Monotone Classification with Relative Approximations
This addresses a theoretical gap in monotone classification for researchers in computational learning theory, providing a relative approximation framework that is incremental over existing absolute error methods.
The paper tackles the problem of monotone classification by introducing the first study on the cost required to find a classifier with error at most (1 + ε) times the optimal, presenting nearly matching upper and lower bounds for all ε, whereas prior work only achieved absolute error bounds.
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 $(1 + ε) \cdot k^*$ where $ε\ge 0$ 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 $ε$. All previous work on the problem can only achieve an error higher than the optimal by an absolute factor.