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Reasoning Vectors: Transferring Chain-of-Thought Capabilities via Task Arithmetic

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper argues that reasoning learned through reinforcement learning can be extracted as a parameter vector and transferred to other models by addition.

desk verdict Clean idea with real cross-domain signal, but the missing random-vector control keeps the 'reasoning vector' interpretation from being airtight. read the letter →

arxiv 2509.01363 v1 pith:HIIKQOLG submitted 2025-09-01 cs.CL

Mohammad Zbeeb , Hasan Abed Al Kader Hammoud , Bernard Ghanem This is my paper
classification cs.CL
keywords reasoningvectorstaskarithmeticmodelmergingreinforcementlearningchain-of-thoughtparameter-spacetransferlinearmodeconnectivitylargelanguagemodels
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

This paper tries to show that a complex capability—multi-step reasoning—can be lifted out of a model that learned it through reinforcement learning and dropped into a different model by simple tensor addition. The vector in question is the parameter difference between two identically initialized models trained on the same data, one by supervised fine-tuning and one by reinforcement learning. Adding that difference to an instruction-tuned model yields consistent gains on math, code, science, and logic benchmarks, while subtracting it damages performance. If right, the result turns expensive reinforcement-learning training into a reusable asset: capabilities, not just knowledge, become portable in weight space.

What carries the argument

The reasoning vector v_reason = θ_GRPO − θ_SFT: the element-wise difference between the weights of a GRPO-trained donor model and its SFT-trained twin. This difference is the object claimed to encode the reinforcement-learning reasoning increment, with shared dataset knowledge cancelled by the subtraction; adding it to a target model's weights is the transfer operation. Linear mode connectivity is invoked to justify why this straight-line move in weight space stays in a low-loss region.

What would settle it

Take the same instruction-tuned base model and add a random vector with the same norm as v_reason; if the random vector reproduces the reported benchmark gains, the claim that the difference vector specifically encodes reasoning is falsified.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that v_reason = θ_GRPO − θ_SFT isolates the parameter change reinforcement learning introduces on top of supervised fine-tuning, because the two donor checkpoints share initialization, architecture, and training data. Injecting v_reason (with scale α = 1) into a compatible instruction-tuned model improves accuracy on GSM8K, HumanEval/HumanEval+, SciQ, and BigBenchHard at 1.5B and 7B scale; the strongest reported gain is 12.3 points on BigBenchHard. Removing the vector by subtraction drops GSM8K accuracy by 11.8 points, below the SFT baseline. A math-derived vector also transfers modestly to code generation and a code-derived vector to math, whic

Load-bearing premise

The load-bearing premise is that the two donor models differ only in the reinforcement-learning stage, so subtracting their weights leaves exactly the reasoning gained by RL and nothing else—if that fails, the vector could carry training noise and the gains could be a generic perturbation effect.

Editorial extensions

If this is right

  • Reasoning can be treated as a modular direction in weight space: adding the vector improves a compatible model with no further training, and subtracting it removes the capability.
  • A vector extracted from math training improves performance on code, science QA, and logical deduction, so the transferred component is not just a GSM8K artifact.
  • Practitioners can reuse the computational investment of existing RL runs by storing one difference tensor and applying it to any compatible checkpoint at negligible cost.
  • The calibration result, where α = 1 works without rescaling, suggests the raw vector magnitude is already meaningful for transfer.
  • The effect appears across model scale (1.5B and 7B), indicating it is not a quirk of a single size.

