# An Explicit VC-Theorem for Low-Degree Polynomials

@article{Chattopadhyay2012AnEV,
title={An Explicit VC-Theorem for Low-Degree Polynomials},
journal={Electron. Colloquium Comput. Complex.},
year={2012},
volume={19},
pages={127}
}
• Published 15 August 2012
• Mathematics, Computer Science
• Electron. Colloquium Comput. Complex.
Let X ⊆ R n and let $${\cal C}$$ be a class of functions mapping ℝ n → { − 1,1}. The famous VC-Theorem states that a random subset S of X of size $$O(\frac{d}{\epsilon^{2}} \log \frac{d}{\epsilon})$$, where d is the VC-Dimension of $${\cal C}$$, is (with constant probability) an e-approximation for $${\cal C}$$ with respect to the uniform distribution on X. In this work, we revisit the problem of constructing S explicitly. We show that for any X ⊆ ℝ n and any Boolean function class \({\cal C…
3 Citations
Constructing Hard Functions Using Learning Algorithms
• Computer Science, Mathematics
2013 IEEE Conference on Computational Complexity
• 2013
The proofs regarding exact and mistake-bounded learning are simple and self-contained, yield explicit hard functions, and show how to use mistake- bounded learners to "diagonalize"' over families of polynomial-size circuits.
Unconditional Lower Bounds in Complexity Theory
This work investigates the hardness of solving natural computational problems according to different complexity measures, and gives near-optimal lower bounds for pseudorandom functions, error-correcting codes, hardcore predicates, randomness extractors, and small-bias generators.
Constructing Hard Functions from Learning Algorithms
• Computer Science, Mathematics
Electron. Colloquium Comput. Complex.
• 2013
The consequences for PAC learning lead to new proofs of Karp-Lipton-style collapse results, and the lower bounds from SQ learning make use of recent work relating combinatorial discrepancy to the existence of hard-on-average functions.

## References

SHOWING 1-10 OF 33 REFERENCES
On the Density of Families of Sets
• N. Sauer
• Mathematics
J. Comb. Theory, Ser. A
• 1972
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
• Mathematics
STOC '97
• 1997
A pseudo-random generator which produces n instances of a problem for which the analog of the XOR lemma holds is given, and it is shown that if any problem in E = DTIAl E(2°t”j) has circuit complexity 2Q(”), then P = BPP.
A unified framework for approximating and clustering data
• Computer Science
STOC '11
• 2011
A unified framework for constructing coresets and approximate clustering for general sets of functions, and shows how to generalize the results of the framework for squared distances, distances to the qth power, and deterministic constructions.
On linear-time deterministic algorithms for optimization problems in fixed dimension
• Mathematics, Computer Science
SODA '93
• 1993
It is shown that with recently developed derandomization techniques, one can convert Clarkson’s randomized algorithm for linear programming in fixed dimension into a linear-time deterministic algorithm, which works in a fairly general abstract setting, which allows us to solve various other problems.
Rl <= Sc
• N. Nisan
• Computer Science, Mathematics
Comput. Complex.
• 1994
It is shown that any randomized logspace algorithm can be simulated deterministically in polynomial time andO(log2 n) space and this puts RL in SC, “Steve's Class”.
Geometric Discrepancy: An Illustrated Guide
1. Introduction 1.1 Discrepancy for Rectangles and Uniform Distribution 1.2 Geometric Discrepancy in a More General Setting 1.3 Combinatorial Discrepancy 1.4 On Applications and Connections 2.
Pseudorandom generators for space-bounded computations
• N. Nisan
• Computer Science, Mathematics
STOC '90
• 1990
Pseudorandom generators are constructed which convertO(SlogR) truly random bits toR bits that appear random to any algorithm that runs inSPACE(S) to simulated using onlyO(Slogn) random bits.
The discrepancy method - randomness and complexity
This book tells the story of the discrepancy method in a few short independent vignettes. It is a varied tale which includes such topics as communication complexity, pseudo-randomness, rapidly mixing
Combinatorial geometry
• Mathematics
Wiley-Interscience series in discrete mathematics and optimization
• 1995
Algorithms
• Computer Science
• 1992
Most of the articles appearing in this column are oriented toward Common Lisp. However, a wider community of Lisp dialects still exists. One that is of particular interest is GNU Emacs Lisp---the