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New Perspectives on the Complexity of Computational Learning, and Other Problems in Theoretical Computer Science
, 2009
"... In this thesis we present the following results. • Learning theory, and in particular PAC learning, was introduced by Valiant [CACM 1984] and has since become a major area of research in theoretical and applied computer science. One natural question that was posed at the very inception of the field ..."
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In this thesis we present the following results. • Learning theory, and in particular PAC learning, was introduced by Valiant [CACM 1984] and has since become a major area of research in theoretical and applied computer science. One natural question that was posed at the very inception of the field is whether there are classes of functions that are hard to learn. PAC learning is hard under widely held conjectures such as the existence of oneway functions, and on the other hand it is known that if PAC learning is hard then P ̸ = NP. We further study sufficient and necessary conditions for PAC learning to be hard, and we prove that: 1. ZK ̸ = BPP implies that PAC learning is hard. 2. It is unlikely using standard techniques that one can prove that PAC learning is hard implies that ZK ̸ = BPP. 3. It is unlikely using standard techniques that one can prove that P ̸ = NP implies that ZK ̸ = BPP.
• Mikhail Belkin (Ohio State University)
"... Copyright © 2010 These materials are owned by their respective copyright owners. All rights reserved. ISBN number: 9780982252925 ..."
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Copyright © 2010 These materials are owned by their respective copyright owners. All rights reserved. ISBN number: 9780982252925
Learning to create is as hard as learning to appreciate
, 2010
"... We explore the relationship between a natural notion of “learning to create ” (LTC) studied by Kearns et al. (STOC ’94) and the standard PAC model of Valiant (CACM ’84), which can be thought of as a formalization of “learning to appreciate”. Our main theorem states that “if learning to appreciate is ..."
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We explore the relationship between a natural notion of “learning to create ” (LTC) studied by Kearns et al. (STOC ’94) and the standard PAC model of Valiant (CACM ’84), which can be thought of as a formalization of “learning to appreciate”. Our main theorem states that “if learning to appreciate is hard, then so is learning to create”. More formally, we prove that if there exists a concept class for which PAC learning with respect to efficiently samplable input distributions is hard, then there exists another (possibly richer) concept class for which the LTC problem is hard. We also show that our theorem is tight in two senses, by proving that there exist concrete concept classes for which PAC learning is hard but LTC is easy, and by showing that it is unlikely our main theorem can be improved to the case of PAC learning with respect to unsamplable input distributions. 1