Disentangling Geometry, Performance, and Training in Language Models

Arjun Subramonian; Atharva Kulkarni; Jacob Mitchell Springer; Swabha Swayamdipta

REVIEW 2 major objections 5 minor 2 cited by

A language model's output-layer geometry mostly reflects training choices, not how well it performs.

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2026-08-02 21:20 UTC pith:YAGCUA4C

load-bearing objection Solid negative result on rank-as-predictor; the 'does not cause' saturation claim outruns the evidence. the 2 major comments →

arxiv 2602.20433 v2 pith:YAGCUA4C submitted 2026-02-24 cs.CL

Disentangling Geometry, Performance, and Training in Language Models

Atharva Kulkarni , Jacob Mitchell Springer , Arjun Subramonian , Swabha Swayamdipta This is my paper

classification cs.CL

keywords effective rankunembedding matrixlanguage model geometrysaturationtraining hyperparametersperformance predictionOLMoisotropy

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper sets out to test whether the 'effective rank' of a language model's unembedding matrix—a geometric summary of how spread out its output weights are—can predict how well the model performs. Training 108 small models under controlled variation of size, token budget, batch size, weight decay, and learning rate, the authors find that effective rank correlates with loss but never reliably predicts it. They argue that low effective rank is not the cause of late-training performance collapse ('saturation') in small models; it co-occurs with it. The deciding factor is training setup: batch size and weight decay shape the geometry of the unembedding matrix, and those same choices move performance. If this is right, geometric metrics belong in the toolbox as diagnostics of training, not as shortcuts for model selection or performance prediction.

Core claim

The paper's central claim is that the effective rank of the unembedding matrix is a symptom of optimization dynamics rather than a driver of performance. Across five evaluation settings—in-distribution loss, out-of-distribution generalization, fine-tuning, catastrophic forgetting, and post-training quantization—the authors observe that high effective rank generally accompanies good performance but with enough counterexamples to rule out a predictive or causal relation. In particular, they challenge prior attributions of 'saturation' to low rank: a low-rank model trained with a small batch and token budget does not degrade, while a model that saturates also shows a sharp rank drop. The author

What carries the argument

The central object is the effective rank of the unembedding matrix, defined as the exponential of the information-theoretic entropy of its normalized singular values. It measures how many 'effective dimensions' the output embedding uses: a high value means the singular values are spread out; a low value means the matrix is concentrated in a few directions. The paper uses this scalar as the primary geometric signature and contrasts it with isotropy (partition-function-based uniformity) and pairwise cosine similarity. Effective rank does the load-bearing work: the causal argument that low rank co-occurs with, but does not cause, saturation is built on comparing its trajectories across architec

Load-bearing premise

The paper's causal conclusion depends on a single low-rank model that avoided degradation, yet that model was trained on only 8 billion tokens at batch size 32—not under the prolonged overtraining in which saturation is observed—so its validity as a counterexample is the load-bearing premise.

What would settle it

Train the same architecture on 128 billion tokens at batch size 32; if loss degrades as effective rank collapses, the counterexample is an artifact of under-training rather than evidence that rank is harmless.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Researchers should stop using effective rank or related geometric metrics as a proxy for downstream performance; in-distribution loss is a more reliable predictor, especially for out-of-distribution behavior.
Interventions that directly target rank (such as rank regularization) will not by themselves improve performance; tuning batch size and weight decay changes both geometry and loss, often in conflicting directions.
Comparisons of model geometry across studies are only meaningful when training hyperparameters (batch size, weight decay, token budget, learning-rate schedule) are matched.
Small-model saturation is not a universal failure mode; architecture and optimization choices, such as attention variant and spectral properties, are more plausible culprits than rank collapse.
Post-training quantization robustness depends in part on extreme rank collapse, so low-rank models may need special care when quantized even if their raw loss is good.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If geometry primarily indexes hyperparameter choices, then the same effective rank could be reached by different hyperparameter combinations with opposite effects on loss; scanning (batch size, weight decay) pairs that produce matched effective rank and checking whether downstream loss still varies would test this directly.
The saturation result suggests a controlled experiment: train the same architecture with sequential versus parallel attention under prolonged overtraining and measure whether rank collapse and loss degradation decouple, isolating architecture from rank as the causal factor.
The paper's logic implies geometric metrics may still be useful as early-warning signs of optimization instability (e.g., sharp rank drops during training), even if they fail as performance predictors; formalizing that diagnostic role on training trajectories is a natural next step.
Because all models are small (at most 75 million parameters), the hyperparameter-to-geometry mapping may shift at scale; a natural check is whether effective rank becomes more or less predictive in large-batch, low-weight-decay regimes common in large-scale training.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Solid negative result on rank-as-predictor; the 'does not cause' saturation claim outruns the evidence.

