D. Helmbold, R. Sloan, and M. K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.  Home/Search Document Not in Database Summary Related Articles Check |
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Noise Tolerant Algorithms for Learning and Searching - Aslam (1995) (3 citations) (Correct)
....[9] While greatly expanding the function classes known to be learnable in the presence of noise, Kearns technique does not constitute a formal reduction from PAC learning to SQ learning. In fact, such a reduction cannot exist: while the class of parity functions is known to be PAC learnable [17], Kearns has shown that this class is provably unlearnable in the SQ model. Kearns technique for converting PAC algorithms to SQ algorithms consists of a few general rules, but each PAC algorithm must be examined in turn and converted to an SQ algorithm individually. Thus, one cannot derive ....
David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240-266, 1992.
Noise Tolerant Algortihms for Learning and Searching - Aslam (1995) (Correct)
....[9] While greatly expanding the function classes known to be learnable in the presence of noise, Kearns technique does not constitute a formal reduction from PAC learning to SQ learning. In fact, such a reduction cannot exist: while the class of parity functions is known to be PAC learnable [17], Kearns has shown that this class is provably unlearnable in the SQ model. Kearns technique for converting PAC algorithms to SQ algorithms consists of a few general rules, but each PAC algorithm must be examined in turn and converted to an SQ algorithm individually. Thus, one cannot derive ....
David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.
Learning with Restricted Focus of Attention - Ben-David, Dichterman (1997) (Correct)
.... An exception is the algorithm for learning the class of parity function (a parity function is a binary functions which computes the parity of some fixed subset of its input binary variables) This class 11 is known to be efficiently PAC learnable (by solving a system of linear equations modulo 2 [15]) but is (provably) not efficiently learnable from statistical queries [17] it is not known whether this class is noise tolerant learnable. 3 Preliminaries 3.1 Classes of Boolean Functions We define here the classes of Boolean functions whose RFA learnability is studied in this work. The ....
David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning integer lattices. SIAM Journal of Computing, 21(2):240--266, 1992.
When Can Two Unsupervised Learners Achieve PAC Separation? - Goldberg (Correct)
....time and sample complexity of the above rule is polynomial in the total number of variables. 8 Paul W. Goldberg The following result distinguishes our learning setting from learnability with uniform misclassi cation noise, or learnability with a restricted focus of attention. A parity function [15] has an associated subset of the variables, and an associated target parity (even or odd) and evaluates to 1 provided that the parity of the number of true elements of that subset agrees with the target parity, otherwise the function evaluates to 0. Corollary 2. The class of parity functions ....
....parity, otherwise the function evaluates to 0. Corollary 2. The class of parity functions is learnable by unsupervised learners. Proof. Once again it is clear that the class is closed under complementation. To learn a parity function from positive examples only, then similar to the algorithm of [15], each unsupervised learner nds the ane subspace of GF (2) n spanned by its examples, and assigns a score of 1 to elements of that subspace and a score of 0 to all elements of the domain. 3 Examples of Concrete Learning Problems The algorithms in this section give an idea of the new technical ....
D. Helmbold, R. Sloan and M.K. Warmuth (1992). Learning Integer Lattices. SIAM Journal on Computing, 21(2), pp. 240-266.
When Can Two Unsupervised Learners Achieve PAC Separation? - Goldberg (2000) (Correct)
....parity, otherwise the function evaluates to 0. This is the only concept class for which there is an obvious algorithm in this framework, 8 since parity functions have the distinction of being both learnable from positive examples only and from negative examples only. Similar to the algorithm of [11], each unsupervised learner nds the ane subspace of GF (2) n spanned by its examples, and assigns a score of 1 to elements of that subspace and a score of 0 to all elements of the domain. Clearly these ane subspaces do not intersect, and if the two subspaces do not cover the whole domain, then ....
D. Helmbold, R. Sloan and M.K. Warmuth (1992). Learning Integer Lattices. SIAM Journal on Computing, 21(2), pp. 240-266.
On Learning Width Two Branching Programs (Extended Abstract) - Bshouty, al. (Correct)
....that f j g, and vice versa. Proof (Sketch) Follows from the above lemma. 4 Properly PAC Learning SW 2 This section presents the details of the proper PAC learning algorithm for SW 2 . We first note that the class of linear (parity) concepts is properly PAC learnable under any distribution, see [HSW92]) Fact 4.1 The concept class Phi = fa T x b j a 2 f0; 1g n ; b 2 f0; 1gg is properly PAC learnable. Proof (Sketch) Suppose that the target concept is f(x) a T x b. We can change each example y to (y; 1) and learn a vector c of length n 1 that satisfies c T (x; 1) a T x b. ....
