Blum, Avrim, Merrick Furst, Je#rey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. 1994. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of the 26th ACM Symposium on the Theory of Computing, pages 253--262.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....consider on line learning algorithms, where learning proceeds in a series of trials . In trial t, an example X t is presented to the learning algorithm A, which makes a prediction 6 This is not surprising since is unlikely that an efficient distribution free DNF learning algorithm exists [2, 1]. In Sections 3 and 4, our results focus on only the current trial, so we omit the subscript t unless it is not clear from context. In this paper we assume # t , # t 1 . X t s label. After this prediction is made, A is told the true label # t , which A uses to update its hypothesis ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of Twenty-sixth ACM Symposium on Theory of Computing, 1994.

....20] uses recent results from [3] on predicting with expert advice, where each possible pruning is an expert. If there is an e#cient way to make predictions, then the expert based algorithm s mistake bound yields an It is unlikely that an e#cient distribution free DNF learning algorithm exists [2, 1]. 85 e#cient algorithm. We take a similar approach, but for simplicity we use the more straightforward WM. To predict nearly as well as the best pruning, we place every possible pruning in a pool (so N = 2 ) and run WM for binary predictions. We start by computing t and W t , which are, ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proceedings of Twenty-sixth ACM Symposium on Theory of Computing, 1994.

....estimates are required within some specified relative error. We show that a class is learnable with relative error statistical queries if and only if it is learnable with (standard) additive error statistical queries. Thus, known learnability and hardness results for statistical query learning [6, 19] also hold in this variant. We demonstrate general bounds on the complexity of relative error SQ learning, and we show that many learning algorithms can naturally be written as highly efficient, relative error SQ algorithms. We further provide simulations of relative error SQ algorithms in both ....

....to be PAC learnable are learnable with additive error statistical queries. By the above theorem, these classes are also learnable with relative error statistical queries. In addition, the hardness results of Kearns [19] for learning parity functions and the general hardness results of Blum et al. [6] based on Fourier analysis also hold for relative error statistical query learning. 4.3 A Natural Example of Relative Error Q Learning In this section we examine a learning problem which has both a simple additive error SQ algorithm and a simple relative error SQ algorithm. We consider the ....

Avrim Blum, Merrick Furst, Jeffery Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of the 26 tn Annual A CM Symposium on the Theory of Computing, May 1994.

....estimates are required within some specified relative error. We show that a class is learnable with relative error statistical queries if and only if it is learnable with (standard) additive error statistical queries. Thus, known learnability and hardness results for statistical query learning [6, 19] also hold in this variant. We demonstrate general bounds on the complexity of relative error SQ learning, and we show that many learning algorithms can naturally be written as highly efficient, relative error SQ algorithms. We further provide simulations of relative error SQ algorithms in both ....

....to be PAC learnable are learnable with additive error statistical queries. By the above theorem, these classes are also learnable with relative error statistical queries. In addition, the hardness results of Kearns [19] for learning parity functions and the general hardness results of Blum et al. [6] based on Fourier analysis also hold for relative error statistical query learning. 4.3 A Natural Example of Relative Error SQ Learning In this section we examine a learning problem which has both a simple additive error SQ algorithm and a simple relative error SQ algorithm. We consider the ....

Avrim Blum, Merrick Furst, Jeffery Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, 1994. To Appear.

....is that there is an efficient algorithm due to Kushilevitz and Mansour [KM93] to find all parities that correlate well with certain Boolean functions assuming that the underlying distribution is uniform. So weakly learning DNF under the uniform distribution is possible by combining these two facts [BFJKMR94]. The third ingredient is a boosting algorithm, developed by Freund [F90] that can turn any weak learning algorithm into a strong learning algorithm. This does not solve the DNF learning problem immediately since the boosting algorithm assumes that the weak learning algorithm works under ....

Avrim Blum, Merrick Furst, Jeffrey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly Learning DNF and Characterizing Statistical Query Learning using Fourier Analysis. In Proceedings of the Twenty Sixth Annual ACM Symposium on Theory of Computing, pages 253--262, 1994.

