On learning sets and functions

Natarajan, B. K.

doi:10.1007/bf00114804

Cited by 116 publications

(117 citation statements)

References 17 publications

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“…In some work on function learning, such as [17,7,11], a dimension known as the graph dimension has proven to be useful. The graph dimension of a class H of functions that map from X to a set Y is the VC-dimension of the class…”

Section: Propositionmentioning

confidence: 99%

Valid Generalisation from Approximate Interpolation

Anthony

Ishai²,

Shawe‐Taylor³

1996

Combinator. Probab. Comp.

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Let H and C be sets of functions from domain X to . We say that H validly generalises C from approximate interpolation if and only if for each η > 0 and , δ ∈ (0, 1) there is m 0 (η, , δ) such that for any function t ∈ C and any probability distribution P on X, if m ≥ m 0 then with P m -probability at least 1 − δ, a sample x = (x 1 , x 2 , . . . , x m ) ∈ X m satisfies ∀h ∈ H, |h(We find conditions that are necessary and sufficient for H to validly generalise C from approximate interpolation, and we obtain bounds on the sample length m 0 (η, , δ) in terms of various parameters describing the expressive power of H.

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Section: Propositionmentioning

confidence: 99%

Valid Generalisation from Approximate Interpolation

Anthony

Ishai²,

Shawe‐Taylor³

1996

Combinator. Probab. Comp.

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“…Given a particular determination, we estimate the size of the function space consistent with the determination. The main result in PAC learning that we use is the dimensionality theorem, which relates the number of training examples needed for successful learning to the size of the function space [Blumer et al, 1989, Natarajan, 1989. Informally, this theorem says that the number of examples sufficient for successful learning varies logarithmically with the asymptotic size of the function space.…”

Section: Informal Overview Of the Approachmentioning

confidence: 99%

“…The relationship of Natarajan's dimension to the more popular Vapnik-Chervonenkis dimension [Blumer et al, 1989] is discussed in [Natarajan, 1989]. To calculate the dimension of the function space B in our boolean function example above, we note that there are 2 2n.…”

Section: Definition 6 a F Unction F Is Consistent With A Set Of Exammentioning

confidence: 99%

Untitled

Mahadevan

Tadepalli

1994

Machine Learning

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Abstract. Prior knowledge, or bias, regarding a concept can reduce the number of examples needed to learn it. Probably Approximately Correct (PAC) learning is a mathematical model of concept learning that can be used to quantify the reduction in the number of examples due to different forms of bias. Thus far, PAC learning has mostly been used to analyze syntactic bias, such as limiting concepts to conjunctions of boolean prepositions. This paper demonstrates that PAC learning can also be used to analyze semantic bias, such as a domain theory about the concept being learned. The key idea is to view the hypothesis space in PAC learning as that consistent with all prior knowledge, syntactic and semantic. In particular, the paper presents an analysis of determinations, a type of relevance knowledge. The results of the analysis reveal crisp distinctions and relations among different determinations, and illustrate the usefulness of an analysis based on the PAC learning model.

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“…As alternative definitions, a variety of notions of dimension to classes of {0,..., m}-valued functions had been proposed [4,7], and Ben-David et al gave a general scheme [2] which unified them. They introduced ~-dimension, where k~ is a family of mappings which translate {0,..., m}-valued functions into {0, 1}-valued ones.…”

Section: Complexity Of ~-Dimension Problemsmentioning

confidence: 99%

“…It is well-known that the Vapnik-Chervonenkis dimension (VC-dimension) which is a combinatorial parameter of a class of binary functions plays the key role to determine whether the class is polynomial-sample learnable or not [3,5,8]. As a natural extension, the learnability of a class of {0,..., m}-valued functions has been characterized by various generalized notions such as pseudodimension [4], graph dimension [7], and Natarajan dimension [7]. Ben-David et al [2] unified them into a general scheme, by introducing a family 9 of mappings which translate {0,..., m}-valued functions into binary ones.…”

Section: Introductionmentioning

confidence: 99%

Complexity of computing generalized VC-dimensions

Shinohara

1994

Machine Learning: ECML-94

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Abstract. In the PAC-learning model, the Vapnik-Chervonenkis (VC) dimension plays the key role to estimate the polynomial-sample learnability of a class of binary functions. For a class of {0,..., m}-valued functions, the notion has been generalized in various ways. This paper investigates the complexity of computing some of generalized VCdimensions: VC*-dimension, ~.-dimension, and ~G-dimension. For each dimension, we consider a decision problem that is, for a given matrix representing a class .T" of functions and an integer K, to determine whether the dimension of .T is greater than K or not. We prove that the VC*-dimension problem is polynomial-time reducible to the satisfiability problem of length J with O(log 2 J) variables, which includes the original VC-dimension problem as a special case. We also show that the k~G-dimension problem is still reducible to the satisfiability problem of length J with O(log 2 J), while the ~.-dimension problem becomes NP-complete.

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On learning sets and functions

Cited by 116 publications

References 17 publications

Valid Generalisation from Approximate Interpolation

Valid Generalisation from Approximate Interpolation

Untitled

Complexity of computing generalized VC-dimensions

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