Dendrograms of Mixing Measures for Softmax-Gated Gaussian Mixture of Experts: Consistency without Model Sweeps

We develop a unified statistical framework for softmax-gated Gaussian mixture of experts (SGMoE) that addresses three long-standing obstacles in parameter estimation and model selection: (i) non-identifiability of gating parameters up to common translations, (ii) intrinsic gate-expert interactions that induce coupled differential relations in the likelihood, and (iii) the tight numerator-denominator coupling in the softmax-induced conditional density. Our approach introduces Voronoi-type loss functions aligned with the gate-partition geometry and establishes finite-sample convergence rates for the maximum likelihood estimator (MLE). In over-specified models, we reveal a link between the MLE’s convergence rate and the solvability of an associated system of polynomial equations characterizing near-nonidentifiable directions. For model selection, we adapt dendrograms of mixing measures to SGMoE, yielding a consistent, sweep-free selector of the number of experts that attains pointwise-optimal parameter rates under overfitting while avoiding multi-size training. Simulations on synthetic data corroborate the theory, accurately recovering the expert count and achieving the predicted rates for parameter estimation while closely approximating the regression function.