read the letter

Two things you should know. First, the paper's main empirical claim—that effective rank and other geometric metrics do not reliably predict downstream performance once you control for hyperparameters—is well supported and worth taking seriously. Second, the causal claim that low rank does not cause Pythia saturation is overreaching, and the reason is exactly the one your stress-test flagged: the low-rank counterexample is an 8B-token run, not an overtrained model.

The genuinely new contribution is the 108-model sweep. Prior work looked at Pythia checkpoints; here they systematically vary batch size, weight decay, learning rate, annealing, and token budget across six sizes, then evaluate on ID loss, OOD, quantization, fine-tuning, and forgetting. The regression analysis is a real step up from scatter plots. Raw Spearman correlations look large, but partial Spearman correlations are near zero and delta-R2 ≤ 0.05 across the tables. That is a clean demonstration that the apparent correlation is confounded by training choices. I credit them for also reporting the Chinchilla-residual correlations, which are large—that makes the partial correlations all the more informative.

The weak point is Section 3.2's handling of saturation. Pythia-14M degrades in the late-training tail. The OLMo-14M-8B run at 8B tokens and batch size 32 is not in that regime. It shows low rank can occur early without immediate harm, but that does not bear on whether rank collapse during extended overtraining contributes to degradation. The matched OLMo-14M run that goes to the same token count as Pythia shows no rank collapse, so there is no instance of low rank without saturation in the relevant regime. The authors' own Appendix C concedes the results are observational, not causal, but the abstract and conclusion still say 'does not cause.' That language should be softened or tested with an overtrained low-rank run or a direct rank intervention.

The paper is not sloppy. It is a careful empirical study, just one where the negative predictive result is stronger than the causal interpretation. For peer review, I'd send it out. The predictive claim is important enough to referee, and the saturation framing is fixable. A serious referee would probably ask for the causal language to be reined in and for at least a few error bars or seeds on the key runs. I'd cite the negative result; I wouldn't cite the causal claim as established.

Referee Report

2 major / 5 minor

Summary. The paper trains 108 OLMo-style small LMs (4M-75M non-embedding parameters) on the Pile with controlled variation of token budget, batch size, weight decay, learning rate, and LR annealing. It measures effective rank, cosine similarity, and isotropy of the unembedding matrix W and the final-token representation H, and evaluates whether these metrics predict in-distribution loss, OOD loss, post-training quantization, fine-tuning, and catastrophic forgetting. The main empirical claims are: (i) geometry metrics show weak, inconsistent predictive power (partial Spearman correlations near zero and predictive ΔR² ≤ 0.05 across most tables); (ii) effective rank is strongly shaped by batch size and weight decay, so hyperparameters confound geometry-performance associations; and (iii) contrary to prior Pythia-based work, low effective rank 'does not cause' late-stage saturation but 'co-occurs' with it. The paper contributes a large controlled observational dataset and a regression framework for separating hyperparameter confounding from geometric signal.

