Stiffness: A New Perspective on Generalization in Neural Networks

By Stanislav Fort, Pawel Krzysztof Nowak, Stanislaw Jastrzebski, Srini Narayanan. 2019

109 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics

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Citations in the corpus

Learning explanations that are hard to vary (2020, background): “Looking at agreement to characterize and improve the representations learned by neural networks is an idea that has been explored in recent work, mostly looking at the cosine similarity between gradients [Du et al., 2018, Eshratifar et al., 2018, Fort et al., 2019, Zhang et al., 2019b].”
Interference and Generalization in Temporal Difference Learning (2020, background): “…learning (Lopez-Paz & Ranzato, 2017; Schaul et al., 2019), it forms the Neural Tangent Kernel (Jacot et al., 2018), it is the Taylor expansion around θ of a gradient update (Achiam et al., 2019), as well as the Taylor expansion of pointwise loss differences (Liu et al., 2019b; Fort et al., 2019).”
The Break-Even Point on Optimization Trajectories of Deep Neural Networks (2020, background): “A natural direction for the future is connecting our observations to recent studies on the relation of measures, such as gradient variance, to the generalization of deep networks (Li et al., 2019; Jiang et al., 2020; Fort et al., 2019).”
Explaining Knowledge Distillation by Quantifying the Knowledge (2020, background): “The stiffness [10] was proposed to diagnose the generalization of a DNN.”
Implicit Regularization via Neural Feature Alignment (2020, background): “Since the geometry we consider is deﬁned through the gradient second moments, our perspectice is closely related to the notions of stiﬀness (Fort et al., 2019) and coherent gradients (Chatterjee, 2020).”
Emergent properties of the local geometry of neural loss landscapes (2019, background): “We note that these measurements are related, but complimentary to the concept of stiffness in [27].”