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Learning Boolean Halfspaces with Small Weights from Membership Queries [article]

Hasan Abasi and Ali Z. Abdi and Nader H. Bshouty
2014 arXiv   pre-print
We consider the problem of proper learning a Boolean Halfspace with integer weights {0,1,...,t} from membership queries only.  ...  with Small Weights Using Membership Queries.  ...  In [4] Abboud et. al. showed that in order to learn boolean Halfspace functions with weights W = {−1, 0, 1}, we need at least O(2 n−o(n) ) membership queries.  ... 
arXiv:1405.1535v1 fatcat:wmj3uolzlbdbrjbsdud4imekni

Learning Functions of Halfspaces using Prefix Covers

Parikshit Gopalan, Adam R. Klivans, Raghu Meka
2012 Journal of machine learning research  
To prove this result, we identify a new structural property of Boolean functions that yields learnability with queries: that of having a small prefix cover.  ...  We present a simple query-algorithm for learning arbitrary functions of k halfspaces under any product distribution on the Boolean hypercube.  ...  Theorem 1 (Learning Functions of Halfspaces) The concept class of arbitrary Boolean functions of k halfspaces can be PAC learned with membership queries under the uniform distribution {0, 1} n to accuracy  ... 
dblp:journals/jmlr/GopalanKM12 fatcat:pzwe6l34bnad3at5fkhwzwb5om

Learning intersections and thresholds of halfspaces

A KLIVANS
2004 Journal of computer and system sciences (Print)  
Finally, we also observe that any function of a constant number of polynomial-weight halfspaces can be learned in polynomial time in the model of exact learning from membership and equivalence queries.  ...  We also give the first quasipolynomial time algorithm for learning any Boolean function of a polylog number of polynomial-weight halfspaces under any distribution on the Boolean hypercube.  ...  Learning in the exact model We also give results for learning an intersection of k weight-w halfspaces in the model of exact learning from membership and equivalence queries [4] .  ... 
doi:10.1016/s0022-0000(03)00181-8 fatcat:iijhmzczcveebbwu2banbmykva

Learning intersections and thresholds of halfspaces

Adam R. Klivans, Ryan O'Donnell, Rocco A. Servedio
2004 Journal of computer and system sciences (Print)  
Finally, we also observe that any function of a constant number of polynomial-weight halfspaces can be learned in polynomial time in the model of exact learning from membership and equivalence queries.  ...  We also give the first quasipolynomial time algorithm for learning any Boolean function of a polylog number of polynomial-weight halfspaces under any distribution on the Boolean hypercube.  ...  Learning in the exact model We also give results for learning an intersection of k weight-w halfspaces in the model of exact learning from membership and equivalence queries [4] .  ... 
doi:10.1016/j.jcss.2003.11.002 fatcat:le4ezu5pwjdkveehqwpncmxgje

Near-Optimal Statistical Query Lower Bounds for Agnostically Learning Intersections of Halfspaces with Gaussian Marginals [article]

Daniel Hsu, Clayton Sanford, Rocco Servedio, Emmanouil-Vasileios Vlatakis-Gkaragkounis
2022 arXiv   pre-print
We prove two variants of our lower bound, each of which combines ingredients from Diakonikolas et al. (2021) with (an extension of) a different earlier approach for agnostic SQ lower bounds for the Boolean  ...  We consider the well-studied problem of learning intersections of halfspaces under the Gaussian distribution in the challenging agnostic learning model.  ...  small low-degree Hermite weight.  ... 
arXiv:2202.05096v1 fatcat:jpmmq7m6qnacvj3gf7zqyafhpi

Page 5022 of Mathematical Reviews Vol. , Issue 96h [page]

1996 Mathematical Reviews  
Summary: “We study the learnability of Boolean functions from membership and equivalence queries.  ...  Summary: “An algorithm is a weak learning algorithm if with some small probability it outputs a hypothesis with error slightly below 50%.  ... 

Unconditional lower bounds for learning intersections of halfspaces

Adam R. Klivans, Alexander A. Sherstov
2007 Machine Learning  
Our main result is that any statistical-query algorithm for learning the intersection of √ n halfspaces in n dimensions must make 2 Ω( √ n) queries.  ...  We also show that the intersection of two majorities (low-weight halfspaces) cannot be computed by a polynomial threshold function (PTF) with fewer than n Ω(log n/ log log n) monomials.  ...  We note here that intersections of a constant number of halfspaces are learnable with membership and equivalence queries in polynomial time via Angluin's algorithm for learning finite automata.  ... 
doi:10.1007/s10994-007-5010-1 fatcat:y554mbcivjf6xdikd224bpujdq

