Stiffness: A New Perspective on Generalization in Neural Networks

TL;DR

This paper introduces the concept of neural network stiffness, measuring how parameter updates on one example affect loss on others, linking stiffness to generalization and exploring its dependence on training dynamics across various models and datasets.

Contribution

The paper proposes the neural network stiffness as a new perspective on understanding generalization, analyzing its dependence on class, input distance, training iteration, and learning rate.

Findings

Higher stiffness correlates with better generalization.

Stiffness varies with class membership and input distance.

CIFAR-100 stiffness matrix reveals super-class awareness.

Abstract

In this paper we develop a new perspective on generalization of neural networks by proposing and investigating the concept of a neural network stiffness. We measure how stiff a network is by looking at how a small gradient step in the network's parameters on one example affects the loss on another example. Higher stiffness suggests that a network is learning features that generalize. In particular, we study how stiffness depends on 1) class membership, 2) distance between data points in the input space, 3) training iteration, and 4) learning rate. We present experiments on MNIST, FASHION MNIST, and CIFAR-10/100 using fully-connected and convolutional neural networks, as well as on a transformer-based NLP model. We demonstrate the connection between stiffness and generalization, and observe its dependence on learning rate. When training on CIFAR-100, the stiffness matrix exhibits a coarse-grained behavior indicative of the model's awareness of super-class membership. In addition, we measure how stiffness between two data points depends on their mutual input-space distance, and establish the concept of a dynamical critical length -- a distance below which a parameter update based on a data point influences its neighbors.