R. E. SCHAPIRE AND L. M. SELLIE, Learning sparse multivariate polynomials over a field with queries and counterexamples, J. Comput. Systems Sci., 52 (1996), pp. 201--213.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....via the Fourier Spectrum. Jackson in [J94] extended the result to learning DNF under the uniform distribution. The output hypothesis is a majority of parities. Bshouty gave in [Bs94] a technique for learning decision trees under any distribution via the Monotone Theory. Schapire and Sellie gave in [SS93] a Lattice based algorithm for learning multivariate polynomials over the binary field under any distribution. In the former the output hypothesis for the decision tree is depth 3 formulas. Jackson [J95] generalizes his DNF learning algorithm from uniform distribution to any fixed constant bounded ....

....PAC learnable with membership queries under any distribution that supports small terms. The output hypotheses of the learning algorithm is a multivariate polynomial. It is known that multivariate polynomials (with monotone terms) are PAC learnable with membership queries under any distribution [SS93]. Our contribution is to show the same when the terms are not nesasary monotone. It is also known that any DNF is PAC learnable with membership queries under constant bounded product distribution [J94] In [J94] the output hypothesis is a majority of parities. Our contribution for j disjoint DNF ....

R. E. Schapire, L. M. Sellie. Learning sparse multivariate polynomial over a field with queries and counterexamples. In Proceedings of the Sixth Annual ACM Workshop on Computational Learning Theory. July, 1993.

....the value of the function at a. In [3] Bshouty, Hancock and Hellerstein gave a polynomial time algorithm for interpolating arithmetic read once formula. Other arithmetic classes that can be interpolated in polynomial time are the classes of sparse polynomials and sparse rational functions. [1,2,4,5,7,8,10]. This work is a nontrivial generalization of the algorithm of Bshouty, Hancock and Hellerstein for polynomial time interpolating arithmetic read once formulas over the basis of addition, subtraction, multiplication and division. An arithmetic read once formula with exponentiation is a read once ....

R. E. Schapire and L. M. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. In Proceedings of the Sixth Annual Workshop on Computational Learning Theory, pages 17-26, 1993.

....some goal node. Angluin [A88] gave the first lattice based algorithm for learning monotone DNF. Bshouty [Bs93] developed the monotone theory, which gives a technique for learning decision trees under any distribution. The output hypothesis in that case is depth 3 formulas. Schapire and Sellie [SS93] gave a lattice based algorithm for learning multivariate polynomials over a finite field under any distribution. Their algorithm depends polynomially on the size of the monotone polynomial that describes the function. Multiplicity Automata theory is a well studied field in Automata theory. ....

R. E. Schapire, L. M. Sellie. Learning sparse multivariate polynomial over a field with queries and counterexamples. In Proceedings of the Sixth Annual ACM Workshop on Computational Learning Theory. July, 1993.

....constant bounded if there is a constant 0 c 1=2, that is independent of the number of variables n, such that for any variable x i , c P rob[x i = 1] 1 Gamma c. Bshouty [Bs93] gave a technique for learning decision trees under any distribution via the Monotone Theory. Schapire and Sellie [SS93] gave a Lattice based algorithm for learning multivariate polynomials over the binary field under any distribution. In the former the output hypothesis for the decision tree is depth 3 formulas. Both techniques, the Fourier Spectrum and the Lattice based algorithms give also learnability of many ....

....of Xor of them are PAC learnable with membership queries under any distribution that supports small terms. The output hypotheses of the learning algorithm is a multivariate polynomial. Learning multivariate polynomials (with monotone terms) with membership and equivalence queries is shown in [SS93], thus multivariate polynomials are PAC learnable under any distribution. Our contribution is to show the learnability when the terms are not monotone. It is also known that any DNF is PAC learnable with membership queries under constant bounded product distribution [J95] where the output ....

R. E. Schapire, L. M. Sellie. Learning sparse multivariate polynomial over a field with queries and counterexamples. In Proceedings of the Sixth Annual ACM Workshop on Computational Learning Theory. July, 1993.