David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning Integer Lattices. In SIAM Journal on Computing, 21(2):240--266,1992.
Learning Nested Differences in the Presence of Malicious Noise - Peter Auer (1997) (1 citation) (Correct)
....noise rate of at most j . After defining the learning models we turn to nested differences of intersectionclosed classes. A class C is intersection closed if T C2C 0 C 2 C for any subclass C 0 C, and if ; 2 C. Intersection closed classes can be learned using the Closure Algorithm (ClosAlg) [HLW88, HSW90, Nat91, HSW92], which uses as hypothesis the closure of all positive (counter)examples seen so far. For any concept class C the closure operator CL C : 2 X 2 X is defined as CL C (S) T C2C;S C C. If not stated otherwise we assume from now on that C is an intersection closed concept class and for ....
D. Helmbold, R. Sloan, and M. K. Warmuth. Learning integer lattices. SIAM J. Comput., 21(2):240--266, 1992.
On the Sample Complexity of Weakly Learning - Goldman, Kearns, al. (1992) (Correct)
....n) unapproximable. Specifically, let P n be the class of concepts c on domain f0; 1g n of the form c(x) x i 1 Phi Delta Delta Delta Phi x i k . Thus, each concept is just the parity function computed on a subset of the n variables. It is known that P n is learnable in polynomial time [10, 15]. It is not hard to show that vcd(P n ) n. Also, note that each concept in P n can be represented by a vector in f0; 1g n . Each vector c 2 f0; 1g n is associated with the parity function c defined by c( x) j c Delta x = n M i=1 c i x i : We use this vector representation throughout ....
David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.
Efficient Noise-Tolerant Learning From Statistical Queries - Kearns (1993) (100 citations) (Correct)
.... false, by showing that the class of all parity functions (where each potential target function is the parity of some unknown subset of the boolean variables x1 ; xn ) which is known to be efficiently learnable in the Valiant model via the solution of a system of linear equations modulo 2 [9], is not efficiently learnable from statistical queries. The fact that the separation of the two models comes via this class is of particular interest, since the parity class has no known efficient noise tolerant algorithm. Theorem 5 Let Fn be the class of all parity functions over n boolean ....
David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.
On-line Learning with Malicious Noise and the Closure Algorithm - Auer, Cesa-Bianchi (1994) (2 citations) (Correct)
....sequences. This approach can be compared to the ideas and results contained in [3, 6, 18] where general (nonefficient) conversion strategies to make an on line learning algorithm robust to adversarial noise were proposed. We consider a very general on line strategy known as Closure Algorithm [8, 10, 11, 20] for learning intersection closed concept classes in the noise free model. We extend this strategy to our noisy learning setting and show a worst case mistake bound of m (d 1)K for learning an arbitrary intersection closed concept class C, where K is the number of noisy labels, d is a ....
....of C) and m is the worst case mistake bound of the Closure Algorithm for learning C in the noise free model. For several concept classes our extension is efficient and in some cases it can tolerate a noise rate equal to the information theoretic upper bound for that class. Using the results of [7, 11] we show that the classes of monotone monomials, k CNF functions, parity functions, integer lattices, and k ring sum expansions can be efficiently learned on line with adversarial noise. We also propose a general technique for showing upper bounds on the noise rate tolerated by any on line learner ....
[Article contains additional citation context not shown here]
D.P. Helmbold, R. Sloan, and M.K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.
Learning Fallible Deterministic Finite Automata - Ron, al. (1995) (3 citations) (Correct)
....any queries directly to the expert. Instead, it always queries the corrector. A simple example of an application of the above observation is a learning algorithm (using membership queries) for noisy parity functions. This algorithm is composed of an algorithm for learning parity functions [13] [19] by solving a system of linear equations over the field of integers modulo 2, and a self correcting algorithm [9] 26] for the same family of functions. We do not know of any other self correcting algorithm that has been directly applied to a related learning problem, but the possibility exists ....
