Sample Complexity of Branch-length Estimation by Maximum Likelihood

TL;DR

This paper provides the first theoretical explanation for the effectiveness of simple coordinate maximization in estimating branch lengths in phylogenetic trees, showing strong concavity and fast convergence under certain conditions.

Contribution

It proves that in the Kesten-Stigum regime, the likelihood landscape is strongly concave, enabling exponential convergence of coordinate maximization to the MLE.

Findings

Likelihood landscape is strongly concave in the Kesten-Stigum regime.

Coordinate maximization converges exponentially fast to the MLE.

Estimation error is within O(1/√m) with polynomial samples.

Abstract

We consider the branch-length estimation problem on a bifurcating tree: a character evolves along the edges of a binary tree according to a two-state symmetric Markov process, and we seek to recover the edge transition probabilities from repeated observations at the leaves. This problem arises in phylogenetics, and is related to latent tree graphical model inference. In general, the log-likelihood function is non-concave and may admit many critical points. Nevertheless, simple coordinate maximization has been known to perform well in practice, defying the complexity of the likelihood landscape. In this work, we provide the first theoretical guarantee as to why this might be the case. We show that deep inside the Kesten-Stigum reconstruction regime, provided with polynomially many $m$ samples (assuming the tree is balanced), there exists a universal parameter regime (independent of the size of the tree) where the log-likelihood function is strongly concave and smooth with high probability. On this high-probability likelihood landscape event, we show that the standard coordinate maximization algorithm converges exponentially fast to the maximum likelihood estimator, which is within $O(1/\sqrt{m})$ from the true parameter, provided a sufficiently close initial point.