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Comparing the Power of Probabilistic Learning and Oracle Identification Under Monotonicity Constraints

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Algorithmic Learning Theory (ALT 1998)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1501))

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Abstract

In the setting of learning indexed families, probabilistic learning under monotonicity constraints is more powerful than deterministic learning under monotonicity constraints even if the probability is close to 1 provided the learning machines are restricted to proper or class preserving hypothesis spaces (cf. [19]). In this paper, we investigate the relation between probabilistic learning and oracle identification under monotonicity constraints. In particular, we deal with the question how much “additional information” provided by oracles is necessary in order to compensate the additional power of probabilistic learning.

If the oracle machines have access to K-oracle, then they can compensate the power of monotonic (conservative) probabilistic machines completely, provided the probability p is greater than 2/3 (1/2). Furthermore, we show that for every recursively enumerable oracle A, there exists a learning problem which is strong-monotonically learnable by an oracle machine having access to A, but not conservatively or monotonically learnable with any probability p < 0. A similar result holds for Peano-complete oracles. However, probabilistic learning under monotonicity constraints is ”rich“ enough to encode every recursively enumerable set in a characteristic learning problem, i.e., for every recursively enumerable set A, and every p < 2/3, there exists a learning problem la which is monotonically learnable with probability p, and monotonically learnable with oracle B if and only if A is Turing-reducible to B. The same result holds for conservative probabilistic learning with p < 1/2, and strong-monotonic learning with probability p = 2/3. In particular, it follows that probabilistic learning under monotonicity constraints cannot be characterized in terms of oracle identification. Moreover, we close an open problem that appeared in [19] by showing that the probabilistic hierarchies of class preserving monotonic and conservative probabilistic learning are dense.

Finally, we show that these probability bounds are strict, i.e., in the case of monotonic probabilistic learning with probability p = 2/3, conservative probabilistic learning with probability p = 1/2, and strong-monotonic probabilistic learning with probability p = 1/2, K is not suficient to compensate the power of probabilistic learning under monotonicity constraints.

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Author information

Authors and Affiliations

  1. Institut für Informatik und Gesellschaft, Albert-Ludwigs-UniversitÄt Freiburg, 79098, Freiburg, Germany

    Léa Meyer

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  1. Léa Meyer
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Editor information

Editors and Affiliations

  1. AG Künstliche Intelligenz - Expertensysteme, UniversitÄt Kaiserslautern, Postfach 3049, D-67653, Kaiserslautern, Germany

    Michael M. Richter

  2. Department of Computer Science, University of Maryland, College Park, MD, 20742, USA

    Carl H. Smith

  3. AG Algorithmischesn Lernen, UniversitÄt Kaiserslautern, Postfach 3049, D-67653, Kaiserslautern, Germany

    Rolf Wiehagen

  4. Graduate School of Information Science and Electrical Engineering Department of Informatics, Kyushu University, Kassuga, 816-8580, Japan

    Thomas Zeugmann

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© 1998 Springer-Verlag Berlin Heidelberg

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Meyer, L. (1998). Comparing the Power of Probabilistic Learning and Oracle Identification Under Monotonicity Constraints. In: Richter, M.M., Smith, C.H., Wiehagen, R., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 1998. Lecture Notes in Computer Science(), vol 1501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49730-7_23

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  • DOI: https://doi.org/10.1007/3-540-49730-7_23

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65013-3

  • Online ISBN: 978-3-540-49730-1

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