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REVIEW 3 major objections 5 minor 52 references

Grokking vs. Learning: Same Features, Different Encodings

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that grokking and ordinary steady training converge on the same features in modular addition and Ising-phase classification, but that steady training can encode those features far more efficiently—most strikingly in a…

desk verdict The compressive regime is a genuine new result, but the 'same features' headline overstates the paper's own Ising appendix. read the letter →

arxiv 2502.01739 v1 pith:SGERGN6V submitted 2025-02-03 cs.LG cond-mat.dis-nncs.AI

Dmitry Manning-Coe , Jacopo Gliozzi , Alexander G. Stapleton , Edward Hirst , Giuseppe De Tomasi , Barry Bradlyn , David S. Berman This is my paper · ORCID
classification cs.LGcond-mat.dis-nncs.AI
keywords grokkingsteadylearningmodelcompressionmagnitudepruningmodularadditionFisherinformationmetricprogressmeasuresIsing
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

Grokking—training that overfits for a long time and then suddenly generalizes—typically ends at the same loss as ordinary, 'steady' training. The paper asks whether the two paths produce fundamentally different models, and answers: no at the level of features, yes at the level of encoding efficiency. In two test beds—classifying snapshots of the 2D Ising model and solving modular addition—interpretability probes show that grokked and steadily trained networks converge on the same feature (energy for Ising, Fourier modes for modular addition), yet pruning reveals that steady training can encode those features far more compactly. The strongest concrete result is a newly identified 'compressive regime' in modular addition, reached by lowering the weight scale at initialization and increasing weight decay, in which terminal train loss trades off linearly with compressibility and pruning removes up to 25x as many weights as from a baseline model. The paper also shows that grokking dynamics are task-dependent and introduces Fisher-information-geometric measures under which grokking trajectories look straight.

What carries the argument

Several objects carry the argument. Grokking is induced and controlled by a single dial: the multiplier $w_0$ applied to the initialization weights, which smoothly interpolates between steady learning and grokking. Features are read out by task-specific probes—pre-activation correlation coefficients against energy and magnetization for the Ising classifier, and the inverse participation ratio in the Fourier basis, $\mathrm{IPR}^{(k)} = \sum_{\nu} |\tilde{W}_{\nu k}|^4 / (\sum_{\nu} \tilde{W}_{\nu k}^2)^2$, for modular addition. Compressibility is quantified by pruning each layer's smallest weights by a fraction $p$, integrating the resulting accuracy curve $a(p)$, and setting $c = 1/(1-a)$; the compressive regime is the region where $c$ and end train loss trade off linearly. Finally, the trajectory analysis uses the Fisher information metric $g^{\mathrm{FIM}}_{ij}$ to define a step magnitude $|s|_{\mathrm{FIM}}$ and a cosine similarity $S_{C-\mathrm{FIM}}(s,s')$ between consecutive parameter-space steps, which reveal the straight-line grokking path.

What would settle it

A concrete check: apply a pruning rule that does not rank weights by magnitude within each layer—for instance, pruning by diagonal Fisher information—to the modular addition models in the compressive regime. If the 25x compression advantage disappears, or if a non-magnitude-pruned grokked model prunes as well as the steady one, then the claimed encoding-efficiency difference is a property of the magnitude-pruning metric rather than of the learned representations; alternatively, a synthetic task with two provably distinct solution families could test whether the feature probes ever miss a genuinely different strategy in one of the two training regimes.

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Extended reading notes

Core claim

The central claim is that grokked and steadily trained models learn exactly the same dataset features, but the efficiency with which those features are encoded can differ by a large factor. The authors establish this by correlating final-layer pre-activations with the physically meaningful features in each task: energy and magnetization for the Ising CNN, and Fourier-basis localization (inverse participation ratio) for the modular addition MLP. They then introduce a compressibility measure based on the area under the magnitude-pruning curve and show that, in modular addition, models trained in a parameter range they call the compressive regime obey a linear trade-off between end train loss and compressibility, reaching compression factors of 25x the base model and 5x the best grokked compression. In the Ising task no compressive regime appears, which the authors take to show that the phenomenon is a distinct, parameter-dependent regime of steady learning rather than a generic feature of training. Dynamically, the paper reports that feature formation during the grokking plateau is task-dependent (visible in modular addition, absent in Ising), that compressibility peaks immediately after the plateau, and that the grokking trajectory follows an approximately straight line in model space as measured by a Fisher-information-based cosine similarity.

