While the phenomenon of grokking, i.e., delayed generalization, has been studied extensively, it remains an open problem whether there is a mathematical framework to characterize what kind of features will emerge, how and in which conditions it happens from training, for complex structured inputs. We propose a novel framework, named $\mathbf{Li_2}$, that captures three key stages for the grokking behavior of 2-layer nonlinear networks: (I) \underline{\textbf{L}}azy learning, (II) \underline{\textbf{i}}ndependent feature learning and (III) \underline{\textbf{i}}nteractive feature learning. At the lazy learning stage, top layer overfits to random hidden representation and the model appears to memorize. Thanks to lazy learning and weight decay, the \emph{backpropagated gradient} $G_F$ from the top layer now carries information about the target label, with a specific structure that enables each hidden node to learn their representation \emph{independently}. Interestingly, the independent dynamics follows exactly the \emph{gradient ascent} of an energy function $E$, and its local maxima are precisely the emerging features. We study whether these local-optima induced features are generalizable, their representation power, and how they change on sample size, in group arithmetic tasks. When hidden nodes start to interact in the later stage of learning, we provably show how $G_F$ changes to focus on missing features that need to be learned. Our study sheds lights on roles played by key hyperparameters such as weight decay, learning rate and sample sizes in grokking, leads to provable scaling laws of memorization and generalization, and reveals the underlying cause why recent optimizers such as Muon can be effective, from the first principles of gradient dynamics. Our analysis can be extended to multi-layer architectures.

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