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Knowledge That Is Linear Before Fine-Tuning Generalizes After It

Faridoun Mehri, Hinrich Schütze

arXiv:2610.22300v1cs.CLcs.AIcs.LG

TL;DR

The paper asks which known facts support learning new facts, beyond simply recalling them. It examines whether linear readability in pretrained representations predicts generalization after fine-tuning and finds a strong association across models and relations.

  • Problem

    The paper asks which known facts support generalization to new facts, a question distinct from whether a model can recall them.

    Source: §1 · pp. 1–2

  • Method

    The study measures pretrained relations’ linear readability with a cross-validated probe and tests whether it predicts generalization of new relations built on them after fine-tuning.

    Source: §3 · p. 6, §6 · p. 9

  • Results

    Across 11 models and 70 relations, Pre-linearity and Generalization correlate at r = 0.87 to 0.92, and the association remains after jointly controlling for Recall, relation size, and subject frequency.

    Source: §5 · p. 8

  • Takeaways & Limitations

    Pre-linearity predicts Generalization better than Recall, indicating that whether facts are active is not the same as whether they are known.

    Source: §1 · p. 3, §6 · p. 10

Abstract

from arXiv · show

Not all facts a language model can recall support further learning equally well. We distinguish active facts, which support generalization beyond their own recall, from inert facts, which can be recalled but do not support such generalization in a given task. We study this distinction by constructing new relations from existing ones. Take the relation that maps a place to its continent (Paris -> Europe) and assign each of its answers an arbitrary new label (Europe -> crab, Asia -> zebra). The new relation then maps Paris -> crab. We fine-tune the model on the new relation for a subset of subjects, without telling it the original relation or the relabeling rule, and test whether it assigns the corresponding labels to held-out subjects that share an answer in the original relation. Having learned Paris -> crab, does it answer Munich -> crab? Across eleven language models and 70 relations, how linearly readable the original relation is in pretrained representations strongly predicts this generalization. Within each model, the two correlate across relations at Pearson r = 0.87 to 0.92. This association persists when controlling, separately and jointly, for the number of fine-tuning subjects and answers, the model's accuracy on the original relation, and how often the subjects occur in pretraining text. It is also robust to re-randomizing the label assignment and fine-tuning seed, and to replacing the animal labels with labels from thirteen other semantic categories. When we map a fine-tuned model's wrong label back to the original answer it was assigned to, Munich -> zebra becomes Munich -> Asia. Answers recovered this way are more often among the linear probe's top answers for the subject than among the pretrained model's own. Whether a new relation generalizes depends less on whether its facts can be recalled than on how readily their shared structure can be read out linearly.

1 INTRODUCTION

The paper asks which known facts support learning new facts and hypothesizes that linear readability predicts whether a new relation built on them will generalize. It tests this through fine-tuning and finds a strong association between pre-fine-tuning linear readability and generalization.

1 INTRODUCTION

  • The paper distinguishes active facts, which support generalization beyond recall, from inert facts, which can be recalled but do not support generalization in a given task.A model may state a fact reliably yet be unable to use it when learning something new.Source: §1 · p. 1
  • Figure 1 contrasts equally recalled relations: fine-tuning generalizes to held-out subjects for the linearly readable relation, but not the other.For the latter, the wrong label matches the answer selected by the linear probe rather than the true answer.Source: Figure 1 · p. 2
  • The study addresses which property of pretrained knowledge determines whether new knowledge built on it generalizes, proposing linear readability as the key predictor.The authors call a relation’s linear readability Pre-linearity.Source: §1 · p. 2
  • The experiment randomly relabels answers in a pretrained relation, fine-tunes on some subjects without revealing the source relation or mapping, and tests labels on held-out subjects.The design tests whether the new relation inherits structure from the pretrained one.Source: §1 · p. 2
  • Across 11 models and 70 relations, Pre-linearity correlates with Generalization at Pearson r = 0.87 to 0.92; jointly controlled correlations remain r = 0.70 to 0.82.The association persists when controlling for Recall, relation size, and subject frequency.Source: §1 · pp. 2–3

2 RELATED WORK

Prior work studies entailment, factual recall, linguistic generalization, and linear relational structure, but the paper identifies a gap: which known facts support learning new facts. It situates its fine-tuning study against these neighboring questions and distinctions.

