Occam learning
Machine learning and data mining |
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In computational learning theory, Occam learning is a model of algorithmic learning where the objective of the learner is to output a succinct representation of received training data. This is closely related to probably approximately correct (PAC) learning, where the learner is evaluated on its predictive power of a test set.
Occam learnability implies PAC learning, and for a wide variety of concept classes, the converse is also true: PAC learnability implies Occam learnability.
Contents
Introduction
Occam Learning is named after Occam's razor, which is a principle stating that, given all other things being equal, a shorter explanation for observed data should be favored over a lengthier explanation. The theory of Occam learning is a formal and mathematical justification for this principle. It was first shown by Blumer, et al.^{[1]} that Occam learning implies PAC learning, which is the standard model of learning in computational learning theory. In other words, parsimony (of the output hypothesis) implies predictive power.
Definition of Occam learning
The succinctness of a concept in concept class can be expressed by the length of the shortest bit string that can represent in . Occam learning connects the succinctness of a learning algorithm's output to its predictive power on unseen data.
Let and be concept classes containing target concepts and hypotheses respectively. Then, for constants and , a learning algorithm is an -Occam algorithm for using iff, given a set of samples labeled according to a concept , outputs a hypothesis such that
- is consistent with on (that is, ), and
- ^{[2]}^{[1]}
where is the maximum length of any sample . An Occam algorithm is called efficient if it runs in time polynomial in , , and We say a concept class is Occam learnable with respect to a hypothesis class if there exists an efficient Occam algorithm for using
The relation between Occam and PAC learning
Occam learnability implies PAC learnability, as the following theorem of Blumer, et al.^{[2]} shows:
Theorem (Occam learning implies PAC learning)
Let be an efficient -Occam algorithm for using . Then there exists a constant such that for any , for any distribution , given samples drawn from and labelled according to a concept of length bits each, the algorithm will output a hypothesis such that with probability at least .
Here, is with respect to the concept and distribution . This implies that the algorithm is also a PAC learner for the concept class using hypothesis class . A slightly more general formulation is as follows:
Theorem (Occam learning implies PAC learning, cardinality version)
Let . Let be an algorithm such that, given samples drawn from a fixed but unknown distribution and labeled according to a concept of length bits each, outputs a hypothesis that is consistent with the labeled samples. Then, there exists a constant such that if , then is guaranteed to output a hypothesis such that with probability at least .
While the above theorems show that Occam learning is sufficient for PAC learning, it doesn't say anything about necessity. Board and Pitt show that, for a wide variety of concept classes, Occam learning is in fact necessary for PAC learning.^{[3]} They proved that for any concept class that is polynomially closed under exception lists, PAC learnability implies the existence of an Occam algorithm for that concept class. Concept classes that are polynomially closed under exception lists include Boolean formulas, circuits, deterministic finite automata, decision-lists, decision-trees, and other geometrically-defined concept classes.
A concept class is polynomially closed under exception lists if there exists a polynomial-time algorithm such that, when given the representation of a concept and a finite list of exceptions, outputs a representation of a concept such that the concepts and agree except on the set .
Proof that Occam learning implies PAC learning
We first prove the Cardinality version. Call a hypothesis bad if , where again is with respect to the true concept and the underlying distribution . The probability that a set of samples is consistent with is at most , by the independence of the samples. By the union bound, the probability that there exists a bad hypothesis in is at most , which is less than if . This concludes the proof of the second theorem above.
Using the second theorem, we can prove the first theorem. Since we have a -Occam algorithm, this means that any hypothesis output by can be represented by at most bits, and thus . This is less than if we set for some constant . Thus, by the Cardinality version Theorem, will output a consistent hypothesis with probability at least . This concludes the proof of the first theorem above.
Improving sample complexity for common problems
Though Occam and PAC learnability are equivalent, the Occam framework can be used to produce tighter bounds on the sample complexity of classical problems including conjunctions,^{[2]} conjunctions with few relevant variables,^{[4]} and decision lists.^{[5]}
Extensions
Occam algorithms have also been shown to be successful for PAC learning in the presence of errors,^{[6]}^{[7]} probabilistic concepts,^{[8]} function learning^{[9]} and Markovian non-independent examples.^{[10]}
See also
References
- ^ ^{a} ^{b} Blumer, A., Ehrenfeucht, A., Haussler, D., & Warmuth, M. K. (1987). Occam's razor. Information processing letters, 24(6), 377-380.
- ^ ^{a} ^{b} ^{c} Kearns, M. J., & Vazirani, U. V. (1994). An introduction to computational learning theory, chapter 2. MIT press.
- ^ Board, R., & Pitt, L. (1990, April). On the necessity of Occam algorithms. In Proceedings of the twenty-second annual ACM symposium on Theory of computing (pp. 54-63). ACM.
- ^ Haussler, D. (1988). Quantifying inductive bias: AI learning algorithms and Valiant's learning framework. Artificial intelligence, 36(2), 177-221.
- ^ Rivest, R. L. (1987). Learning decision lists. Machine learning, 2(3), 229-246.
- ^ Angluin, D., & Laird, P. (1988). Learning from noisy examples. Machine Learning, 2(4), 343-370.
- ^ Kearns, M., & Li, M. (1993). Learning in the presence of malicious errors. SIAM Journal on Computing, 22(4), 807-837.
- ^ Kearns, M. J., & Schapire, R. E. (1990, October). Efficient distribution-free learning of probabilistic concepts. In Foundations of Computer Science, 1990. Proceedings., 31st Annual Symposium on (pp. 382-391). IEEE.
- ^ Natarajan, B. K. (1993, August). Occam's razor for functions. In Proceedings of the sixth annual conference on Computational learning theory (pp. 370-376). ACM.
- ^ Aldous, D., & Vazirani, U. (1990, October). A Markovian extension of Valiant's learning model. In Foundations of Computer Science, 1990. Proceedings., 31st Annual Symposium on (pp. 392-396). IEEE.