Representations Shape Weak-to-Strong Generalization: Theoretical Insights and Empirical Predictions

Baharan Mirzasoleiman; Jiping Li; Yihao Xue

REVIEW 4 major objections 4 minor 1 cited by

Representations Shape Weak-to-Strong Generalization: Theoretical Insights and Empirical Predictions

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Weak-to-strong generalization reduces to an overlap of two representation kernels.

desk verdict A genuinely new theoretical handle on W2SG, but the empirical validation doesn't verify the key assumption, so treat the headline correlations as suggestive not conclusive. read the letter →

arxiv 2502.00620 v4 pith:4YDLCXU5 submitted 2025-02-02 cs.LG cs.AI

Yihao Xue , Jiping Li , Baharan Mirzasoleiman This is my paper · ORCID

classification cs.LGcs.AI

keywords weak-to-stronggeneralizationrepresentationlearningprincipalcomponentanalysiskernelmethodsbenignoverfittinglargelanguagemodelslabel-freeevaluationlinearprobing

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

Weak-to-strong generalization asks whether a stronger model trained on labels produced by a weaker one can still outperform its teacher. This paper claims that the resulting performance gap is governed not by the teacher's overall error but by a geometric overlap between what the two models' principal representations can express. The main theorem states that, up to vanishing terms, the squared prediction gap between the weakly supervised strong model and the same model trained on true labels equals $\|\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)\hat{\boldsymbol{y}}/\sqrt{\hat{n}}\|^2$, where $\boldsymbol{P}_w$ and $\boldsymbol{P}_s$ are scaled projections onto the spans of the principal kernels of the weak and strong models. In words, the mistakes a strong student copies from a weak teacher are exactly those that live in the strong model's principal-representation space while lying outside the weak model's; the same geometry explains benign overfitting, since errors lying in the non-principal part can be fitted without surfacing at test time. Because the metric needs no task labels, the paper uses it to predict weak-to-strong performance across molecular-property regression and five NLP tasks with dozens of language models.

What carries the argument

The load-bearing object is the pair of matrices $\boldsymbol{P}_w$ and $\boldsymbol{P}_s$, defined as $\frac{1}{\hat{n}}\hat{\boldsymbol{K}}(\Pi_{V_w} h_w)(\frac{1}{\hat{n}}\hat{\boldsymbol{K}}(\Pi_{V_w} h_w)+(\beta_w+\tilde{\gamma}_w)\boldsymbol{I})^{-1}$, and similarly for $s$ with $\beta_s+\hat{\gamma}_s$. Each is a regularized projection onto the span of a principal kernel, i.e., the kernel matrix formed by evaluating the model's representation restricted to its well-concentrated principal subspace. The key identity carrying the argument is Theorem 3.8, which expresses the prediction gap as the norm of the weak model's error after the strong model's projection. The mechanism is an overlap: errors in the weak model's output that fall in the span of the strong model's principal kernel are replicated by the weak-to-strong model, while errors landing in the strong model's non-principal directions are fitted or ignored without affecting test error.

What would settle it

Run the linear-probing setup of Experiment I on a single dataset, compute the left-hand side $\mathrm{PredGap}$ directly from the two fitted models' predictions and the right-hand side $\|\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)\hat{\boldsymbol{y}}/\sqrt{\hat{n}}\|^2$ from the same representations; if the difference does not shrink toward zero as $\hat{n}$ grows, or is not small at practical sample sizes within the assumed regime, the theorem's central equality is false. A cheaper check is to hold $\|\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)\|_{\mathrm{op}}$ fixed across weak teachers whose label-noise structures differ and see whether W2SG error stays constant, as the theory requires.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 3.8: under a $(\delta,\hat{\gamma},\tilde{\gamma})$-decomposability assumption on both representations, the prediction gap between the weak-to-strong model and the strong ceiling model decomposes as $\mathrm{PredGap} = \|\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)\hat{\boldsymbol{y}}/\sqrt{\hat{n}}\|^2 \pm o(1)$ with high probability. Here $\hat{\boldsymbol{y}}$ collects the true finetuning labels, while $\boldsymbol{P}_w$ and $\boldsymbol{P}_s$ are regularized scaled projections onto the spans of the principal kernels, meaning the kernel matrices computed from the parts of the representations that are well concentrated on their principal subspaces. The operator $\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)$ thus selects precisely what the weak model is unable to learn but the strong model can learn through its principal representations. The paper further proves a general condition for benign overfitting in this setting (Theorem 4.1) and derives two label-free upper bounds on the prediction gap, both governed by $\|\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)\|_{\mathrm{op}}$ or the $\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)\boldsymbol{P}_s$ variant, which it then validates empirically.