.... Theta to mean both O and Omega Gamma This asymptotic notation, read soft O, soft Omega, and soft Theta, is convenient for expressing bounds while ignoring lower order factors. Note that it is somewhat different than the standard soft order notation. 4 hardness results of Blum, et al. [4] based on Fourier analysis. The advantages of the new model are then demonstrated by the simulations of relative error SQ algorithms in the noise free PAC model, the malicious error PAC model and the classification noise PAC model. In each case, we determine the complexity of the PAC algorithm as ....

Avrim Blum, Merrick Furst, Jeffery Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of the 26 th Annual ACM Symposium on the Theory of Computing, May 1994.

....in his seminal 1984 paper introducing the PAC learning model [37] more than fteen years later this question is widely regarded as one of the most important open problems in learning theory. While many partial results have been given for restricted versions of the DNF learning problem (see e.g. [8, 9, 21, 23, 24, 26, 27, 33, 38, 39]) the diculty of the unrestricted DNF learning problem is evidenced by the fact that, prior to the current work, only two algorithms were known which improve on the naive 2 n time bound [11, 36] The rst subexponential time algorithm for learning DNF was due to Bshouty [11] who gave an ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis, in \Proc. 26th Ann. Symp. on Theory of Computing" (1994), 253-262.

....DNF with a bounded number of terms. A SAT k DNF is a DNF in which each truth assignment satis es at most k terms. Khardon [22] gave a polynomial time membership query algorithm for learning polynomial size SAT 1 DNF under the uniform distribution; this result was later strengthened by Blum et al. [4] to SAT k DNF for any constant k: Bellare [6] gave a polynomial time membership query algorithm for learning O(log n) term DNF under the uniform distribution. This result was strengthened by Blum and Rudich [7] who gave a polynomial time algorithm for exact learning O(log n) term DNF using ....

....s m = x 1 xm : Then m 2B 2 2 ln 2 implies that Pr h s m m p i : 2. 1 The Learning Model Our learning model is a distribution speci c version of Valiant s Probably Approximately Correct (PAC) model [30] which has been studied by many researchers, e.g. [4, 6, 11 13, 16, 18, 22, 24, 26, 27, 31, 32]. Let C be a class of Boolean functions over f0; 1g n ; let D be a probability distribution over f0; 1g n ; and let f 2 C be an unknown target function. A learning algorithm A for C takes as input an accuracy parameter 0 1 and a con dence parameter 0 1: During its execution the ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis, in \Proc. 26th Ann. Symp. on Theory of Computing" (1994), 253-262.

....20] uses recent results from [3] on predicting with expert advice, where each possible pruning is an expert. If there is an e#cient way to make predictions, then the expert based algorithm s mistake bound yields an 2 It is unlikely that an e#cient distribution free DNF learning algorithm exists [2, 1]. 4 e#cient algorithm. We take a similar approach, but for simplicity we use the more straightforward WM. To predict nearly as well as the best pruning, we place every possible pruning in a pool (so N = 2 n ) and run WM for binary predictions. We start by computing W t and W t , which ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proceedings of Twenty-sixth ACM Symposium on Theory of Computing, 1994.

....c t that the algorithm tries to learn. Since Valiant s seminal work, there were several attempts to relax these assumptions, by introducing models of noise. The first such noise model, called the Random Classification Noise model, was introduced in [2] and was extensively studied, e.g. in [1, 6, 9, 12, 13, 16]. In this model the adversary, before providing each example (x; c t (x) to the learning algorithm tosses a biased coin; whenever the coin shows H , which happens with probability j, the classification of the example is flipped and so the algorithm is provided with the, wrongly classified, ....

A. Blum, M. Furst, J. Jackson, M. J. Kearns, Y. Mansour, and S. Rudich, "Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis", STOC94, 1994.

