In spite of their prevalence, the behaviour of Neural Networks when extrapolating far from the training distribution remains poorly understood, with existing results limited to specific cases. In this work, we prove general results -- the first of their kind -- by applying Neural Tangent Kernel (NTK) theory to analyse infinitely-wide neural networks trained until convergence and prove that the inclusion of just one Layer Norm (LN) fundamentally alters the induced NTK, transforming it into a bounded-variance kernel. As a result, the output of an infinitely wide network with at least one LN remains bounded, even on inputs far from the training data. In contrast, we show that a broad class of networks without LN can produce pathologically large outputs for certain inputs. We support these theoretical findings with empirical experiments on finite-width networks, demonstrating that while standard NNs often exhibit uncontrolled growth outside the training domain, a single LN layer effectively mitigates this instability. Finally, we explore real-world implications of this extrapolatory stability, including applications to predicting residue sizes in proteins larger than those seen during training and estimating age from facial images of underrepresented ethnicities absent from the training set.

Existing analyses of optimization in deep learning are either continuous, focusing on (variants of) gradient flow, or discrete, directly treating (variants of) gradient descent. Gradient flow is amenable to theoretical analysis, but is stylized and disregards computational efficiency. The extent to which it represents gradient descent is an open question in deep learning theory. The current paper studies this question. Viewing gradient descent as an approximate numerical solution to the initial value problem of gradient flow, we find that the degree of approximation depends on the curvature along the latter's trajectory. We then show that over deep neural networks with homogeneous activations, gradient flow trajectories enjoy favorable curvature, suggesting they are well approximated by gradient descent. This finding allows us to translate an analysis of gradient flow over deep linear neural networks into a guarantee that gradient descent efficiently converges to global minimum almost surely under random initialization. Experiments suggest that over simple deep neural networks, gradient descent with conventional step size is indeed close to the continuous limit. We hypothesize that the theory of gradient flows will be central to unraveling mysteries behind deep learning.

Recently fair division theory has emerged as a promising approach for allocation of multiple computational resources among agents. While in reality agents are not all present in the system simultaneously, previous work has studied static settings where all relevant information is known upfront. Our goal is to better understand the dynamic setting. On the conceptual level, we develop a dynamic model of fair division, and propose desirable axiomatic properties for dynamic resource allocation mechanisms. On the technical level, we construct two novel mechanisms that provably satisfy some of these properties, and analyze their performance using real data. We believe that our work informs the design of superior multiagent systems, and at the same time expands the scope of fair division theory by initiating the study of dynamic and fair resource allocation mechanisms.