  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.
