T. Hancock and Y. Mansour. Learning Monotone k DNF Formulas on Product Distributions. In Pr4oceedings of the 4th Annual Workshop on Computational Learning Theory, pages 179-183, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... =1 i a i =0 (1 Gamma i ) The distribution D is called constant bounded if there exists a constant c 2 (0; 1) independent of n such that for all i we have i 2 [c; 1 Gamma c] The standard deviation of x i is defined as oe i = i (1 Gamma i ) The influence of variable x i on f (see [KKL88, HM91]) over a product distribution D is defined as the probability that f(x) differs from f(x Phi e i ) when x is chosen according to D. Here x Phi e i means x with its i th bit flipped. We will use the notation I D;i (f) to denote the above probability. Often we will use the restriction notation of ....

Thomas Hancock and Yishay Mansour. Learning Monotone k- DNF Formulas on Product Distributions. In Proceedings of the Fourth Annual Workshop on Computational Learning Theory, pages 179--183, 1991.

....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 ....

T. Hancock and Y. Mansour. Learning monotone k- DNF formulas on product distributions, in \Proc. 4th Ann. Workshop on Comp. Learning Theory" (1991), 179-183.

....these results to the class of functions representable as decision trees. Furst, Jackson, and Smith [13] have elaborated and extended these results in a number of directions. We learn the lesson that the spectral characteristics of f need to be understood and controlled. Hancock and Mansour [17] have similarly shown that monotone k DNF formulae (that is, monotone DNF formulae in which each variable appears at most k times) are learnable from examples drawn randomly according to the uniform distribution. Although monotone formula are not really useful in this shift register application, ....

Thomas Hancock and Yishay Mansour. Learning monotone k DNF formulas on product distributions. In Proceedings of the Fourth Annual Workshop on Computational Learning Theory, pages 179--183, Santa Cruz, California, August 1991.

....learning monotone DNF is equivalent to learning general DNF [20] However this equivalence does not hold for restricted distributions such as the uniform distribution, and many researchers have studied the problem of learning monotone DNF under restricted distributions. Hancock and Mansour [16] gave a polynomial time algorithm for learning monotone read k DNF under constant bounded product distributions. Verbeurgt [32] gave a polynomial time uniform distribution algorithm for learning poly disjoint oneread once monotone DNF and read once factorable monotone DNF. Kucera et al. 24] gave ....

.... as real valued functions which map f0; 1g n to f1; 1g: A Boolean function f : f0; 1g n f1; 1g is monotone if changing the value of an input bit from 0 to 1 never causes the value of f to change from 1 to 1: If D is a distribution and f is a Boolean function on f0; 1g n ; then as in [12, 16] we say that the in uence of x i on f with respect to D is the probability that f(x) di ers from f(y) where y is x with the i th bit ipped and x is drawn from D: For ease of notation let f i;0 denote the function obtained from f by xing x i to 0 and let f i;1 be de ned similarly. We thus ....

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T. Hancock and Y. Mansour. Learning monotone k- DNF formulas on product distributions, in \Proc. 4th Ann. Workshop on Comp. Learning Theory" (1991), 179-183.

....method for testing the equivalence of a read k monotone CNF formula C and an arbitrary monotone DNF D. The algorithm runs in time polynomial in the sum of the sizes of C and D. Before giving the main theorem of this section, we need the following definition and results adapted from the ones in [20]. Definition 1. A function f : f0; 1g n f0; 1g depends on the variable x i if there exists an assignment y 2 f0; 1g n such that f x i 0 (y) 6= f x i 1 (y) Definition 2. Let f be a monotone function over variable set fx 1 ; xn g. We say that A fx 1 ; xn g Gamma fx i g ....

T. Hancock and Y. Mansour. Learning monotone k-DNF formulas on product distributions. Proc. 4th Annu. Workshop on Comput. Learning Theory, pages 179--183, San Mateo, CA, 1991. Morgan Kaufmann.

....in time n O(log n) and Linial et al. 22] gave an algorithm for learning any AC 0 circuit (constant depth, polynomial size, unbounded fanin AND OR gates) under the uniform distribution in n poly(log n) time. A monotone DNF is a DNF with no negated variables. Hancock and Mansour [13] gave a polynomial time algorithm for learning monotone read k DNF under constant bounded product distributions. Verbeurgt [28] gave a polynomial time uniform distribution algorithm for learning poly disjoint one read once monotone DNF and read once factorable monotone DNF. Kucera et 1 al. 21] ....

