Flat Loss, Evolving Features: Grokking in Modular Addition

Key points

• 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.1
• 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.1
• 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.1
• A one-block transformer trained with the same updates and a slowly decaying output gain keeps every training pair correct at every logged checkpoint while its training cross-entropy rises and its held-out accuracy approaches one; with two blocks this holds through the held-out rise, after which training pairs begin to flip.1

Sources (1)

• [1]Flat Loss, Evolving Features: Grokking in Modular Addition
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 15, 07:43 PM

  Can a network keep learning while its training cross-entropy stays nearly flat at a positive level?

  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.

Extractive summary: sentences quoted from the sources.