....Since learning polynomials over small fields with membership queries only is not possible, it raises the question whether with the help of equivalence queries this becomes possible. Before the current work, it was known how to learn the class of multi linear polynomials and polynomials over GF(2) [52]. We further show the learnability of the class of XOR of terms, which is an open problem in [52] the class of polynomials over finite fields, which is an open problem in [52, 19] and the class of bounded degree polynomials over infinite fields (as well as other classes of functions over finite ....

....the question whether with the help of equivalence queries this becomes possible. Before the current work, it was known how to learn the class of multi linear polynomials and polynomials over GF(2) 52] We further show the learnability of the class of XOR of terms, which is an open problem in [52], the class of polynomials over finite fields, which is an open problem in [52, 19] and the class of bounded degree polynomials over infinite fields (as well as other classes of functions over finite and infinite fields) Techniques: We use an algebraic approach for learning multiplicity ....

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R. E. Schapire and L. M. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. J. of Computer and System Sciences, 52(2):201--213, 1996.

....of [40] with a better time and query complexity, and prove the learnability of various concept classes. These include: ffl The class of disjoint DNF, and more generally satisfy O(1) DNF (improving over results of [18, 2, 16] ffl The class of polynomials over finite fields (open problem in [44, 19]) ffl The class of bounded degree polynomials over infinite fields. This paper contains results from 3 conference papers. It is mostly based on the FOCS96 paper by the authors. The rest of the results are from the STOC96 paper of Bergadano, Catalano and Varricchio, and the EuroCOLT97 paper of ....

....32000, Israel. E mail: eyalk cs.technion.ac.il. http: www.cs.technion.ac.il eyalk. This research was supported by Technion V.P.R. Fund 120 872 and by Japan Technion Society Research Fund. k Universit a di L Aquila. E mail: varricch univaq.it. ffl The class of XOR of terms (open problem in [44]) ffl Certain classes of boxes in high dimensions, including a certain class of decisiontrees (which is an open problem in [18] In addition, we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees [18] and polynomials over GF(2) ....

[Article contains additional citation context not shown here]

R. E. Schapire and L. M. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. In Proc. of 6th Annu. ACM Workshop on Comput. Learning Theory, pages 17--26, 1993.

....the additive transform is applied to input variables can be used to decrease the complexity of the interpolation polynomial while speeding up the interpolation algorithm. 1 Introduction Multivariate interpolation over finite fields has applications in symbolic computing [14] learning theories [7], and logic synthesis [12] In the area of logic synthesis, finite field polynomial representations are known as Reed Muller forms, or Ring Sum Expressions (RSE) Boolean and multiple valued functions are often partial, or incompletely specified. This fact can be used to decrease the cost of ....

R. E. Schapire and L. M. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. Symposium on Computational Learning Theory - COLT '93, pages 17--26, May 1993.

....returns the value of the function at a. There are a number of classes of arithmetic formulas that can be interpolated sequentially in polynomial time as well as in parallel in polylogarithmic time (with polynomially many processors) These include sparse polynomials and sparse rational functions ([BT88, BT90, GKS90b, GrKS88, RB89, GKS90b, SS93, M91]) A formula over a variable set V is read once if each variable appears at most once in it. An arithmetic read once formula over a field K is a read once formula over the basic operations of the field K; addition, subtraction, multiplication, division, and constants are also permitted in the ....

R. E. SCHAPIRE AND L. M. SELLIE, Learning sparse multivariate polynomials over a field with queries and counterexamples, J. Comput. Systems Sci., 52 (1996), pp. 201--213.

....via the Fourier Spectrum. Jackson in [J94] extended the result to learning DNF under the uniform distribution. The output hypothesis is a majority of parities. Bshouty gave in [Bs94] a technique for learning decision trees under any distribution via the Monotone Theory. Schapire and Sellie gave in [SS93] a Lattice based algorithm for learning multivariate polynomials over the binary field under any distribution. In the former the output hypothesis for the decision tree is depth 3 formulas. Jackson [J95] generalizes his DNF learning algorithm from uniform distribution to any fixed constant bounded ....

