Small-bias probability spaces: efficient constructions and applications
SIAM Journal on Computing
Decision theoretic generalizations of the PAC model for neural net and other learning applications
Information and Computation
Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
Pseudorandomness for network algorithms
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Toward Efficient Agnostic Learning
Machine Learning - Special issue on computational learning theory, COLT'92
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
Derandomizing Approximation Algorithms Based on Semidefinite Programming
SIAM Journal on Computing
Information Processing Letters
Some optimal inapproximability results
Journal of the ACM (JACM)
Approximate counting by dynamic programming
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Journal of Computer and System Sciences - STOC 2002
Learning intersections and thresholds of halfspaces
Journal of Computer and System Sciences - Special issue on FOCS 2002
Hardness amplification within NP
Journal of Computer and System Sciences - Special issue on computational complexity 2002
Noise stability of functions with low in.uences invariance and optimality
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Balanced max 2-sat might not be the hardest
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Agnostically Learning Halfspaces
SIAM Journal on Computing
Optimal algorithms and inapproximability results for every CSP?
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Some topics in analysis of boolean functions
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Learning Geometric Concepts via Gaussian Surface Area
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Gaussian Bounds for Noise Correlation of Functions and Tight Analysis of Long Codes
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Conditional hardness for satisfiable 3-CSPs
Proceedings of the forty-first annual ACM symposium on Theory of computing
Conditional Hardness for Approximate Coloring
SIAM Journal on Computing
Agnostic Learning of Monomials by Halfspaces Is Hard
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Pseudorandom generators for polynomial threshold functions
Proceedings of the forty-second ACM symposium on Theory of computing
Bounding the average sensitivity and noise sensitivity of polynomial threshold functions
Proceedings of the forty-second ACM symposium on Theory of computing
An invariance principle for polytopes
Proceedings of the forty-second ACM symposium on Theory of computing
Fooling Functions of Halfspaces under Product Distributions
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Lower Bounds for Testing Function Isomorphism
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Inapproximability of hypergraph vertex cover and applications to scheduling problems
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Towards Sharp Inapproximability for Any 2-CSP
SIAM Journal on Computing
Bounded Independence Fools Degree-2 Threshold Functions
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
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Let X be randomly chosen from {-1,1}n, and let Y be randomly chosen from the standard spherical Gaussian on ℝn. For any (possibly unbounded) polytope P formed by the intersection of k halfspaces, we prove that |Pr[X ∈ P] - Pr[Y ∈ P]| ≤ log8/5k ⋅ Δ, where Δ is a parameter that is small for polytopes formed by the intersection of “regular” halfspaces (i.e., halfspaces with low influence). The novelty of our invariance principle is the polylogarithmic dependence on k. Previously, only bounds that were at least linear in k were known. The proof of the invariance principle is based on a generalization of the Lindeberg method for proving central limit theorems and could be of use elsewhere. We give two important applications of our invariance principle, one from learning theory and the other from pseudorandomness. (1) A bound of logO(1)k ⋅ ε1/6 on the Boolean noise sensitivity of intersections of k “regular” halfspaces (previous work gave bounds linear in k). This gives a corresponding agnostic learning algorithm for intersections of regular halfspaces. (2) A pseudorandom generator (PRG) for estimating the Gaussian volume of polytopes with k faces within error δ and seed-length O(log n poly(log k,1/δ)). We also obtain PRGs with similar parameters that fool polytopes formed by intersection of regular halfspaces over the hypercube. Using our PRG constructions, we obtain the first deterministic quasi-polynomial time algorithms for approximately counting the number of solutions to a broad class of integer programs, including dense covering problems and contingency tables.