Load-bearing premise

The load-bearing assumption is that the hand-built interpretability scores—pre-activation correlation with energy and magnetization for Ising, Fourier-basis inverse participation ratio for modular addition—faithfully report the features the network actually uses, so that the 'same features' conclusion would fail if these probes miss the true computational strategy.

Editorial extensions

If this is right

  • The practical upside of grokking is not that it discovers better features: at least in these tasks, steady training finds the same ones, and can find them in a more prunable form.
  • In modular addition, initialization scale becomes a dial that trades final train loss against model size after pruning, so a practitioner can pick a point on that trade-off curve.
  • The compressive regime is parameter-specific and can be switched off (e.g., by raising batch size), so compression advantages must be reported together with the training hyperparameters that produce them.
  • Peak compressibility occurs just after the grokking plateau, and a transient compressibility spike appears even in runs without the compressive regime, so pruning dynamics and terminal compressibility are separate phenomena.
  • Fisher-information-based trajectory measures give a task-general signature of grokking—nearly straight paths in model space—that does not depend on hand-built feature probes.

Reading between the lines

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

  • If the same-features conclusion generalizes beyond toy tasks, then the main remaining reason to prefer grokking would be some other dynamic benefit, and the paper's own evidence would argue for eliminating grokking whenever possible—a claim the authors put forward as an open question.
  • The linear loss–compressibility trade-off looks like a rate–distortion curve indexed by the initial weight scale; viewing it this way suggests the compressive regime might be generically predicted by an information-theoretic account of how initialization scale limits the model's effective capacity.
  • The two-root signature in the gradient of the FIM step magnitude could be tested as a universal grokking detector on tasks where no interpretable 'progress measure' exists; if it marks the plateau start and end across many architectures, it would make hand-built probes unnecessary for detecting grokking.
  • The reported absence of a correlation between compressibility and the local learning coefficient is surprising under the usual intuition that more degenerate models are more compressible, and could motivate a re-examination of what the local learning coefficient measures.
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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

3 major / 5 minor

Summary. The paper compares neural networks trained via grokking and via ordinary 'steady' learning on two controlled tasks: classifying phases of 2D Ising snapshots with a CNN, and modular addition with a one-hidden-layer MLP. The authors sweep only the initialization weight scale to interpolate between regimes, keeping architecture, data, and optimizer fixed. Their central claims are: (1) grokked and steadily trained models learn the same dataset features (energy/magnetization for Ising; Fourier modes for modular addition), (2) the encodings differ, most strikingly in a 'compressive regime' of steady training in modular addition where there is a linear trade-off between end train loss and compressibility, reaching roughly 25x compression of the base model under higher weight decay, (3) model development during the grokking plateau is task-dependent, with modular addition showing progress measures and Ising showing none, and (4) information-geometric measures based on the Fisher Information Metric show that grokking trajectories are approximately straight in model space. The paper includes per-seed data in appendices and provides code.

Significance. The compressive regime is the strongest contribution: it is a reproducible, controlled finding with potential practical value for model compression, and the claim is supported by per-seed plots and a check that increasing batch size removes the effect. The FIM-based trajectory analysis is a novel measurement tool that is general and, if validated, could be useful beyond these two tasks. The authors are to be credited for fixing architecture/data and varying only the initialization scale, for reporting per-seed distributions, and for releasing code. However, the headline 'same features' conclusion is currently overstated: the Ising claim is contradicted by the paper's own Appendix C, and the compressive-regime results in Appendix E show that the steady model's Fourier localization can break down, complicating the feature-identity statement. If the authors can resolve the Ising discrepancy and qualify the feature-identity claim, the paper would be a solid empirical contribution.

