Monotone Classification with Relative Approximations

Abstract:In monotone classification, the input is a multi-set $P$ of points in $\mathbb{R}^d$, each associated with a hidden label from $\{-1, 1\}$. The goal is to identify a monotone function $h$, which acts as a classifier, mapping from $\mathbb{R}^d$ to $\{-1, 1\}$ with a small {\em error}, measured as the number of points $p \in P$ whose labels differ from the function values $h(p)$. The cost of an algorithm is defined as the number of points having their labels revealed. This article presents the first study on the lowest cost required to find a monotone classifier whose error is at most $(1 + \epsilon) \cdot k^*$ where $\epsilon \ge 0$ and $k^*$ is the minimum error achieved by an optimal monotone classifier -- in other words, the error is allowed to exceed the optimal by at most a relative factor. Nearly matching upper and lower bounds are presented for the full range of $\epsilon$. All previous work on the problem can only achieve an error higher than the optimal by an absolute factor.

Machine Learning (cs.LG)

arXiv:2506.10775 [cs.LG]

(or arXiv:2506.10775v3 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2506.10775 Focus to learn more arXiv-issued DOI via DataCite

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