D. Helmbold, R. Sloan, and M. K. Warmuth. (1992). Learning integer lattices. SIAM Journal on Computing, 21 , 240--266.
Efficient Learning from Faulty Data - Decatur (1995) (1 citation) (Correct)
....While greatly expanding the function classes known to be learnable in the presence of noise, Kearns technique does not make use of a formal reduction from PAC learning to SQ learning. In fact, such a reduction cannot exist: while the class of parity functions is known to be PAC learnable (Helmbold, Sloan, and Warmuth, 1992), Kearns shows that this class is provably not learnable in the SQ model. Kearns method for converting PAC algorithms to SQ algorithms consists of a few general strategies, but each PAC algorithm must be examined in turn and converted into an SQ algorithm individually. Thus, one cannot derive ....
....noise. This problem is illustrated by the class of parity functions. The class of parity functions has a malicious error tolerant algorithm yielded by the technique of multiple runs (Kearns and Li, 1988) discussed in Section 6 3. 1, on the noise free PAC algorithm for learning parity functions (Helmbold, Sloan, and Warmuth, 1992). Yet it is not known how to learn parity functions even in the presence of classification noise alone. Conversely, such a transformation does exist when starting with an algorithm which tolerates classification noise. In Section 6 3.1, we describe the technique that Kearns and Li (1988) use to ....
Helmbold, David, Robert Sloan, and Manfred K. Warmuth. (1992). Learning integer lattices. SIAM Journal on Computing, 21(2):240--266.
Learning with Limited Visibility - Dichterman (1998) (3 citations) (Correct)
.... An exception is the algorithm for learning the class of all parity function (a parity function is a binary functions which computes the parity of some fixed subset of its input binary variables) This class is known to be efficiently PAC learnable (by solving a system of linear equations modulo 2 [22]) but is (provably) not efficiently learnable from statistical queries [25] it is not known whether this class is noise tolerant learnable. Note that if it is not, then the SQ model is strictly more restrictive than the model of learning with classification noise) 4 Preliminaries 4.1 Classes ....
D. Helmbold, R. Sloan, and M. K. Warmuth. Learning integer lattices. SIAM J. Comput., 21(2):240-- 266, 1992.
On Learning Width Two Branching Programs - Bshouty, Tamon (1996) (3 citations) (Correct)
....f j g, and vice versa. Proof Follows from the proof of the above lemma. 4 Properly PAC Learning SW 2 This section presents the details of the proper PAC learning algorithm for SW 2 . We first note that the class of linear (parity) concepts is properly PAC learnable under any distribution (see [HSW92]) Fact 4.1 The concept class Phi = fa T x b j a 2 f0; 1g n ; b 2 f0; 1gg is properly PAC learnable. Proof (Sketch) Suppose that the target concept is f(x) a T x b. We can change each example y to (y; 1) and learn a vector c of length n 1 that satisfies c T (x; 1) a T x b. ....
David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning Integer Lattices. In SIAM Journal on Computing, 21(2):240--266,1992.
General Bounds on Statistical Query Learning and PAC Learning .. - Aslam, Decatur (1993) (28 citations) (Correct)
....expanding the class of functions known to be learnable in the presence of classification noise, Kearns technique does not constitute a formal reduction from PAC learning to SQ learning. In fact such a reduction cannot exist since, while the class of parity functions is known to be PAC learnable [11], Kearns has shown that this class is provably unlearnable in the SQ model. Kearns technique for converting PAC algorithms to SQ algorithms consists of a few general rules, but each PAC algorithm must be examined in turn and converted to an SQ algorithm individually. Thus, one cannot derive ....
David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.
Characterizing PAC-learnability of Semilinear Sets - Abe (1998) (1 citation) (Correct)
.... unary is also as hard as that for DNF, if one considers the dimension of the class to be a variable ( variable dimension ) The class of concepts for which we show the analogous results is the submodules of Z d , the learning problem of which has been investigated by Helmbold, Sloan and Warmuth [HSW92]. Linear sets and modules are algebraically closely related. A linear set is a finitely generated semigroup under addition with a constant offset, while a module is indeed a finitely generated module under addition and subtraction. The former is a much more complicated class of concepts, however, ....