Significance. If the predictive and confounding findings are taken as the contribution, this is a useful negative result: it provides a broad, controlled demonstration that effective rank and related geometric metrics add little predictive power beyond scaling-law baselines and are substantially confounded by training choices. The partial-Spearman and predictive-ΔR² analyses are a rigorous way to quantify this confounding, and the planned release of 108 checkpoints is a strength. The causal subclaim about saturation is the weakest part: the supporting evidence lacks a low-rank run in the overtrained regime, and the authors' own Appendix C disclaims causality. With that claim revised or supplemented, the core observation — geometry primarily reflects training choices rather than reliably predicting performance — is credible and relevant to the interpretability community.

major comments (2)

[§3.2, Figure 5] The claim that low effective rank does not cause late-stage saturation rests on the OLMo-14M-8B run (right panel), trained for only 8B tokens with batch size 32. Pythia-14M's saturation and rank drop occur in the 1e10–1e11 token range (left panel), so the 8B run is not an overtraining test and cannot serve as a counterexample to a late-stage causal link. The middle-panel OLMo-14M run, which matches Pythia's schedule, never exhibits low rank. Hence there is no example of "low rank without saturation" in the relevant regime; the evidence reduces to a single architectural comparison confounded with training schedule. The abstract's and takeaway 4's causal-negative statements are therefore unsupported. Please add an overtrained low-rank run (e.g., prolonged training with small batch / low weight decay) or soften the claim to "low rank can appear early without immediate degradation; its causa
[Abstract, Conclusion, Appendix C] The causal framing ("does not cause", "co-occurs") contradicts the manuscript's own Appendix C, which states that the results are "observational rather than fully causal." No intervention on rank is performed; the comparisons vary architecture and multiple hyperparameters simultaneously. Because this causal claim is load-bearing — it is the headline of the abstract and the basis of Practical Takeaway 5 in Appendix B — the manuscript must either add an interventional experiment (e.g., spectral norm capping or explicit rank regularization) under prolonged overtraining, or revise the causal assertions to strictly correlational statements. As written, this is an internal inconsistency between the stated limitations and the conclusions.

minor comments (5)

[§3.1] The weight-decay grid is written "{0.1, 0.5, 0.01}" but Figures 9 and 13-14, as well as the legends, use 0.1, 0.05, 0.01. Please correct the typo.
[§2, Eq. (1)] State the value of epsilon and the summation index r (presumably min(v,d)). The definition p_k = sigma_k / ||sigma||_1 + epsilon is sensitive to epsilon and differs subtly from the standard effective rank of Roy & Vetterli; reporting the exact value is needed for comparability.
[§2, Eq. (2)] "U is the eigenvector of WᵀW" is ambiguous. Should be "the eigenvectors" or "the principal eigenvector" as intended by Arora et al. (2016).
[Abstract / §3.2] The abstract mentions "adversarial cases" (plural) of low-rank models without saturation, but §3.2 describes only one such model (OLMo-14M-8B). If there are additional cases, cite them explicitly; otherwise use singular.
[Appendix A.3, Tables 2-9] The "Residual Spearman (Chinchilla)" rows report correlations with residuals from fitted Chinchilla scaling laws, but the fitted coefficients, fitting procedure, and whether the fit is on the same data are not reported. Given that these values are large (≈−0.7) while partial Spearman correlations are near zero, the estimation details matter for reproducibility.

Circularity Check

0 steps flagged

No circular derivation: the paper is an empirical measurement study with independent benchmarks; analytic choices and a potential confound are evidentiary, not definitional.

full rationale

The paper does not contain a derivation chain in which an output is equivalent to an input by construction. Its central claims are empirical: it trains 108 OLMo-style models and correlates measured effective rank, isotropy, cosine similarity, and representation geometry with held-out loss, quantization robustness, fine-tuning, and forgetting. The effective rank metric (Eq. 1) is a standard definition taken from Roy & Vetterli and is not defined in terms of performance. The 'scaled loss' (loss normalized by the lowest observed loss per model size) and the Chinchilla-residual regressions are analytic normalizations used to remove confounds; neither embeds the paper's conclusion that geometry primarily reflects hyperparameters. The predictive-delta-R2 analysis explicitly cross-validates and finds near-zero added variance, which is an honest negative result rather than a forced prediction. The saturation argument based on OLMo-14M-8B is possibly confounded (8B tokens vs. prolonged overtraining; sequential vs. parallel attention), but a confound is a threat to causal validity, not a circular reduction of the conclusion to its premises. Appendix C even states the results are 'observational rather than fully causal.' Self-citations (Finlayson et al. 2024/2026, Springer et al. 2025) appear only as background or as examples of small-model studies; they are not load-bearing for the main empirical findings. No uniqueness theorem, imported ansatz, or renamed known result carries the argument. Therefore the appropriate finding is no significant circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The paper introduces no new entities and no fitted parameters aimed at producing the central result. The free parameters listed are analytic choices (epsilon, loss normalization, scaling-law baselines) that affect the quantitative comparisons but are not used to fit the conclusion. The axioms are the background assumptions needed to turn a 108-model observational sweep into general statements about geometry, performance, and saturation.

