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GrokAlign: Geometric Characterisation and Acceleration of Grokking

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Grokking happens when a network's Jacobians align with the training data, and a Jacobian-norm regularizer called GrokAlign makes that happen far sooner.

desk verdict Solid practical contribution, but the causal story overreaches; the paper is worth reviewing with major revisions. read the letter →

arxiv 2506.12284 v2 pith:Z5APCI36 submitted 2025-06-14 cs.LG stat.ML

Thomas Walker , Ahmed Imtiaz Humayun , Randall Balestriero , Richard Baraniuk This is my paper · ORCID
classification cs.LGstat.ML MSC 68T07
keywords grokkingJacobianalignmentregularizationdelayedgeneralizationrobustnesscentroidneuraltangentkernelfunctionalgeometry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that grokking---the sudden late emergence of test accuracy and robustness after training accuracy saturates---is caused by the alignment of a network's Jacobian matrices at training points. A network whose Jacobians at each training point $x$ take the form $c x^\top$ (rows are scalar multiples of the input) is shown to be an optimal minimizer of cross-entropy or squared loss under a bounded Jacobian-norm constraint, and when the Jacobian is low rank this aligned form is optimally robust to $\ell_2$ perturbations. Because deep-network training dynamics bias Jacobians toward low rank, the authors conclude that the grokked state is the Jacobian-aligned state. They introduce GrokAlign, a regularizer on the average Frobenius norm of training-point Jacobians, and show it accelerates grokking and induces robustness, with speeds up to about 7.6 times fewer steps and 6.4 times less wall-clock time on the MNIST cross-entropy setting. A cheaper summary statistic, the centroid $\mu_x = (J_x(f))^\top \mathbf{1}$, tracks alignment and identifies the memorization, generalization, and robustification phases of training.

What carries the argument

The load-bearing object is the Jacobian matrix $J_x(f)$ of the network at an input, together with the alignment condition $J_x(f) = c x^\top$. The paper proves that this aligned form is the constrained optimum of the training loss (Theorem 2) and, under the low-rank bias of training, the optimally robust rank-one local map (Theorem 3). The tractable proxy is the centroid $\mu_x = (J_x(f))^\top \mathbf{1}$, which for continuous piecewise affine networks is the centroid of the linear region containing $x$ and can be computed with a single Jacobian-vector product; its dynamics connect to the neural tangent kernel through $\partial_t \langle x, \mu_x \rangle = \frac{\eta}{m} \sum_p \Theta(x, x_p) m_{x_p}$. GrokAlign is the training-side mechanism: it adds the average Frobenius norm of the training-point Jacobians to the loss, enforcing the constraint that makes alignment the optimum.

What would settle it

Train a network in a grokking regime while measuring the effective rank of its training-point Jacobians, for example the fraction of Jacobian variance explained by the first principal component. If test accuracy and robustness emerge while that fraction stays far below one, or if holding Jacobian norms low without allowing rank collapse prevents grokking, the low-rank assumption is violated and the claimed causal mechanism is not doing the work.

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Extended reading notes

Core claim

The central claim is that Jacobian alignment is the cause of grokking. For a continuous piecewise affine network with zero bias, minimizing cross-entropy or mean-squared error under the constraint $\|J_{x_p}(f)\|_F^2 \le \alpha$ forces the Jacobian at every training point to be aligned, $J_{x_p}(f) = c x_p^\top$ (Theorem 2). Among rank-one local maps, $A = c x^\top$ with the largest entry of $c$ at the true class maximizes the $\ell_2$ perturbation needed to flip the prediction, so aligned low-rank Jacobians are optimally robust (Theorem 3). Since training dynamics are biased toward low-rank weight matrices, and hence low-rank Jacobians, the delayed emergence of generalization and robustness is the delayed emergence of this alignment. GrokAlign appends the average Frobenius norm of training Jacobians to the loss to force the constraint directly; experimentally it induces robustness on CIFAR10, inhibits grokking when the norm is held high, and reaches the grokked state much faster than weight decay on MNIST. The centroid $\mu_x = (J_x(f))^\top \mathbf{1}$, equal to the sum of the Jacobian rows, is a Jacobian-vector-product-computable proxy: Jacobian alignment implies centroid alignment, and centroid alignment tracks feature learning through its neural-tangent-kernel dynamics.

Load-bearing premise

The load-bearing premise is that deep-network training dynamics push Jacobians toward low rank. If that rank-minimization bias gives way---if a network groks while its training-point Jacobians stay high-rank---then the optimal-robustness theorem no longer connects to training and Jacobian alignment is not the cause.

