STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Exact learning of DNF formulas using DNF hypotheses
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Journal of Computer and System Sciences - STOC 2001
Exact learning of DNF formulas using DNF hypotheses
Journal of Computer and System Sciences - Special issue on COLT 2002
Solving linear constraints over real and rational fields
Cybernetics and Systems Analysis
SIAM Journal on Computing
Exact learning composed classes with a small number of mistakes
COLT'06 Proceedings of the 19th annual conference on Learning Theory
On PAC learning algorithms for rich boolean function classes
TAMC'06 Proceedings of the Third international conference on Theory and Applications of Models of Computation
Making polynomials robust to noise
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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Distribution-free learnability and unlearnability of polynomial-size Disjunctive Normal Form formulae is discussed. Recently, Bshouty showed that {DNF} is distribution-free learnable in $2^{O(\sqrt n (\log n)^2)}$ time. In this paper we show that Bshouty's learning time is attained by naively searching weak hypotheses among short symmetric functions and appealing to widely-know boosting techniques investigated by Shapire and also by Freund. To obtain this learnability result, we show that a given polynomial-size {DNF} formula can be approximated by a conjunction of length $O(\sqrt n \log n)$ with accuracy at least $2^{-O(\sqrt n \log^2 n)}$. We also obtain a similar lower bound for learning {DNF$ under a cetain joint-distribution. In more precise, for any $0 \le \varepsilon \le 1/2$ and any Boolean conjunction $f$ of length $\Theta(n)$, we construct a joint-distribution over $(x,y) \in \{0,1\}^n \times \{0,1\}$ such that $f(x) = y$ holds with probability at least $1-\varepsilon$ but $h(x) = y$ happens with probability exactly $1/2$ for any Boolean function $h : \{0,1\}^n \to \{0,1\}$ that depends on at most $O(\sqrt{n\varepsilon})$ variables. Therefore, under such a joint-distribution, any naive search must enumerate at least $2^{O(\sqrt{n\varepsilon} \log n)}$ number of symmetric functions for getting a better hypothesis than guessing at random.