The Dynamics of Generalization in Deep Learning

Abstract

We derive a differential equation that governs the evolution of the generalization gap when a model is trained by gradient descent-based methods. This differential equation is driven by two key quantities, a contraction factor that brings together trajectories corresponding to slightly different datasets, and a perturbation factor that accounts for them training on different datasets. The coupled decay of contraction and perturbation guarantees a controlled accumulation of generalization gap during training. We analyze this differential equation to show that the generalization gap is given by a quadratic form that consists of an ``effective Gram matrix'' that depends upon the training trajectory and a certain residual of the predictor at initialization. Our framework is applicable to general deep networks and smooth loss functions. In numerical experiments on different neural network architectures, datasets and sample sizes, we show that this quadratic form accurately captures the actual generalization gap. We also show how to instantiate our framework in a number of examples via analytical calculations. For example, for high-dimensional linear regression, our framework matches existing calculations of generalization gap in the literature exactly in under-parameterized, over-parameterized and critical regimes.