....as we have just mentioned, and the Vapnik Chervonenkis dimension grows nearly linearly even with the unary encoding. The evaluation problem for modules is efficiently solvable in polynomial time, and the Vapnik Chervonenkis dimension grows logarithmically with respect to the unary encoding [HSW92]. Helmbold, Sloan and Warmuth show that not only is the class of submodules of Z d encoded in binary efficiently learnable, but so is the class of nested differences of members of this class [HSW90] In contrast, we show that the prediction problem for the class of finite unions of modules ....
[Article contains additional citation context not shown here]
D. Helmbold, R. Sloan, and M. K. Warmuth. Learning integer lattices. Siam J. Comput. , 21(2):240 -- 266, April 1992.
PAC Learning from Positive Statistical Queries - Denis (1998) (5 citations) (Correct)
....in time bounded by p(1=ffl; 1=ffi; size(f) n) Remark that if a concept class C is PAC learnable and if there exists a learning algorithm for C which does not use negative examples of the target, then C is PAC learnable from positive examples. Therefore, k CNF ( Val84] and integer lattices ([HSW92]) are learnable from positive examples. A similar approach has been taken in [BDL97] A model of unsupervised learning is defined in which the task of the learner is to identify a probability distribution or more precisely, its high probability density areas, from unlabeled examples. Then, a ....
....UNL( with probability 1 3 and return the result according to the labelling defined above. POSEX PAC: the oracles UNL( and EX(f; f ) can easily be simulated using the oracle EX(f; Remark that the class of parity functions can be learned in PAC model using positive examples uniquely ([HSW92], Kea93] It is proved in [Kea93] that it is not learnable with statistical queries. Therefore, the class of parity functions is in POSEX but not in Q. We can t prove that POSEX (resp. POSQ) is strictly included into PAC (resp. Q) We conjecture that the class composed of complementary sets ....
D. Helmbold, R. Sloan, and M. K. Warmuth. Learning integer lattices. SIAM J. COMPUT., 21(2):240--266, 1992.
Sample compression, learnability, and the Vapnik-Chervonenkis .. - Floyd, Warmuth (1995) (6 citations) Self-citation (Warmuth) (Correct)
....T fc : c 2 C and S cg. A concept class C is intersection closed 2 if whenever S is a finite subset of some concept in C then the closure of S, denoted as CLOS(S) is a concept of C. Clearly axis parallel rectangles in R n are intersection closed and there are many other examples given by Helmbold, Sloan, and Warmuth (1990, 1992) such as monomials, vector spaces in R n and integer lattices. For any set of examples labeled consistently with some concept in the intersection closed class C, consider the closure of the positive examples. This concept, the smallest concept in C containing those positive examples, is ....
Helmbold, D., Sloan, R., and Warmuth, M. (1992). Learning integer lattices. Siam Journal on Computing, Vol.
Pac Learning, Noise, and Geometry - Sloan Self-citation (Sloan) (Correct)
....regions in the plane has VC dimension 4. The class B n on E n has VC dimension 2n. The class of all spheres in E n has dimension n 1. For any finite concept class C, we have V Cdim(C) log(jCj) x There are now a number of papers which calculate the VC dimension of different classes [4, 9, 14, 16]. The VC dimension was first studied in connection with statistical pattern recognition. Pollard and Vapnik have both written good books discussing it from that point of view [22, 28] The first source that I am aware of to point out that it has some connection to efficient concept learning is ....
David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.
Journal of Machine Learning Research 7 (2006) 283--306.. - Paul Goldberg Pwg (Correct)
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D. Helmbold, R. Sloan, and M. K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.
Toward Attribute Efficient Learning of Decision Lists and.. - Klivans, Servedio (2006) (Correct)
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D. Helmbold, R. Sloan, and M. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.
Proceedings of the 26th Annual ACM Symposium on Theory of.. - Extended Michael (Correct)
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David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.
Efficient Noise-Tolerant Learning From Statistical Queries - Kearns (1998) (100 citations) (Correct)
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David Helmbold, Robert Sloan, and Manfred K. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992. 25
Learning Juntas - Mossel, O'Donnell, Servedio (2003) (1 citation) (Correct)
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D. Helmbold, R. Sloan, and M. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266., 1992.
Computational Applications of Noise Sensitivity - O'Donnell (2003) (Correct)
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D. Helmbold, R. Sloan, and M. Warmuth. Learning integer lattices. SIAM Journal on Computing, 21(2):240--266, 1992.
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