....DNF with a bounded number of terms. A SAT k DNF is a DNF in which each truth assignment satisfies at most k terms. Khardon [19] gave a polynomial time membership query algorithm for learning polynomial size SAT 1 DNF under the uniform distribution; this result was later strengthened by Blum et al. [3] to SAT k DNF for any constant k: Bellare [5] gave a polynomial time membership query algorithm for learning O(log n) term DNF under the uniform distribution (a somewhat more general result was given by Blum and Rudich [6] Mansour [23] gave a n O(log log n) time membership query algorithm ....

....: Then m 2B 2 ffl 2 ln 2 ffi implies that Pr fi fi fi fi s m m Gamma p fi fi fi fi ffl ffi: 2. 1 The Learning Model Our learning model is a distribution specific version of Valiant s Probably Approximately Correct (PAC) model [26] and has been studied by many researchers, e.g. [3, 5, 8, 9, 10, 13, 15, 19, 21, 22, 23, 27, 28]. Let C be a class of Boolean functions over f0; 1g n ; let D be a probability distribution over f0; 1g n ; and let f 2 C be an unknown target function. A learning algorithm A for C takes as input an accuracy parameter 0 ffl 1 and a confidence parameter 0 ffi 1: During its execution ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis, in "Proc. 26th Ann. Symp. on Theory of Computing" (1994), 253-262.

....in his seminal 1984 paper introducing the PAC learning model [36] more than fifteen years later this question is widely regarded as one of the most important open problems in learning theory. While many partial results have been given for restricted versions of the DNF learning problem (see e.g. [8, 9, 21, 23, 24, 26, 27, 32, 37, 38]) the difficulty of the unrestricted DNF learning problem is evidenced by the fact that, prior to the current work, only two algorithms were known which improve on the naive 2 n time bound [11, 35] The first subexponential time algorithm for learning DNF was due to Bshouty [11] who gave an ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis, in "Proc. 26th Ann. Symp. on Theory of Computing" (1994), 253-262.

....distribution using membership queries in time O(ns 8 = 12 ) where s is the DNF size of the target function f and is the accuracy parameter. The algorithm outputs as its final hypothesis a majority of parity circuit. At the heart of Jackson s Harmonic Sieve algorithm is a procedure WDNF [1] which uses queries to MEM(f) as well as calls to the example oracle EX(f; D) for weakly learning DNF (see Appendix A for a more detailed description of the WDNF algorithm) Jackson proves the following: Lemma 22 [19] For any boolean function f of DNF size s over f0; 1g n and any distribution ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In "26th Annual Symposium on Theory of Computing, " (1994), pp. 253-262.

....is that there is an efficient algorithm due to Kushilevitz and Mansour [KM93] to find all parities that correlate well with certain Boolean functions assuming that the underlying distribution is uniform. So weakly learning DNF under the uniform distribution is possible by combining these two facts [BFJKMR94]. The third ingredient is a boosting algorithm, developed by Freund [F90] that can turn any weak learning algorithm into a strong learning algorithm. This does not solve the DNF learning problem immediately since the boosting algorithm assumes that the weak learning algorithm works under ....

Avrim Blum, Merrick Furst, Jeffrey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly Learning DNF and Characterizing Statistical Query Learning using Fourier Analysis. In Proceedings of the Twenty Sixth Annual ACM Symposium on Theory of Computing, pages 253--262, 1994.

....to learn approximatively. Unfortunately, more negative results have been proved than positive results. The requirement that the learning algorithm must learn under all distributions seems too strong. Several authors have proposed to restrict this model to particuliar distributions ( 3] 8] 11] [4]) or to allow the learner to ask questions ( 2] In many learning situations, teachers provide simpler examples first. Moreover simplicity of examples is defined with respect of the target concept that knows the teacher. Therefore, following the ideas of Li and Vit anyi ( 9] we restrict the ....

