The Strength of Weak Learnability
Machine Learning
Boosting a weak learning algorithm by majority
COLT '90 Proceedings of the third annual workshop on Computational learning theory
A decision-theoretic generalization of on-line learning and an application to boosting
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
Parallel Optimization: Theory, Algorithms and Applications
Parallel Optimization: Theory, Algorithms and Applications
Boosting and Hard-Core Set Construction
Machine Learning
Hard-core distributions for somewhat hard problems
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Tracking the best linear predictor
The Journal of Machine Learning Research
Smooth boosting and learning with malicious noise
The Journal of Machine Learning Research
List-Decoding Using The XOR Lemma
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
On uniform amplification of hardness in NP
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Key agreement from weak bit agreement
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Approximately List-Decoding Direct Product Codes and Uniform Hardness Amplification
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Uniform direct product theorems: simplified, optimized, and derandomized
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Theory and application of trapdoor functions
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
Pseudorandom generators from one-way functions: a simple construction for any hardness
TCC'06 Proceedings of the Third conference on Theory of Cryptography
General hardness amplification of predicates and puzzles
TCC'11 Proceedings of the 8th conference on Theory of cryptography
Computational randomness from generalized hardcore sets
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Characterizing pseudoentropy and simplifying pseudorandom generator constructions
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
Differential privacy for the analyst via private equilibrium computation
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We give a simple, more efficient and uniform proof of the hard-core lemma, a fundamental result in complexity theory with applications in machine learning and cryptography. Our result follows from the connection between boosting algorithms and hard-core set constructions discovered by Klivans and Servedio [11]. Informally stated, our result is the following: suppose we fix a family of boolean functions. Assume there is an efficient algorithm which for every input length and every smooth distribution (i.e. one that doesn't assign too much weight to any single input) over the inputs produces a circuit such that the circuit computes the boolean function noticeably better than random. Then, there is an efficient algorithm which for every input length produces a circuit that computes the function correctly on almost all inputs. Our algorithm significantly simplifies previous proofs of the uniform and the non-uniform hard-core lemma, while matching or improving the previously best known parameters. The algorithm uses a generalized multiplicative update rule combined with a natural notion of approximate Bregman projection. Bregman projections are widely used in convex optimization and machine learning. We present an algorithm which efficiently approximates the Bregman projection onto the set of high density measures when the Kullback-Leibler divergence is used as a distance function. Our algorithm has a logarithmic runtime over any domain from which we can efficiently sample. High density measures correspond to smooth distributions which arise naturally, for instance, in the context of online learning. Hence, our technique may be of independent interest.