N.H. Bshouty, S.A. Goldman, H.D. Mathias, S.Suri and H. Tamaki (1998). NoiseTolerant Distribution-Free Learning of General Geometric Concepts. Journal of the ACM 45(5), pp. 863-890.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the case analysis (on the di erent ways that a partition element may intersect the region with the opposite class label) may need to be extended. If so it should generalize to unions of boxes 3 in xed dimension (as studied in [6] in the setting of query learning, a generalization is studied in [5] in PAC learning) Finally, if boxes are PAC learnable with two unsupervised learners in time polynomial in the dimension, then this would imply learnability of monomials, considered previously. 3.4 Linear Separators in the Plane Given a set S of points in the plane, it would be valid for an ....

N.H. Bshouty, S.A. Goldman, H.D. Mathias, S.Suri and H. Tamaki (1998). NoiseTolerant Distribution-Free Learning of General Geometric Concepts. Journal of the ACM 45(5), pp. 863-890.

....the case analysis (on the di erent ways that a partition element may intersect the region with the opposite class label) may need to be extended. If so it should generalize to unions of boxes 3 in xed dimension (as studied in [4] in the setting of query learning, a generalization is studied in [3] in PAC learning) Finally, if boxes are PAC learnable with two unsupervised learners in time polynomial in the dimension, then this would imply learnability of monomials, considered previously. 3 A box means the intersection of a set of halfspaces whose bounding hyperplanes are axis aligned, ....

N.H. Bshouty, S.A. Goldman, H.D. Mathias, S.Suri and H. Tamaki (1998). NoiseTolerant Distribution-Free Learning of General Geometric Concepts. Journal of the ACM 45(5), pp. 863-890.

....intersecting) n dimensional boxes over the reals in time polynomial in n. Keywords: Computational learning, PAC learning, Membership Queries, Boxes, Geometric objects. 1 Introduction The learnability (under various learning models) of geometric concept classes was studied in many papers (e.g. [7, 8, 9, 5]) A particular attention was given to the case of continuous domains of points (i.e. n ) and concept classes which are de ned as boxes and unions of boxes in this domain (e.g. 7, 12, 15, 11, 2] An axes parallel box is a basic geometric object; it can also be thought of as a conjunction ....

....the domain. Without this property our transformation would not work. We do not know if there is a generic transformation of any algorithm that learns unions of boxes over nite domains into an algorithm that works over in nite domains. There are many results on learning geometric objects (e.g. [9, 5, 7, 15, 16, 14, 4]) These results di er in the learning model employed, the number of dimensions (i.e. constant dimension versus high dimension) the type of domain (i.e. nite discrete domain versus in nite continuous domain) the number of geometric objects, the complexity of the objects (e.g. halfspaces, ....

N. H. Bshouty, S. A. Goldman, H. D. Mathias, S. Suri, and H. Tamaki. Noise-tolerant distribution-free learning of general geometric concepts. In Proc. of 28th Symp. on the Theory of Computing, pages 151-160, 1996.

....output from the noise process is all that the learner can observe. In the subsequent sections we describe the actual outputs that the learner obtains from various noisy oracles: Random Misclassi cation Noise This model was rst introduced by Angluin and Laird [6] and was extensively studied (e.g. [9, 25, 36, 78]) The examples are generated by the noisy oracle in the following way: with probability 1 the noisy oracle returns hx; c t (x)i from EX (c t ; D) with probability the noisy oracle returns D x; c t (x) E from EX (c t ; D) Essentially this is a white noise, a ecting only the label ....

N. H. Bshouty, S. A. Goldman, H. D. Mathias, S. Suri, and H. Tamaki. Noise-tolerant distribution-free learning of general geomtric concepts. In In the proceedings of the 28'th Annual ACM Symposium of Theory of Computing, 1996.

....shown to be polynomial. 1.3 Related Work on Linear Thresholds and Noise tolerant Learning The domain R d (for constant d) is a widely considered domain in the learning theory literature. Examples of learning problems over this domain include PAC learning of boolean combinations of halfspaces [15], query based learning of unions of boxes [16] and unions of halfspaces [9, 4, 13] A technique of [9] generalized by [15] involves generating a set of functions that realise all linear partitions of a sample of input vectors. If m is the sample size then the set of partitions has size O(m d ) ....

....d) is a widely considered domain in the learning theory literature. Examples of learning problems over this domain include PAC learning of boolean combinations of halfspaces [15] query based learning of unions of boxes [16] and unions of halfspaces [9, 4, 13] A technique of [9] generalized by [15] involves generating a set of functions that realise all linear partitions of a sample of input vectors. If m is the sample size then the set of partitions has size O(m d ) Our algorithm uses this technique, which requires d to be constant. Extending the above learning results to general (non ....

N.H. Bshouty, S.A. Goldman, H.D. Mathias, S.Suri and H. Tamaki (1998). NoiseTolerant Distribution-Free Learning of General Geometric Concepts. Journal of the ACM 45(5), pp. 863-890.