2 RELATED WORK

  • Entailment makes a new fact necessarily follow from known facts, whereas statistical generalization relies on a rule that is not perfectly accurate.The paper places its focus on the latter rather than necessary inference.Source: §2 · p. 3
  • Paraphrase and construction-acquisition work concerns variation in linguistic expression, which the authors distinguish from statistical generalization of facts.The cited work includes cases with little or no pretraining-corpus evidence.Source: §2 · p. 3
  • Linear relational embeddings were proposed as a mechanism for unseen relational facts, but recent work on them does not consider statistical generalization of facts.Factual-recall and knowledge-editing research asks how stated facts are learned or edited, not how learned facts support new learning; related fine-tuning studies also leave that question open.Source: §2 · pp. 3–4
  • Some hallucination framing treats unstated claims as undesirable, while the authors note that statistical inferences are not guaranteed to be true.They argue that statistical generalization is part of language models’ effectiveness.Source: §2 · pp. 3–4
  • The authors identify an unaddressed question: which known facts a model can build new facts on, studied here in a fine-tuning setting.They find that whether facts are active is better predicted by linear readability than by recall.Source: §2 · p. 4

3 METHOD

The method tests whether fine-tuning on a relabeled version of a pretrained relation generalizes to held-out subjects. It compares this Generalization with pretrained Recall and the linear readability of the original relation.

New relations and fine-tuning

  • A new relation randomly maps each pretrained answer to a distinct label, so subjects sharing an original answer share a new label.The model is fine-tuned on some subjects without being told the original relation or relabeling rule, then evaluated on held-out subjects.Source: §3 · pp. 4–5

Measures

  • Generalization is candidate accuracy on held-out subjects under the held-out evaluation template, while Recall is pretrained candidate accuracy on fine-tuning subjects.Recall measures how well the pretrained model states the original relation; the two metrics use different subjects and templates.Source: §3 · p. 5

Pre-linearity

  • Pre-linearity measures how accurately a cross-entropy probe predicts pretrained-relation answers from dimension-reduced subject representations.Subject activations are reduced to 16 dimensions with PCA and whitening before the probe is fit.Source: §3 · p. 5
  • Nested cross-validation selects the probe layer and produces held-out predictions for fine-tuning subjects, with a most-frequent-answer baseline subtracted.The inner loop selects the layer; outer folds ensure each subject is predicted by a probe that did not train on it.Source: §3 · p. 6

Controls

  • The analysis tests whether Pre-linearity predicts Generalization after accounting for Recall, answer count, fine-tuning-subject count, and subject frequency.Each control is considered separately and all four are also controlled jointly.Source: §3 · p. 6

4 EXPERIMENTAL SETUP

The experiments use 70 pretrained relations across 11 open-weight models, with randomized answer labels for new relations. Repeated runs and alternate label categories test whether the results depend on those choices.

Pretrained relations

  • 70 relations qualify under requirements for answer count, subjects per answer, and enough fine-tuning subjects for probe cross-validation.The relations are screened from Hernandez, Town, and BEAR sources; spelling relations are excluded because answers can be read from subject letters.Source: §4 · pp. 6–7

Models

  • The study uses 11 open-weight models, including base and post-trained checkpoints, spanning 2B to 27B parameters.Qwen3.5-9B is included in both base and post-trained forms.Source: §4 · p. 7

Label sets

  • The default animal label set contains 179 names, and robustness tests use labels from 13 other qualifying WordNet categories.The label-set comparison reruns each relation on MiniCPM5-2B-Base once per category.Source: §4 · p. 7

Re-randomization

  • Re-randomization tests vary the answer-to-label bijection and fine-tuning seed while holding the subject split, label subset, templates, and exposure fixed.Every relation is rerun three times on MiniCPM5-2B-Base; selected Hernandez and Town relations are rerun across other models.Source: §4 · p. 8

5 RESULTS

Across the tested models, relations that are more linearly readable before fine-tuning tend to yield better generalization afterward. This association survives controls, outperforms Recall as a predictor, and is robust across label sets and reruns.

Main result

  • Across all 11 models, Pre-linearity and Generalization correlate at Pearson r = 0.87 to 0.92 and Spearman ρ = 0.85 to 0.90 across relations.No 95% interval falls below 0.72; the models span base and post-trained checkpoints from 2B to 27B parameters.Source: §5 · p. 8

Controls

  • With Recall, answer count, subject count, and subject frequency controlled jointly, Pre-linearity still correlates with Generalization at r = 0.70 to 0.82.No 95% interval falls below 0.56.Source: §5 · p. 8

Pre-linearity versus Recall

  • Controlling for Recall, Pre-linearity correlates with Generalization at r = 0.79 to 0.86, while Recall’s association largely reflects its overlap with Pre-linearity.Controlling for Pre-linearity instead, Recall’s correlation drops to r = 0.40 to 0.58.Source: §5 · pp. 8–9

