We study the optimization landscape of maximum likelihood estimation for a binary latent tree model with hidden variables at internal nodes and observed variables at the leaves. This model, known as the Cavender-Farris-Neyman (CFN) model in statistical phylogenetics, is also a special case of the ferromagnetic Ising model. While the likelihood function is known to be non-concave with multiple critical points in general, gradient descent-type optimization methods have proven surprisingly effective in practice. We provide theoretical insights into this phenomenon by analyzing the population likelihood landscape in a neighborhood of the true parameter vector. Under some conditions on the edge parameters, we show that the expected log-likelihood is strongly concave and smooth in a box around the true parameter whose size is independent of both the tree topology and number of leaves. The key technical contribution is a careful analysis of the expected Hessian, showing that its diagonal entries are large while its off-diagonal entries decay exponentially in the graph distance between the corresponding edges. These results provide the first rigorous theoretical evidence for the effectiveness of optimization methods in this setting and may suggest broader principles for understanding optimization in latent variable models on trees.

翻译：本文研究了具有内部节点隐变量和叶节点观测变量的二元隐树模型的最大似然估计优化景观。该模型在统计系统发育学中被称为Cavender-Farris-Neyman（CFN）模型，同时也是铁磁伊辛模型的特例。尽管已知似然函数通常是非凹的且存在多个临界点，但梯度下降类优化方法在实践中却表现出惊人的有效性。我们通过分析真实参数向量邻域内的总体似然景观，为这一现象提供了理论解释。在边参数满足特定条件的情况下，我们证明期望对数似然在真实参数周围的一个盒状区域内具有强凹性和光滑性，且该区域的大小与树拓扑结构和叶节点数量均无关。关键的技术贡献在于对期望海森矩阵的精细分析，证明了其对角元占主导地位而非对角元随对应边在图中的距离呈指数衰减。这些结果为优化方法在此类场景中的有效性提供了首个严格的理论依据，并可能为理解树结构隐变量模型的优化问题提供更广泛的理论框架。