Load-bearing premise

Everything rests on the assumption that each model's representations separate cleanly into a concentrated 'principal' part and a nearly structureless residual part; if real LLM representations do not separate this way, the formula and the metric have no guarantee.

Editorial extensions

If this is right

Prediction gaps can be bounded without labels: Corollary 5.1 gives $\mathrm{PredGap} \leq C\|\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)\|_{\mathrm{op}}^2+o(1)$, so ranking candidate weak teachers by this operator norm should track weak-to-strong quality.
Which weak-supervision errors are corrected is determined by geometry: errors in the strong model's principal span are replicated, errors outside it are not, regardless of whether the strong model overfits them.
Benign overfitting in W2SG has a single driving quantity: when the strong model's non-principal kernel is nearly isotropic ($\delta_s=o(\hat{\gamma}_s)$) and $\|\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)\hat{\boldsymbol{y}}/\sqrt{\hat{n}}\|^2$ is small, the W2S model can fit the weak labels almost perfectly yet beat the weak teacher.
A stronger weak model shrinks the gap, all else equal: because a larger span for $\boldsymbol{P}_w$ leaves less room for $\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)$, the formula predicts monotone improvement in W2SG as weak representations cover more of the label space.
The metric transfers beyond the linear-probing setting: the empirical correlations hold for CE-trained classifiers and for full LLM fine-tuning when activation-map representations are used heuristically, not just for the linear-regression setup of the theorem.

Reading between the lines

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

Inference: because the metric measures only representation geometry, it suggests a practical selection rule—among candidate weak supervisors with similar downstream accuracy, choose the one with the smallest $\|\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)\|_{\mathrm{op}}$—which the paper does not explicitly propose.
Inference: the same two-kernel overlap may describe other forms of weak supervision, such as knowledge distillation or human preference feedback, whenever the teacher's errors can be expressed through representations; this is an extrapolation, not a claim of the paper.
Inference: the empirical correlation in end-to-end fine-tuning could in principle be driven by confounds such as model size or pretraining quality rather than the stated kernel overlap; a controlled experiment that equalizes these dimensions while varying only $\|\boldsymbol{P}_s(\boldsymbol{I}-\boldsymbol{P}_w)\|_{\mathrm{op}}$ would sharpen the causal reading.
Inference: replacing activation-map heuristics with gradient-based kernels (e.g., NTK-style linearizations of fine-tuning) could make the metric applicable during training itself and may improve its accuracy for full-model fine-tuning.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The one thing to know: this is a real theory paper, not a repackaged empirical observation. It derives a clean characterization of W2SG: PredGap equals the squared norm of Ps(I-Pw)y/√n up to o(1), where Pw and Ps are ridge hat matrices on the principal kernels of the weak and strong representations. That quantity is genuinely new, and the paper uses it to get a label-free metric that correlates with W2SG error across 8 datasets, 150 small transformers, and 52 LLMs. I walked through the proof in the appendix; it is a standard ridge-regression hats-matrix argument and it appears internally consistent under Assumption 3.7.

What is good: the decomposition into principal and non-principal representations is a useful lens, the theory covers benign overfitting as a special case, and the authors are honest that the empirical metric is a heuristic, especially the activation maps in Exp. III. The paper positions itself properly against Charikar, Lang, Shin, and Wu & Sahai. The correlations are strong, and the metric does something model size alone cannot do.

