  Studying the properties of stochastic noise to optimize complex non-convex
functions has been an active area of research in the field of machine learning.
Prior work has shown that the noise of stochastic gradient descent improves
optimization by overcoming undesirable obstacles in the landscape. Moreover,
injecting artificial Gaussian noise has become a popular idea to quickly escape
saddle points. Indeed, in the absence of reliable gradient information, the
noise is used to explore the landscape, but it is unclear what type of noise is
optimal in terms of exploration ability. In order to narrow this gap in our
knowledge, we study a general type of continuous-time non-Markovian process,
based on fractional Brownian motion, that allows for the increments of the
process to be correlated. This generalizes processes based on Brownian motion,
such as the Ornstein-Uhlenbeck process. We demonstrate how to discretize such
processes which gives rise to the new algorithm fPGD. This method is a
generalization of the known algorithms PGD and Anti-PGD. We study the
properties of fPGD both theoretically and empirically, demonstrating that it
possesses exploration abilities that, in some cases, are favorable over PGD and
Anti-PGD. These results open the field to novel ways to exploit noise for
training machine learning models.