Reading between the lines

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

  • Inference: Because the paper does not compare against a norm-matched random vector, a natural next experiment is to test whether a random direction of the same magnitude produces similar gains; this would distinguish reasoning transfer from a generic perturbation effect.
  • Inference: If reasoning directions can be combined, one could build 'skill arithmetic' by adding several such vectors to a base model; the paper's composition figure gestures at this but does not evaluate it.
  • Inference: The method's dependence on identical initialization suggests a testable boundary: transfer should degrade as donor and target diverge in initialization, and measuring that decay could map the 'compatibility region' in parameter space.
  • Inference: A practical extension would be to extract reasoning vectors from smaller donors and apply them to larger targets of the same family, to see whether the capability transfers across scale as well as within it.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes that reasoning capabilities learned via reinforcement learning can be extracted as a 'reasoning vector' by subtracting the parameters of an SFT checkpoint from a GRPO checkpoint trained on the same data (v_reason = θ_GRPO − θ_SFT), and then transferred to a compatible instruction-tuned model by simple addition (θ_enhanced = θ_target + α·v_reason). Experiments on Qwen2.5 1.5B and 7B models report consistent gains on GSM8K, HumanEval, SciQ, and BigBenchHard, with the largest claimed improvement on BBH (+12.3% for 1.5B). The paper also includes robustness experiments on perturbed GSM8K and ablations showing that subtracting the vector degrades performance. The central claim is that the vector captures a modular, transferable reasoning capability rather than acting as an arbitrary parameter perturbation.

Significance. If the central claim held up, the result would be practically significant: it would allow reusing the computational investment in RL-trained checkpoints through inexpensive tensor arithmetic, and it would support a modular view of high-level capabilities in parameter space. The paper is clearly written and uses publicly available checkpoints, which is a strength for reproducibility. However, the evidence is currently underdetermined: the absence of a random-vector/permutation control means the observed improvements are equally consistent with a generic perturbation effect, and the single-run evaluation with small gains and a custom BBH subset does not yet establish statistical reliability. The theoretical grounding via linear mode connectivity is also not directly applicable to the transfer operation used.

major comments (4)
  1. [Section 3.2, Eqs. (1)–(2), Table 1] The central interpretation of v_reason as a reasoning-specific direction is not tested against a null model. Adding any sufficiently large, structured direction to the weights could shift benchmark accuracies in the observed pattern. The paper provides no control of the form: add a random vector with the same L2 norm as v_reason, a sign-randomized copy, or a permuted version, and measure the same benchmarks. Without such baselines, the gains (+2.6 to +12.3 across benchmarks) and the symmetric degradation from subtraction do not establish that the direction encodes reasoning rather than acting as a generic perturbation. This control is missing from the limitations list in Section 5, and it is load-bearing for the paper's main claim.
  2. [Section 4.1, Table 1, Appendix A.1] All results are from a single run, with no error bars or significance tests. Several reported gains are small (e.g., +2.6 GSM8K, +2.2 HumanEval, +1.5 SciQ for 1.5B) and could be within run-to-run variance. In addition, the BigBenchHard evaluation uses a custom subset of five tasks (Table 4) with no stated selection criterion, and the 1.5B baseline is 6.7%, near random. The large +12.3% BBH improvement may be partly a floor effect or an artifact of task selection. The authors should provide multiple runs or bootstrap confidence intervals and justify/advertise the BBH subset selection procedure.
  3. [Section 4.4, scaling analysis] The scaling factor α was tuned on GSM8K by testing a grid {0.5, 1.0, 1.5, 2.0} and selecting α=1.0 as optimal. The subsequent headline GSM8K result (+4.9%) is therefore partly a selection artifact: the reported number is the best of several values on the test set. To report a fair transfer result, α should be selected on a validation set (or the sensitivity across α should be reported without claiming the selected value as the expected gain). This is a free parameter that affects the central performance numbers.
  4. [Section 3.3, Eq. (4)] The theoretical foundation uses linear mode connectivity, but Eq. (4) applies to interpolation between two models (λθ_A + (1−λ)θ_B), whereas the transfer operation in Eq. (2) is a translation θ_target + v_reason, i.e., extrapolation from θ_target along a direction not necessarily in the same low-loss basin. The LMC argument therefore does not, as written, justify why adding the difference vector to a different target model should remain in a low-loss region. Additionally, the claim that θ_SFT and θ_GRPO are 'identically initialized' and share the same pre-training history is asserted but not demonstrated for the public checkpoints used. The authors should either supply evidence of the checkpoint relationship or explicitly weaken the theoretical claim to a heuristic.
minor comments (5)
  1. [Section 3.1] Typo: 'indepedent' should be 'independent'.
  2. [Throughout] Inconsistent model naming: 'Qwen 2.5' and 'QWEN 2.5' are used interchangeably. Please unify to the official capitalization.
  3. [Table 2] The baseline row reports 45.7 for the original GSM8K condition, while Table 1 reports 45.1 for the same model/benchmark. This inconsistency should be explained or corrected.
  4. [Section 4.1, references] Reference [6] is cited for HumanEval, but that reference is about class-level code generation (ICSE 2024), whereas HumanEval is the function-level benchmark of Chen et al. (2021). Please cite the original HumanEval paper.
  5. [Figure 3] The text states that standardized prompt templates are 'shown in Figure 3', but no Figure 3 appears in the manuscript. Either include the figure or remove the reference.