.... functions which map f0; 1g n to f Gamma1; 1g: A Boolean function f : f0; 1g n f Gamma1; 1g is monotone if changing the value of an input bit from 0 to 1 never causes the value of f to change from 1 to Gamma1: If D is a distribution and f is a Boolean function on f0; 1g n ; then as in [9, 13] we say that the influence of x i on f with respect to D is the probability that f(x) differs from f(y) where y is x with the i th bit flipped and x is drawn from D: For ease of notation let f i;0 denote the function obtained from f by fixing x i to 0 and let f i;1 be defined similarly. We thus ....

[Article contains additional citation context not shown here]

T. Hancock and Y. Mansour. Learning monotone k- DNF formulas on product distributions, in "Proc. 4th Ann. Workshop on Comp. Learning Theory" (1991), 179-183.

....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 ....

T. Hancock and Y. Mansour. Learning monotone k- DNF formulas on product distributions, in "Proc. 4th Ann. Workshop on Comp. Learning Theory" (1991), 179-183.

.... i Q a i =0 (1 Gamma i ) The distribution D is called constant bounded if there exists a constant c 2 (0; 1) independent of n such that for all i we have i 2 [c; 1 Gamma c] The standard deviation of x i is defined as oe i = q i (1 Gamma i ) The influence of variable x i on f (see [KKL88, HM91]) over a product distribution D is defined as the probability that f(x) differs from f(x Phi e i ) when x is chosen according to D. Here x Phi e i means x with its i th bit flipped. We will use the notation I D;i (f) to denote the above probability. Often we will use the restriction notation of ....

Thomas Hancock and Yishay Mansour. Learning Monotone k- DNF Formulas on Product Distributions. In Proceedings of the Fourth Annual Workshop on Computational Learning Theory, pages 179--183, 1991.

....q bounded distributions, the ratio of the measure of any two points in the sample space is bounded by a quantity q possibly different from 1. See [15, 30] for proper learnability results for k term DNF formula in this setting. Weaker results (for the purpose of our paper) are obtained in [19] for product distributions. In this case, homogeneity comes from 5 the assumption that the coordinates of the sample space points are stochastically independent. However, it should be remarked that learning in this context is not proper. All distributions mentioned so far are p smooth, in the ....

....constant k. b) q bounded distribution [6] For a given q 1, a distribution P on Xn is said to be q bounded if, for any v 1 , v 2 Xn, P(v 1 ) qP(v 2 ) A q bounded distribution is a p smooth distribution. In particular P is but(k,q2 k ) for q2 k = poly(n) and a = 0. c) product distribution [19] A set Xn is said to have a product distribution P if (Xn,f,P) is described by a random vector B = B 1 , B n ) where B i , i=1, n, are independent Bernoullian variables of parameters r i , where 0 r r i 1 r 1. A product distribution is a p smooth distribution. In particular P is ....

[Article contains additional citation context not shown here]

HANCOCK T., MANSOUR Y. Learning monotone kµ-DNF formulas on product distributions. Proc. of the 4th Workshop on Comput. Learning Th., pp. 179-183, Morgan Kaufmann, San Mateo, CA, 1991.

.... to Furst, Jackson and Smith [5] several efficient algorithms for learning restricted forms of DNF with respect to the uniform distribution in the Valiant model [12] and efficient algorithms for learning unbounded depth readonce circuits with respect to product distributions in the Valiant model [21, 7]. For all of these classes we can obtain efficient algorithms for learning with noise by Theorem 3; in this list, only for conjunctions [1] and Schapire s work on read once circuits [21] were there previous noise analyses. As further evidence for the generality of the statistical query model and ....

Thomas Hancock and Yishay Mansour. Learning monotone k¯ DNF formulas on product distributions. In Proceedings of the Fourth Annual Workshop on Computational Learning Theory, pages 179--183, August 1991.

....that used by Kearns et al. 15] for learning the class of read once formulas in disjunctive normal form (DNF) against the uniform distribution. A similar result, though based on a different method, was obtained by Pagallo and Haussler [18] These results were extended by Hancock and Mansour [10], and by Schapire [21] as described below. Also, Linial, Mansour and Nisan [17] used a technique based on Fourier spectra to learn the class of constant depth circuits (constructed from gates of unbounded fan in) against the uniform distribution. Furst, Jackson and Smith [7] generalized this ....

Thomas Hancock and Yishay Mansour. Learning monotone k¯ DNF formulas on product distributions. In Proceedings of the Fourth Annual Workshop on Computational Learning Theory, pages 179--183, August 1991.

