Auer, P., Long, P., and Srinivasan, A. (1997). Approximating hyper-rectangles: Learning and pseudo-random sets. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing. To appear.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....supervised learning algorithms such as C4.5 decision tree and BP neural network is very poor. Long and Tan [8] described a polynomial time theoretical algorithm and showed that if the instances in the bags are independent drawn from product distribution, then the APR is PAC leamable. Auer et al. [3] showed that if the instances in the bags are not independent then APR learning under the multi instance learning framework is NP hard. Moreover, they presented a theoretical algorithm that does not require product distribution. Later, the theoretical algorithm was transformed to a practical ....

P. Auer, P. M. Long, and A. Srinivasan. "Approximating hyper-rectangles: learning and pseudo-random sets," Journal of Computer and System Sciences, vol. 57, no.3, pp.376-388, 1998.

....the single tuple bias. In particular, Dietterich et al. 8] have designed APR algorithms to solve the task of predicting whether a molecule is musky or not. Other APR algorithms such MULTIINST [3] have been tested on this learning task, and many interesting learnability results have been obtained [5, 2]. More recently, Maron et al. proposed a new multiple instance algorithm called Diverse Density [12] which they applied to image classification. Finally, the lazy learning approach to multiple instance learning has been investigated by Jun et al. 16] The algorithms mentioned here do not ....

P. Auer, P. Long, and Ashwin Srinivasan. Approximating hyper-rectangles: Learning and pseudo-random sets. In Annual ACM Symposium on Theory of Computing, 1997.

....on ffl from our algorithm s decisions, since in practical use, no value for ffl is given as input to the algorithm. Instead, we searched the parameter space for the most appropriate values given the input data. These changes are similar to changes made by Auer [10] to an algorithm by Auer et al. [11] to learn the concept class of axis parallel boxes when multi instance examples are provided (i.e. an example is positive if any of its points lies in the target box) Auer searched the parameter space for the value that caused his algorithm to produce the box with the lowest error on the training ....

....Subsequently, Long and Tan [65] described an efficient PAC algorithm for learning a single axis parallel box in Q d (where Q denotes the set of rationals) from multiple instance examples if each instance is drawn independently from a product distribution and d need not be constant. Auer et al. [11] gave an efficient PAC algorithm for learning a single axis parallel box in d from multiple instance examples if each instance is drawn independently from an arbitrary distribution over d . This algorithm is also polynomial in d. Later, Auer [10] modified that algorithm (making it more ....

P. Auer, P. M. Long, and A. Srinivasan. Approximating hyper-rectangles: Learning and pseudo-random sets. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 314--323. ACM, 1997.

....another. The description of object is often referred to as an instance of object . Recent research has shown that this traditional framework could be too limited for complex learning problems (Zucker and Ganascia 1994; Dietterich, Lathrop et al. 1996; Long and Tan 1996; Zucker and Ganascia 1996; Auer 1997). This is particularly the case when several descriptions of the same object are associated with the same result, baptized a multiple instance problem (MIP) by Dietterich et al. Dietterich, Lathrop et al. 1996) Thus the term multiple instance characterizes the case where the result f(object ) is ....

....the number # i can vary depending on object and that the suffix j of 1 to # i given to instances instance i,j is purely arbitrary. Note that in the limited theoretical research that has been done on the PAClearnability of this problem, the number vi is equal to a constant r (Long and Tan 1996; Auer 1997; Auer, Long et al. 1997; Blum and Kalai 1997) In the multiple instance framework, Dietterich et al. 1997) suggest that if the result of f is positive for an object it is because at least one of its instances ij has produced this result. If the result is negative it means that none of its ....

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Auer, P., Long, P. 1997. Approximating hyper-rectangles: Learning and pseudo-random sets. Proceeding of the 29th Annual ACM Symposium on Theory of Computation.

....if each molecule can adopt 50 di erent conformations the cognitive strain of analyzing the data is great. In machine learning this situation gives rise to the multiple instance problem rst de ned by Dietterich, Lathrop, and Lozano Perez [7] and subsequently addressed by other researchers [3, 2, 25, 26, 10, 39]. Combinatorial chemistry is a major advance of this decade that makes it possible to synthesize a very large number of diverse compounds in a short time. While the details of combinatorial chemistry techniques are beyond the scope of this proposal, one aspect of these techniques is signi cant to ....

P. Auer, P. Long, and A. Srinivasan. Approximating hyper-rectangles: learning and pseudorandom sets. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computation. ACM, 1997.

.... O(log(1 #) were presented in [7] and were improved to poly(m n (1 #) O( # log(1 #) in [19] These approaches introduce certain new ideas and also build on [15, 18] An # approximation for C n m,k with cardinality poly(logn m O(log(1 #) 1 #) log(#k log(1 #)#) is given in [8], by extending an idea of [11] The following reductions are also shown in [8, 19] Suppose that for all (n, m, k, #) an # approximation S # for C n m,k with log S # = O(log log n k log m log(1 #) can be e#ciently constructed. Then, for all (n, m, k, #) the following ....

