Concentration Bounds for the Collision Estimator

TL;DR

This paper establishes strong concentration bounds for the collision estimator, a key tool in uniformity testing and entropy estimation, demonstrating its reliability without additional boosting techniques.

Contribution

It provides the first high-probability concentration bounds for the collision estimator, extending beyond variance to higher moments using novel techniques.

Findings

Achieves high-probability guarantees without boosting

Bounds higher moments of the estimator

Enhances understanding of collision estimator's reliability

Abstract

We prove a strong concentration result about the natural collision estimator, which counts the number of collisions that occur within an iid sample. This estimator is at the heart of algorithms used for uniformity testing and entropy assessment. While the prior works were limited to only variance, we use elegant techniques of independent interest to bounds higher moments and conclude concentration properties. As an immediate corollary we show that the estimator achieves high-probability guarantee on its own and there is no need for boosting (aka median/majority trick).