Learning on a Razor's Edge: the Singularity Bias of Polynomial Neural Networks
Vahid Shahverdi, Giovanni Luca Marchetti, Kathlén Kohn
TL;DR
The paper studies the implicit bias of deep networks toward subnetworks through the geometry of the neuromanifold for polynomial activations. Using algebraic-geometry techniques, it shows subnetworks correspond to singularities and can be associated with critical points of training in MLPs, while convolutional nets exhibit milder singularities and lack the same bias. The authors prove a dimension formula for the neuromanifold, $\ ext{dim}(\\mathcal{M}_{\mathbf{d}, \sigma}) = \sum_{i=1}^L d_i d_{i-1}$, and show that subnetworks yield singular points (with MLPs displaying a stronger exposedness bias, unlike CNNs). They introduce the notion of exposedness and demonstrate that strict subnetworks are critically exposed under $\\sigma(0)=0$, whereas CNNs do not exhibit such exposedness, even though their neuromanifolds can be singular via nodal points. The work offers a geometric lens on sparsity, connects to the lottery-ticket paradigm, and outlines open problems, including characterizing all singularities and extending results to non-polynomial activations.
Abstract
Deep neural networks often infer sparse representations, converging to a subnetwork during the learning process. In this work, we theoretically analyze subnetworks and their bias through the lens of algebraic geometry. We consider fully-connected networks with polynomial activation functions, and focus on the geometry of the function space they parametrize, often referred to as neuromanifold. First, we compute the dimension of the subspace of the neuromanifold parametrized by subnetworks. Second, we show that this subspace is singular. Third, we argue that such singularities often correspond to critical points of the training dynamics. Lastly, we discuss convolutional networks, for which subnetworks and singularities are similarly related, but the bias does not arise.