A new PAC bound for intersection-closed concept classes
Abstract For hyper-rectangles in $$\mathbb{R}^{d}$$ Auer (1997) proved a PAC bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$, where $$\varepsilon$$ and $$\delta$$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for interse...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Auer, Peter [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2006 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science + Business Media, LLC 2007 |
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| Übergeordnetes Werk: |
Enthalten in: Machine learning - Kluwer Academic Publishers, 1986, 66(2006), 2-3 vom: 08. Mai, Seite 151-163 |
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| Übergeordnetes Werk: |
volume:66 ; year:2006 ; number:2-3 ; day:08 ; month:05 ; pages:151-163 |
| Links: |
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| DOI / URN: |
10.1007/s10994-006-8638-3 |
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| 520 | |a Abstract For hyper-rectangles in $$\mathbb{R}^{d}$$ Auer (1997) proved a PAC bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$, where $$\varepsilon$$ and $$\delta$$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $$d$$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $$O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $$O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$$ and $$O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$. For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $$\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$$ examples to learn some particular maximum intersection-closed concept classes. | ||
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10.1007/s10994-006-8638-3 doi (DE-627)OLC2026521336 (DE-He213)s10994-006-8638-3-p DE-627 ger DE-627 rakwb eng 150 004 VZ Auer, Peter verfasserin aut A new PAC bound for intersection-closed concept classes 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, LLC 2007 Abstract For hyper-rectangles in $$\mathbb{R}^{d}$$ Auer (1997) proved a PAC bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$, where $$\varepsilon$$ and $$\delta$$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $$d$$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $$O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $$O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$$ and $$O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$. For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $$\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$$ examples to learn some particular maximum intersection-closed concept classes. PAC bounds Intersection-closed classes Ortner, Ronald aut Enthalten in Machine learning Kluwer Academic Publishers, 1986 66(2006), 2-3 vom: 08. Mai, Seite 151-163 (DE-627)12920403X (DE-600)54638-0 (DE-576)014457377 0885-6125 nnns volume:66 year:2006 number:2-3 day:08 month:05 pages:151-163 https://doi.org/10.1007/s10994-006-8638-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT GBV_ILN_21 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_70 GBV_ILN_100 GBV_ILN_130 GBV_ILN_2006 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_4012 GBV_ILN_4046 GBV_ILN_4307 GBV_ILN_4318 AR 66 2006 2-3 08 05 151-163 |
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10.1007/s10994-006-8638-3 doi (DE-627)OLC2026521336 (DE-He213)s10994-006-8638-3-p DE-627 ger DE-627 rakwb eng 150 004 VZ Auer, Peter verfasserin aut A new PAC bound for intersection-closed concept classes 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, LLC 2007 Abstract For hyper-rectangles in $$\mathbb{R}^{d}$$ Auer (1997) proved a PAC bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$, where $$\varepsilon$$ and $$\delta$$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $$d$$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $$O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $$O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$$ and $$O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$. For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $$\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$$ examples to learn some particular maximum intersection-closed concept classes. PAC bounds Intersection-closed classes Ortner, Ronald aut Enthalten in Machine learning Kluwer Academic Publishers, 1986 66(2006), 2-3 vom: 08. Mai, Seite 151-163 (DE-627)12920403X (DE-600)54638-0 (DE-576)014457377 0885-6125 nnns volume:66 year:2006 number:2-3 day:08 month:05 pages:151-163 https://doi.org/10.1007/s10994-006-8638-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT GBV_ILN_21 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_70 GBV_ILN_100 GBV_ILN_130 GBV_ILN_2006 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_4012 GBV_ILN_4046 GBV_ILN_4307 GBV_ILN_4318 AR 66 2006 2-3 08 05 151-163 |
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10.1007/s10994-006-8638-3 doi (DE-627)OLC2026521336 (DE-He213)s10994-006-8638-3-p DE-627 ger DE-627 rakwb eng 150 004 VZ Auer, Peter verfasserin aut A new PAC bound for intersection-closed concept classes 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, LLC 2007 Abstract For hyper-rectangles in $$\mathbb{R}^{d}$$ Auer (1997) proved a PAC bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$, where $$\varepsilon$$ and $$\delta$$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $$d$$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $$O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $$O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$$ and $$O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$. For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $$\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$$ examples to learn some particular maximum intersection-closed concept classes. PAC bounds Intersection-closed classes Ortner, Ronald aut Enthalten in Machine learning Kluwer Academic Publishers, 1986 66(2006), 2-3 vom: 08. Mai, Seite 151-163 (DE-627)12920403X (DE-600)54638-0 (DE-576)014457377 0885-6125 nnns volume:66 year:2006 number:2-3 day:08 month:05 pages:151-163 https://doi.org/10.1007/s10994-006-8638-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT GBV_ILN_21 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_70 GBV_ILN_100 GBV_ILN_130 GBV_ILN_2006 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_4012 GBV_ILN_4046 GBV_ILN_4307 GBV_ILN_4318 AR 66 2006 2-3 08 05 151-163 |
