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1. 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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2. 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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3. Towards Testing Monotonicity of Distributions Over General Posets

   Alaikbarpour, Maryam ; Gouleakis, Themis ; Peebles, John ; Rubinfeld, Ronitt ; Yodpinyanee, Anak ( January 2019 , Proceedings of the Thirty-Second Conference on Learning Theory, PMLR)

   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 {\em 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 \emph{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/logn) for testing bigness of distributions on domains of size n. We then build on these lower bounds to give Ω(n/logn) 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. 

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4. Sample-Optimal Identity Testing with High Probability

   Diakonikolas, Ilias ; Gouleakis, Themis ; Peebles, John ; Price, Eric ( July 2018 , 45th International Colloquium on Automata, Languages, and Automata)

   We study the problem of testing identity against a given distribution with a focus on the high confidence regime. More precisely, given samples from an unknown distribution p over n elements, an explicitly given distribution q, and parameters 0< epsilon, delta < 1, we wish to distinguish, with probability at least 1-delta, whether the distributions are identical versus epsilon-far in total variation distance. Most prior work focused on the case that delta = Omega(1), for which the sample complexity of identity testing is known to be Theta(sqrt{n}/epsilon^2). Given such an algorithm, one can achieve arbitrarily small values of delta via black-box amplification, which multiplies the required number of samples by Theta(log(1/delta)). We show that black-box amplification is suboptimal for any delta = o(1), and give a new identity tester that achieves the optimal sample complexity. Our new upper and lower bounds show that the optimal sample complexity of identity testing is Theta((1/epsilon^2) (sqrt{n log(1/delta)} + log(1/delta))) for any n, epsilon, and delta. For the special case of uniformity testing, where the given distribution is the uniform distribution U_n over the domain, our new tester is surprisingly simple: to test whether p = U_n versus d_{TV} (p, U_n) >= epsilon, we simply threshold d_{TV}({p^}, U_n), where {p^} is the empirical probability distribution. The fact that this simple "plug-in" estimator is sample-optimal is surprising, even in the constant delta case. Indeed, it was believed that such a tester would not attain sublinear sample complexity even for constant values of epsilon and delta. An important contribution of this work lies in the analysis techniques that we introduce in this context. First, we exploit an underlying strong convexity property to bound from below the expectation gap in the completeness and soundness cases. Second, we give a new, fast method for obtaining provably correct empirical estimates of the true worst-case failure probability for a broad class of uniformity testing statistics over all possible input distributions - including all previously studied statistics for this problem. We believe that our novel analysis techniques will be useful for other distribution testing problems as well. 

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5. Agnostic Proper Learning of Monotone Functions: Beyond the Black-box Correction Barrier

   Lange, Jane ; Vasilyan, Arsen ( November 2023 , 64th IEEE Symposium on Foundations of Computer Science, FOCS 2023)

   We give the first agnostic, efficient, proper learning algorithm for monotone Boolean functions. Given 2O~(n√/ε) uniformly random examples of an unknown function f:{±1}n→{±1}, our algorithm outputs a hypothesis g:{±1}n→{±1} that is monotone and (opt +ε)-close to f, where opt is the distance from f to the closest monotone function. The running time of the algorithm (and consequently the size and evaluation time of the hypothesis) is also 2O~(n√/ε), nearly matching the lower bound of [13]. We also give an algorithm for estimating up to additive error ε the distance of an unknown function f to monotone using a run-time of 2O~(n√/ε). Previously, for both of these problems, sample-efficient algorithms were known, but these algorithms were not run-time efficient. Our work thus closes this gap in our knowledge between the run-time and sample complexity.This work builds upon the improper learning algorithm of [17] and the proper semiagnostic learning algorithm of [40], which obtains a non-monotone Boolean-valued hypothesis, then “corrects” it to monotone using query-efficient local computation algorithms on graphs. This black-box correction approach can achieve no error better than 2 opt +ε information-theoretically; we bypass this barrier bya)augmenting the improper learner with a convex optimization step, andb)learning and correcting a real-valued function before rounding its values to Boolean. Our real-valued correction algorithm solves the “poset sorting” problem of [40] for functions over general posets with non-Boolean labels. 

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