  A typical assumption for the analysis of first order optimization methods is
the Lipschitz continuity of the gradient of the objective function. However,
for many practical applications this assumption is violated, including loss
functions in deep learning. To overcome this issue, certain extensions based on
generalized proximity measures known as Bregman distances were introduced. This
initiated the development of the Bregman proximal gradient (BPG) algorithm and
an inertial variant (momentum based) CoCaIn BPG, which however rely on problem
dependent Bregman distances. In this paper, we develop Bregman distances for
using BPG methods to train Deep Linear Neural Networks. The main implications
of our results are strong convergence guarantees for these algorithms. We also
propose several strategies for their efficient implementation, for example,
closed form updates and a closed form expression for the inertial parameter of
CoCaIn BPG. Moreover, the BPG method requires neither diminishing step sizes
nor line search, unlike its corresponding Euclidean version. We numerically
illustrate the competitiveness of the proposed methods compared to existing
state of the art schemes.
