In plain words: Grokking—when a network memorizes first and only generalizes much later—is explained as the network compressing itself. A new complexity count of the network's linear pieces tracks this: it drops in a straight line with test error, unlike the usual weight-size measure.
Abstract
We attribute grokking, the phenomenon where generalization is much delayed after memorization, to compression. To do so, we define linear mapping number (LMN) to measure network complexity, which is a generalized version of linear region number for ReLU networks. LMN can nicely characterize neural network compression before generalization. Although the $L_2$ norm has been a popular choice for characterizing model complexity, we argue in favor of LMN for a number of reasons: (1) LMN can be naturally interpreted as information/computation, while $L_2$ cannot. (2) In the compression phase, LMN has linear relations with test losses, while $L_2$ is correlated with test losses in a complicated nonlinear way. (3) LMN also reveals an intriguing phenomenon of the XOR network switching between two generalization solutions, while $L_2$ does not. Besides explaining grokking, we argue that LMN is a promising candidate as the neural network version of the Kolmogorov complexity since it explicitly considers local or conditioned linear computations aligned with the nature of modern artificial neural networks.
Ziming Liu, Ziqian Zhong, Max Tegmark
arXiv:2310.05918 · cs.LG, cs.AI, stat.ML · submitted Oct 9, 2023
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