.... # log(1 #) in [19] These approaches introduce certain new ideas and also build on [15, 18] An # approximation for C n m,k with cardinality poly(logn m O(log(1 #) 1 #) log(#k log(1 #)#) is given in [8] by extending an idea of [11] The following reductions are also shown in [8, 19]. Suppose that for all (n, m, k, #) an # approximation S # for C n m,k with log S # = O(log log n k log m log(1 #) can be e#ciently constructed. Then, for all (n, m, k, #) the following constructions of # approximations S ## for C n m,k are possible, as shown respectively ....

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P. Auer, P. M. Long, and A. Srinivasan. Approximating hyper-rectangles: learning and pseudorandom sets. Journal of Computer and System Sciences, 57:376--388, 1998.

.... the L# norm, because in C finite k,n,d , we allow 2 It is believed that there is no algorithm for learning this class that runs in time poly(d) since this would yield an algorithm for efficiently learning DNF formulas, solving a major open problem in learning theory (Goldman et al. 2000; Auer et al. 1997). the boxes sizes to vary. 2.3 Fuzzy d Dimensional Patterns Goldman and Scott (1999a) then generalized C finite k,n,d . In Section 2.2, we assumed that each target box contained or did not contain a point from a given bag, i.e. containment was measured by a binary function. Now we instead ....

....by the problem of predicting whether a molecule would bind at a particular site. They argued empirically that axis parallel rectangles are good hypotheses for this and other similar learning problems. Subsequently, multi instance learning has been extensively studied (e.g. Long Tan, 1998; Auer et al. 1997; Auer, 1997; Blum Kalai, 1998; Maron LozanoP erez, 1998; Maron, 1998; Maron Ratan, 1998; Blum Kalai, 1998) In several of these papers, the target concept is a single axis parallel box and a bag is classified as positive if at least one of the points in the bag is inside the target box. ....

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Auer, P., Long, P. M., & Srinivasan, A. (1997). Approximating hyper-rectangles: Learning and pseudo-random sets. Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing (pp. 314--323). ACM.

....ffl log log 1 ffl by ffl, we have an ffl generator for R(m; d) using O(log m log d log 1 ffl log log 1 ffl ) bits. We don t know yet how to construct such an explicit ffl generator for R(m; d; k) using O(log k log d log m log 1=ffl) bits. Using an idea of Auer, Long, and Srinivasan [2], we can derive one using O(log k log m log 3=2 1=ffl) bits, which improves their upper bound, but does not serve our purpose here. 5 The Pseudorandom Generator for B(m; d; t) Recall that B(m; d; t) denotes the class of rectangles from R(m; d) with at most t nontrivial dimensions and each ....

P. Auer, P. Long, and A. Srinivasan, Approximating hyper-rectangles: learning and pseudo-random sets, In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 1997.

....2 8, we can see that the number of examples needed by MULTINST are two orders of magnitude greater than the number of examples needed by maxDD. In addition, there are 200 instances per bag in the experiments of Figure 2 8, making it a harder task. The other development was less fortunate: Auer et al. 1996 ] showed that Long and Tan s independence assumption is necessary for any e#ciently PAC learnable algorithm. Specifically, they showed that if there is an algorithm to e#ciently PAClearn from multiple instance examples that are distributed arbitrarily, then it is also possible to e#ciently ....

....find the global Diverse Density maximum over concept space in polynomial time. However, in most cases examined in this thesis, maxDD found the best concept or one which performed well. 7.1. 1 Overview of the MULTINST algorithm We describe a simplified version of the MULTINST algorithm, based on [ Auer et al. 1996 ] We assume there is a distribution D over a d dimensional feature space. Every instance is drawn independently from D. Each contains exactly r instances, and is labeled positive if at least one of the instances falls within the target APR. The target APR is denoted BOX = Y 1#k#d [a i , b i ....

Peter Auer, Phil M. Long, and A. Srinivasan. Approximating hyper-rectangles: learning and pseudorandom sets. In Proceedings of the 1996 Conference on Computational Learning Theory, 1996.

....in a positively labeled bag (the noise ratio) can be arbitrarily high. The multiple instance learning model was only recently formalized by [ Dietterich et al. 1997 ] where they develop algorithms for the drug activity prediction problem. This work was followed by [ Long and Tan, 1996, Auer et al. 1996, Blum and Kalai, 1998 ] who showed that it is difficult to PAC learn in the Multiple Instance model unless very restrictive independence assumptions are made about the way in which examples are generated. Auer, 1997 ] shows that despite these assumptions, the MULTINST algorithm performs ....

