Combining learning rate decay and weight decay with complexity gradient descent - Part I

The role of $L^2$ regularization, in the specific case of deep neural networks rather than more traditional machine learning models, is still not fully elucidated. We hypothesize that this complex interplay is due to the combination of overparameterization and high dimensional phenomena that take place during training and make it unamenable to standard convex optimization methods. Using insights from statistical physics and random fields theory, we introduce a parameter factoring in both the level of the loss function and its remaining nonconvexity: the \emph{complexity}. We proceed to show that it is desirable to proceed with \emph{complexity gradient descent}. We then show how to use this intuition to derive novel and efficient annealing schemes for the strength of $L^2$ regularization when performing standard stochastic gradient descent in deep neural networks.