International Journal of Game Theory
Learnability with respect to fixed distributions
Theoretical Computer Science
Active Learning Using Arbitrary Binary Valued Queries
Machine Learning
On the Size of Weights for Threshold Gates
SIAM Journal on Discrete Mathematics
Toward Efficient Agnostic Learning
Machine Learning - Special issue on computational learning theory, COLT'92
Vector analysis of threshold functions
Information and Computation
Simulating Threshold Circuits by Majority Circuits
SIAM Journal on Computing
On Restricted-Focus-of-Attention Learnability of Boolean Functions
Machine Learning - Special issue on the ninth annual conference on computational theory (COLT '96)
Learning with restricted focus of attention
Journal of Computer and System Sciences
Chow Parameters in Threshold Logic
Journal of the ACM (JACM)
On Small Depth Threshold Circuits
SWAT '92 Proceedings of the Third Scandinavian Workshop on Algorithm Theory
On Small Depth Threshold Circuits
SWAT '92 Proceedings of the Third Scandinavian Workshop on Algorithm Theory
A Note on the Simulation of Exponential Threshold Weights
COCOON '96 Proceedings of the Second Annual International Conference on Computing and Combinatorics
Noise-tolerant learning, the parity problem, and the statistical query model
Journal of the ACM (JACM)
Agnostically Learning Halfspaces
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
SIAM Journal on Discrete Mathematics
Every Linear Threshold Function has a Low-Weight Approximator
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Hardness of Learning Halfspaces with Noise
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
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
Threshold Gate Approximations Based on Chow Parameters
IEEE Transactions on Computers
A new algorithm for minimizing convex functions over convex sets
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Bounding the average sensitivity and noise sensitivity of polynomial threshold functions
Proceedings of the forty-second ACM symposium on Theory of computing
Approximating linear threshold predicates
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Bounded Independence Fools Halfspaces
SIAM Journal on Computing
Approximating Linear Threshold Predicates
ACM Transactions on Computation Theory (TOCT)
An efficient heuristic to identify threshold logic functions
ACM Journal on Emerging Technologies in Computing Systems (JETC)
The inverse shapley value problem
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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In the 2nd Annual FOCS (1961), C. K. Chow proved that every Boolean threshold function is uniquely determined by its degree-0 and degree-1 Fourier coefficients. These numbers became known as the Chow Parameters. Providing an algorithmic version of Chow's theorem --- i.e., efficiently constructing a representation of a threshold function given its Chow Parameters --- has remained open ever since. This problem has received significant study in the fields of circuit complexity, game theory and the design of voting systems, and learning theory. In this paper we effectively solve the problem, giving a randomized PTAS with the following behavior: Theorem: Given the Chow Parameters of a Boolean threshold function f over n bits and any constant ε 0, the algorithm runs in time O(n2 log2 n) and with high probability outputs a representation of a threshold function f' which is ε-close to f. Along the way we prove several new results of independent interest about Boolean threshold functions. In addition to various structural results, these include the following new algorithmic results in learning theory (where threshold functions are usually called "halfspaces"): An ~O(n2)-time uniform distribution algorithm for learning halfspaces to constant accuracy in the "Restricted Focus of Attention" (RFA) model of Ben-David et al. [3]. This answers the main open question of [6]. An O(n2)-time agnostic-type learning algorithm for halfspaces under the uniform distribution. This contrasts with recent results of Guruswami and Raghavendra [21] who show that the learning problem we solve is NP-hard under general distributions. As a special case of the latter result we obtain the fastest known algorithm for learning halfspaces to constant accuracy in the uniform distribution PAC learning model. For constant ε our algorithm runs in time ~O(n2), which substantially improves on previous bounds and nearly matches the Ω(n2) bits of training data that any successful learning algorithm must use.