Editorial extensions

If this is right

  • Under the paper's MNIST cross-entropy grokking setup, GrokAlign reaches 85% test accuracy about 7.6 times sooner in steps and about 6.4 times sooner in wall-clock time than the weight-decay baseline; with mean-squared error the speed-ups are about 1.7 times and 1.5 times.
  • Since aligned low-rank Jacobians are optimally robust, delayed robustness is part of the same grokked state; GrokAlign-induced alignment yields robustness on a CIFAR10 convolutional network where weight-decay training loses robustness.
  • Centroid alignment, computable from a single Jacobian-vector product, identifies the memorization, generalization, and robustification phases of training and signals when further training will no longer help.
  • By maintaining Jacobian norms at a high level, GrokAlign can inhibit grokking, and on modular addition it biases a transformer toward a classification-style solution rather than the algorithm it would otherwise implement.
  • Compared with a gradient-amplification baseline and adversarial training, GrokAlign is the only tested method that both accelerates grokking and aligns the Jacobian, closing the gap between speed and robustness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: grokking onset should be predictable from the centroid geometry of early training; scaling the output to shrink centroid norms while keeping the neural tangent kernel dynamic may accelerate grokking without explicit regularization.
  • Editorial inference: the centroid/NTK identity supplies a scalable diagnostic for feature learning, since centroids cost one Jacobian-vector product and can be monitored in architectures where full Jacobians are too expensive.
  • Editorial inference: if Jacobian alignment is the cause, then explicit low-rank parameterizations combined with GrokAlign should reduce or remove the generalization lag; this is a direct test of the rank-bias premise.
  • Editorial inference: the transformer experiment suggests the alignment account fits classification-style grokking better than algorithmic grokking; tasks that admit both solution types should show intermediate dynamics and could arbitrate between the two mechanisms.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes that grokking — delayed generalization and delayed robustness in deep networks — is caused by the alignment of the network's Jacobian matrices with the training data. It introduces GrokAlign, a Jacobian-norm regularizer, and centroid alignment, a cheaper proxy based on the sum of Jacobian rows. The theoretical core is Theorem 2, which states that a continuous piecewise affine network minimizing cross-entropy or MSE under a Frobenius-norm constraint on the Jacobian at training points is Jacobian-aligned, and Theorem 3, which states that among rank-one Jacobians the aligned one is optimally robust to ℓ2 perturbations. The paper then argues that because training dynamics bias Jacobians toward low rank, Jacobian alignment is the cause of grokking. Empirically, GrokAlign accelerates grokking on an MNIST setup, induces robustness on CIFAR-10, and can inhibit grokking by keeping Jacobian norms high; centroid alignment is shown to track feature learning and robustness onset. A transformer experiment on modular addition, however, shows that GrokAlign does not accelerate grokking in that setting.

Significance. If the central claim were established, the paper would unify delayed generalization and delayed robustness under a single geometric mechanism and provide a practical, interpretable regularizer for accelerating grokking. The paper has genuine strengths: it ships code and a companion webpage, proposes a tractable monitoring metric (centroid alignment) with an explicit connection to the neural tangent kernel, and reports large speedups on a standard MNIST grokking benchmark (Table 1). The KKT derivation in Appendix E is a coherent characterization of a constrained optimum. However, the causal claim in Section 5 is substantially stronger than what the theory and experiments support. The main gap is the bridge from static constrained optima to training dynamics, which rests on an explicitly acknowledged but unproven low-rank bias assumption. In addition, the paper's own transformer experiment contradicts the universal version of the causal claim. These issues are fixable by reframing the contribution as a sufficient-condition characterization for classification-style tasks under an explicit low-rank hypothesis, but they are load-bearing for the paper's main conclusion.

