On restricted-focus-of-attention learnability of Boolean functions

Birkendorf, Andreas; Dichterman, Eli; Jackson, Jeffrey C.; Klasner, Norbert; Simon, Hans Ulrich

doi:10.1145/238061.238098

Cited by 15 publications

(23 citation statements)

References 22 publications

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“…, x n ) used for learning (see Ben-David & Dichterman 1994 for details). As observed by Birkendorf et al (1998);Goldberg (2001), the class of linear threshold functions over {−1, 1} n is uniform-distribution information-theoretically learnable from cc 16 (2007) Every LTF has a low-weight approximator 199 poly(n) many examples in this framework if and only if any linear threshold function is information-theoretically specified to high accuracy from Chow parameter estimates which are accurate to an additive ±1/poly(n). With this motivation Birkendorf et al gave the following result:…”

Section: Approximating An Ltf From Noisy Versions Of Its Low-degree Fmentioning

confidence: 92%

“…Theorem 6.1 gives a strong bound on the precision required in the Chow parameters if f has low weight, but a weak bound for arbitrary LTFs since W may need to be 2 Ω(n log n) . Subsequently Goldberg (2001) gave an incomparable result which can be rephrased as follows: In contrast, our bound in Theorem 1.2 has a worse dependence on but has a 1/n rather than 1/quasipoly(n) dependence on n. Theorem 1.2 yields an affirmative answer (at least for constant ) to the open question of whether arbitrary linear threshold functions can be learned in the uniform distribution 1-RFA model with polynomial sample complexity: Thus far we have followed the proof from Birkendorf et al (1998) (which is itself closely based on Bruck 1990), and indeed it is not difficult to complete the proof of Theorem 6.1 from here. Instead we will use our ideas from Section 4.…”

Section: Approximating An Ltf From Noisy Versions Of Its Low-degree Fmentioning

confidence: 93%

“…The second application is to the problem of reconstructing a linear threshold function from (approximations to) its low-degree Fourier coefficients. Various cc 16 (2007) Every LTF has a low-weight approximator 183 forms of this problem have been studied since the 1960s (see Birkendorf et al 1998;Bruck 1990;Chow 1961;Goldberg 2001;Kaszerman 1963;Winder 1971; we give a detailed description of prior work in Section 6). We show that for any constant > 0, any linear threshold function f is information-theoretically specified to within error by estimates of its degree-0 and degree-1 Fourier coefficients (sometimes known as its Chow parameters) which are accurate to within an additive ±1/O(n):…”

Section: Applicationsmentioning

confidence: 99%

“…Theorem 6.1 (Birkendorf et al 1998). Theorem 6.1 gives a strong bound on the precision required in the Chow parameters if f has low weight, but a weak bound for arbitrary LTFs since W may need to be 2 Ω(n log n) .…”

Section: Approximating An Ltf From Noisy Versions Of Its Low-degree Fmentioning

confidence: 99%

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Every Linear Threshold Function has a Low-Weight Approximator

Servedio

2007

comput. complex.

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Abstract. Given any linear threshold function f on n Boolean variables, we construct a linear threshold function g which disagrees with f on at most an fraction of inputs and has integer weights each of magnitude at most √ n · 2Õ (1/ 2 ) . We show that the construction is optimal in terms of its dependence on n by proving a lower bound of Ω( √ n) on the weights required to approximate a particular linear threshold function. We give two applications. The first is a deterministic algorithm for approximately counting the fraction of satisfying assignments to an instance of the zero-one knapsack problem to within an additive ± . The algorithm runs in time polynomial in n (but exponential in 1/ 2 ). In our second application, we show that any linear threshold function f is specified to within error by estimates of its Chow parameters (degree 0 and 1 Fourier coefficients) which are accurate to within an additive ±1/(n · 2Õ (1/ 2 ) ). This is the first such accuracy bound which is inverse polynomial in n, and gives the first polynomial bound (in terms of n) on the number of examples required for learning linear threshold functions in the "restricted focus of attention" framework.

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Section: Approximating An Ltf From Noisy Versions Of Its Low-degree Fmentioning

confidence: 92%

Section: Approximating An Ltf From Noisy Versions Of Its Low-degree Fmentioning

confidence: 93%

Section: Applicationsmentioning

confidence: 99%

Section: Approximating An Ltf From Noisy Versions Of Its Low-degree Fmentioning

confidence: 99%

See 2 more Smart Citations

Every Linear Threshold Function has a Low-Weight Approximator

Servedio

2007

comput. complex.

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“…In the field of machine learning, there are also many algorithms introduced for learning Boolean functions [22][23][24][25][26][27][28][29][30]. If these algorithms are modified to learning BNs of a bounded indegree of k, i.e., n Boolean functions with k inputs, their complexities are at least O(N · n k+1 ).…”

Section: Introductionmentioning

confidence: 99%

Improved Time Complexities for Learning Boolean Networks

Zheng

Kwoh

2013

Entropy

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Existing algorithms for learning Boolean networks (BNs) have time complexities of at least O(N · n 0.7(k+1) ), where n is the number of variables, N is the number of samples and k is the number of inputs in Boolean functions. Some recent studies propose more efficient methods with O(N · n 2 ) time complexities. However, these methods can only be used to learn monotonic BNs, and their performances are not satisfactory when the sample size is small. In this paper, we mathematically prove that OR/AND BNs, where the variables are related with logical OR/AND operations, can be found with the time complexity of O(k·(N + logn)·n 2 ), if there are enough noiseless training samples randomly generated from a uniform distribution. We also demonstrate that our method can successfully learn most BNs, whose variables are not related with exclusive OR and Boolean equality operations, with the same order of time complexity for learning OR/AND BNs, indicating our method has good efficiency for learning general BNs other than monotonic BNs. When the datasets are noisy, our method can still successfully identify most BNs with the same efficiency. When compared with two existing methods with the same settings, our method achieves a better comprehensive performance than both of them, especially for small training sample sizes. More importantly, our method can be used to learn all BNs. However, of the two methods that are compared, one can only be used to learn monotonic BNs, and the other one has a much worse time complexity than our method. In conclusion, our results demonstrate that Boolean networks can be learned with improved time complexities.Entropy 2013, 15 3763

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Learning Fixed-Dimension Linear Thresholds from Fragmented Data

Goldberg

2001

Information and Computation

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On restricted-focus-of-attention learnability of Boolean functions

Cited by 15 publications

References 22 publications

Every Linear Threshold Function has a Low-Weight Approximator

Every Linear Threshold Function has a Low-Weight Approximator

Improved Time Complexities for Learning Boolean Networks

Learning Fixed-Dimension Linear Thresholds from Fragmented Data

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