Dimensional Criticality at Grokking Across MLPs and Transformers
For researchers studying emergent phenomena in deep learning, this provides a robust macroscopic signature of the grokking transition, enabling early detection and deeper understanding of generalization dynamics.
The paper introduces a macroscopic observable, the effective cascade dimension D(t), that reveals a dynamical crossing of a critical point precisely at the grokking transition in Transformers and MLPs, with task-dependent crossing directions. The observable diverges from ungrokked runs 100-200 epochs before the behavioral transition.
Abrupt transitions between distinct dynamical regimes are a hallmark of complex systems. Grokking in deep neural networks provides a striking example -- an abrupt transition from memorization to generalization long after training accuracy saturates -- yet robust macroscopic signatures of this transition remain elusive. Here we introduce \textbf{TDU--OFC} (Thresholded Diffusion Update--Olami-Feder-Christensen), an offline avalanche probe that converts gradient snapshots into cascade statistics and extracts a \emph{macroscopic observable} -- the time-resolved effective cascade dimension $D(t)$ -- via grokking-aligned finite-size scaling. Across Transformers trained on modular addition and MLPs trained on XOR, we discover a localized dynamical crossing of the Gaussian diffusion baseline $D=1$ precisely at the generalization transition. The crossing direction is task-dependent: modular addition descends through $D=1$ (approaching from $D>1$), while XOR ascends (from $D<1$). This opposite-direction convergence is consistent with attraction toward a candidate shared critical manifold, rather than trivial residence near $D \approx 1$. Negative controls confirm this picture: ungrokked runs remain supercritical ($D>1$) and never enter the post-transition regime. In addition, avalanche distributions exhibit heavy tails and finite-size scaling consistent with the dimensional exponent extracted from $D(t)$. Shadow-probe controls ($α_{\mathrm{train}}=0$) confirm that $D(t)$ is non-invasive, and grokked trajectories diverge from ungrokked ones in $D(t)$ some $100$--$200$ epochs before the behavioral transition.