The soft spot is exactly the one in the stress-test note: the theorem needs the (δ,γ̂,γ̃)-decomposability assumption, and the experiments never check it on the actual representations. A real LLM's residual subspace after PCA thresholding is almost surely not kernel-isotropic; it contains syntax, token identity, and style features. If that is true, the o(1) terms in Theorem 3.8 could dominate, and the observed correlation might just be tracking weak-model quality. That is a validation gap, not a proven flaw—the theory stands on its own terms. It is a gap you can close: measure the residual kernel's deviation from isotropy, report correlations with hyperparameters chosen on a validation split rather than the best on the test set, and release code. The paper's own cross-model transfer experiment (Table 2) helps, but it is underplayed; it could be the headline validation. Also, the hyperparameter tuning on the test set in Exp. III is genuinely worrisome.

Who should read it: anyone working on W2SG, AI alignment, or weak-label training. It deserves a serious referee; the theoretical result is important enough that the empirical shortfalls should be fixed, not used to desk-reject. If I were the editor, I would send it out and ask for the decomposability check and cleaner hyperparameter handling.

Overall: a solid theoretical core with an empirical evaluation that overreaches slightly. Worth engaging with and citing.

Referee Report

4 major / 4 minor

Summary. The paper studies weak-to-strong generalization (W2SG) in a fixed-representation linear-regression setting. It introduces a decomposability condition (Def. 3.3) on representation kernels and proves (Thm. 3.8) that the prediction gap between the strong model trained on weak labels and the strong ceiling is, up to o(1), the squared norm of P_s(I-P_w)(1/√n)y_hat, where P_w and P_s are ridge-regularized projection matrices onto the principal kernels of the weak and strong representations. From this it derives upper bounds (Cors. 5.1, 5.2) and a practical metric ||P_s(I-P_w)||_op, with heuristics substituting PCA thresholds and an effective regularization beta_eff for the theoretical quantities. Experiments on molecular transformers, embedding models, and 52 LLMs report strong Spearman correlations between the metric and W2S test error, and a benign-overfitting case study is developed in Sec. 4.

Significance. If the main theorem is correct, it gives a mechanistic, representation-level account of when a strong model will or will not replicate its weak supervisor's errors, and it connects W2SG to benign overfitting. The derivation is self-contained: P_w and P_s are defined from the data, and the proof proceeds through the closed-form ridge solution and hat-matrix lemmas (A.12, A.13). The metric is a norm of a product of these matrices, not a fitted value, and the empirical scope (8 datasets, 150 transformers, 52 LLMs) is a genuine strength. The main weaknesses are empirical: the key decomposability assumption is not checked on the actual models, and the reported correlations for Exp. III are best-case over hyperparameters tuned per dataset. These weaknesses are fixable with diagnostics and a cleaner evaluation protocol, so I regard the work as promising but not yet fully supported.

major comments (4)

