用多参数对数构造新镜面下降算法，可灵活适应数据分布。
Mirror Descent Using the Tempesta Generalized Multi-parametric Logarithms
- 以泰梅斯塔多参数对数为链接函数，构建广义镜面下降框架。
- 通过调节超参数，使算法适应不同数据几何与分布特性。
- 适用于需自适应优化的机器学习场景，尤其适合复杂分布建模。
本文提出一类广义镜面下降（Mirror Descent, MD）算法，其核心是利用泰梅斯塔多参数变形对数作为Bregman散度的链接函数。该链接函数定义了原空间与对偶空间间的映射，关联着一个理论上无限大的广义迹形式熵族。为推导新的MD更新规则，我们估计了逼近该多参数对数逆函数的广义指数函数。泰梅斯塔对数及其逆指数函数的形状与性质可通过多个超参数调节。通过学习这些超参数，算法可自适应训练数据的分布或几何结构，并实现期望的优化性质。该方法构建了一类新型、高度灵活的镜面下降及无镜面下降更新机制。
原文摘要 · Abstract (English)
In this paper, we develop a wide class Mirror Descent (MD) algorithms, which play a key role in machine learning. For this purpose we formulated the constrained optimization problem, in which we exploits the Bregman divergence with the Tempesta multi-parametric deformation logarithm as a link function. This link function called also mirror function defines the mapping between the primal and dual spaces and is associated with a very-wide (in fact, theoretically infinite) class of generalized trace-form entropies. In order to derive novel MD updates, we estimate generalized exponential function, which closely approximates the inverse of the multi-parametric Tempesta generalized logarithm. The shape and properties of the Tempesta logarithm and its inverse-deformed exponential functions can be tuned by several hyperparameters. By learning these hyperparameters, we can adapt to distribution or geometry of training data, and we can adjust them to achieve desired properties of MD algorithms. The concept of applying multi-parametric logarithms allow us to generate a new wide and flexible family of MD and mirror-less MD updates.
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