18 research outputs found
Queries revisited
AbstractWe begin with a brief tutorial on the problem of learning a finite concept class over a finite domain using membership queries and/or equivalence queries. We then sketch general results on the number of queries needed to learn a class of concepts, focusing on the various notions of combinatorial dimension that have been employed, including the teaching dimension, the exclusion dimension, the extended teaching dimension, the fingerprint dimension, the sample exclusion dimension, the Vapnik–Chervonenkis dimension, the abstract identification dimension, and the general dimension
Improved Bounds on Quantum Learning Algorithms
In this article we give several new results on the complexity of algorithms
that learn Boolean functions from quantum queries and quantum examples.
Hunziker et al. conjectured that for any class C of Boolean functions, the
number of quantum black-box queries which are required to exactly identify an
unknown function from C is ,
where is a combinatorial parameter of the class C. We
essentially resolve this conjecture in the affirmative by giving a quantum
algorithm that, for any class C, identifies any unknown function from C using
quantum black-box
queries.
We consider a range of natural problems intermediate between the exact
learning problem (in which the learner must obtain all bits of information
about the black-box function) and the usual problem of computing a predicate
(in which the learner must obtain only one bit of information about the
black-box function). We give positive and negative results on when the quantum
and classical query complexities of these intermediate problems are
polynomially related to each other.
Finally, we improve the known lower bounds on the number of quantum examples
(as opposed to quantum black-box queries) required for -PAC
learning any concept class of Vapnik-Chervonenkis dimension d over the domain
from to . This new lower bound comes
closer to matching known upper bounds for classical PAC learning.Comment: Minor corrections. 18 pages. To appear in Quantum Information
Processing. Requires: algorithm.sty, algorithmic.sty to buil
Tight Bounds on Proper Equivalence Query Learning of DNF
We prove a new structural lemma for partial Boolean functions , which we
call the seed lemma for DNF. Using the lemma, we give the first subexponential
algorithm for proper learning of DNF in Angluin's Equivalence Query (EQ) model.
The algorithm has time and query complexity , which
is optimal. We also give a new result on certificates for DNF-size, a simple
algorithm for properly PAC-learning DNF, and new results on EQ-learning -term DNF and decision trees
Conjunctions of Unate DNF Formulas: Learning and Structure
AbstractA central topic in query learning is to determine which classes of Boolean formulas are efficiently learnable with membership and equivalence queries. We consider the class Rkconsisting of conjunctions ofkunate DNF formulas. This class generalizes the class ofk-clause CNF formulas and the class of unate DNF formulas, both of which are known to be learnable in polynomial time with membership and equivalence queries. We prove that R2can be properly learned with a polynomial number of polynomial-size membership and equivalence queries, but can be properly learned in polynomial time with such queries if and only if P=NP. Thus the barrier to properly learning R2with membership and equivalence queries is computational rather than informational. Few results of this type are known. In our proofs, we use recent results of Hellersteinet al.(1997,J. Assoc. Comput. Mach.43(5), 840–862), characterizing the classes that are polynomial-query learnable, together with work of Bshouty on the monotone dimension of Boolean functions. We extend some of our results to Rkand pose open questions on learning DNF formulas of small monotone dimension. We also prove structural results for Rk. We construct, for any fixedk⩾2, a class of functionsfthat cannot be represented by any formula in Rk, but which cannot be “easily” shown to have this property. More precisely, for any functionfonnvariables in the class, the value offon any polynomial-size set of points in its domain is not a witness thatfcannot be represented by a formula in Rk. Our construction is based on BCH codes