Robustness

  • Across 13 alternate WordNet label categories, the Pre-linearity–Generalization correlation stays between r = 0.91 and 0.93.Relation rankings also closely match those produced with animal labels, with ρ = 0.95 to 0.97.Source: §5 · p. 9
  • Across three randomized runs on MiniCPM5-2B-Base, the correlation is r = 0.92, 0.93, and 0.92.Across reruns of 17 Hernandez and Town relations on ten models, each run’s r lies between 0.78 and 0.95.Source: §5 · p. 9

Generalization failures

  • The probe’s three highest-ranked answers account for 21.5% of held-out errors versus 8.8% under shuffling, more than the pretrained model’s top answers.The pretrained model’s three highest-ranked answers account for 11.5% versus 7.3% shuffled.Source: §5 · p. 9

6 CONCLUSION

Across eleven models and 70 relations, how linearly readable a pretrained relation is predicts how well a new relation built on it generalizes after fine-tuning. Pre-linearity uses nested subject splits to measure this readability without probing held-out subjects.

6 CONCLUSION

  • Across three MiniCPM5-2B-Base runs with new random bijections and fine-tuning seeds, the reported correlations are r = 0.92, 0.93, and 0.92.The table reports a mean of 0.93 ±.03.Source: §6 · p. 9, Table 2
  • Pre-linearity predicts Generalization across eleven models and 70 relations, and remains a better predictor than Recall after controlling for Recall, relation size, and subject frequency.The conclusion describes Pre-linearity as how linearly readable a relation is before fine-tuning.Source: §6 · pp. 9–10

A PRE-LINEARITY CROSS-VALIDATION

  • Pre-linearity excludes held-out subjects H from the probe and uses nested splits of fine-tuning subjects T, with the inner loop selecting the layer.Each fine-tuning subject receives an outer-loop prediction from a probe that did not see it.Source: Figure 5 · p. 14, A PRE-LINEARITY CROSS-VALIDATION · p. 14

B FINE-TUNING RECIPE

Models are fine-tuned on all parameters with a shared recipe, and every model reaches accuracy 1.00 on its fine-tuning subjects after the final epoch.

B FINE-TUNING RECIPE

  • Fine-tuning updates all parameters with AdamW for 20 epochs, using cross-entropy on only the label’s first token; the learning rate warms to 10^-5 then cosine-decays to 10^-6.Each subject-template record is used once per epoch, for 20 presentations total. Each fine-tuning fact is seen 80 times across four wordings, 20 times in each wording.Source: B FINE-TUNING RECIPE · p. 14
  • Every entry in Table 3 is 1.00 for accuracy on fine-tuning subjects after the last epoch.The table reports candidate accuracy under four fine-tuning templates, averaged within subjects and answers.Source: Table 3 · p. 15

C PER-RELATION RESULTS

The per-relation figures compare Pre-linearity, Generalization, and pretrained Recall across 70 relations and eleven models. They show relation-level summaries and a model-by-model view, with relations ordered by mean Generalization.

C PER-RELATION RESULTS

  • Figure 6 orders relations by mean Generalization and shows the median, mean, and range across eleven models for Pre-linearity, Generalization, and Recall.The range is not a confidence interval; symbols identify the Hernandez, Town, and BEAR sets.Source: C PER-RELATION RESULTS · pp. 16, 18, Figure 6 · p. 17
  • Figure 7 places relations in rows and models in columns, with separate blocks for Pre-linearity, Generalization, and Recall.Square colors encode within-block quantile ranks, with equal numbers of squares at each color step.Source: Figure 7 · p. 19

D CORRELATION PER REPEAT

The repeat-level correlation tables report how the association between Pre-linearity and Generalization varies across fine-tuning runs. They cover MiniCPM5-2B-Base across all relations and repeated runs on the 17 Hernandez and Town relations.

D CORRELATION PER REPEAT

  • Table 2 gives the correlation for MiniCPM5-2B-Base on all 70 relations, while Tables 4 and 5 report correlations for each of three runs on the 17 Hernandez and Town relations.The three runs differ in random bijection and fine-tuning seed.Source: D CORRELATION PER REPEAT · p. 20, Table 4 · p. 20, Table 5 · p. 20
  • Table 4 reports Pearson r for three fine-tuning runs on 17 Hernandez and Town relations across ten models, while Table 5 reports Spearman ρ for the same setup.Table 4 also reports the mean relation-level Pre-linearity correlation against Generalization averaged over the three runs.Source: Table 4 · p. 20, Table 5 · p. 20

E TOWN RELATIONS

The Town relations assign real towns answers based on region, geographic bands, and country language. Towns without a unique or available answer are excluded from the relevant relation.

