Universal Lower Bounds and Optimal Rates: Achieving Minimax Clustering Error in Sub-Exponential Mixture Models
This work provides theoretical guarantees for clustering algorithms in various mixture models, addressing a foundational problem in unsupervised learning, but it is incremental as it extends existing results to broader distributions.
The paper establishes a universal lower bound for clustering error in mixture models using Chernoff divergence and shows that iterative algorithms achieve this bound for sub-exponential mixtures, including Laplace-distributed errors, with Bregman hard clustering being optimal for exponential family mixtures.
Clustering is a pivotal challenge in unsupervised machine learning and is often investigated through the lens of mixture models. The optimal error rate for recovering cluster labels in Gaussian and sub-Gaussian mixture models involves ad hoc signal-to-noise ratios. Simple iterative algorithms, such as Lloyd's algorithm, attain this optimal error rate. In this paper, we first establish a universal lower bound for the error rate in clustering any mixture model, expressed through a Chernoff divergence, a more versatile measure of model information than signal-to-noise ratios. We then demonstrate that iterative algorithms attain this lower bound in mixture models with sub-exponential tails, notably emphasizing location-scale mixtures featuring Laplace-distributed errors. Additionally, for datasets better modelled by Poisson or Negative Binomial mixtures, we study mixture models whose distributions belong to an exponential family. In such mixtures, we establish that Bregman hard clustering, a variant of Lloyd's algorithm employing a Bregman divergence, is rate optimal.