1,740 research outputs found
Learning k-term DNF Formulas with an Incomplete Oracle
We consider the problem of learning k-term DNF formulas using equivalence queries and incomplete membership queries as defined by Angluin and Slonim. We demonstrate the this model can be applied to non-monotone classes. Namely, we describe a polynomial algorithm that exactly identifies a k-term DNF formula with a k-term DNF hypothesis using incomplete membership queries and equivalence queries from the class of DNF formulas
An Interactive Model of Teaching
Previous teaching models in the learning theory community have been batch models. That is, in these models the teacher has generated a single set of helpful examples to present to the learner. In this paper we present an interactive model in which the learner has the ability to ask queries as in the query learning model of Angluin [1]. We show that this model is at least as powerful as previous teaching models. We also show that anything learnable with queries, even by a randomized learner, is teachable in our model. In all previous teaching models, all classes shown to be teachable are known to be efficiently learnable. An important concept class that is not known to be learnable is DNF formulas. We demonstrate the power of our approach by providing a determinstic teacher and learner for the class of DNF formulas. The learner makes only equivalence queries and all hypotheses are also DNF formulas
A fuzzified BRAIN algorithm for learning DNF from incomplete data
Aim of this paper is to address the problem of learning Boolean functions
from training data with missing values. We present an extension of the BRAIN
algorithm, called U-BRAIN (Uncertainty-managing Batch Relevance-based
Artificial INtelligence), conceived for learning DNF Boolean formulas from
partial truth tables, possibly with uncertain values or missing bits.
Such an algorithm is obtained from BRAIN by introducing fuzzy sets in order
to manage uncertainty. In the case where no missing bits are present, the
algorithm reduces to the original BRAIN
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
Learning to Reason: Leveraging Neural Networks for Approximate DNF Counting
Weighted model counting (WMC) has emerged as a prevalent approach for
probabilistic inference. In its most general form, WMC is #P-hard. Weighted DNF
counting (weighted #DNF) is a special case, where approximations with
probabilistic guarantees are obtained in O(nm), where n denotes the number of
variables, and m the number of clauses of the input DNF, but this is not
scalable in practice. In this paper, we propose a neural model counting
approach for weighted #DNF that combines approximate model counting with deep
learning, and accurately approximates model counts in linear time when width is
bounded. We conduct experiments to validate our method, and show that our model
learns and generalizes very well to large-scale #DNF instances.Comment: To appear in Proceedings of the Thirty-Fourth AAAI Conference on
Artificial Intelligence (AAAI-20). Code and data available at:
https://github.com/ralphabb/NeuralDNF
A novel Boolean kernels family for categorical data
Kernel based classifiers, such as SVM, are considered state-of-the-art algorithms and are widely used on many classification tasks. However, this kind of methods are hardly interpretable and for this reason they are often considered as black-box models. In this paper, we propose a new family of Boolean kernels for categorical data where features correspond to propositional formulas applied to the input variables. The idea is to create human-readable features to ease the extraction of interpretation rules directly from the embedding space. Experiments on artificial and benchmark datasets show the effectiveness of the proposed family of kernels with respect to established ones, such as RBF, in terms of classification accuracy
Learning using Local Membership Queries
We introduce a new model of membership query (MQ) learning, where the
learning algorithm is restricted to query points that are \emph{close} to
random examples drawn from the underlying distribution. The learning model is
intermediate between the PAC model (Valiant, 1984) and the PAC+MQ model (where
the queries are allowed to be arbitrary points).
Membership query algorithms are not popular among machine learning
practitioners. Apart from the obvious difficulty of adaptively querying
labelers, it has also been observed that querying \emph{unnatural} points leads
to increased noise from human labelers (Lang and Baum, 1992). This motivates
our study of learning algorithms that make queries that are close to examples
generated from the data distribution.
We restrict our attention to functions defined on the -dimensional Boolean
hypercube and say that a membership query is local if its Hamming distance from
some example in the (random) training data is at most . We show the
following results in this model:
(i) The class of sparse polynomials (with coefficients in R) over
is polynomial time learnable under a large class of \emph{locally smooth}
distributions using -local queries. This class also includes the
class of -depth decision trees.
(ii) The class of polynomial-sized decision trees is polynomial time
learnable under product distributions using -local queries.
(iii) The class of polynomial size DNF formulas is learnable under the
uniform distribution using -local queries in time
.
(iv) In addition we prove a number of results relating the proposed model to
the traditional PAC model and the PAC+MQ model
Learning pseudo-Boolean k-DNF and Submodular Functions
We prove that any submodular function f: {0,1}^n -> {0,1,...,k} can be
represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a
natural generalization of DNF representation for functions with integer range.
Each term in such a formula has an associated integral constant. We show that
an analog of Hastad's switching lemma holds for pseudo-Boolean k-DNFs if all
constants associated with the terms of the formula are bounded.
This allows us to generalize Mansour's PAC-learning algorithm for k-DNFs to
pseudo-Boolean k-DNFs, and hence gives a PAC-learning algorithm with membership
queries under the uniform distribution for submodular functions of the form
f:{0,1}^n -> {0,1,...,k}. Our algorithm runs in time polynomial in n, k^{O(k
\log k / \epsilon)}, 1/\epsilon and log(1/\delta) and works even in the
agnostic setting. The line of previous work on learning submodular functions
[Balcan, Harvey (STOC '11), Gupta, Hardt, Roth, Ullman (STOC '11), Cheraghchi,
Klivans, Kothari, Lee (SODA '12)] implies only n^{O(k)} query complexity for
learning submodular functions in this setting, for fixed epsilon and delta.
Our learning algorithm implies a property tester for submodularity of
functions f:{0,1}^n -> {0, ..., k} with query complexity polynomial in n for
k=O((\log n/ \loglog n)^{1/2}) and constant proximity parameter \epsilon
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