Learning large softmax mixtures with warm start EM

TL;DR

This paper analyzes the EM algorithm for softmax mixture models in high dimensions, establishing identifiability, convergence conditions, and proposing methods for effective warm starts to improve parameter recovery.

Contribution

It provides the first theoretical results on local and full identifiability of SSMs with $L > 1$, and introduces new moment-based methods for warm start initialization.

Findings

Proves local and full identifiability of SSMs with high-dimensional features.

Characterizes initialization radius for EM convergence.

Develops moment-based methods for warm start construction.

Abstract

Softmax mixture models (SMMs) are discrete $K$-mixtures introduced to model the probability of choosing an attribute $x_j \in \RR^L$ from $p$ candidates, in heterogeneous populations. They have been known as mixed multinomial logits in the econometrics literature, and are gaining traction in the LLM literature, where single softmax models are routinely used in the final layer of a neural network. This paper provides a comprehensive analysis of the EM algorithm for SMMs in high dimensions. Its population-level theoretical analysis forms the basis for proving (i) local identifiability, in SSMs with generic features and, further, via a stochastic argument, (ii) full identifiability in SSMs with random features, when $p$ is large enough. These are the first results in this direction for SSMs with $L > 1$. The population-level EM analysis characterizes the initialization radius for algorithmic convergence. This also guides the construction of warm starts of the sample level EM. Under suitable initialization, the EM algorithm is shown to recover the mixture atoms of the SSM at near-parametric rate. We provide two main directions for warm start construction, both based on a new method for estimating the moments of the mixing measure underlying an SSM with random design. First, we construct a method of moments (MoM) estimator of the mixture parameters, and provide its first theoretical analysis. While MoM can enjoy parametric rates of convergence, and thus can serve as a warm-start, the estimator's quality degrades exponentially in $K$. Our recommendation, when $K$ is not small, is to run the EM algorithm several times with random initializations. We again make use of the novel latent moments estimation method to estimate the $K$-dimensional subspace of the mixture atoms. Sampling from this subspace reduces substantially the number of required draws.