Revisiting Instance-Optimal Cluster Recovery in the Labeled Stochastic Block Model

TL;DR

This paper introduces IAC, a scalable, instance-adaptive clustering algorithm for the Labeled Stochastic Block Model that achieves optimal recovery performance without prior knowledge of model parameters.

Contribution

The paper presents IAC, the first algorithm matching instance-specific lower bounds in expectation and high probability, with a simple spectral plus iterative refinement approach.

Findings

IAC achieves near-optimal misclassification bounds.

IAC operates with $O(n ext{polylog}(n))$ complexity.

The method does not require prior knowledge of model parameters.

Abstract

In this paper, we investigate the problem of recovering hidden communities in the Labeled Stochastic Block Model (LSBM) with a finite number of clusters whose sizes grow linearly with the total number of nodes. We derive the necessary and sufficient conditions under which the expected number of misclassified nodes is less than $ s $, for any number $ s = o(n) $. To achieve this, we propose IAC (Instance-Adaptive Clustering), the first algorithm whose performance matches the instance-specific lower bounds both in expectation and with high probability. IAC is a novel two-phase algorithm that consists of a one-shot spectral clustering step followed by iterative likelihood-based cluster assignment improvements. This approach is based on the instance-specific lower bound and notably does not require any knowledge of the model parameters, including the number of clusters. By performing the spectral clustering only once, IAC maintains an overall computational complexity of $ \mathcal{O}(n\, \text{polylog}(n)) $, making it scalable and practical for large-scale problems.