Deep Networks Always Grok and Here is Why
This work addresses the phenomenon of grokking for researchers and practitioners in deep learning, revealing its broader occurrence and linking it to robustness, though it is incremental in extending prior findings.
The paper demonstrates that grokking, or delayed generalization, is widespread in practical deep learning settings beyond controlled experiments, and introduces the concept of delayed robustness where networks become robust to adversarial examples after training. It provides an analytical explanation based on local complexity and linear regions, showing a phase transition during training that leads to grokking.
Grokking, or delayed generalization, is a phenomenon where generalization in a deep neural network (DNN) occurs long after achieving near zero training error. Previous studies have reported the occurrence of grokking in specific controlled settings, such as DNNs initialized with large-norm parameters or transformers trained on algorithmic datasets. We demonstrate that grokking is actually much more widespread and materializes in a wide range of practical settings, such as training of a convolutional neural network (CNN) on CIFAR10 or a Resnet on Imagenette. We introduce the new concept of delayed robustness, whereby a DNN groks adversarial examples and becomes robust, long after interpolation and/or generalization. We develop an analytical explanation for the emergence of both delayed generalization and delayed robustness based on the local complexity of a DNN's input-output mapping. Our local complexity measures the density of so-called linear regions (aka, spline partition regions) that tile the DNN input space and serves as a utile progress measure for training. We provide the first evidence that, for classification problems, the linear regions undergo a phase transition during training whereafter they migrate away from the training samples (making the DNN mapping smoother there) and towards the decision boundary (making the DNN mapping less smooth there). Grokking occurs post phase transition as a robust partition of the input space thanks to the linearization of the DNN mapping around the training points. Website: https://bit.ly/grok-adversarial