How many queries are needed to learn?

Hellerstein, Lisa; Raghavan, Vasanthan; Pillaipakkamnatt, Krishnan; Wilkins, Dawn

doi:10.1145/225058.225114

Cited by 37 publications

(55 citation statements)

References 22 publications

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“…By directly combining results (not techniques) of Umans in [34,35] with an observation of Bshouty mentioned in [19], one can immediately obtain weaker versions of some of our negative results.…”

Section: Combinatorial Consequences Of Complexity Theoretic Assumptionsmentioning

confidence: 68%

“…Proposition 2 (Hellerstein et al [19]). Let C be a class of Boolean formulas with associated size function s such that the problem of generating all elements in C m,n , given m and n, is computable.…”

Section: The Connection Between Certificates and Learningmentioning

confidence: 99%

“…Hellerstein et al [19] and Hegedüs [17] showed independently that there are certain combinatorial objects called certificates that give a precise characterization of polynomial-query learnability in the exact model of membership and proper equivalence queries. Most of the results in this paper are based on proofs of the existence or non-existence of certificates of a given size.…”

Section: A Gap Theorem and Its Consequencementioning

confidence: 99%

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Exact learning of DNF formulas using DNF hypotheses

Hellerstein¹,

Raghavan

2005

Journal of Computer and System Sciences

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We show the following: (a) For any > 0, log (3+ ) n-term DNF cannot be polynomial-query learned with membership and strongly proper equivalence queries. (b) For sufficiently large t, t-term DNF formulas cannot be polynomialquery learned with membership and equivalence queries that use t 1+ -term DNF formulas as hypotheses, for some < 1 (c) Read-thrice DNF formulas are not polynomial-query learnable with membership and proper equivalence queries. (d) log n-term DNF formulas can be polynomial-query learned with membership and proper equivalence queries. (This complements a result of Bshouty, Goldman, Hancock, and Matar that √ log n-term DNF can be so learned in polynomial time.)Versions of (a)-(c) were known previously, but the previous versions applied to polynomial-time learning and used complexity theoretic assumptions. In contrast, (a)-(c) apply to polynomial-query learning, imply the results for polynomial-time learning, and do not use any complexity-theoretic assumptions.

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Section: Combinatorial Consequences Of Complexity Theoretic Assumptionsmentioning

confidence: 68%

Section: The Connection Between Certificates and Learningmentioning

confidence: 99%

Section: A Gap Theorem and Its Consequencementioning

confidence: 99%

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Exact learning of DNF formulas using DNF hypotheses

Hellerstein¹,

Raghavan

2005

Journal of Computer and System Sciences

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“…Whether or not a counterexample can be obtained without the assumption that P = NP is an interesting question. Recent results of [36], [34] address a related question. Before describing the functions represented by F , we first describe the encoding of any given φ.…”

Section: Proofmentioning

confidence: 99%

Complexity theoretic hardness results for query learning

et al. 1998

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We investigate the complexity of learning for the well-studied model in which the learning algorithm may ask membership and equivalence queries. While complexity theoretic techniques have previously been used to prove hardness results in various learning models, these techniques typically are not strong enough to use when a learning algorithm may make membership queries. We develop a general technique for proving hardness results for learning with membership and equivalence queries (and for more general query models). We apply the technique to show that, assuming NP = co-NP, no polynomial-time membership and (proper) equivalence query algorithms exist for exactly learning read-thrice DNF formulas, unions of k ≥ 3 halfspaces over the Boolean domain, or some other related classes. Our hardness results are representation dependent, and do not preclude the existence of representation independent algorithms.The general technique introduces the representation problem for a class F of representations (e.g., formulas), which is naturally associated with the learning problem for F . This problem is related to the structural question of how to characterize functions representable by formulas in F , and is a generalization of standard complexity problems such as Satisfiability. While in general the representation problem is in Σ P 2 , we present a theorem demonstrating that for "reasonable" classes F , the existence of a polynomial-time membership and equivalence query algorithm for exactly learning F implies that the representation problem for F is in fact in co-NP. The theorem is applied to prove hardness results such as the ones mentioned above, by showing that the representation problem for specific classes of formulas is NP-hard.

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“…A second motivation comes from the use of certificates [13,12] to study query complexity. It is known [13,12] that a class C is learnable from equivalence and membership queries if and only if the class C has polynomial certificates. Previous work in [5] developed certificates for propositional classes.…”

Section: Introductionmentioning

confidence: 99%

The subsumption lattice and query learning

Khardon

Arias

2006

Journal of Computer and System Sciences

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The paper identifies several new properties of the lattice induced by the subsumption relation over first-order clauses and derives implications of these for learnability. In particular, it is shown that the length of subsumption chains of function free clauses with bounded size can be exponential in the size. This suggests that simple algorithmic approaches that rely on repeating minimal subsumption-based refinements may require a long time to converge. It is also shown that with bounded size clauses the subsumption lattice has a large branching factor. This is used to show that the class of first-order length-bounded monotone clauses is not properly learnable from membership queries alone. Finally, the paper studies pairing, a generalization operation that takes two clauses and returns a number of possible generalizations. It is shown that there are clauses with an exponential number of pairing results which are not related to each other by subsumption. This is used to show that recent pairing-based algorithms can make exponentially many queries on some learning problems.

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How many queries are needed to learn?

Cited by 37 publications

References 22 publications

Exact learning of DNF formulas using DNF hypotheses

Exact learning of DNF formulas using DNF hypotheses

Complexity theoretic hardness results for query learning

The subsumption lattice and query learning

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