free parameters (3)

epsilon in effective rank = not reported
Added to normalized singular-value probabilities in Eq. (1) to avoid log(0); value affects effective rank, especially for collapsed matrices, but is never stated.
per-model-size scaled-loss normalizer = lowest observed ID loss per size
Used to define 'scaled loss' so models of different sizes can be pooled; this ad hoc normalization shapes the batch-size and weight-decay curves in Figures 8-12.
Chinchilla scaling-law coefficients (E, A, B, alpha, beta) = fitted, values not reported
Fitted to the same 108 models to compute residual Spearman correlations and the ΔR2 baseline; values are not disclosed, so the residual analyses are not independently checkable.

axioms (5)

standard math Singular value decomposition and Shannon entropy define effective rank (Roy & Vetterli 2007)
Eq. (1) is taken as given; no theorem is proved in this paper.
domain assumption OLMo-style 4M-75M models trained on Pile tokenized by allenai/gpt-neox-olmo-dolma-v1 are representative of small LMs
Used to generalize from 108 runs to claims about geometry-performance relationship; authors flag size limitation in Appendix C.
domain assumption Pythia-14M and OLMo-14M are matched except sequential vs parallel attention
Saturation comparison in §3.2 attributes the difference to architecture; data ordering, tokenizer, optimizer state, and seeds are not controlled.
domain assumption Single-run training is sufficient: observed counterexamples are not noise
No seeds/repeats/error bars are reported; scatter plots treat each dot as deterministic.
ad hoc to paper Scaled loss normalization by the lowest observed per-size loss is a valid basis for cross-configuration comparison
This choice changes the shape of the batch-size/weight-decay curves in Figures 8-12; not justified beyond convenience.

pith-pipeline@v1.3.0-alltime-deepseek · 34830 in / 10662 out tokens · 94786 ms · 2026-08-02T21:20:05.087833+00:00 · methodology

0 comments

read the original abstract

Geometric properties of Transformer weights, particularly the unembedding matrix, have been widely useful in language model interpretability research. Yet, their utility for estimating downstream performance remains unclear. In this work, we systematically investigate the relationship between model performance and the unembedding matrix geometry, particularly its effective rank. Our experiments, involving a suite of 108 OLMo-style language models trained under controlled variation, reveal several key findings. While the best-performing models often exhibit a high effective rank, this trend is not universal across tasks and training setups. Contrary to prior work, we find that low effective rank does not cause late-stage performance degradation in small models, but instead co-occurs with it; we find adversarial cases where low-rank models do not exhibit saturation. Moreover, we show that effective rank is strongly influenced by pre-training hyperparameters, such as batch size and weight decay, which in-turn affect the model's performance. Lastly, extending our analysis to other geometric metrics and final-layer representation, we find that these metrics are largely aligned, but none can reliably predict downstream performance. Overall, our findings suggest that the model's geometry, as captured by existing metrics, primarily reflects training choices rather than performance.

Figures

Figures reproduced from arXiv: 2602.20433 by Arjun Subramonian, Atharva Kulkarni, Jacob Mitchell Springer, Swabha Swayamdipta.