....are PAC learnable with membership queries under any distribution that supports small terms. The output hypotheses of the learning algorithm is a multivariate polynomial. It is known that multivariate polynomials (with monotone terms) are PAC learnable with membership queries under any distribution [SS93]. Our contribution is to show the same when the terms are not nesasary monotone. It is also known that any DNF is PAC learnable with membership queries under constant bounded product distribution [J94] In [J94] the output hypothesis is a majority of parities. Our contribution for j disjoint ....

R. E. Schapire, L. M. Sellie. Learning sparse multivariate polynomial over a field with queries and counterexamples. In Proceedings of the Sixth Annual ACM Workshop on Computational Learning Theory. July, 1993.

....http: www.cs.technion.ac.il eyalk. Supported by Technion V.P.R. Fund 120 872, by Japan Technion Society Research Fund, by the fund for the promotion of research at the Technion, and by the German Israeli Foundation for scientific research and development (GIF) Ang87b, AFP92, RS93, Bsh93, BR95, SS96, BCV96, BBB 96, Bsh97] and many others) In some of the above, and in several related papers (e.g. Ang88, GM92, AHK93, Bsh93, AS94, BCGS95, Bsh97, AKST97] the following common approach is used: the input space, f0; 1g n , is viewed as a lattice with the natural partial order (i.e. for ....

R. E. Schapire and L. M. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. J. of Computer and System Sciences, 52(2):201--213, 1996.

....the spirit of [24] with a better query complexity. As a result, we improve the complexity for all classes that use the algorithms of [8, 24] and also obtain the best query complexity for several classes known to be learnable by other methods such as decision trees [13] and polynomials over GF(2) [26]. ffl We prove the learnability of some new classes that were not known to be learnable before. Most notably, the class of polynomials over finite fields (open problem in [26, 14] the class of bounded degree polynomials over infinite fields, the class of XOR of terms (open problem in [26] and ....

....for several classes known to be learnable by other methods such as decision trees [13] and polynomials over GF(2) 26] ffl We prove the learnability of some new classes that were not known to be learnable before. Most notably, the class of polynomials over finite fields (open problem in [26, 14]) the class of bounded degree polynomials over infinite fields, the class of XOR of terms (open problem in [26] and a certain class of decision trees (open problem in [13] ffl While multiplicity automata were shown to be useful to prove the learnability of some subclasses of DNF ....

[Article contains additional citation context not shown here]

R. E. Schapire and L. M. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. In Proc. of 6th Annu. ACM Workshop on Comput. Learning Theory, pages 17--26, 1993.

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R. E. SCHAPIRE AND L. M. SELLIE, Learning sparse multivariate polynomials over a field with queries and counterexamples, J. Comput. Systems Sci., 52 (1996), pp. 201--213.

No context found.

R. E. Schapire and L. M. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. In Proceedings of the Sixth Annual Workshop on Computational Learning Theory, pages 17-26, 1993.

No context found.

R. Schapire and L. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. In Proceedings of the Sixth Annual ACM Workshop on Computational Learning Theory, pages 17--26. ACM Press, 1993.

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Robert Schapire and Linda Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. In Proceedings of the Sixth Annual ACM Workshop on Computational Learning Theory, pages 17--26. ACM Press, 1993.

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R. E. Schapire and L. M. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. In Proceedings of the Sixth Annual Workshop on Computational Learning Theory, pages 17-26, 1993.

No context found.

Robert E. Schapire and Linda M. Sellie, Learning sparse multivariate polynomials over a field with queries and counterexamples, Proceeding of the Sixth ACM Workshop on Computational Learning Theory, 1993.

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Robert Schapire and Linda Selle, Learning Sparse Multivariate Polynomials over a Field with Queries and Counterexamples. In Journal of Computer and System Sciences, 52, 201-213, 1996.

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Robert E. Schapire and Linda M. Sellie, Learning sparse multivariate polynomials over a field with queries and counterexamples, Proceeding of the Sixth ACM Workshop on Computational Learning Theory, 1993.

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R. E. Schapire and L. M. Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. In Proceedings of the Sixth Annual Workshop on Computational Learning Theory, pages 17-26, 1993.

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