Dimensional Criticality at Grokking Across MLPs and Transformers

• URL: http://arxiv.org/abs/2604.16431v1
• Date: Mon, 06 Apr 2026 13:43:20 GMT
• Title: Dimensional Criticality at Grokking Across MLPs and Transformers
• Abstract summary: Abrupt transitions between distinct dynamical regimes are a hallmark of complex systems.<n>We introduce textbfTDU--OFC (Thresholded Diffusion Update--Olami-Feder-Christensen), an offline avalanche probe.<n>Across Transformers trained on modular addition and generalizations trained on XOR, we discover a localized crossing of the diffusion baseline $D=1$.
• Score: 2.652953665748039
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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