Understanding parameter estimation of softmax gating Gaussian mixture of experts has remained a long-standing open problem in the literature. It is mainly due to three fundamental theoretical challenges associated with the softmax gating: (i) the identifiability only up to the translation of the parameters; (ii) the intrinsic interaction via partial differential equation between the softmax gating and the expert functions in Gaussian distribution; (iii) the complex dependence between the numerator and denominator of the conditional density of softmax gating Gaussian mixture of experts. We resolve these challenges by proposing novel Vononoi loss functions among parameters and establishing the convergence rates of the maximum likelihood estimator (MLE) for solving parameter estimation in these models. When the number of experts is unknown and over-specified, our findings show a connection between the rate of MLE and a solvability problem of a system of polynomial equations.

翻译：理解Softmax门控高斯混合专家模型中的参数估计一直是文献中长期存在的开放性问题。这主要源于Softmax门控带来的三个基本理论挑战：（i）参数仅具有平移可辨识性；（ii）Softmax门控与高斯分布专家函数之间通过偏微分方程产生的内在相互作用；（iii）Softmax门控高斯混合专家条件密度中分子与分母之间的复杂依赖关系。我们通过提出参数间新颖的Voronoi损失函数，并建立这些模型中用于参数估计的最大似然估计（MLE）的收敛速度来攻克这些挑战。当专家数量未知且过设定时，我们的研究揭示了MLE速率与多项式方程组可解性问题之间的联系。