Organizations: University Paris City & University of Reunion, Paris, France · Dipartimento di Fisica e Chimica Emilio Segrè, Università degli Studi di Palermo, Palermo, Italy · Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA · EnergyLab, University of Reunion, Saint-Denis, France · PEACCEL, AI for Biologics, Paris, France
Abstract
Neural networks trained on modular arithmetic exhibit grokking, a delayed transition from memorisation to generalisation known to depend on model capacity: too little and the network memorises slowly or not at all, too much and it generalises almost immediately. What happens at the extreme of this spectrum, when the architecture's expressible function class collapses to a finite-dimensional algebraic variety? We study two-layer networks with a holomorphic monomial activation sigma(z)=z^k, trained on modular tasks encoded via roots of unity. Here the network output, regardless of hidden width, is confined to a (k+1)-dimensional subspace of characters of (Z_p)^2, an O(k/p^2) slice of the full function space. We give a complete algebraic characterisation of this subspace: a task is representable if and only if its discrete Fourier support lies on the diagonal u+v = k (mod p), which for linear-phase targets reduces to the arithmetic criterion m+n=k. This is not merely a constraint on eventual generalisation but on memorisation itself: because the outputs are algebraically confined, a non-representable target cannot be fit even on the training set, and we prove a positive lower bound on the training loss, independent of width. Across 585 runs the algebraic prediction matches the observed outcome with 99.8% accuracy, with no memorisation regime and no grokking; outcomes split cleanly into instant success and outright failure. This binary behaviour is the limiting case of the capacity-grokking relationship: when the expressible class shrinks to a fixed algebraic object, the question of when a network will grok dissolves into whether it can represent the target at all. A bottleneck ablation connects this extreme to standard networks, tracing a continuous path from representational failure, through memorisation without generalisation, to grokking with a shrinking gap as capacity grows.
Neural networks trained past memorization frequently undergo a delayed transition to generalization, a phenomenon known as grokking. Despite theoretical progress on \emph{why} this transition occurs, the quantitative structure of \emph{when} it occurs in hyperparameter space remains uncharacterized. We map the memorization-to-generalization boundary across 384 configurations of two-hidden-layer MLPs on modular arithmetic, fitting a power-law scaling relation for generalization onset time: Tgrok∝H−0.27D−2.04η−0.50λ−0.64 (R2=0.732; 0.821 with interactions). The exponent hierarchy reveals that data complexity (D−2.04) is the dominant driver of regime transition, not model capacity (H−0.27): doubling data accelerates generalization by ∼4×, while doubling width yields only ∼1.2×. A sharp phase boundary at weight decay λ≳1.0 separates grokking from non-grokking configurations, and weight norm trajectories show monotonic compression during the transition, consistent with implicit regularization selecting low-complexity solutions. These results provide a quantitative foundation for predicting and controlling regime transitions in overparameterized networks.
Existing accounts of grokking explain the phenomena in terms of mechanistic frameworks such as circuit efficiency or lazy-to-rich transitions. However, despite a known dependence between grokking and model size, how model capacity shapes grokking remains an open question. We give an information-theoretic account of this relationship on the task of modular arithmetic, showing that grokking does not immediately occur when a model becomes large enough to memorise the training set, but rather emerges as the outcome of a competition between two measurable timescales: a memorisation speed Tmem(P) and a generalisation speed Tgen(P), both of which are functions of model parameter count P. Adapting the information capacity framework of Morris et al. (2025), we estimate Tmem(P) on random-label data of equivalent complexity and Tgen(P) on the modular task itself, and show that grokking emerges close to the parameter scale where these timescales intersect. The framework also suggests an empirical model for predicting memorisation speed given model capacity and dataset complexity, recovering the previously reported empirical observation that larger models memorise faster. Overall, we motivate the formalisation of different learning timescales as important abstractions to study when explaining how model capacity shapes grokking on algorithmic tasks.
We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and consistent topological signature of grokking: a sharp increase in both the maximum and total persistence of first homology (H1). Persistence diagrams reveal the emergence of a dominant long-lived topological feature together with increasingly structured secondary features, reflecting the underlying cyclic structure of the task. Compared to existing spectral and geometric diagnostics -- specifically, Fourier analysis and local intrinsic dimension -- persistent homology provides a unified geometric and topological characterization of representation learning, capturing both local and global multi-scale structure. Ablations across data regimes and control settings show that these topological transitions are tied to generalization rather than memorization. Our results suggest that persistent homology offers a principled and interpretable framework for analyzing how neural networks internalize latent structure during training.