Stiffness: A New Perspective on Generalization in Neural Networks

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.

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References in corpus (14)

• Deep Residual Learning for Image Recognition
• Neural Tangent Kernel: Convergence and Generalization in Neural Networks
• Understanding deep learning requires rethinking generalization
• On the Number of Linear Regions of Deep Neural Networks
• Gradient Descent Provably Optimizes Over-parameterized Neural Networks
• A Closer Look at Memorization in Deep Networks
• Exponential expressivity in deep neural networks through transient chaos
• Sensitivity and Generalization in Neural Networks: an Empirical Study
• On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization
• An Investigation into Neural Net Optimization via Hessian Eigenvalue Density
• On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Length
• Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians
• Deep Information Propagation
• Emergent properties of the local geometry of neural loss landscapes

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