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Property Testing of Boolean Function
The field of property testing has been studied for decades, and Boolean functions are among the most classical subjects to study in this area.
In this thesis we consider the property testing of Boolean functions: distinguishing whether an unknown Boolean function has some certain property (or equivalently, belongs to a certain class of functions), or is far from having this property. We study this problem under both the standard setting, where the distance between functions is measured with respect to the uniform distribution, as well as the distribution-free setting, where the distance is measured with respect to a fixed but unknown distribution.
We obtain both new upper bounds and lower bounds for the query complexity of testing various properties of Boolean functions:
- Under the standard model of property testing, we prove a lower bound of \Omega(n^{1/3}) for the query complexity of any adaptive algorithm that tests whether an n-variable Boolean function is monotone, improving the previous best lower bound of \Omega(n^{1/4}) by Belov and Blais in 2015. We also prove a lower bound of \Omega(n^{2/3}) for adaptive algorithms, and a lower bound of \Omega(n) for non-adaptive algorithms with one-sided errors that test unateness, a natural generalization of monotonicity. The latter lower bound matches the previous upper bound proved by Chakrabarty and Seshadhri in 2016, up to poly-logarithmic factors of n.
- We also study the distribution-free testing of k-juntas, where a function is a k-junta if it depends on at most k out of its n input variables. The standard property testing of k-juntas under the uniform distribution has been well understood: it has been shown that, for adaptive testing of k-juntas the optimal query complexity is \Theta(k); and for non-adaptive testing of k-juntas it is \Theta(k^{3/2}). Both bounds are tight up to poly-logarithmic factors of k. However, this problem is far from clear under the more general setting of distribution-free testing. Previous results only imply an O(2^k)-query algorithm for distribution-free testing of k-juntas, and besides lower bounds under the uniform distribution setting that naturally extend to this more general setting, no other results were known from the lower bound side. We significantly improve these results with an O(k^2)-query adaptive distribution-free tester for k-juntas, as well as an exponential lower bound of \Omega(2^{k/3}) for the query complexity of non-adaptive distribution-free testers for this problem. These results illustrate the hardness of distribution-free testing and also the significant role of adaptivity under this setting.
- In the end we also study distribution-free testing of other basic Boolean functions. Under the distribution-free setting, a lower bound of \Omega(n^{1/5}) was proved for testing of conjunctions, decision lists, and linear threshold functions by Glasner and Servedio in 2009, and an O(n^{1/3})-query algorithm for testing monotone conjunctions was shown by Dolev and Ron in 2011. Building on techniques developed in these two papers, we improve these lower bounds to \Omega(n^{1/3}), and specifically for the class of conjunctions we present an adaptive algorithm with query complexity O(n^{1/3}). Our lower and upper bounds are tight for testing conjunctions, up to poly-logarithmic factors of n
Almost Optimal Distribution-Free Junta Testing
We consider the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1}^n. Chen, Liu, Servedio, Sheng and Xie [Zhengyang Liu et al., 2018] showed that the distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes O~(k^2)/epsilon queries. In this paper, we give a simple two-sided error adaptive algorithm that makes O~(k/epsilon) queries
Distribution-Free Proofs of Proximity
Motivated by the fact that input distributions are often unknown in advance,
distribution-free property testing considers a setting in which the algorithmic
task is to accept functions having a certain property
and reject functions that are -far from , where the
distance is measured according to an arbitrary and unknown input distribution
. As usual in property testing, the tester is required to do so
while making only a sublinear number of input queries, but as the distribution
is unknown, we also allow a sublinear number of samples from the distribution
.
In this work we initiate the study of distribution-free interactive proofs of
proximity (df-IPP) in which the distribution-free testing algorithm is assisted
by an all powerful but untrusted prover. Our main result is a df-IPP for any
problem , with communication, sample, query,
and verification complexities, for any proximity parameter
. For such proximity parameters, this result matches the
parameters of the best-known general purpose IPPs in the standard uniform
setting, and is optimal under reasonable cryptographic assumptions.
For general values of the proximity parameter , our
distribution-free IPP has optimal query complexity but the
communication complexity is , which
is worse than what is known for uniform IPPs when . With
the aim of improving on this gap, we further show that for IPPs over
specialised, but large distribution families, such as sufficiently smooth
distributions and product distributions, the communication complexity can be
reduced to (keeping the query
complexity roughly the same as before) to match the communication complexity of
the uniform case
On sample-based testers
The standard definition of property testing endows the tester with the ability to make arbitrary queries to “elements ” of the tested object. In contrast, sample-based testers only obtain independently distributed elements (a.k.a. labeled samples) of the tested object. While samplebased testers were defined by Goldreich, Goldwasser, and Ron (JACM 1998), most research in property testing is focused on query-based testers. In this work, we advance the study of sample-based property testers by providing several general positive results as well as by revealing relations between variants of this testing model. In particular: • We show that certain types of query-based testers yield sample-based testers of sublinear sample complexity. For example, this holds for a natural class of proximity oblivious testers. • We study the relation between distribution-free sample-based testers and one-sided error sample-based testers w.r.t the uniform distribution. While most of this work ignores the time complexity of testing, one part of it does focus on this aspect. The main result in this part is a sublinear-time sample-based tester for k-Colorability, for any k ≥ 2
A Nonparametric Test For Homogeneity Of Variances: Application To GPAs Of Students Across Academic Majors
We propose a nonparametric (or distribution-free) procedure for testing the equality of several population variances (or scale parameters). The proposed test is a modification of Bakir’s (1989, Commun. Statist., Simul-Comp., 18, 757-775) analysis of means by ranks (ANOMR) procedure for testing the equality of several population means. A proof is given to establish the distribution-free property of the modified procedure. The proposed procedure is then applied to test whether or not the variability in the grade point averages (GPAs) of students differs across five business academic majors. We collect the GPAs (observations) of a random sample of students from each major under study. The absolute deviations of the observations from the overall median of the combined sample are then calculated and ranked from least to largest. The average ranks and two decision lines are then plotted on a graph paper to detect not only the existence of significant differences among variances, but also to pinpoint which variances are causing those differences
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