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10.1007/s10994-006-8638-3 doi (DE-627)OLC2026521336 (DE-He213)s10994-006-8638-3-p DE-627 ger DE-627 rakwb eng 150 004 VZ Auer, Peter verfasserin aut A new PAC bound for intersection-closed concept classes 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, LLC 2007 Abstract For hyper-rectangles in $$\mathbb{R}^{d}$$ Auer (1997) proved a PAC bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$, where $$\varepsilon$$ and $$\delta$$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $$d$$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $$O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $$O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$$ and $$O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$. For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $$\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$$ examples to learn some particular maximum intersection-closed concept classes. PAC bounds Intersection-closed classes Ortner, Ronald aut Enthalten in Machine learning Kluwer Academic Publishers, 1986 66(2006), 2-3 vom: 08. Mai, Seite 151-163 (DE-627)12920403X (DE-600)54638-0 (DE-576)014457377 0885-6125 nnns volume:66 year:2006 number:2-3 day:08 month:05 pages:151-163 https://doi.org/10.1007/s10994-006-8638-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT GBV_ILN_21 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_70 GBV_ILN_100 GBV_ILN_130 GBV_ILN_2006 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_4012 GBV_ILN_4046 GBV_ILN_4307 GBV_ILN_4318 AR 66 2006 2-3 08 05 151-163 |
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10.1007/s10994-006-8638-3 doi (DE-627)OLC2026521336 (DE-He213)s10994-006-8638-3-p DE-627 ger DE-627 rakwb eng 150 004 VZ Auer, Peter verfasserin aut A new PAC bound for intersection-closed concept classes 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, LLC 2007 Abstract For hyper-rectangles in $$\mathbb{R}^{d}$$ Auer (1997) proved a PAC bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$, where $$\varepsilon$$ and $$\delta$$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $$d$$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $$O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $$O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$$ and $$O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$. For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $$\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$$ examples to learn some particular maximum intersection-closed concept classes. PAC bounds Intersection-closed classes Ortner, Ronald aut Enthalten in Machine learning Kluwer Academic Publishers, 1986 66(2006), 2-3 vom: 08. Mai, Seite 151-163 (DE-627)12920403X (DE-600)54638-0 (DE-576)014457377 0885-6125 nnns volume:66 year:2006 number:2-3 day:08 month:05 pages:151-163 https://doi.org/10.1007/s10994-006-8638-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT GBV_ILN_21 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_70 GBV_ILN_100 GBV_ILN_130 GBV_ILN_2006 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_4012 GBV_ILN_4046 GBV_ILN_4307 GBV_ILN_4318 AR 66 2006 2-3 08 05 151-163 |
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a new pac bound for intersection-closed concept classes |
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A new PAC bound for intersection-closed concept classes |
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Abstract For hyper-rectangles in $$\mathbb{R}^{d}$$ Auer (1997) proved a PAC bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$, where $$\varepsilon$$ and $$\delta$$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $$d$$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $$O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $$O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$$ and $$O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$. For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $$\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$$ examples to learn some particular maximum intersection-closed concept classes. © Springer Science + Business Media, LLC 2007 |
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Abstract For hyper-rectangles in $$\mathbb{R}^{d}$$ Auer (1997) proved a PAC bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$, where $$\varepsilon$$ and $$\delta$$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $$d$$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $$O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $$O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$$ and $$O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$. For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $$\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$$ examples to learn some particular maximum intersection-closed concept classes. © Springer Science + Business Media, LLC 2007 |
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Abstract For hyper-rectangles in $$\mathbb{R}^{d}$$ Auer (1997) proved a PAC bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$, where $$\varepsilon$$ and $$\delta$$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $$d$$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $$O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $$O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$$ and $$O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $$O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$$. For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $$\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$$ examples to learn some particular maximum intersection-closed concept classes. © Springer Science + Business Media, LLC 2007 |
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