A multiobjective continuation method to compute the regularization path of deep neural networks

Abstract

Sparsity is a highly desired feature in deep neural networks (DNNs) since it ensures numerical efficiency, improves the interpretability (due to the smaller number of relevant features), and robustness. For linear models, it is well known that there exists a regularization path connecting the sparsest solution in terms of the ${\ell }^1$ norm, i.e., zero weights and the non-regularized solution. Recently, there was a first attempt to extend the concept of regularization paths to DNNs by means of treating the empirical loss and sparsity (${\ell }^1$ norm) as two conflicting criteria and solving the resulting multiobjective optimization problem. However, due to the non-smoothness of the ${\ell }^1$ norm and the large number of parameters, this approach is not very efficient from a computational perspective. To overcome this limitation, we present an algorithm that allows for the approximation of the entire Pareto front for the above-mentioned objectives in a very efficient manner for high-dimensional DNNs with millions of parameters. We present numerical examples using both deterministic and stochastic gradients. We furthermore demonstrate that knowledge of the regularization path allows for a well-generalizing network parametrization.