Sally A. Goldman, Michael J. Kearns, and Robert E. Schapire. On the sample complexity of weak learning. In Proceedings of the Third Annual Workshop on Computational Learning Theory, pages 217-231. Morgan Kaufmann, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....In the malicious error model, an adversary is allowed, with some fixed probability, to substitute a labelled example of his choosing for the labelled example the learner would ordinarily see. While a limited number of eificient PAC algorithms had been developed which tolerate classification noise [2, 16, 26], no general framework for eflcient learning 1 in the presence of classification noise was known until Kearns introduced the Statistical Query model [19] 1Angluin and Laird [2] introduced a general framework for learning in the presence of classification noise. However, their methods do not ....

Sally A. Goldman, Michael J. Kearns, and Robert E. Schapire. On the sample complexity of weak learning. In Proceedings of the Third Annual Workshop on Computational Learning Theory, pages 217-231. Morgan Kaufmann, 1990.

....In the malicious error model, an adversary is allowed, with some fixed probability, to substitute a labelled example of his choosing for the labelled example the learner would ordinarily see. While a limited number of efficient PAC algorithms had been developed which tolerate classification noise [2, 16, 26], no general framework for efficient learning in the presence of classification noise was known until Kearns introduced the Statistical Query model [19] Angluin and Laird [2] introduced a general framework for learning in the presence of classification noise. However, their methods do not ....

Sally A. Goldman, Michael J. Kearns, and Robert E. Schapire. On the sample complexity of weak learning. In Proceedings of COLT '90, pages 217--231. Morgan Kaufmann, 1990.

.... sense [87, 86] Other useful variants not discussed here are those in which the distribution and target are permitted to change a little between observations, as in [23, 53, 54] models REFERENCES 60 of weak learning in which the learner only has to do slightly better than random guessing [48, 90, 55], and variants in which the learning algorithm has access to the predictions of experts [36] It is hoped that the reader has gained a flavour of this subject. There are many theoretical problems still to be solved within this framework. Furthermore, there is still much work to be done in ....

S. A. Goldman, M. J. Kearns, and R. E. Schapire. On the sample complexity of weak learning. In Proc. 3rd Annu. Workshop on Comput. Learning Theory, pages 217--231. Morgan Kaufmann, San Mateo, CA, 1990.

....[HW92b] requires a large number of consistency oracle queries to make each prediction. In Section 8 we show using our algorithm that sample size 2d Gamma p dlogd) suffices for weak learning (not necessarily polynomial weak learning) of concept classes of VC dimension d. In Goldman et al. [GKS90] it was shown that no algorithm can weakly learn some concept classes of VC dimension d from d Gamma O(log(d) examples. Section 9 compares our prediction algorithm with several of the standard prediction algorithms, as well as the weak prediction algorithm of [HW92b] This comparison includes an ....

....is to determine using our methods the smallest sample size (as a function of the VC dimension of the concept class) that implies weak learning. Since here we are not interested in computational resources we omit the parameterization during this section. Even though the results in Goldman et al. [GKS90] suggest the sample complexity for weak learning is not well correlated with the VC dimension, the following theorem and corollary gives the lowest sample size known to us (2d Gamma O( p d log d) for which there is a general weak learning algorithm. Theorem 8.1: Let d be the VC dimension of ....

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Sally A. Goldman, Michael J. Kerns, and Robert E. Schapire. On the sample complexity of weak learning. In Proceedings of the 1990 Workshop on Computational Learning Theory, pages 217--231, San Mateo, CA, August 1990. Morgan Kaufmann.

....In the malicious error model, an adversary is allowed, with some fixed probability, to substitute a labelled example of his choosing for the labelled example the learner would ordinarily see. While a limited number of efficient PAC algorithms had been developed which tolerate classification noise [2, 11, 16], no general framework for efficient learning 1 in the presence of classification noise was known until Kearns introduced the Statistical Query model [12] In the SQ model, the example oracle of the standard PAC model is replaced by a statistics oracle. An SQ algorithm queries this new oracle ....

....simulations. For additive error, one may consider any interval [a; a M ] and simply translate . 6 A weak learning algorithm is one which outputs an hypothesis whose accuracy is just slightly better than random guessing. ing queries are also be probabilistic. Goldman, Kearns and Schapire [11] show that by allowing a weak learning algorithm to output a probabilistic hypothesis, the complexity of learning is reduced. Therefore this this generalization gives the algorithm designer more freedom and power. Furthermore, the ability to efficiently simulate these algorithms in the PAC model ....