major comments (4)
  1. [Section 4.3 / Figure 8] The transformer experiment on modular addition is an internal disconfirmation of the universal causal claim. Section 5 states that 'Jacobian alignment is the cause for grokking,' which implies that enforcing alignment should accelerate grokking wherever grokking occurs. Yet Figure 8 shows that applying GrokAlign does not accelerate grokking on a canonical benchmark, and the authors attribute this to the model learning an algorithmic solution that is 'not entirely compatible with the Jacobian and centroid aligned perspective.' The paper should either restrict the causal claim to classification-style tasks, or provide a mechanism explaining why algorithmic grokking is outside the scope. As written, the central claim is contradicted by the paper's own evidence.
  2. [Section 2 / Limitations] The bridge from Theorems 2 and 3 to training dynamics is an unproven assumption. Theorem 2 characterizes the minimizer of a loss under an explicit Jacobian-norm constraint, and Theorem 3 characterizes robustness among rank-one Jacobians; neither theorem describes the training trajectory. The paper's conclusion that 'the cause of grokking is the alignment of the deep network's Jacobian matrices' depends on the claim that 'deep network training dynamics bias toward low rank weight matrices... and thus low rank Jacobians.' This is acknowledged in the Limitations as a dependency of 'some of our reasoning.' Because the low-rank bias is the only link between the static theorems and actual training, the causal conclusion does not follow without it. The paper should state the low-rank bias as an explicit conditional hypothesis and present the results as a sufficient-condition characterization rather than a cause.
  3. [Appendix E, Theorem 2] The proof of Theorem 2 solves for a specific ansatz (rows equal to a x_p for the correct class and b x_p for incorrect classes) under an equality constraint on the Jacobian norm. The theorem statement asserts that a network minimizing the loss under the constraints 'is Jacobian-aligned,' but the proof does not establish uniqueness or that every global minimizer has this form. It also assumes zero bias, which is stated but not reflected in the theorem's main text (B_{ω_{x_p}} = 0 appears only in the proof context). The statement should be qualified as 'there exists a minimizer of this form' unless global optimality and uniqueness are proven, and the zero-bias assumption should be explicit in the theorem statement in the main text.
  4. [Section 4.3 / Table 1] The empirical evidence for accelerating grokking is limited to a single MNIST setup, and the cross-entropy baseline row reports no absolute steps or time, only the placeholder '–', which makes the 7.56× speedup difficult to verify against the raw data. Given that Figure 8 shows no acceleration on modular addition, the paper's 'Accelerating Grokking' claim should be qualified as applying to the studied classification-style settings, and the authors should provide the underlying baseline numbers or a pointer to the code artifact for direct reproduction.
minor comments (6)
  1. [Section 2] There is a typo in 'the the cause of grokking' in the contributions paragraph; also 'the the' appears in Section 4.3 ('the natural solution solution').
  2. [Appendix E] The word 'Lagrange' is misspelled as 'Langrange' twice in the proof of Theorem 2.
  3. [Theorem 2] The constraint bound α is used in the theorem and proof but is not defined in the main text; please define α as the allowed Frobenius-norm bound before the theorem statement.
  4. [Appendix E, MSE proof] In the mean-squared-error proof, the vector e_i is introduced as a standard basis vector of R^d, but the output space is R^C; this notation should be corrected (e_{y_p} ∈ R^C).
  5. [Figure 7] The constrained case uses a target Jacobian norm of 5.0; the choice of this value and its sensitivity should be reported, as it is a free parameter of the experiment.
  6. [References] Reference [31] contains a typo: 'Meural Information Processing Systems' should be 'Neural Information Processing Systems.'

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the core derivations are self-contained; the causal 'cause' claim overreaches but does not reduce to its inputs by construction.

full rationale

The paper's central theoretical chain is not circular. Theorem 2 solves a constrained convex optimization problem via KKT conditions and identifies the minimizer's Jacobian form; Theorem 3 is an independent Cauchy-Schwarz robustness bound; Theorem 9 is an analytical neural-tangent-kernel/centroid identity. GrokAlign is the direct enforcement of Theorem 2's norm constraint, so observing alignment under GrokAlign is a consequence of the theorem, not a fitted parameter renamed as a prediction, and the reported acceleration (Table 1) and monitoring claims are empirical and falsifiable. The main caveat is the Section 5 causal statement ('Jacobian alignment is the cause for grokking'): it relies on the low-rank-bias assumption, which the Limitations section explicitly acknowledges, and Figure 8 shows GrokAlign does not accelerate the modular-addition transformer, an internal disconfirmation of the universal causal claim. Those are evidentiary and scope problems, not circular reductions: no equation or fitted quantity is equivalent to its input by construction. Self-citations ([3], [7]) supply background geometry and region-migration interpretation but are not load-bearing for the core derivation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a hand-set regularization strength, the unproven low-rank bias assumption, and the zero-bias restriction. These are the main 'unpaid' inputs the reader must accept. No new physical entities are introduced; the centroid is a known quantity from spline theory, and GrokAlign is a method, not an entity.

free parameters (2)
  • lambda_Jac (GrokAlign regularization weight) = 0.001, 0.0001, 1.0 depending on experiment
    The strength of the Jacobian-norm penalty is hand-chosen per experiment. It is central to GrokAlign and is not derived from theory.
  • Target Jacobian norm 5.0 in the constrained case of Figure 7 = 5.0
    A hand-set anchor used to keep Jacobian norms high and inhibit grokking; no principled criterion is given for choosing 5.
assumptions (4)
  • domain assumption Deep network training dynamics bias Jacobians toward low rank
    Load-bearing for connecting Theorem 3 to actual training. Cited from refs 22-26, but not proven in the studied settings and acknowledged as a limitation.
  • domain assumption Zero bias in the network (B = 0)
    Required by Theorems 2 and 3. The paper states it will 'unless stated otherwise, omit bias terms in the deep networks we consider.'
  • standard math Continuous piecewise affine network representation
    Standard spline theory for ReLU-type networks; limits the scope of the theoretical claims, though the centroid computation itself does not require it.
  • standard math Convexity of the constrained loss and sufficiency of KKT conditions
    Used in the proofs of Theorem 2 for both cross-entropy and MSE. The convexity claim holds on the constrained set.