**Figure 1.** Figure 1: (A) The effective rank of a model’s unembedding matrix, W, tends to correlate with its performance, but does not guarantee it. (B) The effective rank of the unembedding matrix is influenced by hyperparameter choices (e.g. batch size). (C) These hyperparameters in turn can affect model performance. (D) The effective rank of the last token’s final-layer representation, H shows little correlation with that o… view at source ↗

**Figure 2.** Figure 2: In-distribution evaluation: Variation in effective rank does not translate to equivalent change in in-domain loss. Out-of-distribution performance tracks in-distribution loss, not effective rank. We use the Paloma benchmark (Magnusson et al., 2024) to assess OOD performance [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** Figure 4: Post-training quantization evaluation: Highly collapsed models are heavily impacted, while high-rank models remain largely intact but not entirely consistent. Saturation is not caused by low effective rank alone. Smaller Pythia models (≤ 160M parameters) are known to exhibit performance degradation toward the end of pretraining (Biderman et al., 2023; van der Wal et al., 2025), a phenomenon commonly ref… view at source ↗

**Figure 3.** Figure 3: shows results on the Dolma-100 programminglanguages subset. Overall, OOD loss closely mirrors ID loss, while variation in effective rank R(W) is not consistently correlated with OOD losses. We also observe an agreementon-the-line phenomenon (Miller et al., 2021; Baek et al., 2022), where models exhibit a near-linear relationship between ID and OOD loss. This suggests that ID loss is a more reliable pred… view at source ↗

**Figure 6.** Figure 6: , we find that the best-performing models tend to exhibit high R(W). However, we observe many counterexamples in which models with comparable or even higher effective rank incur substantially worse fine-tuning loss. Moreover, unlike the pre-training evaluation results, we do not observe a strong agreement-on-the-line between finetuning and pre-training losses. We observe consistent trends across addition… view at source ↗

**Figure 5.** Figure 5: Saturation evaluation: Not all small language models experience saturation. Low effective rank alone does not cause it. Instead, saturation may result from other training or architectural factors that can also induce rank collapse. Takeways: 1) Models with ↑ R(W) have better ID and OOD performance, albeit it is not predictive across all training setups. 2) ID loss is a better indicator of OOD behaviour th… view at source ↗

**Figure 8.** Figure 8: Effect of batch size on R(W) and performance: Larger batch sizes preserve higher effective rank in the unembedding matrix. However, performance exhibits a U-shaped dependence on batch size, with extremely large batches not always improving loss, explaining why higher R(W) does not necessarily translate to better performance. High regularization prompts high effective rank, but varied performance [PITH_FUL… view at source ↗

**Figure 9.** Figure 9: Effect of weight decay on R(W) and performance: Stronger regularization consistently preserves higher effective rank. Its impact on loss depends on model size, yielding both good and poor performance at similar R(W). Learning Rate affects effective rank and performance inconsistently. As mentioned in §3.1, we choose the optimal learning rate for each model based on the scaling laws of Porian et al. (2024)… view at source ↗

**Figure 11.** Figure 11: Effect of LR decay on R(W) and performance: Annealing the learning rate toward 0 improves performance, while changes in R(W) remain minimal. hence, with varying batch sizes. Prior work suggests that pre-training on tokens way beyond the Chinchilla optimal (Hoffmann et al., 2022) can continue to improve performance (Sardana et al., 2024; Gadre et al., 2025). Our results from [PITH_FULL_IMAGE:figures/ful… view at source ↗

**Figure 10.** Figure 10: Effect of learning rate on R(W) and performance: Smaller models perform better with higher learning rates, whereas larger models benefit from lower learning rates. However, variations in R(W) remain minimal. High learning rate annealing leads to high effective rank and low performance [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗

**Figure 14.** Figure 14: Pareto front of similarity and effective rank metrics for the unembedding matrix vs last layer representation. Representations show higher effective rank and less variation than the unembedding matrix, with low cosine similarity. Embedding’s geometric variation does not directly transfer to representations. 6. Conclusion In this work, we present a systematic study of the relationship between language mo… view at source ↗

**Figure 13.** Figure 13: Pareto fronts of unembedding geometry metrics. Effective rank and cosine similarity follow a non-monotonic trend, while isotropy varies mainly with weight decay. Representation geometry varies less than the unembedding geometry. From [PITH_FULL_IMAGE:figures/full_fig_p008_13.png] view at source ↗

**Figure 15.** Figure 15: Effect of batch size on in-distribution performance and different geometric metrics for unembedding matrix as well as final-token’s last-layer representations [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗

