Generalized Exponentiated Gradient Algorithms Using the Euler Two-Parameter Logarithm
This work is incremental, offering a method to adapt gradient algorithms to data distributions using a specific class of entropies.
The authors tackled the problem of designing adaptive gradient descent algorithms by introducing a new class of Generalized Exponentiated Gradient algorithms using Mirror Descent with a two-parameter deformed logarithm, achieving tunable properties through learned hyperparameters.
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.