IIn this paper we propose and investigate a new class of Generalized Exponentiated Gradient (GEG) algorithms using Mirror Descent (MD) updates, and applying the Bregman divergence with a two--parameter deformation of the logarithm as a link function. This link function (referred here to as the Euler logarithm) is associated with a relatively wide class of trace--form entropies. In order to derive novel GEG/MD updates, we estimate a deformed exponential function, which closely approximates the inverse of the Euler two--parameter deformed logarithm. The characteristic shape and properties of the Euler logarithm and its inverse--deformed exponential functions, are tuned by two hyperparameters. By learning these hyperparameters, we can adapt to the distribution of training data and adjust them to achieve desired properties of gradient descent algorithms. In the literature, there exist nowadays more than fifty mathematically well-established entropic functionals and associated deformed logarithms, so it is impossible to investigate all of them in one research paper. Therefore, we focus here on a class of trace-form entropies and the associated deformed two--parameters logarithms.

翻译：本文提出并研究了一类新的广义指数梯度算法，该算法采用镜像下降更新，并应用以双参数变形对数作为链接函数的布雷格曼散度。该链接函数（本文称为欧拉对数）与一类相对广泛的迹形式熵相关联。为推导新颖的GEG/MD更新规则，我们估计了一个变形指数函数，其紧密逼近欧拉双参数变形对数的逆函数。欧拉对数及其逆变形指数函数的特征形态与性质由两个超参数调节。通过学习这些超参数，我们可以适应训练数据的分布，并调整它们以实现梯度下降算法的期望特性。目前文献中已存在超过五十种数学上完备的熵泛函及相关变形对数，因此无法在一篇研究论文中全面考察所有类型。鉴于此，本文聚焦于一类迹形式熵及其关联的双参数变形对数。