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Cite this review

Pith. "Pith review of GrokAlign: Geometric Characterisation and Acceleration of Grokking." pith.science (2026). https://pith.science/paper/Z5APCI36

@misc{pith2026250612284,
  author       = {Pith},
  title        = {Pith review of: GrokAlign: Geometric Characterisation and Acceleration of Grokking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z5APCI36}},
  note         = {Machine review of arXiv:2506.12284}
}
read the original abstract

A key challenge for the machine learning community is to understand and accelerate the training dynamics of deep networks that lead to delayed generalisation and emergent robustness to input perturbations, also known as grokking. Prior work has associated phenomena like delayed generalisation with the transition of a deep network from a linear to a feature learning regime, and emergent robustness with changes to the network's functional geometry, in particular the arrangement of the so-called linear regions in deep networks employing continuous piecewise affine nonlinearities. Here, we explain how grokking is realised in the Jacobian of a deep network and demonstrate that aligning a network's Jacobians with the training data (in the sense of cosine similarity) ensures grokking under a low-rank Jacobian assumption. Our results provide a strong theoretical motivation for the use of Jacobian regularisation in optimizing deep networks -- a method we introduce as GrokAlign -- which we show empirically to induce grokking much sooner than more conventional regularizers like weight decay. Moreover, we introduce centroid alignment as a tractable and interpretable simplification of Jacobian alignment that effectively identifies and tracks the stages of deep network training dynamics. Accompanying webpage (https://thomaswalker1.github.io/blog/grokalign.html) and code (https://github.com/ThomasWalker1/grokalign).

Figures

Figures reproduced from arXiv: 2506.12284 by the authors.

Figure 1
Figure 1. For a deep network to grok, its Jacobians should align such that the sum of their rows are cosine￾similar to the point at which they were computed; we dub this condition centroid aligned. We train a ReLU network on the MNIST dataset [1] using GrokAlign. We take three training data points, left, and observe the linear regions (using SplineCam [2]) of the deep network along with the centroids [3] of the three data poi… view at source ↗
Figure 2
Figure 2. Under weight-decay and GrokAlign, the effective rank of the Jaco￾bian matrices evaluated at the training data tends towards rank one. Here we trained ReLU networks on the MNIST classifica￾tion task [1] under the mean-squared error and cross-entropy loss functions using the AdamW optimizer [29]. Throughout training, we recorded the average explained variance of the first principal component of the Ja￾cobians evaluate… view at source ↗
Figure 3
Figure 3. Theorem 9 holds in practice: we observe that a changing inner product indeed corresponds to a feature learning regime. Here we train a two-layer scalar-output ReLU network using the binary-cross-entropy loss function to distinguish between the zero and one class of the MNIST dataset [1]. We train the model using full-batch gradient descent for 4000 steps at a learning rate of 0.01. At the beginning of training we fi… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Centroid alignment of a point from the training set identifies the generalisation and robustness of a deep network. We study a two-layer network of width 2048 learning the XOR classification task described in Section 4.2. In the top row we train the network using full-…
Figure 5
Figure 5. Figure 5: GrokAlign capitalises on the low rank implicit bias of deep network training to induce robustness by aligning the Jacobians of the deep network. Here we train a convolutional neural network with six convolutional layers and two linear layers, with no bias terms, on a 1…
Figure 6
Figure 6. Figure 6: The geometry of the function of a centroid aligned deep network is more resemblant of the geometry of the training data, indicating that the transformation its applies to local regions of data points is more representative of their features. We compare the two convolut…
Figure 7
Figure 7. Figure 7: By maintaining the Frobenius norm of the Jacobians at the training data relatively high we can keep the norms of the centroids relatively high which prevents grokking. We take the MNIST grokking set up of Liu et al. [6]. In the minimising case we impose GrokAlign with …
Figure 8
Figure 8. Figure 8: GrokAlign biases a single layer transformer model to learn a classification style solution. Here we obtain the centroid alignment statistics for a single layer transformer trained on modular addition [4]. We use the same training pipeline as in Nanda et al. [15], with …
Figure 9
Figure 9. Figure 9: Without the ability to manipulate its embedding matrices, a single layer transformer learns the classification style solution for performing modular addition and thus benefits from being trained with GrokAlign. We adopt the same configurations as in [PITH_FULL_IMAGE:f…
Figure 10
Figure 10. Figure 10: When no regularisation is used, minimising the centroid norm at initialisation is essential for ensuring the alignment of the centroid during training, and output scaling ensures this more effectively than scaling the weights at initialisation. Here we repeat the expe…
Figure 11
Figure 11. Figure 11: Here we corroborate the right plot of [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.