Stochastic Weighted Function Norm Regularization

Deep neural networks (DNNs) have become increasingly important due to their excellent empirical performance on a wide range of problems. However, regularization is generally achieved by indirect means, largely due to the complex set of functions defined by a network and the difficulty in measuring function complexity. There exists no method in the literature for additive regularization based on a norm of the function, as is classically considered in statistical learning theory. In this work, we propose sampling-based approximations to weighted function norms as regularizers for deep neural networks. We provide, to the best of our knowledge, the first proof in the literature of the NP-hardness of computing function norms of DNNs, motivating the necessity of a stochastic optimization strategy. Based on our proposed regularization scheme, stability-based bounds yield a $\mathcal{O}(N^{-\frac{1}{2}})$ generalization error for our proposed regularizer when applied to convex function sets. We demonstrate broad conditions for the convergence of stochastic gradient descent on our objective, including for non-convex function sets such as those defined by DNNs. Finally, we empirically validate the improved performance of the proposed regularization strategy for both convex function sets as well as DNNs on real-world classification and segmentation tasks.