In this paper we propose and investigate a wide class of Mirror Descent updates (MD) and associated novel Generalized Exponentiated Gradient (GEG) algorithms by exploiting various trace-form entropies and associated deformed logarithms and their inverses - deformed (generalized) exponential functions. The proposed algorithms can be considered as extension of entropic MD and generalization of multiplicative updates. In the literature, there exist nowadays over fifty mathematically well defined generalized entropies, so impossible to exploit all of them in one research paper. So we focus on a few selected most popular entropies and associated logarithms like the Tsallis, Kaniadakis and Sharma-Taneja-Mittal and some of their extension like Tempesta or Kaniadakis-Scarfone entropies. The shape and properties of the deformed logarithms and their inverses are tuned by one or more hyperparameters. By learning these hyperparameters, we can adapt to distribution of training data, which can be designed to the specific geometry of the optimization problem, leading to potentially faster convergence and better performance. The using generalized entropies and associated deformed logarithms in the Bregman divergence, used as a regularization term, provides some new insight into exponentiated gradient descent updates.

本研究解决了当前文献中大量广义熵的使用问题，提出了一类基于迹形熵和变形对数的镜像下降更新和广义指数梯度算法。通过调整超参数，使算法适应训练数据的分布，提升优化问题的收敛速度和性能，为指数梯度下降更新提供了新视角。

使用迹形熵和变形对数的镜像下降和新颖的指数梯度算法