Norm-based Generalization Bounds for Compositionally Sparse Neural Networks

In this paper, we investigate the Rademacher complexity of deep sparse neural networks, where each neuron receives a small number of inputs. We prove generalization bounds for multilayered sparse ReLU neural networks, including convolutional neural networks. These bounds differ from previous ones, as they consider the norms of the convolutional filters instead of the norms of the associated Toeplitz matrices, independently of weight sharing between neurons. As we show theoretically, these bounds may be orders of magnitude better than standard norm-based generalization bounds and empirically, they are almost non-vacuous in estimating generalization in various simple classification problems. Taken together, these results suggest that compositional sparsity of the underlying target function is critical to the success of deep neural networks.

On the Implicit Bias Towards Minimal Depth of Deep Neural Networks

We study the implicit bias of gradient based training methods to favor low-depth solutions when training deep neural networks. Recent results in the literature suggest that penultimate layer representations learned by a classifier over multiple classes exhibit a clustering property, called neural collapse. We demonstrate empirically that the neural collapse property extends beyond the penultimate layer and tends to emerge in intermediate layers as well. In this regards, we hypothesize that gradient based methods are implicitly biased towards selecting neural networks of minimal depth for achieving this clustering property.