Circularity Check

1 steps flagged · score 3.0 of 10

Partial circularity: the scaling factor α is fit to GSM8K, so the headline GSM8K gain is in-sample; core vector extraction and other benchmarks remain independent.

  1. fitted input called prediction [Section 4.4 Scaling Analysis; Table 1; Section 3.2]
    "We investigated the effect of the scaling factor α from Equation 2, testing values in {0.5, 1.0, 1.5, 2.0}. We found that α = 1.0 achieved the optimal performance (50.0%)"

    Equation (2) defines the enhanced model as θ_target + α·v_reason, with α a free scaling coefficient. The authors tune α on GSM8K accuracy, selecting α=1.0 because it yields 50.0% on GSM8K. That same 50.0% (+4.9%) GSM8K result is then reported in Table 1 and the abstract as evidence that the reasoning vector transfers. The GSM8K improvement is therefore not an independent prediction—it is the selection criterion for α. The §3.2 claim that 'α = 1 was consistently effective' is a post-hoc description of this selection. Other benchmarks (HumanEval, SciQ, BigBenchHard) were not used to select α and remain out-of-sample, so this circularity is partial.

full rationale

The paper is largely self-contained: v_reason = θ_GRPO − θ_SFT is computed from two public checkpoints rather than fitted to the target benchmarks, and the transfer results on HumanEval, SciQ, and BigBenchHard are genuine out-of-sample evidence. No self-citations are load-bearing, and the LMC argument in §3.3 cites external prior work rather than the authors' own uniqueness theorem. The main circularity is the α=1.0 scaling factor: it is selected by maximizing GSM8K accuracy, and then the GSM8K improvement is advertised as a finding, making that number in-sample. The absence of a random-vector/permutation control is a serious underdetermination of the reasoning-specific interpretation, but that is an experimental-design/correctness concern rather than a circular derivation.

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

The central claim rests on the assumption that the weight difference between two fine-tuned checkpoints isolates a capability, plus the linear mode connectivity assumption that this difference can be added to other models. These are domain assumptions not verified in the paper. The free parameters include the scaling coefficient and the BBH task subset selection.

free parameters (2)
  • scaling factor α = 1.0
    Selected from {0.5, 1.0, 1.5, 2.0} as optimal on GSM8K in the scaling ablation (Section 4.4); then reused for all other benchmarks.
  • BigBenchHard task subset = 5 tasks listed in Table 4
    The BigBenchHard evaluation uses a custom subset of tasks without a stated selection criterion; this choice directly affects the reported +12.3% gain.
assumptions (4)
  • domain assumption Linear Mode Connectivity holds between θ_SFT and θ_GRPO, so the interpolation path stays in a low-loss region (Eq. 4).
    Invoked in Section 3.3 as the theoretical foundation; the authors do not verify the inequality for the actual checkpoints.
  • domain assumption θ_SFT and θ_GRPO are identically initialized, use the same dataset, and differ only in the final optimization objective.
    Stated in Section 3.1; required for subtraction to isolate RL reasoning; the paper does not verify training histories beyond the model card.
  • domain assumption Adding v_reason to θ_target keeps the parameters in a compatible low-loss basin.
    Implied by Section 3.3; no loss measurements are provided for the enhanced model.
  • domain assumption Benchmark accuracy differences are a valid measure of reasoning capability.
    The paper interprets accuracy gains on GSM8K, HumanEval, SciQ, and BBH as evidence of reasoning transfer (Section 4.2).

