### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 19-53 |

Number of pages | 35 |

Journal | Computational Complexity |

Volume | 7 |

Issue number | 1 |

State | Published - 1998 |

### Fingerprint

### Keywords

- Complexity theory
- Computational learning theory
- Equivalence queries
- Membership queries
- Query learning
- Read-thrice DNF
- Threshold functions

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Mathematics(all)
- Computational Mathematics
- Theoretical Computer Science

### Cite this

*Computational Complexity*,

*7*(1), 19-53.

**Complexity theoretic hardness results for query learning.** / Aizenstein, Howard; Hegedus, Tibor; Hellerstein, Lisa; Pitt, Leonard.

Research output: Contribution to journal › Article

*Computational Complexity*, vol. 7, no. 1, pp. 19-53.

}

TY - JOUR

T1 - Complexity theoretic hardness results for query learning

AU - Aizenstein, Howard

AU - Hegedus, Tibor

AU - Hellerstein, Lisa

AU - Pitt, Leonard

PY - 1998

Y1 - 1998

N2 - 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 ΣP2, 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.

AB - 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 ΣP2, 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.

KW - Complexity theory

KW - Computational learning theory

KW - Equivalence queries

KW - Membership queries

KW - Query learning

KW - Read-thrice DNF

KW - Threshold functions

UR - http://www.scopus.com/inward/record.url?scp=0032361270&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032361270&partnerID=8YFLogxK

M3 - Article

VL - 7

SP - 19

EP - 53

JO - Computational Complexity

JF - Computational Complexity

SN - 1016-3328

IS - 1

ER -