Deep Neural Networks Tend To Extrapolate Predictably

TL;DR

Neural networks tend to predict a constant value close to the optimal constant solution when faced with out-of-distribution inputs, contradicting the common belief of unpredictable and overconfident extrapolation.

Contribution

This work reveals that neural networks extrapolate predictably towards a constant value near the OCS across various datasets, architectures, and loss functions, supported by empirical and theoretical analysis.

Findings

Neural networks' predictions stabilize to a constant value on OOD inputs.

The constant prediction often approximates the optimal constant solution minimizing training loss.

The observed behavior holds across multiple datasets, architectures, and loss functions.

Abstract

Conventional wisdom suggests that neural network predictions tend to be unpredictable and overconfident when faced with out-of-distribution (OOD) inputs. Our work reassesses this assumption for neural networks with high-dimensional inputs. Rather than extrapolating in arbitrary ways, we observe that neural network predictions often tend towards a constant value as input data becomes increasingly OOD. Moreover, we find that this value often closely approximates the optimal constant solution (OCS), i.e., the prediction that minimizes the average loss over the training data without observing the input. We present results showing this phenomenon across 8 datasets with different distributional shifts (including CIFAR10-C and ImageNet-R, S), different loss functions (cross entropy, MSE, and Gaussian NLL), and different architectures (CNNs and transformers). Furthermore, we present an explanation for this behavior, which we first validate empirically and then study theoretically in a simplified setting involving deep homogeneous networks with ReLU activations. Finally, we show how one can leverage our insights in practice to enable risk-sensitive decision-making in the presence of OOD inputs.