Monotone Classification with Relative Approximations

TL;DR

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.

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 $(1 + ε) \cdot k^*$ 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.

Figures (2)

• Figure 1: An input set $P$ for Problem 1
• Figure 2: Illustration of the dominance width

Theorems & Definitions (6)

• Example 1.1
• Example 1.2
• Example 1.3
• Theorem 1
• Proposition 1
• proof