A Counterexample to Fourier Alignment in Single-Neuron Modular Addition

While neural collapse (NC) predicts that a $K$-class-balanced classifier should organize terminal representations as a $(K-1)$-dimensional simplex equiangular tight frame (ETF), modular addition consistently enters a different regime: networks compress to a two-dimensional cyclic geometry in which both classifier weights and token embeddings lie on circles. We refine the explanation of this phenomenon in three directions. First, we formalize a layerwise non-uniform training mechanism: downstream classifier weights are driven by dense cross-entropy gradients into a rank-2 equiangular configuration before upstream embeddings fully reorganize, and once this classifier plane forms, backpropagated feature gradients constrain embedding motion to the same plane while weight decay suppresses orthogonal components. Second, after this subspace locking, the induced in-plane dynamics admit an entropy-regularized transport interpretation on $S^1$; combined with modular-addition labels, this reduces embedding formation to phase alignment, whose minimizers are single-frequency characters of $\mathbb{Z}/P\mathbb{Z}$ and hence equal-angle points on a circle. Third, we quantify why this solution prevails over NC: a simplex ETF gains only an $O(1)$ advantage in cross-entropy, whereas the cyclic rank-2 solution enjoys a $Θ(K)$ advantage under Schatten or weight-decay surrogates, yielding a critical threshold $λ_{\mathrm{crit}} = Θ(1/K)$. Our results explain both why classifier weights move first and why embeddings subsequently align with them, showing that grokking on modular arithmetic is governed not by maximal separation alone but by a task-structured trade-off between separation, symmetry, and complexity.

Grokking is the delayed generalisation phenomenon where a transformer trained on modular arithmetic abruptly transitions from near-chance to near-perfect validation accuracy. It has been attributed to a Fourier-based algorithmic circuit, but its timing, causal structure, and controllability remain poorly understood. We introduce the Frequency Synchronization Degree (FSD), a normalised, permutation-tested metric for Fourier circuit synchronisation requiring no prior knowledge of the circuit. Across nine modular addition configurations (five primes, three seeds), FSD reaches its post-grokking level 500 to 3000 steps before grokking (mean lead 1722 steps, every configuration positive, sign-test p approx 0.004), and synchronises before a restricted-logit loss baseline in all nine cases, making it the earliest available predictor. We give direct causal evidence that the inter-phase gap is a regularisation phenomenon: forking training at the FSD-ceiling step and varying weight decay lambda produces monotonically earlier grokking, with delta-t proportional to 1/lambda. This law replicates across three primes (R-squared 0.89 to 0.99 on seed-averaged delta-t); per-run R-squared is unstable due to the chaotic transition, so we report error bars rather than single runs. Grokking occurs at a near-constant memorisation norm across lambda, grounding the constant in a threshold mechanism. This is not an artefact of applying a Fourier detector to a Fourier circuit: on the non-abelian group S5, a basis-faithful generalisation of FSD precedes grokking on all six seeds, while the original Fourier FSD does not. Using the FSD ceiling to schedule a weight-decay increase also accelerates grokking over a fixed schedule without destabilising training. An attention-only variant groks with a strong FSD precursor while an MLP-only model never groks.

In this paper, we study the gradient descent dynamics for jointly training both layers of a one-hidden-layer ReLU network to fit a linear target function. Concretely, we consider a realizable setting where inputs are drawn i.i.d. from a Gaussian distribution and labels follow a planted linear model. This stylized framework captures salient features of end-to-end training in inverse problems and certain auto-encoder models. Despite its apparent simplicity, the dynamics remain poorly understood, in part because the loss landscape contains multiple non-strict saddle points, making it unclear why gradient descent from random initialization reliably escapes bad stationary regions. We provide a detailed characterization of the optimization landscape and prove that gradient descent from a moderately small random initialization-simultaneously training both layers-converges to a global minimizer at a linear rate with order-wise optimal sample complexity. Our analysis tracks the trajectory through three phases: an alignment phase in which hidden weights progressively align with the planted direction while the output weights maintain the correct sign pattern; a growth phase in which the norms of both layers increase while preserving alignment; and a local refinement phase in which the aligned neurons rapidly converge to the planted direction, yielding fast local convergence. To rigorously show that GD avoids non-strict saddles, we develop trajectory-level control arguments for the end-to-end dynamics. In addition, we establish novel uniform concentration results that hold along the entire trajectory, and are essential for obtaining order-wise optimal sample complexity. We corroborate our theory with extensive experiments across a range of configurations.