major comments (3)
  1. [§4.1 and Appendix C] Section 4.1 states that 'every neuron in the final layer is either perfectly correlated or anti-correlated with the energy of the input' and that 'for every seed, in both learning and grokking, the energy has the highest correlation' with the majority of neurons in the second CNN and fully connected layer. Appendix C (Fig. 13 and surrounding text) reports the opposite: in the grokking regime, 'three of the seeds have significant deviation from perfect correlation with the energy, and instead correlate to the raw value of the magnetization,' and the caption says 'in some seeds learning the magnetization.' These statements cannot both be true. The same-features claim is the first headline conclusion, so the paper must quantify the per-seed/per-neuron deviations, report whether the three deviating seeds reach the same test accuracy as the others, and either restrict the conclusion to the majority of seeds or explain why magnetization-correlated neurons are consistent with 'learning the same feature.' The deliberate construction of the dataset to favor energy (§4.1) additionally means the near-universal energy correlation is partly a property of the task design.
  2. [§4.2, §5, Appendix D/E] The same-features conclusion is also in tension with the compressive regime. In §4.2, IPR localization is used as evidence that grokked and steadily trained models learn the same Fourier features (comparable mean IPRs of 0.43 and 0.53). Yet Appendix D reports that the compressive regime 'leads to an improvement in compressibility through a decrease in Fourier basis localization,' and Appendix E states that in the highly compressive regime 'the mean IPR falls by close to an order of magnitude, indicating that the model's Fourier representation is breaking down.' If IPR is a feature-identity measure, then a steady model in the compressive regime does not share the same localized Fourier features as a grokked model. The main text should state this caveat explicitly and clarify what 'same features' means for delocalized representations.
  3. [Abstract and §6.3] The abstract claims that 'models undergoing grokking follow a straight path in information space,' but the evidence in §6.3 is a peak cosine similarity that is near one only for the Ising grokking regime; for modular addition the peak is noticeably lower, and in all regimes the consecutive-step cosine similarity decays to near zero after learning. The straight-line statement should be restricted to the pre-/during-grokking phase and to the Ising task, or additional per-seed evidence should be provided showing straightness in modular addition grokking.
minor comments (5)
  1. [Figure 13 caption] The caption of Figure 13 says 'the dominant behaviour is still to learn the magnetization,' which contradicts the body text's claim that energy is the dominant feature; please correct the likely typo or reconcile the statement with §4.1.
  2. [§5 and Appendix E] The text alternates between the area-based compressibility c of Eq. (5) and the weight fraction needed to maintain 95% accuracy ('cut 95% of the weights'); the relationship between these two quantities should be stated explicitly so that '25x compression' is unambiguous.
  3. [Table 1] The Ising task uses weight decay 0.1 while modular addition uses 3e-5; because weight decay is the mechanism behind the straight-to-the-origin trajectory in §6.3, the large asymmetry between the two tasks should be discussed when comparing their dynamics.
  4. [Appendix F] The FIM is approximated by its diagonal, citing Amari et al. (2019), but no numerical check of the diagonal approximation is reported for these specific architectures; a brief validation would strengthen the FIM-based conclusions.
  5. [§2.2] The one-hot vector notation in §2.2 has typographical artifacts (e.g., '0 1, ...,1a, ...,0P'); please clean up the notation.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: central claims are empirical measurements and parameter scans; self-cited FIM tools are measurement devices, not inputs.

full rationale

The paper is an empirical study, not a derivation, and I find no step in which a claimed result is identical to an input by construction. The same-features conclusions are supported by independently measured correlation coefficients (Section 4.1) and Fourier IPRs (Section 4.2), both computed on trained models; the design choice to favor energy is an input, but the equal-energy conclusion across grokking and steady training is not definitional. The compressive-regime result (Section 5) is a parameter scan: the weight multiplier w0 is varied, and end train loss and pruning-area compressibility are measured separately; the observed linear trade-off is an empirical curve, not a fitted constraint. The FIM trajectory measures (Section 6.3) are self-cited to Berman et al., but the FIM is a standard object and the cited works are used as a measurement toolkit, not to force the straight-line conclusion. I therefore score no circularity. I do flag one internal inconsistency that is a correctness concern rather than circularity: Appendix C (Fig. 13) reports that in the grokking Ising regime 'three of the seeds have significant deviation from perfect correlation with the energy, and instead correlate to the raw value of the magnetization,' which weakens the main-text claim in Section 4.1 that 'every neuron in the final layer is either perfectly correlated or anti-correlated with the energy.' This per-seed variability should be reconciled, but it does not make the conclusion an input of the analysis.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The analysis is empirical, not a derivation. The central phenomena rest on hand-chosen controls (weight multiplier, batch size, weight decay), on the assumption that correlation and IPR probes reveal true features, on the engineered Ising dataset, and on the FIM diagonal approximation for trajectory measures. No new physical entities are postulated.

