Grokking and Generalization Collapse: Insights from \texttt{HTSR} theory

Abstract

We study the well-known grokking phenomena in neural networks (NNs) using a 3-layer MLP trained on 1 k-sample subset of MNIST, with and without weight decay, and discover a novel third phase -- \emph{anti-grokking} -- that occurs very late in training and resembles but is distinct from the familiar \emph{pre-grokking} phases: test accuracy collapses while training accuracy stays perfect. This late-stage collapse is distinct, from the known pre-grokking and grokking phases, and is not detected by other proposed grokking progress measures. Leveraging Heavy-Tailed Self-Regularization HTSR through the open-source WeightWatcher tool, we show that the HTSR layer quality metric $α$ alone delineates all three phases, whereas the best competing metrics detect only the first two. The \emph{anti-grokking} is revealed by training for $10^7$ and is invariably heralded by $α< 2$ and the appearance of \emph{Correlation Traps} -- outlier singular values in the randomized layer weight matrices that make the layer weight matrix atypical and signal overfitting of the training set. Such traps are verified by visual inspection of the layer-wise empirical spectral densities, and by using Kolmogorov--Smirnov tests on randomized spectra. Comparative metrics, including activation sparsity, absolute weight entropy, circuit complexity, and $l^2$ weight norms track pre-grokking and grokking but fail to distinguish grokking from anti-grokking. This discovery provides a way to measure overfitting and generalization collapse without direct access to the test data. These results strengthen the claim that the \emph{HTSR} $α$ provides universal layer-convergence target at $α\approx 2$ and underscore the value of using the HTSR alpha $(α)$ metric as a measure of generalization.