[Definition 3.3 / Theorem 3.8, App. A.7] Theorem 3.8's equality is conditional on Assumption 3.7: both representations must be (δ, γhat, γtilde)-decomposable, which requires the residual (non-principal) kernel to satisfy near-isotropy (condition (c)), small cross-sample inner products (condition (d)), and vanishing population covariance (condition (e)). The empirical metric in Sec. 5 replaces V with a PCA or variance threshold and uses activation maps for Exp. III, but the paper never reports any of the quantities in Def. 3.3 for the actual models. On real transformer and LLM activations, the residual subspace after eigenvalue thresholding is likely to contain structured, task-relevant features whose kernel is far from γhat I, in which case the o(1) terms in Thm. 3.8 can dominate and the correlation between ||P_s(I-P_w)||_op and Err_w2s need not reflect the mechanism in the theorem. I request a concrete diagnostic: for the representations used in Exps. I-III, report the relative isotropy error ||(1/n)K(Π_{V⊥}h) - γhat I||_op / γhat, the cross-sample norm in condition (d), and an empirical proxy for ||Σ(Π_{V⊥}h)||_op across the weak models, and show they are small, or provide a perturbation argument showing Thm. 3.8 is robust to their failure.
[Sec. 5.4 and App. D.2] The empirical claim that the metric is 'without requiring labels' is weakened by the evaluation protocol. App. D.2 states that in Exp. III the hyperparameters α and β are tuned per dataset 'reporting the best result', i.e., selected by maximizing the correlation with Err_w2s, which uses the labels whose absence is advertised. The main-text figures for Exp. III therefore report best-case correlations, not predictions under a fixed or held-out hyperparameter choice. App. D.2 does provide a cross-model transfer experiment, but only for Exp. II (Table 2). I request that all three experiments report correlations under a pre-specified or cross-validated hyperparameter choice (e.g., selected on half the weak models and evaluated on the other half), and that the text qualify the 'label-free' claim accordingly.
[Example 3.5 and Theorem 3.6, App. C.2.1] The stated scalings do not in general imply condition (c) of Def. 3.3. For the sub-Gaussian spiked model with residual variance σ²/(d-k), the residual Gram matrix satisfies ||(1/n)K(Π_{V⊥}h) - (σ²/n)I||_op = (σ²/n)·||(1/(d-k))Z^T Z - I||_op, which is of order (σ²/n)·√(n/(d-k)) by Lemma C.3. Condition (c) with δ=0 requires this to be o((σ²/n)²). With d=ω(n²) and σ²=O(n), this need not hold when σ² is small (e.g., σ²=1); one would need d-k = ω(n³/σ⁴) or σ²=Ω(n). The same issue affects Theorem 3.6, where m=ω(n²) is assumed. The examples should either be restated with a stronger scaling that actually satisfies Def. 3.3, or condition (c) must be relaxed with a correspondingly revised proof.
[Sec. 5.4, Figs. 2-5] The paper positions ||P_s(I-P_w)||_op as capturing the specific geometric overlap between the complement of the weak principal kernel and the strong principal kernel. The reported correlations, however, could conceivably be driven by a generic 'better weak representation' factor: P_w appears directly in the metric, and any improvement in the weak model both reduces the norm and improves Err_w2s. The paper compares against dimension-based baselines (Fig. 5) but not against the weak model's own test error, ||P_w||_op, or the principal-subspace dimension of the weak model. I request baseline comparisons or partial correlations that control for these simpler quantities, so that the specificity of the P_s(I-P_w) mechanism is supported.

minor comments (4)

[Lemma A.12] In the statement and proof of Lemma A.12, the second factor of P_w uses 1/˜n, but for consistency with the definition in Sec. 3.4 and with the push-through step later in the proof, it should be 1/ˆn; please correct this typo.
[Sec. 2] The sentence 'we explicitly links W2SG performance' in Sec. 2 should read 'we explicitly link W2SG performance'.
[Sec. 5.4] The Spearman correlations are reported without uncertainty intervals or p-values; given that Exps. II and III have only 22-28 points per dataset, I recommend adding bootstrap confidence intervals or a permutation-based p-value.
[Abstract and Sec. 5.1] The phrase 'without requiring labels' is too strong because App. D.2 admits that labels are used to select the hyperparameters. Please qualify the claim as, for example, 'without labels for the metric itself once its few hyperparameters are fixed'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is a self-contained derivation from ridge-regression algebra, and the empirical metric is an approximation of the theorem's hat matrices rather than a fitted surrogate.

full rationale

The central claim is an algebraic consequence of ridge-regression closed forms. Under Assumption 3.7, Lemma A.12 rewrites the weak model's normalized error on the strong finetuning set as (I - P_w) * yhat / sqrt(nhat) + o(1), and Lemma A.13 rewrites the strong model's error-propagation operator as P_s, so Theorem 3.8 follows by substitution into the PredGap expression of Eq. (5). The quantities P_w and P_s are defined directly from the principal kernel matrices and the regularization constants; they are not fitted values, and the o(1) terms are controlled by the stated decomposability conditions rather than by calibration to the predicted gap. Corollaries 5.1 and 5.2 are norm bounds derived from the same theorem, and the empirical metric in Eq. (2) is an approximation of this algebraic object, not a fitted surrogate. The paper's use of hyperparameters alpha and beta_eff is a standard estimation choice, and the cross-model transfer experiment in Table 2 indicates the correlations are not imposed by construction. No load-bearing step is justified by a self-citation or by a uniqueness/ansatz citation; concerns about whether real LLM activations satisfy Assumption 3.7 concern the scope of the theorem, not the circularity of the derivation.