....DNF are PACS learnable A k term DNF formula is a Boolean formula consisting of a disjunction of at most k monomials. The class of k term DNF formulas is not PAC learnable unless RP=NP. Recently weak learnability of DNF formulas w.r.t. the uniform distribution using membership queries was proved ([4]) A Poly term DNF formula is a Boolean formula in disjunctive normal form with a polynomial bound on the number of monomials. More precisely, let F n;d be the class of Boolean formulas over n variables in disjunctive normal form with a number of monomials bounded by n d . We consider a ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, S. Rudich, Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis, Proc. th 26th ACM Symposium on Theory of Computing, (1994) 253-262.

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Avrim Blum, Merrick Furst, Jeff Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, 1994.

No context found.

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In STOC 1994.

....is the reason the model was developed) and (b) nearly all machine learning algorithms can be phrased as SQ algorithms. What makes the SQ model especially interesting is that one can information theoretically prove lower bounds on the ability of SQ algorithms to learn certain classes of functions [22, 3, 20, 31, 32]. The relationship between the standard model and the EV model for quantum computation is quite similar to that between the PAC model and the SQ model in machine learning, which motivates our definition of the Statistical Query Sampling problem. In particular, the SQ sampling problem can be ....

....polynomial in n. Furthermore, if a completely random x is picked, the probability it is a positive input is 1=p. Thus even exponentially many queries may only help the sampling by an exponentially small margin. Proof: Our proof strategy is similar to that used by Kearns [22] and Blum et al. [3] in the context of SQ learning. We describe an adversarial SQS oracle g SQS that does not commit to any particular predicate at the beginning. Rather, the oracle maintains a candidate predicate set P , which initially includes all predicates in the class L n;p (a total p of them) Each time ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In STOC 1994.

No context found.

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proceedings of Twenty-sixth ACM Symposium on Theory of Computing, 1994.

....section we apply our techniques to developing noise tolerant versions of two well known membership query learning algorithms. The rst of these is an algorithm originally presented by Goldreich and Levin [9] which we call the Weak Parity, or WP, algorithm (it has also been called the KM algorithm [2,21] by researchers in learning theory because it was rst applied to prove learnability results by Kushilevitz and Mansour [18] WP is essentially an agnostic learning algorithm [14] that nds the parity functions that are best correlated (with respect to the uniform distribution) with a given ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing (1994) 253-262.

....1 if and only if f 0 (z) b 0 and 34 A. BIRKENDORF, E. DICHTERMAN, J. JACKSON, N. KLASNER, AND H. U. SIMON for all i in I , z i = x i . Then clearly E z2D [g I;x;b 0 (z; f 0 (z) pS (I ; x; b) for every I , x, and b. Writing pS this way allows us to apply an observation of Blum et al. [7] to our analysis. They showed that for any f0; 1g valued function f 0 with corresponding f 2 f1; 1g and any function g(z; f 0 (z) E z2D [g(z; f 0 (z) X a g(a0)E z2D [ a (z) X a g(a1)E z2D [f(z) a (z) where a 2 f0; 1g n . Now each of the g I;x;b is a deterministic ....

Blum, A., Furst, M., Jackson, J. C., Kearns, M., Mansour, Y., and Rudich, S. (1994). Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 253-262.

.... a technique originally due to Goldreich and Levin [20] and rst applied within a learning algorithm (KM) by Kushilevitz and Mansour [26] can be used to nd such a parity (using membership queries) Combining these two facts gives that the KM algorithm weakly learns DNF with respect to uniform [9]. An obvious method to consider for turning this weak learner into a strong learner is some form of hypothesis boosting [29, 18, 17, 19] In fact, HS is based on a particularly simple and ecient version of boosting discovered by Freund [18] Each stage i of Freund s boosting algorithm explicitly ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich, Weakly learning DNF and characterizing statistical query learning using Fourier analysis, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, 1994, pp. 253-262.

No context found.

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 253--262, 1994.

