Almost Asymptotically Optimal Active Clustering Through Pairwise Observations

TL;DR

This paper introduces a new theoretical framework and an asymptotically optimal algorithm for active clustering of items into unknown groups using noisy pairwise queries, with proven bounds on query complexity.

Contribution

It provides a fundamental lower bound on query complexity and designs an asymptotically optimal active clustering algorithm based on the GLR statistic.

Findings

Established a lower bound on expected queries for clustering accuracy.

Designed an asymptotically optimal active clustering algorithm.

Showed the performance gap to the lower bound is within a constant factor.

Abstract

We propose a new analysis framework for clustering $M$ items into an unknown number of $K$ distinct groups using noisy and actively collected responses. At each time step, an agent is allowed to query pairs of items and observe bandit binary feedback. If the pair of items belongs to the same (resp.\ different) cluster, the observed feedback is $1$ with probability $p>1/2$ (resp.\ $q<1/2$). Leveraging the ubiquitous change-of-measure technique, we establish a fundamental lower bound on the expected number of queries needed to achieve a desired confidence in the clustering accuracy, formulated as a sup-inf optimization problem. Building on this theoretical foundation, we design an asymptotically optimal algorithm in which the stopping criterion involves an empirical version of the inner infimum -- the Generalized Likelihood Ratio (GLR) statistic -- being compared to a threshold. We develop a computationally feasible variant of the GLR statistic and show that its performance gap to the lower bound can be accurately empirically estimated and remains within a constant multiple of the lower bound.