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Cite this review

Pith. "Pith review of Reasoning Vectors: Transferring Chain-of-Thought Capabilities via Task Arithmetic." pith.science (2026). https://pith.science/paper/HIIKQOLG

@misc{pith2026250901363,
  author       = {Pith},
  title        = {Pith review of: Reasoning Vectors: Transferring Chain-of-Thought Capabilities via Task Arithmetic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HIIKQOLG}},
  note         = {Machine review of arXiv:2509.01363}
}
abstract

Large language models often require costly optimization, such as reinforcement learning, to master complex reasoning tasks. This work demonstrates that reasoning ability, once learned, can be extracted and transferred between models as a compact task vector. We source two publicly available, identically initialized Qwen2.5 models, one fine-tuned with supervised fine-tuning (SFT) and the other with group relative policy optimization (GRPO) on the same dataset. From these, we extract a reasoning vector: $v_{\text{reason}} = \theta_{\text{GRPO}} - \theta_{\text{SFT}}$. We hypothesize that this vector captures the reasoning capability instilled by reinforcement learning while factoring out shared knowledge from the SFT process. When added to compatible instruction-tuned models through simple arithmetic, this vector consistently improves performance across diverse reasoning benchmarks: GSM8K (+4.9%), HumanEval (+4.3%), SciQ (+1.7%), and BigBenchHard (+12.3% for the 1.5B model). The performance improvements persist under adversarial conditions. Conversely, subtracting the vector causes significant performance degradation (-11.8% on GSM8K), demonstrating the vector's strong contribution to the model's reasoning abilities. This work shows how reasoning capabilities, typically developed through expensive training, can be extracted from existing open-source models and reused through simple tensor arithmetic, offering a practical way to enhance models by recycling prior computational investments.

Figures

Figures reproduced from arXiv: 2509.01363 by the authors.

Figure 1
Figure 1. Merging the Fine-Tuning and Reasoning Vectors. Let ∆f = θf − θ0 denote the fine￾tuning vector (f) and vreason = θr − θf denote the reasoning vector (r). By adding vreason to a base model, we obtain an enhanced model with improved reasoning capabilities, effectively transferring the outcome of the reinforcement learning phase. ∗This work was completed during the author’s research internship at KAUST. 1 arXiv:2509.013… view at source ↗
Figure 2
Figure 2. Reasoning vector operations in weight space. Each panel illustrates a different transfor￾mation: (1) Vector injection shifts a base model toward improved reasoning. (2) Negation removes the reasoning component, suppressing chain-of-thought behavior. (3) Addition combines multiple skill vectors, enabling multi-skill reasoning. (4) Analogy-style composition transfers capabilities across domains, supporting generalizat… view at source ↗
Figure 3
Figure 3. Evaluation prompt templates for HumanEval and GSM8K. This design allows us to distinguish between improvements from parameter modification alone versus those elicited by explicit reasoning prompts. 0 20 40 60 Accuracy (%) 2.6 2.2 0.5 1.5 10.2 4.9 4.3 3.1 1.6 12.3 Qwen2.5 1.5B GSM8K HumanEval HumanEval+ SciQ BigBenchHard 0 25 50 75 Accuracy (%) 3.6 2.5 3.2 1.5 0.9 5.0 3.7 3.7 1.7 1.4 Qwen2.5 7B Baseline +Vector +Vect… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Accuracy improvements from reasoning vector transfer. Performance of QWEN2.5 models (1.5B left, 7B right) on five benchmarks. Bars compare baseline (blue), vector-enhanced (+Vector, green), and vector with prompt (+Vector+Think, orange). Green annotations show absolute…
Figure 5
Figure 5. Figure 5: Robustness of Qwen2.5 1.5B under four perturbation conditions. Blue bars show baseline, orange is vector-enhanced, and green adds a reasoning prompt. The vector provides consistent gains across all conditions. Cross-Domain Transfer. To assess generalization, we tested …

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.