Exponentiated Gradient Meets Gradient Descent
This work addresses the limitation of multiplicative updates to positive semi-definite matrices, offering a more flexible approach for machine learning practitioners dealing with general matrix parameters.
The paper introduces a new regularization method called hypentropy that unifies additive (gradient descent) and multiplicative updates, enabling application to general rectangular matrices, and provides tight regret bounds with empirical validation in multiclass learning.
The (stochastic) gradient descent and the multiplicative update method are probably the most popular algorithms in machine learning. We introduce and study a new regularization which provides a unification of the additive and multiplicative updates. This regularization is derived from an hyperbolic analogue of the entropy function, which we call hypentropy. It is motivated by a natural extension of the multiplicative update to negative numbers. The hypentropy has a natural spectral counterpart which we use to derive a family of matrix-based updates that bridge gradient methods and the multiplicative method for matrices. While the latter is only applicable to positive semi-definite matrices, the spectral hypentropy method can naturally be used with general rectangular matrices. We analyze the new family of updates by deriving tight regret bounds. We study empirically the applicability of the new update for settings such as multiclass learning, in which the parameters constitute a general rectangular matrix.