**Figure 16.** Figure 16: Effect of weight decay on in-distribution performance and different geometric metrics for unembedding matrix as well as final-token’s last-layer representations. promoting more uniform distribution in the embedding space. While the cosine similarity of the final layer representations A(H) remains nearly constant across weight decay values (with a minor exception for the OLMo-8M model), the effective rank … view at source ↗

**Figure 17.** Figure 17: Effect of learning rate on in-distribution performance and different geometric metrics for unembedding matrix as well as final-token’s last-layer representations. As seen in [PITH_FULL_IMAGE:figures/full_fig_p016_17.png] view at source ↗

**Figure 18.** Figure 18: Effect of learning rate annealing on in-distribution performance and different geometric metrics for unembedding matrix as well as final-token’s last-layer representations. Annealing the learning rate to a higher minimum value consistently results in higher loss compared to annealing to zero, as evidenced in [PITH_FULL_IMAGE:figures/full_fig_p016_18.png] view at source ↗

**Figure 19.** Figure 19: Effect of token budget on in-distribution performance and different geometric metrics for unembedding matrix as well as final-token’s last-layer representations. monotonically with larger token budgets. Both metrics indicate that extended training promotes more orthogonal and uniformly distributed unembedding vectors. However, the geometric metrics of the final layer representations – A(H) and R(H) – exhi… view at source ↗

**Figure 20.** Figure 20 [PITH_FULL_IMAGE:figures/full_fig_p018_20.png] view at source ↗

**Figure 21.** Figure 21: Extended out-of-distribution evaluation in relation to geometric metrics of the unembedding matrix and last-layer final-token representation. Metric Effective Rank R(W) Cosine Similarity A(W) Isotropy I(W) Effective Rank R(H) Cosine Similarity A(H) Raw Spearman -0.687 0.585 -0.170 0.726 0.517 Residual Spearman (linear) -0.229 0.259 -0.214 -0.135 -0.209 Residual Spearman (Chinchilla) -0.725 0.637 -0.300 0.… view at source ↗

**Figure 22.** Figure 22: Extended post-training quantization evaluation on in-domain data in relation to geometric metrics of the unembedding matrix and last-layer final-token representation. Metric Effective Rank R(W) Cosine Similarity A(W) Isotropy I(W) Effective Rank R(H) Cosine Similarity A(H) Raw Spearman -0.659 0.559 -0.064 0.772 0.336 Residual Spearman (linear) -0.071 0.110 -0.132 -0.155 -0.144 Residual Spearman (Chinchill… view at source ↗

**Figure 23.** Figure 23: Extended post-training quantization evaluation on out-of-domain data in relation to geometric metrics of the unembedding matrix and last-layer final-token representation. Metric Effective Rank R(W) Cosine Similarity A(W) Isotropy I(W) Effective Rank R(H) Cosine Similarity A(H) Raw Spearman -0.629 0.510 -0.171 0.907 -0.358 Residual Spearman (linear) 0.044 -0.054 -0.107 -0.142 0.265 Residual Spearman (Hoffm… view at source ↗

**Figure 24.** Figure 24: Extended fine-tuning evaluation (StarCoder-Python) in relation to geometric metrics of the unembedding matrix and last-layer final-token representation. Metric Effective Rank R(W) Cosine Similarity A(W) Isotropy I(W) Effective Rank R(H) Cosine Similarity A(H) Raw Spearman -0.642 0.526 -0.170 0.905 0.651 Residual Spearman (linear) 0.008 -0.021 -0.105 -0.111 -0.245 Residual Spearman (Hoffmann) -0.652 0.532 … view at source ↗

**Figure 25.** Figure 25: Extended fine-tuning evaluation (OpenWebMath) in relation to geometric metrics of the unembedding matrix and last-layer final-token representation. Metric Effective Rank R(W) Cosine Similarity A(W) Isotropy I(W) Effective Rank R(H) Cosine Similarity A(H) Raw Spearman -0.591 0.486 0.017 0.808 0.685 Residual Spearman (linear) 0.024 0.001 0.057 -0.041 0.048 Residual Spearman (Hoffmann) -0.578 0.480 0.026 0.7… view at source ↗

