More Like this

1. Testing k-Monotonicity

   Canonne, C; Grigorescu, E.; Guo, S; Kumar, A; Wimmer, K. (January 2017, ITCS)

   A Boolean {\em $$k$$-monotone} function defined over a finite poset domain $${\cal D}$$ alternates between the values $$0$$ and $$1$$ at most $$k$$ times on any ascending chain in $${\cal D}$$. Therefore, $$k$$-monotone functions are natural generalizations of the classical {\em monotone} functions, which are the {\em $$1$$-monotone} functions. Motivated by the recent interest in $$k$$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of $$k$$-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are $$k$$-monotone (or are close to being $$k$$-monotone) from functions that are far from being $$k$$-monotone. Our results include the following: \begin{enumerate} \item We demonstrate a separation between testing $$k$$-monotonicity and testing monotonicity, on the hypercube domain $$\{0,1\}^d$$, for $$k\geq 3$$; \item We demonstrate a separation between testing and learning on $$\{0,1\}^d$$, for $$k=\omega(\log d)$$: testing $$k$$-monotonicity can be performed with $$2^{O(\sqrt d \cdot \log d\cdot \log{1/\eps})}$$ queries, while learning $$k$$-monotone functions requires $$2^{\Omega(k\cdot \sqrt d\cdot{1/\eps})}$$ queries (Blais et al. (RANDOM 2015)). \item We present a tolerant test for functions $$f\colon[n]^d\to \{0,1\}$$ with complexity independent of $$n$$, which makes progress on a problem left open by Berman et al. (STOC 2014). \end{enumerate} Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid $$[n]^d$, and draw connections to distribution testing techniques. 

   more » « less
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 

   more » « less
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. 

   more » « less
4. Approximating the Distance to Monotonicity of Boolean Functions

   https://doi.org/10.1137/1.9781611975994.123  

   Pallavoor, Ramesh Krishnan; Raskhodnikova, Sofya; Waingarten, Erik (January 2020, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020)

   We design a nonadaptive algorithm that, given a Boolean 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 O-tilde(\sqrt{n})-approximation to the distance of f to monotonicity. Furthermore, we show that for any constant k > 0, approximating the distance to monotonicity up to n^(1/2−k)-factor requires 2^{n^k} nonadaptive queries, thereby ruling out a poly(n, 1/α)-query nonadaptive algorithm for such approximations. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. Approximating the distance to a property is closely related to tolerantly testing that property. Our lower bound stands in contrast to standard (non-tolerant) testing of monotonicity that can be done nonadaptively with O-tilde(n/ε^2) queries. We obtain our lower bound by proving an analogous bound for erasure-resilient testers. An α-erasure-resilient tester for a desired property gets oracle access to a function that has at most an α fraction of values erased. The tester has to accept (with probability at least 2/3) if the erasures can be filled in to ensure that the resulting function has the property and to reject (with probability at least 2/3) if every completion of erasures results in a function that is ε-far from having the property. Our method yields the same lower bounds for unateness and being a k-junta. These lower bounds improve exponentially on the existing lower bounds for these properties. 

   more » « less
5. 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. 

   more » « less