  The rise of deep learning in recent years has brought with it increasingly
clever optimization methods to deal with complex, non-linear loss functions.
These methods are often designed with convex optimization in mind, but have
been shown to work well in practice even for the highly non-convex optimization
associated with neural networks. However, one significant drawback of these
methods when they are applied to deep learning is that the magnitude of the
update step is sometimes disproportionate to the magnitude of the weights (much
smaller or larger), leading to training instabilities such as vanishing and
exploding gradients. An idea to combat this issue is gradient descent with
proportional updates. Gradient descent with proportional updates was introduced
in 2017. It was independently developed by You et al (Layer-wise Adaptive Rate
Scaling (LARS) algorithm) and by Abu-El-Haija (PercentDelta algorithm). The
basic idea of both of these algorithms is to make each step of the gradient
descent proportional to the current weight norm and independent of the gradient
magnitude. It is common in the context of new optimization methods to prove
convergence or derive regret bounds under the assumption of Lipschitz
continuity and convexity. However, even though LARS and PercentDelta were shown
to work well in practice, there is no theoretical analysis of the convergence
properties of these algorithms. Thus it is not clear if the idea of gradient
descent with proportional updates is used in the optimal way, or if it could be
improved by using a different norm or specific learning rate schedule, for
example. Moreover, it is not clear if these algorithms can be extended to other
problems, besides neural networks. We attempt to answer these questions by
establishing the theoretical analysis of gradient descent with proportional
updates, and verifying this analysis with empirical examples.
