Flat Loss, Evolving Features: Grokking in Modular Addition

Akash Kumar, Karthik Sankaranarayanan

Can a network keep learning while its training cross-entropy stays nearly flat at a positive level? We show that it can. On modular addition, a standard testbed for grokking, ReLU networks trained with supported-Polar hidden updates, normalized output-layer updates and weight decay can keep 100% training accuracy while their hidden features change and held-out accuracy rises from zero to above 90%, at relatively flat training cross-entropy. Correct classification need not minimize cross-entropy, which still depends on prediction confidence. We prove this coexistence for one-hidden-layer networks in two finite-window theorems from constructed initializations: one gives a constant gain in the Fourier fraction of the hidden features, and the other holds at arbitrarily low positive loss, where a readout refitted on the training data generalizes before the network. Experiments from random initialization show the same coexistence on a window fixed in advance, and in deeper networks held-out accuracy also rises over post-hoc windows with a net increase in training cross-entropy. With a last-layer rate set per training fraction and head decay scaled with the prime, deeper networks at…