On Least Square Estimation in Softmax Gating Mixture of Experts

Huy Nguyen; Nhat Ho; Alessandro Rinaldo

Home/Archive/arXiv

arXiv · 2402.02952

On Least Square Estimation in Softmax Gating Mixture of Experts

Huy Nguyen·Nhat Ho·Alessandro Rinaldo

View original source ↗Read source document ↗Export metadata ↓

Abstract

Mixture of experts (MoE) model is a statistical machine learning design that aggregates multiple expert networks using a softmax gating function in order to form a more intricate and expressive model. Despite being commonly used in several applications owing to their scalability, the mathematical and statistical properties of MoE models are complex and difficult to analyze. As a result, previous theoretical works have primarily focused on probabilistic MoE models by imposing the impractical assumption that the data are generated from a Gaussian MoE model. In this work, we investigate the performance of the least squares estimators (LSE) under a deterministic MoE model where the data are sampled according to a regression model, a setting that has remained largely unexplored. We establish a condition called strong identifiability to characterize the convergence behavior of various types of expert functions. We demonstrate that the rates for estimating strongly identifiable experts, namely the widely used feed-forward networks with activation functions $\mathrm{sigmoid}(\cdot)$ and $\tanh(\cdot)$, are substantially faster than those of polynomial experts, which we show to exhibit a surprising slow estimation rate. Our findings have important practical implications for expert selection.

Explore related subjects

stat.ML cs.LG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Huy Nguyen, Nhat Ho, Alessandro Rinaldo. 2024-06-24. On Least Square Estimation in Softmax Gating Mixture of Experts. https://arxiv.org/abs/2402.02952

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Robust Mixture Models for Algorithmic Fairness Under Latent Heterogeneity

Machine learning models optimized for average performance can perform poorly on vulnerable subpopulations. Existing approaches often rely on groups specified in advance, yet fairness-relevant subgroup structure may be latent, intersectional, and driven by complex interactions among continuous and discrete attributes. We introduce \textbf{ROME} (\textbf{\underline{RO}}bust \textbf{\underline{M}}ixture \textbf{\underline{E}}nsemble), a framework that learns latent group structure while optimizing worst-group predictive performance. ROME connects latent-variable modeling with distributionally robust optimization (DRO) through two complementary approaches: an Expectation-Maximization formulation with robust aggregation for linear models and a neural Mixture-of-Experts formulation for nonlinear settings. Across simulations and three real-world regression datasets, ROME improves worst-group performance while maintaining competitive overall accuracy, including in comparisons with established group-aware and group-label-free robust learning methods. ROME provides a flexible approach to robust prediction when fairness-relevant attributes are available for subgroup discovery but their direct use in group-specific outcome models is restricted.

stat.ML↗

arXiv2026

Boltzmann generators for amorphous particle systems

Sampling configurations in thermodynamic equilibrium is a long-standing challenge in statistical physics. Boltzmann generators address this problem by employing generative models to propose independent configurations, which are then reweighted via importance sampling using exact likelihood evaluations. Recent Boltzmann Generators based on continuous normalizing flows and flow matching have achieved significant success for particle systems and biomolecules. However, these approaches have not been extended to amorphous materials (glasses), for which equilibrium sampling is notoriously slow. Because of their disordered structure, the invariances and geometrical constraints of amorphous materials differ from those of crystals and biomolecules, preventing the direct use of existing generative models. Here, we develop Boltzmann Generators tailored to amorphous materials by building the required equivariances directly into Riemannian stochastic interpolants. Our framework incorporates periodic boundary conditions and particle symmetries using equivariant graph neural networks. Numerical experiments demonstrate that enforcing physical symmetries significantly improves the accuracy of Boltzmann Generators, but also reveal an intrinsic limitation of the continuous-flow formulation: accumulated numerical errors during likelihood integration break time-reversibility, compromising exact thermodynamic reweighting. These results reveal a fundamental challenge for continuous-flow generative models in statistical mechanics and call for alternative approaches that preserve exact thermodynamic consistency.

stat.ML↗

arXiv2026

Diagonalized Attention for Individualized Regression: Latent-Row Localization and Prediction

Modern text and image representations are often matrix-valued, with rows corresponding to tokens, patches, or other local feature vectors. Predictive information is often sparse but sample-specific, making classical sparse regression methods with a common support poorly suited to this heterogeneity. This paper formalizes an individualized sparse regression framework for matrix-valued covariates in which each observation has its own rows of interest, while the associated regression effects are shared across the population. To estimate this model, we introduce a diagonalized attention mechanism that uses query--key scores to localize sample-specific signal rows and a value matrix for downstream regression. The proposed method has a parameter dimension independent of sample size and can identify rows of interest for new observations without their responses. We establish existence theorems showing that, under suitable score-separation and concentration conditions, single-head and multi-head diagonalized attention models recover the latent rows with high probability, yielding prediction risk bounds. Our theory therefore provides a statistical explanation of how attention-based scoring localizes sample-specific signals in heterogeneous matrix-valued data. Simulations demonstrate strong prediction and localization in regression and misspecified classification across varying sample sizes, dimensions, and signal cardinalities. Real sentiment analyses show improved classification accuracy and interpretable token selection.

stat.ML↗