Joseph Shenouda

Liu Yang, Jifan Zhang, Joseph Shenouda et al.

Weight decay is one of the most widely used forms of regularization in deep learning, and has been shown to improve generalization and robustness. The optimization objective driving weight decay is a sum of losses plus a term proportional to the sum of squared weights. This paper argues that stochastic gradient descent (SGD) may be an inefficient algorithm for this objective. For neural networks with ReLU activations, solutions to the weight decay objective are equivalent to those of a different objective in which the regularization term is instead a sum of products of $\ell_2$ (not squared) norms of the input and output weights associated with each ReLU neuron. This alternative (and effectively equivalent) regularization suggests a novel proximal gradient algorithm for network training. Theory and experiments support the new training approach, showing that it can converge much faster to the sparse solutions it shares with standard weight decay training.

Joseph Shenouda, Yamin Zhou, Robert D. Nowak

Motivated by the growing theoretical understanding of neural networks that employ the Rectified Linear Unit (ReLU) as their activation function, we revisit the use of ReLU activation functions for learning implicit neural representations (INRs). Inspired by second order B-spline wavelets, we incorporate a set of simple constraints to the ReLU neurons in each layer of a deep neural network (DNN) to remedy the spectral bias. This in turn enables its use for various INR tasks. Empirically, we demonstrate that, contrary to popular belief, one can learn state-of-the-art INRs based on a DNN composed of only ReLU neurons. Next, by leveraging recent theoretical works which characterize the kinds of functions ReLU neural networks learn, we provide a way to quantify the regularity of the learned function. This offers a principled approach to selecting the hyperparameters in INR architectures. We substantiate our claims through experiments in signal representation, super resolution, and computed tomography, demonstrating the versatility and effectiveness of our method. The code for all experiments can be found at https://github.com/joeshenouda/relu-inrs.

Julia Nakhleh, Joseph Shenouda, Robert D. Nowak

This paper studies the properties of solutions to multi-task shallow ReLU neural network learning problems, wherein the network is trained to fit a dataset with minimal sum of squared weights. Remarkably, the solutions learned for each individual task resemble those obtained by solving a kernel regression problem, revealing a novel connection between neural networks and kernel methods. It is known that single-task neural network learning problems are equivalent to a minimum norm interpolation problem in a non-Hilbertian Banach space, and that the solutions of such problems are generally non-unique. In contrast, we prove that the solutions to univariate-input, multi-task neural network interpolation problems are almost always unique, and coincide with the solution to a minimum-norm interpolation problem in a Sobolev (Reproducing Kernel) Hilbert Space. We also demonstrate a similar phenomenon in the multivariate-input case; specifically, we show that neural network learning problems with large numbers of tasks are approximately equivalent to an $\ell^2$ (Hilbert space) minimization problem over a fixed kernel determined by the optimal neurons.

Joseph Shenouda, Rahul Parhi, Kangwook Lee et al.

This paper introduces a novel theoretical framework for the analysis of vector-valued neural networks through the development of vector-valued variation spaces, a new class of reproducing kernel Banach spaces. These spaces emerge from studying the regularization effect of weight decay in training networks with activations like the rectified linear unit (ReLU). This framework offers a deeper understanding of multi-output networks and their function-space characteristics. A key contribution of this work is the development of a representer theorem for the vector-valued variation spaces. This representer theorem establishes that shallow vector-valued neural networks are the solutions to data-fitting problems over these infinite-dimensional spaces, where the network widths are bounded by the square of the number of training data. This observation reveals that the norm associated with these vector-valued variation spaces encourages the learning of features that are useful for multiple tasks, shedding new light on multi-task learning with neural networks. Finally, this paper develops a connection between weight-decay regularization and the multi-task lasso problem. This connection leads to novel bounds for layer widths in deep networks that depend on the intrinsic dimensions of the training data representations. This insight not only deepens the understanding of the deep network architectural requirements, but also yields a simple convex optimization method for deep neural network compression. The performance of this compression procedure is evaluated on various architectures.

Joseph Shenouda, Waheed U. Bajwa

Computational reproducibility is a growing problem that has been extensively studied among computational researchers and within the signal processing and machine learning research community. However, with the changing landscape of signal processing and machine learning research come new obstacles and unseen challenges in creating reproducible experiments. Due to these new challenges most computational experiments have become difficult, if not impossible, to be reproduced by an independent researcher. In 2016 a survey conducted by the journal Nature found that 50% of researchers were unable to reproduce their own experiments. While the issue of computational reproducibility has been discussed in the literature and specifically within the signal processing community, it is still unclear to most researchers what are the best practices to ensure reproducibility without impinging on their primary responsibility of conducting research. We feel that although researchers understand the importance of making experiments reproducible, the lack of a clear set of standards and tools makes it difficult to incorporate good reproducibility practices in most labs. It is in this regard that we aim to present signal processing researchers with a set of practical tools and strategies that can help mitigate many of the obstacles to producing reproducible computational experiments.