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

The central claim depends on the decomposability of representations (a domain assumption), the linear-representation model of finetuning (a domain assumption), and standard concentration results. The only free parameters in the empirical metric are the PCA thresholds α and effective regularizations β_eff; these are tuned on the data. No new entities are postulated beyond the mathematical constructs of principal and non-principal representation subspaces.

free parameters (4)

alpha_w (PCA threshold for weak principal representations) = 0.1 (Exp I), 0.001 (Exp II), tuned in {0.02, 0.05} (Exp III)
Threshold for selecting principal components when approximating P_w in Eq 2 (Sec 5.2).
alpha_s (PCA threshold for strong principal representations) = 0.1 (Exp I), 0.05 (Exp II), tuned in {0.02, 0.05} (Exp III)
Threshold for selecting principal components for the strong model's P_s.
beta_eff,w (effective regularization for weak) = 0.1 (Exp I), 0.0001 (Exp II), tuned in {0.2,0.5,0.8,1,2,4,8} (Exp III)
Replaces (β_w+γtilde_w) in P_w, absorbing implicit regularization (Sec 5.2).
beta_eff,s (effective regularization for strong) = 0.1 (Exp I), 0.01 (Exp II), tuned in {0.2,...,8} (Exp III)
Replaces (β_s+γhat_s) in P_s.

assumptions (4)

domain assumption Finetuning is equivalent to learning a linear function on fixed representations h (Sec 3.1).
Justified by linear probing practice and NTK regime; for full LLM finetuning (Exp III) this is an approximation.
domain assumption Weak and strong representations are (δ,γhat,γtilde)-decomposable w.r.t. principal subspaces V_w and V_s (Assumption 3.7 / Def 3.3).
Load-bearing premise of Thm 3.8; examples are given (3.4, 3.5, 3.6) but not verified on real models.
domain assumption In Sec 4, the strong ceiling model is near-perfect (Err_sc = o(1)) and δ_s = o(γhat_s), allowing benign overfitting.
These are conditions for Thm 4.1; they restrict the setting.
ad hoc to paper In Exp III, layer-wise normalized activation maps of pretrained LLMs can serve as the representation h(x) for computing the metric.
Heuristic introduced to apply the theory to full LLM finetuning (Sec 5.2); not derived from the theory.

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Cite this review

Pith. "Pith review of Representations Shape Weak-to-Strong Generalization: Theoretical Insights and Empirical Predictions." pith.science (2026). https://pith.science/paper/4YDLCXU5

@misc{pith2026250200620,
  author       = {Pith},
  title        = {Pith review of: Representations Shape Weak-to-Strong Generalization: Theoretical Insights and Empirical Predictions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4YDLCXU5}},
  note         = {Machine review of arXiv:2502.00620}
}

read the original abstract

Weak-to-Strong Generalization (W2SG), where a weak model supervises a stronger one, serves as an important analogy for understanding how humans might guide superhuman intelligence in the future. Promising empirical results revealed that a strong model can surpass its weak supervisor. While recent work has offered theoretical insights into this phenomenon, a clear understanding of the interactions between weak and strong models that drive W2SG remains elusive. We investigate W2SG through a theoretical lens and show that it can be characterized using kernels derived from the principal components of weak and strong models' internal representations. These kernels can be used to define a space that, at a high level, captures what the weak model is unable to learn but is learnable by the strong model. The projection of labels onto this space quantifies how much the strong model falls short of its full potential due to weak supervision. This characterization also provides insights into how certain errors in weak supervision can be corrected by the strong model, regardless of overfitting. Our theory has significant practical implications, providing a representation-based metric that predicts W2SG performance trends without requiring labels, as shown in experiments on molecular predictions with transformers and 5 NLP tasks involving 52 LLMs.