Cryptographic Hardness for Learning Intersections of Halfspaces

Adam Klivans, Alexander Sherstov
2006 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)  
We also prove that PAC learning intersections of n ε low-weight halfspaces would yield a polynomial-time quantum solution toÕ(n 1.5 )-SVP andÕ(n 1.5 )-SIVP (shortest vector problem and shortest independent  ...  We give the first representation-independent hardness results for PAC learning intersections of halfspaces, a central concept class in computational learning theory.  ...  If, in addition to membership queries, the learner can make equivalence queries, Klivans and Shpilka [16] have shown how to exactly learn restricted types of depth-3 arithmetic circuits via multiplicity  ... 
doi:10.1109/focs.2006.24 dblp:conf/focs/KlivansS06 fatcat:ua2vzd3oczhtramz34sdjvpmjq

On PAC Learning Algorithms for Rich Boolean Function Classes [chapter]

Rocco A. Servedio
2006 Lecture Notes in Computer Science  
We give an overview of the fastest known algorithms for learning various expressive classes of Boolean functions in the Probably Approximately Correct (PAC) learning model.  ...  In addition to surveying previously known results, we use existing techniques to give the first known subexponential-time algorithms for PAC learning two natural and expressive classes of Boolean functions  ...  to the target function, which are often known as membership queries.  ... 
doi:10.1007/11750321_42 fatcat:iqkiulnpd5at7a7enenxwxxjgy

On PAC learning algorithms for rich Boolean function classes

Lisa Hellerstein, Rocco A. Servedio
2007 Theoretical Computer Science  
We give an overview of the fastest known algorithms for learning various expressive classes of Boolean functions in the Probably Approximately Correct (PAC) learning model.  ...  In addition to surveying previously known results, we use existing techniques to give the first known subexponential-time algorithms for PAC learning two natural and expressive classes of Boolean functions  ...  to the target function, which are often known as membership queries.  ... 
doi:10.1016/j.tcs.2007.05.018 fatcat:cfpv3cqkrvbu3ndw7x2vk6j5he

Cryptographic hardness for learning intersections of halfspaces

Adam R. Klivans, Alexander A. Sherstov
2009 Journal of computer and system sciences (Print)  
We also prove that PAC learning intersections of n low-weight halfspaces would yield a polynomial-time quantum solution toÕ (n 1.5 )-SVP andÕ (n 1.5 )-SIVP (shortest vector problem and shortest independent  ...  We give the first representation-independent hardness results for PAC learning intersections of halfspaces, a central concept class in computational learning theory.  ...  If, in addition to membership queries, the learner can make equivalence queries, Klivans and Shpilka [16] have shown how to exactly learn restricted types of depth-3 arithmetic circuits via multiplicity  ... 
doi:10.1016/j.jcss.2008.07.008 fatcat:emanuf7fevdahh5ty6gbmxar6e

Computational learning theory

Dana Angluin
1992 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing - STOC '92  
Hence, predicting 3p-boolean formulas or finite unions of dfas or two-way dfas with membership queries is as hard as predicting boolean formulas with membership queries. 12.3 Implications of  ...  Such a query is called a membership query. In this setting we may define PAC- learning with membership queries in the obvious way.  ... 
doi:10.1145/129712.129746 dblp:conf/stoc/Angluin92 fatcat:7aw3cnd745bellyhu7phywpul4

Optimal bounds for sign-representing the intersection of two halfspaces by polynomials [article]

Alexander A. Sherstov
2010 arXiv   pre-print
learning DNF formulas and read-once Boolean formulas.  ...  Our result shows that the intersection of two halfspaces on 0,1^n only admits a trivial 2^Theta(n)-time learning algorithm based on sign-representation by polynomials, unlike the advances achieved in PAC  ...  Furthermore, if membership queries are allowed, DNF formulas are known to be learnable in polynomial time with respect to the uniform distribution on the hypercube [12] . Our Techniques.  ... 
arXiv:0910.4224v2 fatcat:rde5ogkzbrdsbla22ezfa5r6em

Noise-Tolerant Parallel Learning of Geometric Concepts

Nader H. Bshouty, Sally A. Goldman, H.David Mathias
1998 Information and Computation  
noise-tolerant parallel algorithm to PAClearn the class of geometric concepts defined by a polynomial number of (d&1)-dimensional hyperplanes against an arbitrary distribution where each hyperplane has a slope from  ...  We present several efficient parallel algorithms for PAC-learning geometric concepts in a constant-dimensional space.  ...  that has small weight outside.  ... 
doi:10.1006/inco.1998.2737 fatcat:vp4f6ufyizbyhh4oukdpwudfqa

Optimal bounds for sign-representing the intersection of two halfspaces by polynomials

Alexander A. Sherstov
2013 Combinatorica  
Furthermore, if membership queries are allowed, DNF formulas are known to be learnable in polynomial time with respect to the uniform distribution on the hypercube [12] . Our Techniques.  ...  As a result, f can be PAC learned in time polynomial in N; using any of a variety of halfspace learning algorithms.  ...  degree into Boolean functions with high threshold density, due to Krause and Pudlák [21, Prop. 2.1] .  ... 
doi:10.1007/s00493-013-2759-7 fatcat:76pzqyhbtfhcbnsw2xha2a7mwm
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