E TOWN RELATIONS

  • 300 towns from 20 regions form the shared pool; towns unresolved by a relation’s fixed rule are left out.Each region contributes 15 towns, all real places with English Wikipedia articles.Source: E TOWN RELATIONS · p. 21
  • Five Wikidata-based relations use town items matched by English Wikipedia title or name; 50 towns are unmatched, and ambiguous values are excluded.For multiple population or elevation values, the preferred value is used, or otherwise the latest point-in-time value.Source: E TOWN RELATIONS · p. 21
  • Latitude and longitude are divided into five contiguous bands with 50 towns each; longitude is positive eastward.The latitude and longitude bands span the ranges specified for each coordinate in the source passage.Source: E TOWN RELATIONS · p. 21
  • Elevation bands cover 195 towns and population bands cover 235, with five roughly equal-sized groups for each attribute.Unmatched towns and towns lacking a unique attribute value are excluded from the applicable relation.Source: E TOWN RELATIONS · p. 21
  • The country-language relation retains 123 towns across 11 languages after excluding countries with multiple official languages and languages represented by fewer than five towns.The initial 134 towns with a single official language are reduced by dropping six languages with fewer than five towns.Source: E TOWN RELATIONS · p. 21

F LABEL SETS

The label sets are constructed to provide distinct, plausible labels that avoid overlap with relation subjects and answers. The section also reports that the explanations account for errors well above the shuffled baseline.

F LABEL SETS

  • The animal-label algorithm applies five filters, including WordNet sense, label uniqueness, English frequency, tokenizer distinctness, and embedding similarity.The cosine threshold is 0.446 under MiniCPM5-2B-Base.Source: F LABEL SETS · p. 21
  • 179 animal labels remain, each with its own first token under every model’s tokenizer; that token is trained and scored.Source: Table 6 · p. 22
  • The comparison label sets come from WordNet noun lexicographer files and use single-token labels under MiniCPM5-2B-Base.Table 7 describes each set’s root, size eligibility, median frequency, and mean nearest-neighbor cosine.Source: Table 7 · p. 22
  • Neighbor pruning removes labels that match relation subjects or answers, or whose primary sense lies within four WordNet hypernym steps of them.Source: F LABEL SETS · p. 22
  • Every model’s errors are explained well above the shuffled baseline.Figure 8 compares each explanation’s observed share of errors with its shuffled share and reports 95% intervals.Source: Figure 8 · p. 23

G ATTRIBUTING GENERALIZATION FAILURES

The analysis attributes held-out errors to candidate explanations and compares their coverage with shuffled and chance baselines. It also specifies bootstrap procedures, supplementary materials and licenses, and the selection context for six tiny models.

G ATTRIBUTING GENERALIZATION FAILURES

  • The analysis covers 37,571 held-out errors from 757 fine-tuned models that make errors; the remaining models answer all held-out subjects correctly.Source: G ATTRIBUTING GENERALIZATION FAILURES · p. 23
  • Candidate explanations include probe-ranked and recall-ranked answers, the most frequent answer, nearest-subject answers, and pre-fine-tuning answers.An explanation accounts for an error when the model gives exactly the answer it names; the Shapley split shares credit when multiple explanations match.Source: G ATTRIBUTING GENERALIZATION FAILURES · p. 23
  • The Shapley split allocates each explained error equally among matching explanations; shuffled and chance baselines estimate coincidental matches in different ways.The shuffled baseline preserves each model’s wrong-answer preferences while breaking their links to subjects; the chance baseline samples uniformly among wrong labels.Source: G ATTRIBUTING GENERALIZATION FAILURES · pp. 23–24
  • 5.0% of errors match the most frequent answer observed, versus 6.1% under shuffling and 1.9% by chance.Because the most frequent answer is assigned to every held-out subject, shuffling cannot test its subject-specific explanatory value.Source: G ATTRIBUTING GENERALIZATION FAILURES · p. 24

H CONFIDENCE INTERVALS

I SUPPLEMENTARY MATERIAL AND LICENSES

  • The supplementary repository includes prompts, relations, label sets, model-selection materials, and code; the listed datasets and label sets carry their stated licenses.The passage identifies CC BY-SA 4.0 for adapted BEAR rows, the WordNet 3.0 license for labels, and MIT for the Hernandez relations.Source: I SUPPLEMENTARY MATERIAL AND LICENSES · p. 24

J SELECTION OF SIX TINY MODELS

  • Six tiny open-weight models were chosen from the Artificial Analysis leaderboard inspected on 18 September 2026; K2 Horizon 3.7B was skipped because the code did not support it.Source: J SELECTION OF SIX TINY MODELS · p. 25

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