On the degree of polynomials that approximate symmetric Boolean functions (preliminary version)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
An O(nlog log n) learning algorithm for DNF under the uniform distribution
COLT '92 Proceedings of the fifth annual workshop on Computational learning theory
Learning in the presence of malicious errors
SIAM Journal on Computing
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
Statistical queries and faulty PAC oracles
COLT '93 Proceedings of the sixth annual conference on Computational learning theory
Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
Toward Efficient Agnostic Learning
Machine Learning - Special issue on computational learning theory, COLT'92
An introduction to computational learning theory
An introduction to computational learning theory
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
Matrix computations (3rd ed.)
Communication complexity
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
A linear lower bound on the unbounded error probabilistic communication complexity
Journal of Computer and System Sciences - Complexity 2001
Communication Complexity Lower Bounds by Polynomials
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Limitations of learning via embeddings in euclidean half spaces
The Journal of Machine Learning Research
Learning intersections and thresholds of halfspaces
Journal of Computer and System Sciences - Special issue on FOCS 2002
Agnostically Learning Halfspaces
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Theoretical Computer Science - Algorithmic learning theory(ALT 2002)
Separating AC0 from depth-2 majority circuits
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On Computation and Communication with Small Bias
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Complexity measures of sign matrices
Combinatorica
SIAM Journal on Computing
Bounded Independence Fools Halfspaces
SIAM Journal on Computing
SIAM Journal on Computing
| Hi-index | 0.00 |
We prove that the concept class of disjunctions cannot be pointwise approximated by linear combinations of any small set of arbitrary real-valued functions. That is, suppose there exist functions φ1,...,φr: {-1,1}n → R with the property that every disjunction f on n variables has ∥f -Σi=1 r αiφi∥∞ ≤ 1/3 for some reals α1,..., αr. We prove that then r ≥ 2Ω(√n). This lower bound is tight. We prove an incomparable lower bound for the concept class of linear-size DNF formulas. For the concept class of majority functions, we obtain a lower bound of Ω(2n/n), which almost meets the trivial upper bound of 2n for any concept class. These lower bounds substantially strengthen and generalize the polynomial approximation lower bounds of Paturi and show that the regression-based agnostic learning algorithm of Kalai et al. is optimal. Our techniques involve a careful application of results in communication complexity due to Razborov and Buhrman et al.