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1. Towards Testing Monotonicity of Distributions Over General Posets

   Alaikbarpour, A.; Gouleakis, T.; Peebles, J.; Rubinfeld, R.; Yodpinyanee, A (January 2019, Proceedings of the Thirty-Second Conference on Learning Theory (COLT 2019))

   In this work, we consider the sample complexity required for testing the monotonicity of distributions over partial orders. A distribution p over a poset is monotone if, for any pair of domain elements x and y such that x ⪯ y, p(x) ≤ p(y). To understand the sample complexity of this problem, we introduce a new property called bigness over a finite domain, where the distribution is T-big if the minimum probability for any domain element is at least T. We establish a lower bound of Ω(n/ log n) for testing bigness of distributions on domains of size n. We then build on these lower bounds to give Ω(n/ log n) lower bounds for testing monotonicity over a matching poset of size n and significantly improved lower bounds over the hypercube poset. We give sublinear sample complexity bounds for testing bigness and for testing monotonicity over the matching poset. We then give a number of tools for analyzing upper bounds on the sample complexity of the monotonicity testing problem. The previous lower bound for testing Monotonicity of 

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2. Testing Intersecting and Union-Closed Families

   https://doi.org/10.4230/LIPICS.ITCS.2024.33  

   Chen, Xi; De, Anindya; Li, Yuhao; Nadimpalli, Shivam; Servedio, Rocco A (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Guruswami, Venkatesan (Ed.)

   Inspired by the classic problem of Boolean function monotonicity testing, we investigate the testability of other well-studied properties of combinatorial finite set systems, specifically intersecting families and union-closed families. A function f: {0,1}ⁿ → {0,1} is intersecting (respectively, union-closed) if its set of satisfying assignments corresponds to an intersecting family (respectively, a union-closed family) of subsets of [n]. Our main results are that - in sharp contrast with the property of being a monotone set system - the property of being an intersecting set system, and the property of being a union-closed set system, both turn out to be information-theoretically difficult to test. We show that: - For ε ≥ Ω(1/√n), any non-adaptive two-sided ε-tester for intersectingness must make 2^{Ω(n^{1/4}/√{ε})} queries. We also give a 2^{Ω(√{n log(1/ε)})}-query lower bound for non-adaptive one-sided ε-testers for intersectingness. - For ε ≥ 1/2^{Ω(n^{0.49})}, any non-adaptive two-sided ε-tester for union-closedness must make n^{Ω(log(1/ε))} queries. Thus, neither intersectingness nor union-closedness shares the poly(n,1/ε)-query non-adaptive testability that is enjoyed by monotonicity. To complement our lower bounds, we also give a simple poly(n^{√{nlog(1/ε)}},1/ε)-query, one-sided, non-adaptive algorithm for ε-testing each of these properties (intersectingness and union-closedness). We thus achieve nearly tight upper and lower bounds for two-sided testing of intersectingness when ε = Θ(1/√n), and for one-sided testing of intersectingness when ε = Θ(1). 

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3. Monotone probability distributions over the Boolean cube can be learned with sublinear samples

   Rubinfeld, Ronitt; Vasilyan, Arsen (January 2020, 11th Innovations in Theoretical Computer Science (ITCS 2020))

   A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we give (to our knowledge) the first algorithm that learns an approximation of the distribution in statistical distance using a number of samples that is sublinear in the domain. To do this, we develop a structural lemma describing monotone probability distributions. The structural lemma has further implications to the sample complexity of basic testing tasks for analyzing monotone probability distributions over the Boolean cube: We use it to give nontrivial upper bounds on the tasks of estimating the distance of a monotone distribution to uniform and of estimating the support size of a monotone distribution. In the setting of monotone probability distributions over the Boolean cube, our algorithms are the first to have sample complexity lower than known lower bounds for the same testing tasks on arbitrary (not necessarily monotone) probability distributions. One further consequence of our learning algorithm is an improved sample complexity for the task of testing whether a distribution on the Boolean cube is monotone. 

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4. Monotone Probability Distributions over the Boolean Cube Can Be Learned with Sublinear Samples

   Rubinfeld, Ronitt; Vasilyan, Arsen (January 2020, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020)

   null (Ed.)

   A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we give (to our knowledge) the first algorithm that learns an approximation of the distribution in statistical distance using a number of samples that is sublinear in the domain. To do this, we develop a structural lemma describing monotone probability distributions. The structural lemma has further implications to the sample complexity of basic testing tasks for analyzing monotone probability distributions over the Boolean cube: We use it to give nontrivial upper bounds on the tasks of estimating the distance of a monotone distribution to uniform and of estimating the support size of a monotone distribution. In the setting of monotone probability distributions over the Boolean cube, our algorithms are the first to have sample complexity lower than known lower bounds for the same testing tasks on arbitrary (not necessarily monotone) probability distributions. One further consequence of our learning algorithm is an improved sample complexity for the task of testing whether a distribution on the Boolean cube is monotone. 

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5. The query complexity of certification

   https://doi.org/10.1145/3519935.3519993  

   Blanc, G.; Koch, C.; Lange, J.; Tan, L. (January 2022, 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 22))

   We study the problem of certification: given queries to a function f : {0,1}n → {0,1} with certificate complexity ≤ k and an input x⋆, output a size-k certificate for f’s value on x⋆. For monotone functions, a classic local search algorithm of Angluin accomplishes this task with n queries, which we show is optimal for local search algorithms. Our main result is a new algorithm for certifying monotone functions with O(k8 logn) queries, which comes close to matching the information-theoretic lower bound of Ω(k logn). The design and analysis of our algorithm are based on a new connection to threshold phenomena in monotone functions. We further prove exponential-in-k lower bounds when f is non-monotone, and when f is monotone but the algorithm is only given random examples of f. These lower bounds show that assumptions on the structure of f and query access to it are both necessary for the polynomial dependence on k that we achieve. 

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