EM for Mixture of Linear Regression with Clustered Data

TL;DR

This paper demonstrates that exploiting clustered structures in distributed data significantly reduces the number of EM iterations needed for accurate parameter estimation in mixture of linear regressions, especially with proper initialization.

Contribution

It introduces a novel analysis showing EM converges in a constant number of iterations when data has a clustered structure, improving over the standard logarithmic iteration complexity.

Findings

EM requires only O(1) iterations with structured data

Proper initialization is crucial for fast convergence

The analysis combines asymptotic optimization and generalization guarantees

Abstract

Modern data-driven and distributed learning frameworks deal with diverse massive data generated by clients spread across heterogeneous environments. Indeed, data heterogeneity is a major bottleneck in scaling up many distributed learning paradigms. In many settings however, heterogeneous data may be generated in clusters with shared structures, as is the case in several applications such as federated learning where a common latent variable governs the distribution of all the samples generated by a client. It is therefore natural to ask how the underlying clustered structures in distributed data can be exploited to improve learning schemes. In this paper, we tackle this question in the special case of estimating $d$-dimensional parameters of a two-component mixture of linear regressions problem where each of $m$ nodes generates $n$ samples with a shared latent variable. We employ the well-known Expectation-Maximization (EM) method to estimate the maximum likelihood parameters from $m$ batches of dependent samples each containing $n$ measurements. Discarding the clustered structure in the mixture model, EM is known to require $O(\log(mn/d))$ iterations to reach the statistical accuracy of $O(\sqrt{d/(mn)})$. In contrast, we show that if initialized properly, EM on the structured data requires only $O(1)$ iterations to reach the same statistical accuracy, as long as $m$ grows up as $e^{o(n)}$. Our analysis establishes and combines novel asymptotic optimization and generalization guarantees for population and empirical EM with dependent samples, which may be of independent interest.