free parameters (4)
  • weight multiplier w0 = swept; Ising 1-10, modular addition 0.1-100
    Hand-chosen control that tunes between grokking and steady learning; the compressive regime is defined by the low-w0 end of this sweep.
  • weight decay for high-compression runs = 3e-4 versus 3e-5
    The 25x compression result uses a hand-selected weight decay found by rough search; models with 30% larger decay fail to learn.
  • FIM diagonal threshold = 0.1
    Hand-chosen threshold for the stiff/sloppy split in Appendix F; used in the split trajectory analysis, not in the main same-features claim.
  • batch size = 64 main text, 200 control
    The compressive regime disappears at batch size 200, showing that the phenomenon is parameter-dependent rather than generic.
assumptions (5)
  • domain assumption Pre-activation correlation and Fourier IPR faithfully measure the features the network uses.
    Used to conclude that grokked and steady models learn the same features; if these probes miss the true strategy, the conclusion fails. Sections 4.1 and 4.2, Appendix C.
  • ad hoc to paper Grokking is adequately defined by the 5% train and test accuracy margins in Definition 1.
    The classification of runs into grokking versus steady learning depends on arbitrary margins, and at the margins some seeds mix regimes, for example one Ising seed persistently fails to grok. Section 3.
  • domain assumption The FIM diagonal approximation holds for these networks.
    The trajectory measures use a diagonal approximation of the Fisher Information Metric, following Amari et al. 2019, as invoked in Appendix F. If inaccurate, the straight-path claim may be an artifact.
  • domain assumption The two toy tasks are informative representatives of grokking behavior.
    Generalization to larger models is explicitly left open in Section 7, and the compressive regime and task-dependent plateau development are demonstrated only in these settings.
  • domain assumption The Ising dataset design biases the model toward the energy feature.
    Section 4.1 states that metastable ordered snapshots with low magnetization were deliberately included and a CNN was chosen to detect interfaces, so the same-features result for Ising is partly a consequence of task design.

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Pith. "Pith review of Grokking vs. Learning: Same Features, Different Encodings." pith.science (2026). https://pith.science/paper/SGERGN6V

@misc{pith2026250201739,
  author       = {Pith},
  title        = {Pith review of: Grokking vs. Learning: Same Features, Different Encodings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGERGN6V}},
  note         = {Machine review of arXiv:2502.01739}
}
read the original abstract

Grokking typically achieves similar loss to ordinary, "steady", learning. We ask whether these different learning paths - grokking versus ordinary training - lead to fundamental differences in the learned models. To do so we compare the features, compressibility, and learning dynamics of models trained via each path in two tasks. We find that grokked and steadily trained models learn the same features, but there can be large differences in the efficiency with which these features are encoded. In particular, we find a novel "compressive regime" of steady training in which there emerges a linear trade-off between model loss and compressibility, and which is absent in grokking. In this regime, we can achieve compression factors 25x times the base model, and 5x times the compression achieved in grokking. We then track how model features and compressibility develop through training. We show that model development in grokking is task-dependent, and that peak compressibility is achieved immediately after the grokking plateau. Finally, novel information-geometric measures are introduced which demonstrate that models undergoing grokking follow a straight path in information space.

Figures

Figures reproduced from arXiv: 2502.01739 by the authors.