....that used by Kearns et al. 12] for learning the class of read once formulas in disjunctive normal form (DNF) against the uniform distribution. A similar result, though based on a different method, was obtained by Pagallo and Haussler [15] These results were extended by Hancock and Mansour [6], and by Schapire [18] as described below. Also, Linial, Mansour and Nisan [14] used a technique based on Fourier spectra to learn the class of constant depth formulas (constructed from gates of unbounded fan in) against the uniform distribution. Furst, Jackson and Smith [5] generalized this ....

Thomas Hancock and Yishay Mansour. Learning monotone k¯ DNF formulas on product distributions. In Computation Learning Theory: Proceedings of the Fourth Annual Workshop, August 1991.

....method for testing the equivalence of a read k monotone CNF formula C and an arbitrary monotone DNF D. The algorithm runs in time polynomial in the sum of the sizes of C and D. Before giving the main theorem of this section, we need the following definition and results adapted from the ones in [HM91] Definition 1 A function f : f0; 1g n f0; 1g depends on the variable x i if there exists an assignment y 2 f0; 1g n such that f x i 0 (y) 6= f x i 1 (y) Definition 2 Let f be a monotone function over variable set fx 1 ; x n g. We say that A fx 1 ; x n g Gamma fx i g ....

T. Hancock and Y. Mansour. Learning monotone k¯-DNF formulas on product distributions. In Proc. 4th Annu. Workshop on Comput. Learning Theory, pages 179--183, San Mateo, CA, 1991. Morgan Kaufmann.

....time algorithm for learning k CNF, in the PAC model, and showed how to learn polynomial size monotone DNF formulas using queries. There has been some success in devising algorithms for learning DNF with various restriction on the number of times a variable can appear in the DNF formula (see [KLPV87, Han91, HM91, AP91]) Unfortunately none of the results seem to extend to the general case. Negative results have been shown for learning DNF in the PAC model. In [PV88] it was shown that deciding if a given set of examples can be described as a two term DNF is NPComplete. In [AK91] it was shown that, under some ....

Thomas Hancock and Yishay Mansour. Learning monotone k¯ DNF formulas on product distributions. In COLT, pages 179--183, August 1991.

....in the case where the constraints are the expectation of individual and pairs of attributes. For such a given maximum entropy distribution we develop an efficient learning algorithm for read once DNF. We also show how to extend our results to monotone read k DNF, following the techniques of [HM91] 1 Introduction The PAC learning model [Val84] is the most basic model in computational learning theory. Its introduction brought forward a simple set of assumptions and raised many challenging problems. Initially, the main goal was a computational one, to develop new algorithms within this ....

....expectations. Our algorithm receives the parameters of the maximumentropy distribution, and based on them and random examples from the target concept, it finds a read once DNF that approximates well the target read once DNF. We extend the rusults to monotone read k DNF. A previous result by [HM91] describes an algorithm that efficiently approximates monotone read k DNF over product distributions. We follow thier techniques and prove that a simmilar algorithm works for Maximum Entropy distributions as well. Our algorithm requires only statistical queries, for this reason we use the ....

[Article contains additional citation context not shown here]

Thomas Hancock and Yishay Mansour. Learning Monotone k¯ DNF Formulas on Product Distributions. In Proceedings of COLT '91, pages ?--?. Morgan Kaufmann, 1991.

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T. Hancock and Y. Mansour. Learning Monotone k DNF Formulas on Product Distributions. In Pr4oceedings of the 4th Annual Workshop on Computational Learning Theory, pages 179-183, 1991.

No context found.

Thomas Hancock and Yishay Mansour. Learning monotone k DNF formulas on product distributions. In Proceedings of the Fourth Annual Workshop on Computational Learning Theory, pages 179--183, August 1991.

No context found.

T. Hancock and Y. Mansour. Learning monotone k- DNF formulas on product distributions. In Proceedings of the Fourth Annual Conference on Computational Learning Theory, pages 179-193, 1991.

No context found.

T. Hancock and Y. Mansour. Learning monotone k- DNF formulas on product distributions. In Proceedings of the Fourth Annual Conference on Computational Learning Theory, pages 179--193, 1991.

No context found.

Thomas Hancock and Yishay Mansour. Learning monotone k DNF formulas on product distributions. In Proceedings of the Fourth Annual Workshop on Computational Learning Theory, pages 179-183, Santa Cruz, California, August 1991.

No context found.

T. Hancock and Y. Mansour. Learning monotone k- DNF formulas on product distributions. In Proc. 4th Ann. Workshop on Comp. Learning Theory, pages 179--193, 1991.

No context found.

T. Hancock and Y. Mansour. Learning monotone k- DNF formulas on product distributions. In Proceedings of the Fourth Annual Conference on Computational Learning Theory, pages 179-193, 1991.

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