By Dana Ron

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**Additional info for Property Testing: a Learning Theory Perspective (Foundations and Trends in Machine Learning)**

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A function f : {0, 1}n → {0, 1} is a k-junta for an integer k ≤ n if f is a function of at most k variables. Namely, there exists a set J ⊆ [n], where |J| ≤ k such that f (x) = f (y) for every x, y ∈ {0, 1}n that satisfy xi = yi for each i ∈ J. We say in such a case that J dominates the function f . The main result of [71] is stated next. 6. For every ﬁxed k, the property of being a k-junta is testable using poly(k)/ queries. Fischer et al. 6 by describing and analyzing several algorithms. 2 Juntas 353 the algorithm is nonadaptive or adaptive (that is, queries may depend on answers to previous queries), and whether it has one-sided error or √ ˜ two-sided error.

They prove that for these classes, in contrast to standard testing, where the query complexity does not depend on n, every distribution-free 368 Other Models of Testing testing algorithm must make Ω((n/ log n)1/5 ) queries (for constant ). While there is still a gap between this lower bound and the upper bound implied by learning these classes, a strong dependence on n is unavoidable in the distribution-free case. Finally, we note that Halevy and Kushilevitz [93] also study distribution-free testing of graph properties in sparse graphs, and give an algorithm for distribution-free testing of connectivity, with similar complexity to the standard testing algorithm for this property.

3. 2 Low-Degree Polynomials 337 of V f (x; y1 , . . , y ), taken over all y1 , . . , y , equals the vote of a large fraction of the -tuples y1 , . . , y (assuming η(f ) is suﬃciently small). 13. ,y V f (x; y1 , . . , y ) = g(x) . 25) Then γ(x) ≥ 1 − 2q η(f ). Proof Sketch. 13 it will actually be more convenient to work with another measure of “correctness” (or “consistency”) of a point x. ,z V f (x; y1 , . . , y ) = V f (x; z1 , . . , z ) . 5, γ(x) ≥ δ(x). Hence it suﬃces to obtain a lower bound on δ(x).