## Property Testing: A Learning Theory Perspective

Citations: | 25 - 5 self |

### BibTeX

@MISC{Ron_propertytesting:,

author = {Dana Ron},

title = {Property Testing: A Learning Theory Perspective},

year = {}

}

### OpenURL

### Abstract

Property testing deals with tasks where the goal is to distinguish between the case that an object (e.g., function or graph) has a prespecified property (e.g., the function is linear or the graph is bipartite) and the case that it differs significantly from any such object. The task should be performed by observing only a very small part of the object, in particular by querying the object, and the algorithm is allowed a small failure probability. One view of property testing is as a relaxation of learning the object (obtaining an approximate representation of the object). Thus property testing algorithms can serve as a preliminary step to learning. That is, they can be applied in order to select, very efficiently, what hypothesis class to use for learning. This survey takes the learning-theory point of view and focuses on results for testing properties of functions that are of interest to the learning theory community. In particular, we cover results for testing algebraic properties of functions such as linearity, testing properties defined by concise representations, such as having a small DNF representation, and more. 1

### Citations

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(Show Context)
Citation Context ...l allows queries and the underlying distribution is uniform, then the corresponding testing model is the standard one from Definition 2.1. We first recall what a Probably Approximately Correct (PAC ) =-=[134]-=- learning algorithm is, and in particular what is a proper learning algorithm. In what follows a representation class is a set of functions together with a language for describing the functions in the... |

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(Show Context)
Citation Context ...esting algebraic properties of functions such as linearity, testing properties defined by concise representations, such as having a small DNF representation, and more.1 Introduction Property testing =-=[82, 128]-=- is the study of the following class of problems. Given the ability to perform (local) queries concerning a particular object the problem is to determine whether the object has a predetermined (global... |

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240 |
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(Show Context)
Citation Context ...and computation trees over F . The problem of learning polynomials in various learning models (including using only membership queries, i.e., interpolation), has been considered in many papers (e.g., =-=[31, 35, 38, 41, 49, 51, 55, 76, 89, 115, 127, 129, 138, 139]-=-). In particular, when allowed membership queries and equivalence queries (which implies learnability in the PAC model with membership queries), Beimel et al. [31] show that learning n-variate polynom... |

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(Show Context)
Citation Context ...y (where ɛ is the distance parameter that the algorithm is given as input) is the alternative hypothesis. There are two major mathematical approaches to the study of testing in statistics (see, e.g., =-=[136]-=- and [113]). In the first, the alternative is taken to approach the null hypothesis at a certain rate as a function of the number of data points; when the correct rate is chosen the error probabilitie... |

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(Show Context)
Citation Context ...ass of graph properties (in particular coloring and subgraph-freeness properties). As in [82] their algorithms do not have a dependence on n, but since they build on the Regularity Lemma of Szemerédi =-=[132]-=-, the dependence on 1/ɛ is quite high. Interestingly, Alon [3] proved that for subgraph freeness, if the subgraph is not bipartite, then the dependence on 1/ɛ must be superpolynomial. A sequence of wo... |

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Citation Context ...ons (that is, functions of the form f : F n → F ). We note that by building on [21] it is possible to obtain a linear dependence on d in the case of degree-d polynomials and sufficiently large fields =-=[131]-=-. Class of functions Number of queries References linear functions O(1/ɛ) [42] univariate deg-d O(d +1/ɛ) [128] polynomials deg-d polynomials, O(poly(d)/ɛ) [128] |F |≥d +2 deg-d polynomials, |F | =2 O... |

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55 |
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(Show Context)
Citation Context ...and computation trees over F . The problem of learning polynomials in various learning models (including using only membership queries, i.e., interpolation), has been considered in many papers (e.g., =-=[31, 35, 38, 41, 49, 51, 55, 76, 89, 115, 127, 129, 138, 139]-=-). In particular, when allowed membership queries and equivalence queries (which implies learnability in the PAC model with membership queries), Beimel et al. [31] show that learning n-variate polynom... |

54 | Testing random variables for independence and identity - Batu, Fortnow, et al. - 2001 |

52 | Learning in the presence of finitely or infinitely many irrelevant attributes - Blum, Hellerstein, et al. - 1995 |

51 | Testing k-colorability - Alon, Krivelevich |

48 | Testing juntas - Fischer, Kindler, et al. - 2004 |

48 | Some improvements to total degree tests - Friedl, Sudan - 1995 |

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43 | Every monotone graph property is testable - Alon, Shapira - 2005 |

42 | Approximating the minimum spanning tree weight in sublinear time - Chazelle, Rubinfeld, et al. - 2005 |

40 | Testing polynomials over general fields - Kaufman, Ron |

39 | Testing subgraphs in large graphs - Alon |

38 | Graph limits and parameter testing - Borgs, Chayes, et al. - 2006 |

38 | More efficient PAC learning of DNF with membership queries under the uniform distribution - Bshouty, Jackson, et al. - 1999 |

38 | Boosting and hard-core sets - Klivans, Servedio - 2003 |

37 | Testing basic boolean formulae
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(Show Context)
Citation Context ... noted when the result is discussed. The main results mentioned in this section are summarized in Table 4.1. 4.1 Singletons, Monomials, and Monotone DNF In this subsection, we describe the results of =-=[125]-=-. We give full details for the simplest case of testing the class of singleton functions, and only the high-level ideas for the more complex problems of testing monomials and monotone DNF. The query c... |

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34 | Learning sparse multivariate polynomials over a field with queries and counterexamples
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(Show Context)
Citation Context ...and computation trees over F . The problem of learning polynomials in various learning models (including using only membership queries, i.e., interpolation), has been considered in many papers (e.g., =-=[31, 35, 38, 41, 49, 51, 55, 76, 89, 115, 127, 129, 138, 139]-=-). In particular, when allowed membership queries and equivalence queries (which implies learnability in the PAC model with membership queries), Beimel et al. [31] show that learning n-variate polynom... |

33 | Testing Reed-Muller Codes - Alon, Kaufman, et al. |

32 | Linearity testing over characteristic two - Bellare, Coppersmith, et al. - 1996 |

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30 | A lower bound for testing 3-colorability in bounded-degree graphs - Bogdanov, Obata, et al. - 2002 |

30 | Fast approximation PCPs for multidimensional binpacking problems - Batu, Rubinfeld, et al. - 1999 |

29 | On the strength of comparisons in property testing, manuscript (available as ECCC - Fischer |