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# Towards Testing Monotonicity of Distributions Over General Posets

*Proceedings of the Thirty-Second Conference on Learning Theory*, PMLR 99:34-82, 2019.

#### Abstract

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 $x \preceq y$, $p(x) \leq p(y)$. 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 $\Omega(n/\log n)$ for testing bigness of distributions on domains of size $n$. We then build on these lower bounds to give $\Omega(n/\log{n})$ 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.