Figures

Figures reproduced from arXiv: 2502.00620 by the authors.

**Figure 1.** An illustration of our main result (Thm. 3.8). The path connecting the two highlighted regions represents the overlap b/w the complement of a scaled span of the weak model’s principal kernel and the scaled span of the strong model’s principal kernel, determining the contribution of the weak model’s errors to PredGap. et al. (2023) propose an analogy that can be empirically explored today: can a weak model effectiv… view at source ↗

**Figure 2.** Results of Exp. I: our metric strongly correlates [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** A strong correlation between ∥Ps(I − Pw)∥op and Errw2s is observed in Exp. II where we finetune embedding models. 0.8 0.9 Ps(I Pw) op 0.2 0.3 0.4 0.5 E r r w 2 s SC=0.842 (a) SciQ 0.97 0.98 0.99 Ps(I Pw) op 0.1 0.2 0.3 0.4 E r r w 2 s SC=0.804 (b) Amazon Polarity 0.96 0.98 Ps(I Pw) op 0.20 0.25 0.30 0.35 0.40 0.45 E r r w 2 s SC=0.853 (c) Cosmos-QA [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: A strong correlation between ∥Ps(I − Pw)∥op and Errw2s is observed in Exp. III involving general-purpose LLMs. 0.980 0.985 0.990 0.995 Ps(I Pw) op 0.35 0.40 0.45 E r r w 2 s SC=0.905 5000 4500 4000 3500 -dim(act_map) 0.35 0.40 0.45 E r r w 2 s SC=-0.065 300 200 100 -di…

**Figure 5.** Figure 5: In Exp. III, for models with activation map [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Effect of hyperparameters in Exp. II. Colors indicate Spearman correlation. [PITH_FULL_IMAGE:figures/full_fig_p038_6.png]

**Figure 7.** Figure 7: Effect of hyperparameters in Exp. III. Colors indicate Spearman correlation. [PITH_FULL_IMAGE:figures/full_fig_p038_7.png]

**Figure 8.** Figure 8: Results for ∥Ps(I − Pw)Ps∥op in Exp. I. D.3. Results for ∥Ps(I − Pw)Ps∥op Results for ∥Ps(I − Pw)Ps∥op are presented in Figures 8, 9, and 10. We observe a strong correlation between Errw2s and ∥Ps(I − Pw)Ps∥op across the settings. These correlations are similar to thos…

**Figure 9.** Figure 9: Results for ∥Ps(I − Pw)Ps∥op in Exp. II. 0.7 0.8 0.9 Ps(I Pw)Ps op 0.2 0.3 0.4 0.5 E r r w 2s SC=0.842 (a) SciQ 0.96 0.98 Ps(I Pw)Ps op 0.1 0.2 0.3 0.4 E r r w 2s SC=0.810 (b) Amazon Polarity 0.90 0.95 Ps(I Pw)Ps op 0.20 0.25 0.30 0.35 0.40 0.45 E r r w 2s SC=0.846 (c)…

**Figure 10.** Figure 10: Results for ∥Ps(I − Pw)Ps∥op in Exp. III. 0.85 0.90 0.95 Ps(I Pw) op 0.2 0.3 0.4 0.5 E r r w 2 s SC=0.741 10000 8000 6000 4000 -dim(act_map) 0.2 0.3 0.4 0.5 E r r w 2 s SC=0.272 10000 5000 0 -dim( h) 0.2 0.3 0.4 0.5 E r r w 2 s SC=0.428 0.98 0.99 Ps(I Pw) op 0.05 0.10…

**Figure 11.** Figure 11: The top panel shows results on SciQ for models with sizes ≤ 10000, while the bottom panel shows results on Amazon Polarity for models with sizes ≤ 8000. The patterns observed here are consistent with those discussed in [PITH_FULL_IMAGE:figures/full_fig_p039_11.png]

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Pith tools

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

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" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

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