Learning threshold functions with small weights using membership queries

Abboud, Elias; Agha, Nader; Bshouty, Nader H.; Radwan, Nizar; Saleh, Fathi

doi:10.1145/307400.307483

Cited by 7 publications

(6 citation statements)

References 4 publications

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“…Now the hitting set problem is equivalent to the set cover problem, i.e., find the minimal number of elements S ⊂ X such that ∪ x∈S C x = C. It is known that the greedy algorithm that at each stage, chooses the set that contains the largest number of uncovered elements, achieves an approximation ratio of ln |C|, [84]. Now (8) follows from ( 7), Lemma 1 and 3 in Lemma 5.…”

Section: Any Class C Is Non-adaptively Query-efficiently Learnable In...mentioning

confidence: 99%

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Exact learning from an honest teacher that answers membership queries

Bshouty

2018

Theoretical Computer Science

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Given a teacher that holds a function f : X → R from some class of functions C. The teacher can receive from the learner an element d in the domain X (a query) and returns the value of the function in d, f (d) ∈ R. The learner goal is to find f with a minimum number of queries, optimal time complexity, and optimal resources. In this survey, we present some of the results known from the literature, different techniques used, some new problems, and open problems.

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Section: Any Class C Is Non-adaptively Query-efficiently Learnable In...mentioning

confidence: 99%

“…Abboud et al [8] give a lower bound Ω(2 n / √ n) for BH(−1, 0, 1) (Boolean Halfspaces with weights {−1, 0, +1}). Therefore, BH(−1, 0, 1) is non-adaptive almost optimally learnable.…”

Section: Cnfmentioning

confidence: 99%

Exact learning from an honest teacher that answers membership queries

Bshouty

2018

Theoretical Computer Science

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“…Special case n = 2 is considered in [9]. Learning threshold Boolean functions with small weights is investigated in [1,7].…”

Section: Related Workmentioning

confidence: 99%

Specifying a positive threshold function via extremal points

Razgon¹,

Zamaraev²,

Zamaraeva³

et al. 2017

Preprint

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An extremal point of a positive threshold Boolean function f is either a maximal zero or a minimal one. It is known that if f depends on all its variables, then the set of its extremal points completely specifies f within the universe of threshold functions. However, in some cases, f can be specified by a smaller set. The minimum number of points in such a set is the specification number of f . It was shown in [S.-T. Hu. Threshold Logic, 1965] that the specification number of a threshold function of n variables is at least n + 1. In [M. Anthony, G. Brightwell, and J. Shawe-Taylor. On specifying Boolean functions by labelled examples. Discrete Applied Mathematics, 1995] it was proved that this bound is attained for nested functions and conjectured that for all other threshold functions the specification number is strictly greater than n+1. In the present paper, we resolve this conjecture negatively by exhibiting threshold Boolean functions of n variables, which are non-nested and for which the specification number is n + 1. On the other hand, we show that the set of extremal points satisfies the statement of the conjecture, i.e. a positive threshold Boolean function depending on all its n variables has n + 1 extremal points if and only if it is nested. To prove this, we reveal an underlying structure of the set of extremal points.

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“…For the case where the dimension is not constant, we show that learning this class implies P=NP. We note that there are other applications for learning halfspaces; for example, Hegedűs (1995); Zolotykh and Shevchenko (1995); Abboud et al (1999); Abasi et al (2014).…”

Section: Introductionmentioning

confidence: 99%

Learning Disjunctions of Predicates

Bshouty,

Drachsler-Cohen,

Vechev

et al. 2017

Preprint

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Let F be a set of boolean functions. We present an algorithm for learning F ∨ := {∨ f ∈S f | S ⊆ F} from membership queries. Our algorithm asks at most |F| • OPT(F ∨ ) membership queries where OPT(F ∨ ) is the minimum worst case number of membership queries for learning F ∨ . When F is a set of halfspaces over a constant dimension space or a set of variable inequalities, our algorithm runs in polynomial time.The problem we address has practical importance in the field of program synthesis, where the goal is to synthesize a program that meets some requirements. Program synthesis has become popular especially in settings aiming to help end users. In such settings, the requirements are not provided upfront and the synthesizer can only learn them by posing membership queries to the end user. Our work enables such synthesizers to learn the exact requirements while bounding the number of membership queries. c char∞ 2017 N.

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Learning threshold functions with small weights using membership queries

Cited by 7 publications

References 4 publications

Exact learning from an honest teacher that answers membership queries

Exact learning from an honest teacher that answers membership queries

Specifying a positive threshold function via extremal points

Learning Disjunctions of Predicates

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