Testing unateness nearly optimally

Chen, Xi; Waingarten, Erik

doi:10.1145/3313276.3316351

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…Subsequently, Chen et al [30] gave an adaptive unateness tester with query complexity O d 3/4 /ε 2 for the same class of functions. An exciting recent development is an O d 2/3 /ε 2 -query algorithm for this problem by Chen and Waingarten [28], which only leaves a polylogarithmic (in d) gap between the upper bound and the lower bound.…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

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Baleshzar¹,

Chakrabarty²,

Pallavoor³

et al. 2020

Theory of Comput.

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We study the problem of testing unateness of functions f : {0, 1} d → R. A function f : {0, 1} d → R is unate if for every coordinate i ∈ [d], the function is either nonincreasing in the i th coordinate or nondecreasing in the i th coordinate. We give an O((d/ε) • log(d/ε))-query nonadaptive tester and an O(d/ε)-query adaptive tester and show that both testers are optimal for a fixed distance parameter ε. Previously known unateness testers worked only for Boolean functions, and their query complexity had worse dependence on the dimension d both for the adaptive and the nonadaptive case. Moreover, no lower bounds for testing unateness were known. (Concurrent work by Chen et al. (STOC'17) proved an Ω(d/ log 2 d) lower bound on the nonadaptive query complexity of testing unateness of Boolean functions.) We also generalize our results to obtain optimal unateness testers for functions f : [n] d → R.

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Section: Conclusion and Open Questionsmentioning

confidence: 99%

Untitled

Baleshzar¹,

Chakrabarty²,

Pallavoor³

et al. 2020

Theory of Comput.

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“…Testing monotonicity and unateness (first studied in [41]), as well as

k

‐juntas (first studied in [38]), are among the most widely investigated problems in property testing ([1, 2, 4, 5, 8–10, 13, 15, 16, 19–25, 28, 33, 34, 36, 39, 44, 45, 48, 53, 55, 56] study monotonicity testing, [3, 27–29, 46] study unateness testing, and [11, 12, 17, 26, 30, 60, 61] study

k

‐junta testing). Nearly all the previous work on these properties is in the standard testing model.…”

Section: Introductionmentioning

confidence: 99%

“…Nearly all the previous work on these properties is in the standard testing model. The best bounds on the query complexity of these problems are an

Õ (\sqrt{n})

‐query nonadaptive algorithm of [45] and lower bounds of

\tilde{Ω} (\sqrt{n})

(nonadaptive) and

\tilde{Ω} (n^{1 / 3})

(adaptive) [28] for monotonicity, and tight upper and lower bounds of

\tilde{Θ} (n^{2 / 3})

for unateness testing [27, 28], as well as

Θ (k \log k)

for

k

‐junta testing [11, 60].…”

Section: Introductionmentioning

confidence: 99%

Approximating the distance to monotonicity of Boolean functions

Pallavoor

Raskhodnikova

Waingarten

2021

Random Struct Algorithms

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We design a nonadaptive algorithm that, given oracle access to a function f:{0,1} n→{0,1} which is α‐far from monotone, makes poly(n,1/α) queries and returns an estimate that, with high probability, is an Õ(n)‐approximation to the distance of f to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly(n,1/α)‐query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing that, for all constant κ>0, every nonadaptive n1/2−κ‐approximation algorithm for this problem requires 2nκ queries. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. We obtain our lower bound by proving an analogous bound for erasure‐resilient (and tolerant) testers. Our method also yields the same lower bounds for unateness and being a k‐junta.

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Agnostic proper learning of monotone functions: beyond the black-box correction barrier

Lange,

Vasilyan

2023

2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)

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Testing unateness nearly optimally

Cited by 4 publications

References 15 publications

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Approximating the distance to monotonicity of Boolean functions

Agnostic proper learning of monotone functions: beyond the black-box correction barrier

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