Figure 1
Figure 1. Snapshots of the 16 x 16 Ising lattice, where white and black squares indicate ±1 spin assignments. As temperature in￾creases, snapshots become more disordered, passing through a phase transition at temperature Tc. The two phases are separated by a transition that occurs at a critical temperature, Tc. For T < Tc, the system is in an ordered phase, while for T > Tc, it is in a disordered phase. In our model, we consi… view at source ↗
Figure 2
Figure 2. Ising and modular addition training curves, for varying weight multipliers at initialization (larger values are redder). The dashed lines indicate training accuracies/losses, and the solid lines test accuracies/losses. from which we define grokking as: Definition 2 (Grokking). A training run is said to grok if tgrok > ttrain. Otherwise, we say that the model “steadily learns”. Following (Liu et al., 2023a), we induc… view at source ↗
Figure 4
Figure 4. In the modular addition task, the model learns the Fourier modes of the input data. To see this, we Fourier transform the unembedding matrix and plot its coefficients, which correspond to different Fourier frequencies. The heatmaps for (a) grokking and (b) learning suggest that each neuron predominantly activates on the sine and cosine of a single frequency. This is confirmed by calculating the IPR for all neurons, … view at source ↗
Figures from the paper (31 more)
Figure 5
Figure 5. Figure 5: Pruning curves for (a) Ising and (b) modular addition. In the modular addition task, there is a linear trade-off between the end-loss and the compressibility c, shown in (c), while no such relationship for the Ising task, shown in (d). Data in the top row is averaged o…
Figure 7
Figure 7. Figure 7: Dynamical measures comparing learning and grokking for the Ising task. Unlike modular addition, the model shows no sign of developing through the grokking plateau. the absolute pre-activation, wk wk ≡ P i∈D |z i k | P i∈D P k∈L |z i k | , r˜ ≡ 1 |L| X k∈L wk|rk|, (6) w…
Figure 8
Figure 8. Figure 8: FIM-cosine similarities, SC−F IM(s, s′ ), of the model trajectories through model space during training (averaged over 10 runs). The similarity between consecutive steps, (s e , se+1), is shown in (a) for the Ising task and in (b) for modular addition. The similarity b…
Figure 9
Figure 9. Figure 9: A summary of the outcomes of the runs for the 10 different seeds used for each weight multiplier. We see that all bar one of the runs groks for the higher weight multiplier regime of the plot, and that for the majority of seeds the final test accuracies and losses for …
Figure 10
Figure 10. Figure 10: A summary of the outcomes of the runs for the 5 different seeds used for each weight multiplier in modular addition. B. The Ising classification problem This section gives a more formal definition of the Ising model used in the main text. Generically, the Ising model …
Figure 11
Figure 11. Figure 11: Three extended domain wall configurations of the Ising model. These snapshots are correctly classified as ordered by the energy, but are misclassified if the network only learns its magnetization. Such configurations are common in local update Monte Carlo simulations.…
Figure 12
Figure 12. Figure 12: A summary of the correlation coefficients for the different measures across seeds and all layers for the learning case in Ising. Vertical dashed lines separate different seeds. D. Compressibility We present additional results for compressibility. In the modular additi…
Figure 13
Figure 13. Figure 13: A summary of the correlation coefficients for the different measures across seeds and all layers for the grokking case in Ising. Vertical dashed lines separate different seeds. We can see that although the dominant behaviour is still to learn the magnetization, the gr…
Figure 14
Figure 14. Figure 14: Pruning curves for batch size 64 (left) as in the main text, and 200 (right). All other parameters are as in table 1. We see that changing the batch size eliminates the compressive regime. At batch size 200 there is no significant increase in compressibility at lower …
Figure 15
Figure 15. Figure 15: Correlations between model compressibility and other model properties for the parameters considered in the main text. We note that the although the train loss trades off linearly with the compressibility, there is no relationship with the magnetization. In the compres…
Figure 16
Figure 16. Figure 16: Correlations between model compressibility and other model properties for batch size 200, and remaining parameters as in the main text. Note that here, there is no relationship between grokking and the IPRs. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
Figure 17
