  Explaining the generalization characteristics of deep learning is an emerging
topic in advanced machine learning. There are several unanswered questions
about how learning under stochastic optimization really works and why certain
strategies are better than others. In this paper, we address the following
question: \textit{can we probe intermediate layers of a deep neural network to
identify and quantify the learning quality of each layer?} With this question
in mind, we propose new explainability metrics that measure the redundant
information in a network's layers using a low-rank factorization framework and
quantify a complexity measure that is highly correlated with the generalization
performance of a given optimizer, network, and dataset. We subsequently exploit
these metrics to augment the Stochastic Gradient Descent (SGD) optimizer by
adaptively adjusting the learning rate in each layer to improve in
generalization performance. Our augmented SGD -- dubbed RMSGD -- introduces
minimal computational overhead compared to SOTA methods and outperforms them by
exhibiting strong generalization characteristics across application,
architecture, and dataset.
