of computing
On the Fourier spectrum of monotone functions
Journal of the ACM (JACM)
The harmonic sieve: a novel application of Fourier analysis to machine learning theory and practice
The harmonic sieve: a novel application of Fourier analysis to machine learning theory and practice
On the computational power of depth-2 circuits with threshold and modulo gates
Theoretical Computer Science
New degree bounds for polynomial threshold functions
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Noise-tolerant learning, the parity problem, and the statistical query model
Journal of the ACM (JACM)
A Random Sampling based Algorithm for Learning the Intersection of Half-spaces
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Computational applications of noise sensitivity
Computational applications of noise sensitivity
Learning intersections and thresholds of halfspaces
Journal of Computer and System Sciences - Special issue on FOCS 2002
Learnability and Automatizability
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
New lower bounds for statistical query learning
Journal of Computer and System Sciences - Special issue on COLT 2002
Cryptographic Hardness for Learning Intersections of Halfspaces
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
New Results for Learning Noisy Parities and Halfspaces
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Computational Complexity
A tamper-proof and lightweight authentication scheme
Pervasive and Mobile Computing
Cryptographic hardness for learning intersections of halfspaces
Journal of Computer and System Sciences
Quantum algorithms for highly non-linear Boolean functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
SIAM Journal on Computing
Making polynomials robust to noise
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
A complete characterization of statistical query learning with applications to evolvability
Journal of Computer and System Sciences
SIAM Journal on Computing
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We prove new lower bounds for learning intersections of halfspaces, one of the most important concept classes in computational learning theory. Our main result is that any statistical-query algorithm for learning the intersection of $\sqrt{n}$ halfspaces in n dimensions must make $2^{\varOmega (\sqrt{n})}$ queries. This is the first non-trivial lower bound on the statistical query dimension for this concept class (the previous best lower bound was n 驴(log驴n)). Our lower bound holds even for intersections of low-weight halfspaces. In the latter case, it is nearly tight. We also show that the intersection of two majorities (low-weight halfspaces) cannot be computed by a polynomial threshold function (PTF) with fewer than n 驴(log驴n/log驴log驴n) monomials. This is the first super-polynomial lower bound on the PTF length of this concept class, and is nearly optimal. For intersections of k=驴(log驴n) low-weight halfspaces, we improve our lower bound to $\min\{2^{\varOmega (\sqrt{n})},n^{\varOmega (k/\log k)}\},$ which too is nearly optimal. As a consequence, intersections of even two halfspaces are not computable by polynomial-weight PTFs, the most expressive class of functions known to be efficiently learnable via Jackson's Harmonic Sieve algorithm. Finally, we report our progress on the weak learnability of intersections of halfspaces under the uniform distribution.