....In other words, Alice does have a winning (probabilistic) strategy for the original game above. This improves on two previous DNF learning algorithms in the same uniform withmembership model: Mansour s quasi polynomial time algorithm [43] and a polynomial time weak learner due to Blum et al. [10]. Our algorithm for DNF learning is largely the combination of two powerful tools. One of these is a beautiful technique due to Goldreich and Levin [29] Given a black box that will answer membership queries for a Boolean function f over f0; 1g n , their technique efficiently locates a subset A ....

....in A is a weak approximator for f with respect to uniform, if such an A exists. This technique later formed the basis for a well known learning theoretic result by Kushilevitz and Mansour [37] and has generally been referred to as the KM algorithm in the learning theoretic community (see, e.g. [10]) However, we will refer to the algorithm in this paper as the weak parity algorithm, or WP for short. A useful connection between DNF learning and the WP algorithm was first noted by Blum et al. 10] They showed that for every DNF f , there is a parity function A that weakly approximates f ....

[Article contains additional citation context not shown here]

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich, Weakly learning DNF and characterizing statistical query learning using Fourier analysis, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, 1994, pp. 253--262. 54

....an elusive open problem in computational learning theory. Furthermore, superpolynomial lower bounds for this same problem have been proven for a wide class of algorithms that includes the top down decision tree approach (and also all variants of this approach that have been proposed to date) [2]. The positive results for efficient decision tree learning in computational learning theory all make extensive use of membership queries [11, 5, 4, 9] which provide the learning algorithm with black box access to the target function (experimentation) rather than only an oracle for random ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of the 26th ACM Symposium on the Theory of Computing. ACM Press, New York, NY, 1994.

No context found.

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 253--262, May 1994.

....exist of problems learnable with random classification noise in the PAC model but not learnable by statistical queries. This is especially interesting because one can characterize information theoretically (i.e. without complexity assumptions) what kinds of problems can be learned in the SQ model [4]. For example, the class of parity functions, which can be learned efficiently from non noisy data in the PAC model, provably cannot be learned efficiently in the SQ model under the uniform distribution. Unfortunately, there is also no known efficient non SQ algorithm for learning them in the ....

....unary queries. Thus the seeming generalization of the SQ model to allow for O(log n) wise queries does not close the gap we have demonstrated between what is efficiently learnable in the SQ and noisy PAC models. Note that this result is the best possible with respect to k because the results of [4] imply that for k = log n) there are concept classes learnable from k wise queries but not unary queries. On the other hand, log n) wise queries are in a sense less interesting because it is not clear whether they can in general be simulated in the presence of noise. 1.1 Main ideas The ....

[Article contains additional citation context not shown here]

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 253--262, May 1994.

No context found.

Avrim Blum, Merrick Furst, Jeffrey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 253--262, 1994.

....history. Valiant [18] introduced the problem and gave efficient algorithms for learning certain subclasses of DNF. Since then, learning algorithms have been developed for a number of other subclasses of DNF [13, 4, 3, 11, 2, 1, 7, 16, 9] and recently for the unrestricted class of DNF expressions [6, 12], but almost all of these results and in particular the results for the unrestricted class use membership queries (the learner is told the output value of the target function on learner specified inputs) This has left open the question of to what extent membership queries are necessary for ....

....DNF is learnable without the full power of membership queries. Furthermore, we generalize the notion of classification noise and show that our algorithm learns DNF even if the quantum example oracle exhibits such noise. This is particularly interesting in light of recent results of Blum et al. [6] showing that DNF is not learnable in the statistical query (SQ) model. Because SQ learning is conjectured to be equivalent to the model of PAC learning with classification noise, our result is evidence that quantum learning algorithms may be better able to handle noise than traditional ....

[Article contains additional citation context not shown here]

Blum, A., Furst, M., Jackson, J., Kearns, M., Mansour, Y., and Rudich, S. Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis. in: Proceedings of the 26th Annual ACM Symposium on Theory of Computing. 1994, pp. 253--262.