**Figure 26.** Figure 26: Extended forgetting evaluation (StarCoder-Python) in relation to geometric metrics of the unembedding matrix and last-layer final-token representation. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_26.png] view at source ↗

**Figure 27.** Figure 27: Extended forgetting evaluation (OpenWebMath) in relation to geometric metrics of the unembedding matrix and last-layer final-token representation. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_27.png] view at source ↗

discussion (0)

Forward citations

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Reference graph

Works this paper leans on

10 extracted references · 5 linked inside Pith · cited by 2 Pith papers

[1]

Thus, always measure downstream performance metrics first

Do not use effective rank alone to diagnose model problems.If one observes a low effective rank during training, this doesn’t necessarily indicate the model is failing; it may simply reflect your hyperparameter choices (especially weight decay and batch size). Thus, always measure downstream performance metrics first
[2]

Hyperparameter choices matter more than geometry.Rather than directly optimizing for geometric properties (e.g., adding explicit rank regularization), focus on tuning standard hyperparameters like batch size and weight decay, which naturally influence geometry and have more predictable effects on performance
[3]

Avoid geometry-based early stopping.Halting training solely because the effective rank is decreasing is not a good option, as we observe that some of our best-performing models had relatively low effective rank
[4]

URL https://openreview.net/forum?id= WGXb7UdvTX. Soldaini, L., Kinney, R., Bhagia, A., Schwenk, D., Atkin- son, D., Authur, R., Bogin, B., Chandu, K., Dumas, J., Elazar, Y ., Hofmann, V ., Jha, A., Kumar, S., Lucy, L., Lyu, X., Lambert, N., Magnusson, I., Morrison, 12 Disentangling Geometry, Performance, and Training in Language Models J., Muennighoff, N....

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Zhai, S., Likhomanenko, T., Littwin, E., Busbridge, D., Ramapuram, J., Zhang, Y ., Gu, J., and Susskind, J

URLhttps://arxiv.org/abs/2408.11804. Zhai, S., Likhomanenko, T., Littwin, E., Busbridge, D., Ramapuram, J., Zhang, Y ., Gu, J., and Susskind, J. M. Stabilizing transformer training by preventing attention entropy collapse. In Krause, A., Brunskill, E., Cho, K., Engelhardt, B., Sabato, S., and Scarlett, J. (eds.),Pro- ceedings of the 40th International Con...

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Small model saturation may be avoidable.Unlike Pythia models, our OLMo models do not suffer from late-stage 21 Disentangling Geometry, Performance, and Training in Language Models 20 40 60 80 Effective Rank (W) 7.2 7.4 7.6 FT Loss OLMo-4M (Starcoder Python) 20 40 60 80 Effective Rank (W) 5.6 5.7 5.8 5.9 6.0 OLMo-8M (Starcoder Python) 40 60 80 Effective Ra...
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Future work should use interventional studies (e.g., directly manipulating rank while holding other factors constant) rather than purely correlational analyses

Distinguish correlation from causation.Our results suggest effective rank is asymptomof training dynamics rather than acauseof performance differences. Future work should use interventional studies (e.g., directly manipulating rank while holding other factors constant) rather than purely correlational analyses. C. Limitations While our study provides a sy...

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URLhttps://arxiv.org/abs/1909.08053. Skean, O., Arefin, M. R., Zhao, D., Patel, N. N., Naghiyev, J., LeCun, Y ., and Shwartz-Ziv, R. Layer by layer: Uncov- ering hidden representations in language models. InForty- second International Conference on Machine Learning,

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URL https://openreview.net/forum?id= RvfPnOkPV4. Ethayarajh, K. How contextual are contextualized word rep- resentations? Comparing the geometry of BERT, ELMo, and GPT-2 embeddings. In Inui, K., Jiang, J., Ng, V ., and Wan, X. (eds.),Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conf...

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URL https://openreview.net/forum?id= wg1PCg3CUP. Li, M. Z., Agrawal, K. K., Ghosh, A., Teru, K. K., Santoro, A., Lajoie, G., and Richards, B. A. Tracing the repre- sentation geometry of language models from pretraining to post-training. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025. URL https://openreview.net/forum?id...

