Z. Chen, W. Maass, "On-line learning of rectangles and unions of rectangles", to appear in Machine Learning.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Classic with equivalence and membership queries. However, for reasons that will become apparent in Section 8, we would like to avoid asking membership queries involving only changed vertex labels, so we do not present this approach here. As a second alternative, application of clever techniques (Chen Maass, 1994) 3 would probably allow the same binary search on the AT LEAST and AT MOST constraint values to be performed, but without relying on membership queries. We leave the details as an exercise for the truly motivated reader. Instead, for simplicity, we relax the model and argue that a fully ....

Chen, Z. & Maass, W. (1994). On-line learning of rectangles and unions of rectangles. Machine Learning, 17(2/3), 23--50.

.... polygons are learnable in the PAC model with random classification noise [32] and the agnostic PAC model [33] Learning boxes: There have been a number of papers on learning the simpler class of concepts formed from halfspaces whose bounding hyperplanes are parallel to the coordinate axes ([34, 35, 36, 37, 38, 39, 40]) Blumer et al. 9] show that choosing the smallest bounding box covering the set of positive examples in a given sample yields a PAC learning algorithm that is polynomial in both the number of hyperplanes, and the dimension. Chen and Maass [35] present an efficient on line mistake bounded ....

....to the coordinate axes ( 34, 35, 36, 37, 38, 39, 40] Blumer et al. 9] show that choosing the smallest bounding box covering the set of positive examples in a given sample yields a PAC learning algorithm that is polynomial in both the number of hyperplanes, and the dimension. Chen and Maass [35] present an efficient on line mistake bounded algorithm that learns a single box in R d , while Auer [34] investigates the case where there is noise in the mistake bound model. Ameur [40] improves the space complexity of Chen and Maass algorithm at the cost of increased time complexity. Maass ....

Z. Chen and W. Maass. On-line learning of rectangles and unions of rectangles. Machine Learning, 17:201--223, 1994.

.... 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 that they naturally extend DNF formulae (in other words, in the case = 2 any union of t boxes is equivalent to a DNF formula with t terms) That is, a box can be viewed as a conjunction of non boolean properties of ....

.... ) EQs the algorithm reaches Step (6) That is, the running time is poly(n k ; t k ; log k ) which is poly(n; t; log ) for k = O(1) Remark 5.6 Algorithm fast elimination can be used to learn the class of all unions of O(1) boxes with only equivalence queries. This was an open problem in [15] that was later solved in [24] The idea is that if we have a function that can be represented as a union of O(1) boxes, then its negation can be represented as a union of O(1) degenerate boxes. Let f be a union of k boxes, that is (using the notation of the proof of Lemma 3.2) there exists ....

Z. Chen and W. Maass. On-line learning of rectangles and unions of rectangles. Machine Learning, 17(2/3):23--50, 1994.

....have been a number of papers on learning the boxes. Blumer et al. BEHW89] show that choosing the smallest bounding box covering the set of positive examples in a given sample yields a PAC learning algorithm that is polynomial in both the number of hyperplanes, and the dimension. Chen and Maass [CM94] present an efficient on line mistake bounded algorithm that learns a single box in R d , while Auer [Aue93] investigates the case where there is noise in the mistake bound model. Ameur [Ame95] improves the space complexity of Chen and Maass algorithm at the cost of increase time complexity. ....

Zhixiang Chen and Wolfgang Maass. On-line learning of rectangles and unions of rectangles. Machine Learning, 17:201--223, 1994.

....an integer from 1 to m for some constant m. The concept is an interval of the form [1; h] or [h; m] where h 2 f1; Delta Delta Delta Delta; mg. The classification of a point x by an interval [a; b] denoted by [a; b] x) is positive if and only if a x b. The following result of Chen and Maass [CM94] states that HI can be learned both positive and negative robustly. Lemma 1. CM94] There is a learning algorithm that encounters at most log m valid positive (resp. negative) counterexamples and at most 3L log m 1 negative (resp. positive) counterexamples in the learning process. Here, L is the ....

....h] or [h; m] where h 2 f1; Delta Delta Delta Delta; mg. The classification of a point x by an interval [a; b] denoted by [a; b] x) is positive if and only if a x b. The following result of Chen and Maass [CM94] states that HI can be learned both positive and negative robustly. Lemma 1. [CM94] There is a learning algorithm that encounters at most log m valid positive (resp. negative) counterexamples and at most 3L log m 1 negative (resp. positive) counterexamples in the learning process. Here, L is the number of false negative (positive) lies which occur in the learning process. ut ....

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Z. Chen and W. Maass. On-line learning of rectangles and unions of rectangles. Machine Learning, 17:201--223, 1994.

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Z. Chen, W. Maass, "On-line learning of rectangles and unions of rectangles", to appear in Machine Learning.

....in an injury constructions in recursion theory. When one makes a wrong assignment, then one s goal is injured in the sense that one would never achieve his goal unless the assignment is undone. A solution to the credit assignment problem related to rectangle learning given by Chen and Maass [CMb] is reminiscent of the finite injury priority method. The injury constructions in this paper can be viewed as a new method to solve the credit assignment problem by applying priority methods from recursion theory in order to construct concrete algorithms. Previous work on learning unions of ....

....from recursion theory in order to construct concrete algorithms. Previous work on learning unions of rectangles with equivalence queries was able to overcome the related credit assignment problem by employing local search strategies that could tolerate certain types of one sided errors (see [CMb]) Based on this design technique and certain more powerful local search strategies that can tolerate some twosided errors, Chen [C] exhibited an algorithm for learning unions of two rectangles over the domain [0; n Gamma 1] 2 with O(log 2 n) equivalence queries and using unions of two ....

Z. Chen, W. Maass, "On-line learning of rectangles and unions of rectangles", Machine Learning, 17, pages 201-223, 1994.

.... class 1 CNF in Valiant s seminal paper [V] Another O(2 d log n) learning algorithm was exhibited in [MTa] and [MTb] The question whether there is a learning algorithm for single rectangles whose complexity is O(poly(d; log n) was proposed by David Haussler (see also [MTb] Chen and Maass [CMa, CMb] gave a positive solution to this open question by introducing a new design technique. The learning algorithm in [CMa, CMb] for rectangles consists of 2d separate search strategies which search for the 2d boundaries of the target rectangle. A learning algorithm with this type of modular design ....

....The question whether there is a learning algorithm for single rectangles whose complexity is O(poly(d; log n) was proposed by David Haussler (see also [MTb] Chen and Maass [CMa, CMb] gave a positive solution to this open question by introducing a new design technique. The learning algorithm in [CMa, CMb] for rectangles consists of 2d separate search strategies which search for the 2d boundaries of the target rectangle. A learning algorithm with this type of modular design tends to fail because of the well known credit assignment problem : which of the 2d local search strategies should be ....

Z. Chen, W. Maass, "On-line learning of rectangles and unions of rectangles", to appear in Machine Learning.

....faced the problem of labeling the samples. Techniques such as uncertainty sampling[45] reduce the amount of labeled data needed, but do not eliminate the labeling problem. Clustering techniques do not require labeled inputs, and have been applied successfully to large collections of documents[15]. Indeed, the Web offers fertile ground for document clustering research. However, because clustering techniques take weaker (unlabeled) inputs than other data mining techniques, they produce weaker (unlabeled) output. From the machine learning point of view, one needs a sufficiently large ....

Chen, Z. and Maass, W., "On-line learning of rectangles and unions of rectangles," Machine Learning, Special Issue of the Fifth ACM Annual Conference on Computational Learning Theory, 17, pages 201-223, 1994.

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