HOW NEURAL NETWORKS EXTRAPOLATE: FROM FEEDFORWARD TO GRAPH NEURAL NETWORKS

Abstract

We study how neural networks trained by gradient descent extrapolate, i.e., what they learn outside the support of the training distribution. Previous works report mixed empirical results when extrapolating with neural networks: while feedforward neural networks, a.k.a. multilayer perceptrons (MLPs), do not extrapolate well in certain simple tasks, Graph Neural Networks (GNNs) -structured networks with MLP modules -have shown some success in more complex tasks. Working towards a theoretical explanation, we identify conditions under which MLPs and GNNs extrapolate well. First, we quantify the observation that ReLU MLPs quickly converge to linear functions along any direction from the origin, which implies that ReLU MLPs do not extrapolate most nonlinear functions. But, they can provably learn a linear target function when the training distribution is sufficiently "diverse". Second, in connection to analyzing the successes and limitations of GNNs, these results suggest a hypothesis for which we provide theoretical and empirical evidence: the success of GNNs in extrapolating algorithmic tasks to new data (e.g., larger graphs or edge weights) relies on encoding task-specific non-linearities in the architecture or features. Our theoretical analysis builds on a connection of over-parameterized networks to the neural tangent kernel. Empirically, our theory holds across different training settings.

1. INTRODUCTION

Humans extrapolate well in many tasks. For example, we can apply arithmetics to arbitrarily large numbers. One may wonder whether a neural network can do the same and generalize to examples arbitrarily far from the training data (Lake et al., 2017) . Curiously, previous works report mixed extrapolation results with neural networks. Early works demonstrate feedforward neural networks, a.k.a. multilayer perceptrons (MLPs), fail to extrapolate well when learning simple polynomial functions (Barnard & Wessels, 1992; Haley & Soloway, 1992) . However, recent works show Graph Neural Networks (GNNs) (Scarselli et al., 2009) , a class of structured networks with MLP building blocks, can generalize to graphs much larger than training graphs in challenging algorithmic tasks, such as predicting the time evolution of physical systems (Battaglia et al., 2016) , learning graph algorithms (Velickovic et al., 2020) , and solving mathematical equations (Lample & Charton, 2020). To explain this puzzle, we formally study how neural networks trained by gradient descent (GD) extrapolate, i.e., what they learn outside the support of training distribution. We say a neural network extrapolates well if it learns a task outside the training distribution. At first glance, it may seem that neural networks can behave arbitrarily outside the training distribution since they have high capacity (Zhang et al., 2017) and are universal approximators (Cybenko, 1989; Funahashi, 1989; Hornik et al., 1989; Kurkova, 1992) . However, neural networks are constrained by gradient descent training (Hardt et al., 2016; Soudry et al., 2018) . In our analysis, we explicitly consider such implicit bias through the analogy of the training dynamics of over-parameterized neural networks and kernel regression via the neural tangent kernel (NTK) (Jacot et al., 2018) .