Peter Auer, Phil M. Long, and A. Srinivasan. Approximating hyper-rectangles: learning and pseudorandom sets. In Proceedings of the 1996 Conference on Computational Learning Theory, 1996.

....elements is mapped to positive by the target concept. Long and Tan [27] described an efficient PAC algorithm for learning a single axisparallel box in Q d from multiple instance examples under a product distribution where Q denotes the set of rationals and d need not be constant. Auer et al. [4] gave an efficient PAC algorithm for learning a single axis parallel box in d from multiple instance examples if each instance is drawn independently from an arbitrary distribution over d . This algorithm also 2 Note that while we reduce our problem to learning a disjunction of boxes, ....

....Subsequently, Long and Tan [27] described an efficient PAC algorithm for learning a single axis parallel box in Q d (where Q denotes the set of rationals) from multiple instance examples if each instance is drawn independently from a product distribution and d need not be constant. Auer et al. [4] gave an efficient PAC algorithm for learning a single axis parallel box in d from multiple instance examples if each instance is drawn independently from an arbitrary distribution over d . This algorithm is also polynomial in d. Later, Auer [3] modified that algorithm (making it more ....

P. Auer, P. M. Long, and A. Srinivasan. Approximating hyper-rectangles: Learning and pseudo-random sets. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 314--323. ACM, 1997.

.... We also describe a more efficient (and somewhat technically more involved) reduction to the Statistical Query model that results in a polynomial time algorithm for learning axis parallel rectangles with sample complexity O(d 2 r=ffl 2 ) saving roughly a factor of r over the results of Auer et al. 1997). Keywords: Multiple instance examples, classification noise, statistical queries 1. Introduction and Definitions In the standard PAC learning model, a learning algorithm is repeatedly given labeled examples of an unknown target concept, drawn independently from some probability distribution. ....

....domain. Long and Tan (1996) describe an algorithm that learns axis parallel rectangles in the above PAC setting, under the condition that D is a product distribution (i.e. the coordinates of each single instance are chosen independently) with sample complexity O(d 2 r 6 =ffl 10 ) Auer et al. 1997) give an algorithm that does not require D to be a product distribution and has a much improved sample complexity O(d 2 r 2 =ffl 2 ) and running time O(d 3 r 2 =ffl 2 ) The O notation hides logarithmic factors. Auer (1997) reports on the empirical performance of this ....

[Article contains additional citation context not shown here]

Auer, P., Long, P., and Srinivasan, A. (1997). Approximating hyper-rectangles: Learning and pseudo-random sets. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing.

....Figure 3. Furthermore their analysis relies on learning p concepts [KS94] which in this case gives an unrealistic bound of O(1=ffl 10 ) on the number of examples necessary to guarantee learning with accuracy 1 Gamma ffl. Using an improved algorithm it was shown by Auer, Long, and Srinivasan [ALS97], that boxes can be PAClearned using just O(1=ffl 2 ) examples and without assuming that the instances are drawn from a product distribution. The essential ideas of this algorithm are reviewed in Section 2. In this paper we investigate if the theoretical algorithm of [ALS97] can be turned into a ....

....Long, and Srinivasan [ALS97] that boxes can be PAClearned using just O(1=ffl 2 ) examples and without assuming that the instances are drawn from a product distribution. The essential ideas of this algorithm are reviewed in Section 2. In this paper we investigate if the theoretical algorithm of [ALS97] can be turned into a competitive algorithm for realistic datasets. As a benchmark we will use the datasets and algorithms considered in [DLLP97] In order to do so we have to modify the algorithm of [ALS97] in two ways. First, the modified algorithm has to cope with a varying number of instances ....

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P. Auer, P.M. Long, and A. Srinivasan. Approximating hyper-rectangles: learning and pseudorandom sets. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computation, 1997. To appear.

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Auer, P., Long, P., and Srinivasan, A. (1997). Approximating hyper-rectangles: Learning and pseudo-random sets. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing. To appear.

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P. Auer, P.M. Long, and A. Srinivasan. Approximating hyper-rectangles: learning and pseudo-random sets. Journal of Computer and System Sciences, vol.57, no.3, pp.376--388, 1998.

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Auer, P., Long, P.M., Srinivasan, A.: Approximating hyper-rectangles: learning and pseudo-random sets. Journal of Computer and System Sciences 57 (1998) 376--388

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P. Auer, P.M. Long, and A. Srinivasan. Approximating hyper-rectangles: learning and pseudo-random sets. Journal of Computer and System Sciences, vol.57, no.3, pp.376--388, 1998.

No context found.

Auer, P., Long, P.M., Srinivasan, A.: Approximating hyper-rectangles: learning and pseudo-random sets. Journal of Computer and System Sciences 57 (1998) 376--388

No context found.

P. Auer, P. M. Long, and A. Srinivasan. Approximating hyper-rectangles: Learning and pseudo-random sets. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 314--323. ACM, 1997.

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