Clustering with Queries under Semi-Random Noise

TL;DR

This paper introduces robust clustering algorithms that operate effectively under semi-random noise, matching the performance of fully-random models and providing the first parameter-free algorithm for such settings.

Contribution

It develops the first computationally efficient, semi-random noise-tolerant clustering algorithms with guarantees matching fully-random models, and introduces a parameter-free algorithm for the fully-random case.

Findings

Queries needed scale as O(nk log n / (1-2p)^2).

Algorithms can identify large clusters efficiently under semi-random noise.

First parameter-free clustering algorithm for fully-random noise model.

Abstract

The seminal paper by Mazumdar and Saha \cite{MS17a} introduced an extensive line of work on clustering with noisy queries. Yet, despite significant progress on the problem, the proposed methods depend crucially on knowing the exact probabilities of errors of the underlying fully-random oracle. In this work, we develop robust learning methods that tolerate general semi-random noise obtaining qualitatively the same guarantees as the best possible methods in the fully-random model. More specifically, given a set of $n$ points with an unknown underlying partition, we are allowed to query pairs of points $u,v$ to check if they are in the same cluster, but with probability $p$, the answer may be adversarially chosen. We show that information theoretically $O\left(\frac{nk \log n} {(1-2p)^2}\right)$ queries suffice to learn any cluster of sufficiently large size. Our main result is a computationally efficient algorithm that can identify large clusters with $O\left(\frac{nk \log n} {(1-2p)^2}\right) + \text{poly}\left(\log n, k, \frac{1}{1-2p} \right)$ queries, matching the guarantees of the best known algorithms in the fully-random model. As a corollary of our approach, we develop the first parameter-free algorithm for the fully-random model, answering an open question by \cite{MS17a}.