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[1] [1]

Thus, always measure downstream performance metrics first

Do not use effective rank alone to diagnose model problems.If one observes a low effective rank during training, this doesn’t necessarily indicate the model is failing; it may simply reflect your hyperparameter choices (especially weight decay and batch size). Thus, always measure downstream performance metrics first

[2] [2]

Hyperparameter choices matter more than geometry.Rather than directly optimizing for geometric properties (e.g., adding explicit rank regularization), focus on tuning standard hyperparameters like batch size and weight decay, which naturally influence geometry and have more predictable effects on performance

[3] [3]

Avoid geometry-based early stopping.Halting training solely because the effective rank is decreasing is not a good option, as we observe that some of our best-performing models had relatively low effective rank

[4] [4]

URL https://openreview.net/forum?id= WGXb7UdvTX. Soldaini, L., Kinney, R., Bhagia, A., Schwenk, D., Atkin- son, D., Authur, R., Bogin, B., Chandu, K., Dumas, J., Elazar, Y ., Hofmann, V ., Jha, A., Kumar, S., Lucy, L., Lyu, X., Lambert, N., Magnusson, I., Morrison, 12 Disentangling Geometry, Performance, and Training in Language Models J., Muennighoff, N....

Pith/arXiv arXiv 2024

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Zhai, S., Likhomanenko, T., Littwin, E., Busbridge, D., Ramapuram, J., Zhang, Y ., Gu, J., and Susskind, J

URLhttps://arxiv.org/abs/2408.11804. Zhai, S., Likhomanenko, T., Littwin, E., Busbridge, D., Ramapuram, J., Zhang, Y ., Gu, J., and Susskind, J. M. Stabilizing transformer training by preventing attention entropy collapse. In Krause, A., Brunskill, E., Cho, K., Engelhardt, B., Sabato, S., and Scarlett, J. (eds.),Pro- ceedings of the 40th International Con...

Pith/arXiv arXiv 2023

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Small model saturation may be avoidable.Unlike Pythia models, our OLMo models do not suffer from late-stage 21 Disentangling Geometry, Performance, and Training in Language Models 20 40 60 80 Effective Rank (W) 7.2 7.4 7.6 FT Loss OLMo-4M (Starcoder Python) 20 40 60 80 Effective Rank (W) 5.6 5.7 5.8 5.9 6.0 OLMo-8M (Starcoder Python) 40 60 80 Effective Ra...

[7] [10]

Future work should use interventional studies (e.g., directly manipulating rank while holding other factors constant) rather than purely correlational analyses

Distinguish correlation from causation.Our results suggest effective rank is asymptomof training dynamics rather than acauseof performance differences. Future work should use interventional studies (e.g., directly manipulating rank while holding other factors constant) rather than purely correlational analyses. C. Limitations While our study provides a sy...

2023

[8] [2020]

Skean, O., Arefin, M

URLhttps://arxiv.org/abs/1909.08053. Skean, O., Arefin, M. R., Zhao, D., Patel, N. N., Naghiyev, J., LeCun, Y ., and Shwartz-Ziv, R. Layer by layer: Uncov- ering hidden representations in language models. InForty- second International Conference on Machine Learning,

Pith/arXiv arXiv 1909

[9] [2024]

Ethayarajh, K

URL https://openreview.net/forum?id= RvfPnOkPV4. Ethayarajh, K. How contextual are contextualized word rep- resentations? Comparing the geometry of BERT, ELMo, and GPT-2 embeddings. In Inui, K., Jiang, J., Ng, V ., and Wan, X. (eds.),Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conf...

Pith/arXiv arXiv 2019

[10] [2025]

URL https://openreview.net/forum?id= wg1PCg3CUP. Li, M. Z., Agrawal, K. K., Ghosh, A., Teru, K. K., Santoro, A., Lajoie, G., and Richards, B. A. Tracing the repre- sentation geometry of language models from pretraining to post-training. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025. URL https://openreview.net/forum?id...

Pith/arXiv arXiv 2025