....algorithmic procedure; given a function and a threshold parameter it finds in polynomial time all the Fourier coefficients of the function that are greater than the threshold. Originally the procedure was used to learn decision trees [5] and later it was used for learning polynomial size DNF [8, 2, 4]. The Fourier transform technique applies naturally to the uniform distribution, and, indeed, most of the learnability results based on the Fourier transform are with respect to the uniform distribution. Some of the results were extended to product distribution [1, 3] A great advantage of the ....

Avrim Blum, Merrick Furst, Jeffrey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In The 26th Annual ACM Symposium on Theory of Computing, pages 253 -- 262, 1994.

.... a technique originally due to Goldreich and Levin [20] and first applied within a learning algorithm (KM) by Kushilevitz and Mansour [26] can be used to find such a parity (using membership queries) Combining these two facts gives that the KM algorithm weakly learns DNF with respect to uniform [9]. An obvious method to consider for turning this weak learner into a strong learner is some form of hypothesis boosting [29, 18, 17, 19] In fact, HS is based on a particularly simple and e#cient version of boosting discovered by Freund [18] Each stage i of Freund s boosting algorithm explicitly ....

<F3.797e+05> A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S.<F3.838e+05> Rudich,<F3.808e+05> Weakly learning DNF and characterizing statistical query learning using Fourier<F3.838e+05> analysis, in Proceedings 26th Annual ACM Symp. on Theory of Computing, Montreal, Canada, 1994, pp. 253--262.

....an elusive open problem in computational learning theory. Furthermore, superpolynomial lower bounds for this same problem have been proven for a wide class of algorithms that includes the top down decision tree approach (and also all variants of this approach that have been proposed to date) [2]. The positive results for efficient decision tree learning in computational learning theory all make extensive use of membership queries [14, 5, 4, 11] which provide the learning algorithm with black box access to the target function (experimentation) rather than only an oracle for random ....

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of the 26th ACM Symposium on the Theory of Computing. ACM Press, New York, NY, 1994.

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Blum, Avrim, Merrick Furst, Je#rey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. 1994. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of the 26th ACM Symposium on the Theory of Computing, pages 253--262.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour and S. Rudich. Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis. In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, pages 253-262, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour and S. Rudich. Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis. In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, pages 253-262, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour and S. Rudich. Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis. In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, pages 253-262, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proceedings of the Twenty-Sixth Annual Symposium on Theory of Computing, pages 253-262, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of the Twenty-Sixth Annual Symposium on Theory of Computing, pages 253--262, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour and S. Rudich. Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis. In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, pages 253-262, 1994.

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Avrim Blum, Merrick Furst, Je#rey Jackson, Michael Kearns, Yishay Mansour, Steven Rudich. Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis. In Proceedings of the 26-th Annual ACM Symposium on the Theory of Computing, pp. 253--262, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proc. 26th Ann. ACM Symp. on the Theory of Computing, pages 253--262, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, pages 253--262, Montr'eal, Qu'ebec, Canada, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In ACM, editor, Proc. 26thACM Symposium on Theory of Computing, pages 253{ 262. ACM Press, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of the Twenty-Sixth Annual Symposium on Theory of Computing, pages 253-262, 1994.

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Avrim Blum, Merrick Furst, Jerey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis. STOC 1994.

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Avrim Blum, Merrick Furst, Jeffrey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly Learning DNF and Characterizing Statistical Query using Fourier Analysis. In Proceedings of the Twenty Sixth Annual ACM Symposium on Theory of Computing, pages 253--262, 1994.

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Avrim Blum, Merrick Furst, Jeffrey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 253--262, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using fourier analysis. In Proc. of Twentysixth ACM Symposium on Theory of Computing, 1994.

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Avrim Blum, Merrick Furst, Jeffrey Jackson, Michael Kearns, Yishay Mansour, and Steven Rudich. Weakly Learning DNF and Characterizing Statistical Query using Fourier Analysis. In Proceedings of the Twenty Sixth Annual ACM Symposium on Theory of Computing, pages 253--262, 1994.

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A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. Proceedings of the 26th ACM Symposium on the Theory of Computing, (1994), 382-389.

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