Sally A. Goldman, Michael J. Kearns, and Robert E. Schapire. On the sample complexity of weak learning. In Proceedings of the Third Annual Workshop on Computational Learning Theory, pages 217--231. Morgan Kaufmann, 1990.

....time complexity of learning F is polynomially bounded in all relevant learning parameters. 7 7 Discussion Throughout this paper, we have assumed that the hypothesis classes used by all weak learning algorithms are composed solely of deterministic hypotheses. However, Goldman, Kearns and Schapire [10] have shown that in many cases, algorithms which are allowed to output probabilistic hypotheses are more efficient than algorithms which are required to output deterministic hypotheses. By allowing weak learning algorithms to output probabilistic hypotheses, our boosting algorithm may construct ....

Sally A. Goldman, Michael J. Kearns, and Robert E. Schapire. On the sample complexity of weak learning. In Proceedings of COLT '90, pages 217--231. Morgan Kaufmann, 1990.

.... in a probabilistic sense [97, 96] Other useful variants not discussed here are those in which the distribution and target are permitted to change a little between observations, as in [25, 58, 59] models of weak learning in which the learner only has to do slightly better than random guessing [51, 100, 60], and variants in which the learning algorithm has access to the predictions of experts [38] Something which has not been discussed in any detail here is the use of real output neural networks for classification. Recent work [106, 23] has shown that, here, the scale sensitive fl dimension is ....

S. A. Goldman, M. J. Kearns, and R. E. Schapire. On the sample complexity of weak learning. In Proc. 3rd Annu. Workshop on Comput. Learning Theory, pages 217--231. Morgan Kaufmann, San Mateo, CA, 1990.

....the number of sample points falling in the set 1, 2, 2# 2 N becomes sharply peaked at (2#)#N . The remaining sample points fail to eliminate any of the functions of generalization error # since they all agree with the target function f N on the remaining points. Now it is known (Goldman, Kearns, Schapire, 1990) that in order to eliminate 2 s(#) N parity functions over a uniform distribution, the sample size m must obey m # s(#) N ; P1: rba Machine Learning KL36204(Haus) October 10, 1996 14:3 216 D. HAUSSLER et al. for smaller m, there is a constant probability that at least one parity function ....

Goldman, S.A., Kearns, M.J., & Schapire, R.E. (1990). On the sample complexity of weak learning. In Proceedings of the 3rd Workshop on Computational Learning Theory (pp. 217--231), San Mateo, CA: Morgan Kaufmann.

....of sample points falling in the set f1; 2; 2ffl Delta 2 N g becomes sharply peaked at (2ffl)ffN . The remaining sample points fail to eliminate any of the functions of generalization error ffl since they all agree with the target function fN on the remaining points. Now it is known [11] that in order to eliminate 2 s(ffl) DeltaN parity functions over a uniform distribution, the sample size m must obey m s(ffl) Delta N ; for smaller m, there is a constant probability that at least one parity function remains in the version space. Thus, we obtain that if (2ffl)ffN s(ffl)N ....

S. A. Goldman, M. J. Kearns, and R. E. Schapire. On the sample complexity of weak learning. In Proceedings of the 3rd Workshop on Computational Learning Theory, pages 217--231. Morgan Kaufmann, San Mateo, CA, 1990.

....of sample points falling in the set f1; 2; 2ffl Delta 2 N g becomes sharply peaked at (2ffl)ffN . The remaining sample points fail to eliminate any of the functions of generalization error ffl since they all agree with the target function fN on the remaining points. Now it is known [13] that in order to eliminate 2 s(ffl) DeltaN parity functions over a uniform distribution, the sample size m must obey m s(ffl) Delta N ; for smaller m, there is a constant probability that at least one parity function remains in the version space. Thus, we obtain that if (2ffl)ffN s(ffl)N ....

S. A. Goldman, M. J. Kearns, and R. E. Schapire. On the sample complexity of weak learning. In Proceedings of the 3rd Workshop on Computational Learning Theory, pages 217--231. Morgan Kaufmann, San Mateo, CA, 1990.

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Bibliography 183 Goldman, Sally A., Michael J. Kearns, and Robert E. Schapire. (1990). On the sample complexity of weak learning. In Proceedings of the Third Annual Workshop on Computational Learning Theory, pages 217--231. Morgan Kaufmann.

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