Likelihood landscape of binary latent model on a tree

TL;DR

This paper provides a theoretical analysis of the likelihood landscape for binary latent tree models, showing conditions under which coordinate maximization efficiently finds the MLE.

Contribution

It proves strong concavity of the population log-likelihood in the reconstruction regime and explains the success of simple algorithms in latent tree inference.

Findings

Population log-likelihood is strongly concave near the true parameter.

Coordinate maximization converges exponentially fast to an accurate MLE.

Decay property of the population Hessian explains optimization success.

Abstract

We investigate the optimization landscape of maximum likelihood estimation (MLE) for the Cavender-Farris-Neyman (CFN) model, a two-state latent tree model fundamental to statistical phylogenetics and the ferromagnetic Ising model. Although the log-likelihood function is non-concave and may admit many critical points, simple coordinate maximization algorithms are remarkably effective in practice. We provide the first theoretical justification for this success. We prove that sufficiently deep inside the reconstruction regime, the population log-likelihood is strongly concave and smooth within a box around the true parameter, whose size is independent of tree topology and number of leaves. This fundamental result implies that the empirical landscape shares these regularity properties with high probability given polynomial sample complexity and also that coordinate maximization converges exponentially fast to an $O(1/\sqrt{m})$-consistent MLE. Our analysis centers on a novel decay property of the population Hessian: diagonal entries remain large while off-diagonal entries decay exponentially with graph distance. These results provide rigorous theoretical evidence for the efficacy of likelihood-based tree inference and suggest broader principles for latent variable models.