....the class of depth two linear threshold circuits with constant fan in at the input gates, and as a consequence, obtain a noise tolerant mistake bound algorithm for learning (the class of circuits containing) the class of convex polyhedra in fixed dimension. In the continuous domain, Bshouty et al. [26] present noise tolerant PAC algorithm for learning the class of arbitrary boolean functions of s halfspaces in constant dimension, See also [27] for earlier work) For the discrete domain, Ben David et al. 28] give an exact learning algorithm for the same class using equivalence queries only. ....

N. Bshouty, S. Goldman, D. Mathias, S. Suri, and H. Tamaki. Noise-tolerant distribution-free learning of general geometric concepts. In Proc. 27th Annual ACM Symposium on Theory of Computing, pages 151--160. ACM Press, New York, NY, 1996.

....Phone: 972 4 8294303. Fax: 972 4 8221128. This research was supported by Technion V.P.R. Fund 120 872 and by Japan Technion Society Research Fund. 1 Introduction The learnability (under various learning models) of geometric concept classes was studied in many papers (e.g. [8, 11, 13, 5]) A particular attention was given to the case of discrete domains of points (i.e. f0; Gamma 1g n ) and concept classes which are defined as unions of boxes in this domain (e.g. 22, 23, 15, 2, 16, 17, 24] One of the reasons that unions of boxes seem to be interesting concepts is ....

....to the case of DNF formulae) attempts to learn sub classes of this class. There are two main directions: 1) Sub classes in which the number of dimensions, n, is limited to O(1) In this case unions of boxes (and more general geometric concept classes) are known to be learnable in the PAC model [13] and even in the weaker online model [5] 2) Sub classes in which the number of boxes in the union is limited to O(1) but the number of dimensions in not restricted) Again, this sub class is learnable in the PAC model [21] and in the on line model [24] In this work we generalize some of the ....

N. H. Bshouty, S. A. Goldman, H. D. Mathias, S. Suri, and H. Tamaki. Noisetolerant distribution-free learning of general geometric concepts. In Proc. of the 28th Annu. ACM Symp. on the Theory of Computing, pages 151--160, 1996.

....[L88] to the class of depth two neural network with constant fan in at the input gates, and as a consequence, obtain a noise tolerant mistake bounded algorithm for learning (the class of circuits containing) the class of convex polyhedra in fixed dimension. In the continuous domain, Bshouty et al. [BGMST95] give a noise tolerant PAC algorithm for learning arbitrary boolean functions of s halfspaces in constant dimension. See also [BGM95] for earlier work. Kwek and Pitt [KP95] describes an Occam algorithm that learns intersection of k half spaces in [0; 1] n with membership queries in time ....

N. Bshouty, S. Goldman, D. Mathias, S. Suri, and H. Tamaki. Noise-Tolerant Distribution Free Learning of General Geometric Concepts Unpublished manuscript.

....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, ....

N. H. Bshouty, S. A. Goldman, H. D. Mathias, S. Suri, and H. Tamaki, "Noise-Tolerant Distribution-Free Learning of General Geometric Concepts", STOC96, pp. 151--160, 1996.

....same class by again drawing a sufficiently large sample and constructing a hypothesis consistent with the sample that consists of at most s(2d) s boxes. Both the time and sample complexity of their algorithm depend polynomially on (2d) s ; 1 ffl , and log 1 ffi . Recently, Bshouty et al. [BGMST95] present a noise tolerant PAC algorithm to learn any geometric concept defined by a boolean combination of s halfspaces for d constant, and Kwek and Pitt [KP95] give a algorithm to learn in the PACmemb model the intersection of s halfspaces in d dimensions that has time and sample complexity ....

N. Bshouty, S. Goldman, D. Mathias, S. Suri, and H. Tamaki. Noise-tolerant distribution-free learning of general geometric concepts. Manuscript in preparation.

....depends on m. Similarly, we can efficiently learn the class of constant degree semi algebraic functions (that is, the functions which are defined using a collection of constant degree algebraic surfaces) in a fixed dimension. Results of this strength were known before only in the PAC model [BGMST96]. Organization: In section 2 we provide some preliminary definition and facts. In Section 3 we present the composition theorem. Finally, in Section 4 we present space bounded ( simple ) algorithms for several concept classes. In Appendix A we prove that every class C with VC dim(C) 1 is ....

N. H. Bshouty, S. A. Goldman, H. D. Mathias, S. Suri, and H. Tamaki. "Noise-Tolerant Distribution-Free Learning of General Geometric Concepts" Proc. of the 28th Annu. ACM Symp. on Theory of Computing, pages 151--160, 1996.

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N. H. Bshouty, S. A. Goldman, H. D. Mathias, S. Suri, and H. Tamaki. Noisetolerant distribution-free learning of general geomtric concepts. In In the proceedings of the 28th Annual ACM Symposium of Theory of Computing, 1996.

No context found.

Bshouty, Goldman, Mathias, Suri, and Tamaki. Noise-tolerant distribution-free learning of general geometric concepts. 1996.

No context found.

Bshouty, Goldman, Mathias, Suri, and Tamaki. Noise-tolerant distribution-free learning of general geometric concepts. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing. 1996.

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Bshouty, Goldman, Mathias, Suri, and Tamaki. Noise-tolerant distribution-free learning of general geometric concepts. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing. 1996.

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