Figure 17. Figure 17: In the Ising case, there is no compressive phase, and so there is no trade-off between the end-loss and the model compression accessible by tuning the weight multiplier. 50% 60% 70% 80% 90% 100% 0% 20% 40% 60% 80% 100% 50% 60% 70% 80% 90% 100% 0 1 2 3 4 5 6 7 0.1 0.2 …
Figure 18
Figure 18. Figure 18: Pruning curves for the highly compressed curves in the top left of the modular addition pruning areas curve. There is a very sharp transition to very highly compressed models between weight multipliers w0 = 2 and w0 = 1. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]
Figure 19
Figure 19. Figure 19: Correlations between the compressibility and measures of training dynamics and model features in the highly compressive regime (weight decay 3 × 10−4 . 21 [PITH_FULL_IMAGE:figures/full_fig_p021_19.png]
Figure 20
Figure 20. Figure 20: fig. 20. These spectrums show gaps in all cases, which are approximately at the same positions, and the dashed line marks [PITH_FULL_IMAGE:figures/full_fig_p022_20.png]
Figure 20
Figure 20. Figure 20: FIM diagonal values for the neural network model after training, sorted increasing values, and averaged over the 10 seed runs. Plots showing both the learning and grokking regimes; for both the Ising (a), and ModAdd (b) tasks. The dashed line represent the 0.1 value s…
Figure 21
Figure 21. Figure 21: Accuracies and losses as a function of % parameters pruned for the grokked and steady-learning regimes (Ising and modular arithmetic networks). A natural question one may ask is: “how does Fisher pruning act per-layer?” In order to facilitate such an analysis, we prop…
Figure 22
Figure 22. Figure 22: Losses and accuracies of the modular addition network output as a function of parameters Fisher-pruned 25 [PITH_FULL_IMAGE:figures/full_fig_p025_22.png]
Figure 23
Figure 23. Figure 23: Accuracies of Ising network output as a function of parameters Fisher-pruned 26 [PITH_FULL_IMAGE:figures/full_fig_p026_23.png]
Figure 24
Figure 24. Figure 24: Losses of Ising network output as a function of the number of parameters Fisher-pruned 27 [PITH_FULL_IMAGE:figures/full_fig_p027_24.png]
Figure 27
Figure 27. Figure 27: fig. 27 [PITH_FULL_IMAGE:figures/full_fig_p028_27.png]
Figure 25
Figure 25. Figure 25: Different global magnitude pruning schemes for the two tasks, averaged over 8 seeds and using two weight multipliers. The results of pruning the Ising network are shown on the top row, while the analogous results for modular addition are on the bottom row. (a,d) Paral…
Figure 26
Figure 26. Figure 26: Layer-resolved magnitude pruning of the weights in the Ising network. Results of pruning only (a) The first CNN, (b) The second CNN, (c) The first fully connected layer, and (d) The second fully connected layer. 0.1 0 0.2 0.4 0.6 0.8 1.0 (a) (b) 0 20 40 60 80 100 % pa…
Figure 27
Figure 27. Figure 27 [PITH_FULL_IMAGE:figures/full_fig_p029_27.png]
Figure 28
Figure 28. Figure 28: A summary of the correlation coefficients for the different measures across seeds and the first convolutional layer for the grokking case in Ising. G.2. Comparing Fisher and Euclidean Model Space Geometry One may wonder how the novel FIM-inspired dynamical measures wo…
Figure 29
Figure 29. Figure 29: A summary of the correlation coefficients for the different measures across seeds and the second convolutional layer for the grokking case in Ising. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_29.png]
Figure 30
Figure 30. Figure 30: A summary of the correlation coefficients for the different measures across seeds and the fully connected layer for the grokking case in Ising. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_30.png]
Figure 31
Figure 31. Figure 31: Step magnitudes of the models’ trajectories through model space during training. Measures computed using the FIM |s|F IM, or simply with the Euclidean assumption |s|, and averaged over the 10 runs. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_31.png]
Figure 32
Figure 32. Figure 32: Step magnitudes, |s|F IM, of the models’ trajectories through model space during training. Measures computed using the FIM, averaged over the 10 runs. Plots show the behaviour of the full model (a) & (b), as well as the sloppy and stiff submodels (c) & (d); for both t…
Figure 33
Figure 33. Figure 33: Discrete gradients of FIM steps, d|s|F IM/dt, along the model space learning trajectory, averaged over the 10 seed runs. Plots showing both the learning and grokking regimes; for both the Ising (a), and ModAdd (b) investigations. The axes are restricted to highlight t…
Figure 34
Figure 34. Figure 34: Split FIM-cosine similarities between consecutive steps, SC−F IM(s e , se+1), for the sloppy and stiff submodels. Plots showing both the learning and grokking regimes; for both the Ising (a), and ModAdd (b) investigations. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_34.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.