# Active Learning a Convex Body in Low Dimensions

## TL;DR

This paper investigates the problem of classifying points relative to a convex body in low dimensions, providing query complexity bounds based on geometric properties of the point set.

## Contribution

It introduces new bounds for active learning in low dimensions, linking query complexity to convex position and specific instance parameters.

## Key findings

- In 2D and 3D, the query complexity is $O(h(P) \, \log n)$.
- In 2D, the complexity is also bounded by $O(v(P,C) \, \log^2 n)$, matching lower bounds.
- The results connect geometric properties with active learning efficiency.

## Abstract

Consider a set $P \subseteq \Re^d$ of $n$ points, and a convex body $C$ provided via a separation oracle. The task at hand is to decide for each point of $P$ if it is in $C$ using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using $O( h(P) \log n)$ queries, where $h(P)$ is the largest subset of points of $P$ in convex position. Furthermore, we show that in two dimensions one can solve this problem using $O( v(P,C) \log^2 n )$ oracle queries, where $v(P, C)$ is a lower bound on the minimum number of queries that any algorithm for this specific instance requires.