A General Theory for Softmax Gating Multinomial Logistic Mixture of Experts
This work addresses a theoretical gap for researchers in machine learning, providing foundational insights into classification MoE models, though it is incremental as it builds on existing regression analyses.
The authors tackled the lack of theoretical understanding for mixture-of-experts models in classification by establishing convergence rates for density and parameter estimation in softmax gating multinomial logistic MoE, showing slower-than-polynomial rates when expert parameters vanish, and proposed modified softmax gating functions to improve these rates significantly.
Mixture-of-experts (MoE) model incorporates the power of multiple submodels via gating functions to achieve greater performance in numerous regression and classification applications. From a theoretical perspective, while there have been previous attempts to comprehend the behavior of that model under the regression settings through the convergence analysis of maximum likelihood estimation in the Gaussian MoE model, such analysis under the setting of a classification problem has remained missing in the literature. We close this gap by establishing the convergence rates of density estimation and parameter estimation in the softmax gating multinomial logistic MoE model. Notably, when part of the expert parameters vanish, these rates are shown to be slower than polynomial rates owing to an inherent interaction between the softmax gating and expert functions via partial differential equations. To address this issue, we propose using a novel class of modified softmax gating functions which transform the input before delivering them to the gating functions. As a result, the previous interaction disappears and the parameter estimation rates are significantly improved.