A Comparison of New and Old Algorithms for a Mixture Estimation Problem
We investigate the problem of estimating the proportion vector which maximizes the likelihood of a given sample for a mixture of given densities. We adapt a framework developed for supervised learning and give simple derivations for many of the standard iterative algorithms like gradient projection...
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| Veröffentlicht in: | Machine learning Jg. 27; H. 1; S. 97 - 119 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Springer Nature B.V
1997
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| ISSN: | 0885-6125, 1573-0565 |
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| Abstract | We investigate the problem of estimating the proportion vector which maximizes the likelihood of a given sample for a mixture of given densities. We adapt a framework developed for supervised learning and give simple derivations for many of the standard iterative algorithms like gradient projection and EM. In this framework, the distance between the new and old proportion vectors is used as a penalty term. The square distance leads to the gradient projection update, and the relative entropy to a new update which we call the exponentiated gradient update (EG^sub ^). Curiously, when a second order Taylor expansion of the relative entropy is used, we arrive at an update EM^sub ^ which, for =1, gives the usual EM update. Experimentally, both the EM^sub ^-update and the EG^sub ^-update for > 1 outperform the EM algorithm and its variants. We also prove a polynomial bound on the rate of convergence of the EG^sub ^ algorithm.[PUBLICATION ABSTRACT] |
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| AbstractList | We investigate the problem of estimating the proportion vector which maximizes the likelihood of a given sample for a mixture of given densities. We adapt a framework developed for supervised learning and give simple derivations for many of the standard iterative algorithms like gradient projection and EM. In this framework, the distance between the new and old proportion vectors is used as a penalty term. The square distance leads to the gradient projection update, and the relative entropy to a new update which we call the exponentiated gradient update (EG sub( eta )). Curiously, when a second order Taylor expansion of the relative entropy is used, we arrive at an update EM sub( eta ) which, for eta identical with 1, gives the usual EM update. Experimentally, both the EM sub( eta )-update and the EG sub( eta )-update for eta > 1 outperform the EM algorithm and its variants. We also prove a polynomial bound on the rate of convergence of the EG sub( eta ) algorithm. We investigate the problem of estimating the proportion vector which maximizes the likelihood of a given sample for a mixture of given densities. We adapt a framework developed for supervised learning and give simple derivations for many of the standard iterative algorithms like gradient projection and EM. In this framework, the distance between the new and old proportion vectors is used as a penalty term. The square distance leads to the gradient projection update, and the relative entropy to a new update which we call the exponentiated gradient update (EG^sub ^). Curiously, when a second order Taylor expansion of the relative entropy is used, we arrive at an update EM^sub ^ which, for =1, gives the usual EM update. Experimentally, both the EM^sub ^-update and the EG^sub ^-update for > 1 outperform the EM algorithm and its variants. We also prove a polynomial bound on the rate of convergence of the EG^sub ^ algorithm.[PUBLICATION ABSTRACT] |
| Author | Warmuth, Manfred K. Schapire, Robert E. Helmbold, David P. Singer, Yoram |
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| Cites_doi | 10.1137/1026034 10.1145/225058.225121 10.1137/0135036 10.1111/j.1467-9965.1991.tb00002.x 10.1016/B978-1-55860-213-7.50029-8 10.1137/0135032 10.1023/A:1022869011914 10.1145/225298.225333 |
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| References | D. Helmbold (130973_CR7) 1996 B. Peters (130973_CR14) 1978; 35 J. Bridle (130973_CR2) 1989 T. Cover (130973_CR3) 1991; 1 X. Meng (130973_CR12) 1992 130973_CR1 130973_CR6 130973_CR5 R. Redner (130973_CR16) 1984; 26 130973_CR8 130973_CR11 130973_CR9 N. Littlestone (130973_CR10) 1988; 2 130973_CR17 A. Dempster (130973_CR4) 1977; B39 B. Peters (130973_CR15) 1978; 35 130973_CR13 |
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| Title | A Comparison of New and Old Algorithms for a Mixture Estimation Problem |
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