Recognition: 1 theorem link
Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement
Pith reviewed 2026-05-11 10:10 UTC · model grok-4.3
The pith
Integrating self-improvement across pre-training, post-training, and inference produces math-specialized models with stronger reasoning on competition problems.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The authors claim that a closed self-improvement loop—using the current model to generate data, scoring it with a reward model derived from prior samples, and retraining—yields progressive gains in mathematical capability. This cycle runs through pre-training data creation, multiple rounds of supervised fine-tuning data evolution, reinforcement learning, and reward-guided inference, resulting in models that handle both chain-of-thought and tool-integrated reasoning on English and Chinese math benchmarks.
What carries the argument
The reward model obtained from massive sampling of model outputs, which filters data for iterative supervised fine-tuning, guides reinforcement learning, and steers inference sampling.
If this is right
- The final models support both chain-of-thought and tool-integrated reasoning on grade-school to competition-level problems.
- Iterative reward-model updates allow each stronger supervised fine-tuning model to train an improved reward model for the next cycle.
- Reinforcement learning on the final supervised model uses the ultimate reward model to further refine outputs.
- Reward-guided sampling at inference time improves answer quality on the evaluated English and Chinese datasets.
- The approach covers both Chinese and English mathematical reasoning across ten benchmarks of varying difficulty.
Where Pith is reading between the lines
- The same sampling-plus-filtering loop might be applied to other specialized domains if a reliable reward model can be built for those domains.
- Success would imply that models can bootstrap expertise in reasoning-heavy fields with less dependence on human-labeled data.
- If the cycle can be sustained without mode collapse, it raises the possibility of continued performance gains through repeated self-refinement rounds.
- The bilingual capability suggests the method preserves or enhances cross-lingual transfer when data generation and filtering are applied to mixed-language corpora.
Load-bearing premise
Repeated sampling plus reward-model filtering produces steadily higher-quality math data without compounding errors or narrowing the model's output distribution.
What would settle it
A controlled experiment in which the iterative reward-model data evolution steps are removed and the resulting models show no gain or a loss on held-out competition benchmarks such as AIME24 or MATH would falsify the claim that the self-improvement pipeline drives the observed performance.
read the original abstract
In this report, we present a series of math-specific large language models: Qwen2.5-Math and Qwen2.5-Math-Instruct-1.5B/7B/72B. The core innovation of the Qwen2.5 series lies in integrating the philosophy of self-improvement throughout the entire pipeline, from pre-training and post-training to inference: (1) During the pre-training phase, Qwen2-Math-Instruct is utilized to generate large-scale, high-quality mathematical data. (2) In the post-training phase, we develop a reward model (RM) by conducting massive sampling from Qwen2-Math-Instruct. This RM is then applied to the iterative evolution of data in supervised fine-tuning (SFT). With a stronger SFT model, it's possible to iteratively train and update the RM, which in turn guides the next round of SFT data iteration. On the final SFT model, we employ the ultimate RM for reinforcement learning, resulting in the Qwen2.5-Math-Instruct. (3) Furthermore, during the inference stage, the RM is used to guide sampling, optimizing the model's performance. Qwen2.5-Math-Instruct supports both Chinese and English, and possess advanced mathematical reasoning capabilities, including Chain-of-Thought (CoT) and Tool-Integrated Reasoning (TIR). We evaluate our models on 10 mathematics datasets in both English and Chinese, such as GSM8K, MATH, GaoKao, AMC23, and AIME24, covering a range of difficulties from grade school level to math competition problems.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents the Qwen2.5-Math series (1.5B/7B/72B) whose core contribution is integrating self-improvement across the full pipeline: pre-training data generation from Qwen2-Math-Instruct, post-training iterative RM training from model samples followed by SFT data evolution and RL, and inference-time RM-guided sampling. The models are evaluated on ten English and Chinese math benchmarks spanning grade-school to competition level (GSM8K, MATH, AIME24, etc.).
Significance. If the iterative self-improvement loop demonstrably improves data quality without compounding errors, the approach would provide a scalable, largely automated route to stronger mathematical reasoning models and reduce reliance on human-curated corpora. The manuscript currently supplies only end-to-end benchmark numbers, so the practical significance cannot yet be assessed.
major comments (3)
- [Post-training phase] Post-training phase (abstract and §3): the central claim that iterative RM-SFT evolution produces net gains in data quality rests on the untested assumption that repeated sampling plus RM filtering avoids distributional shift or reward hacking. No per-iteration quality metrics, error rates on generated traces, or control runs that hold total tokens fixed while disabling iteration are reported.
- [Abstract and post-training] Abstract and post-training description: the RM is first trained on samples from Qwen2-Math-Instruct and then used to filter the next SFT round, creating an explicit circular dependency; without reported RM accuracy on held-out human data, agreement statistics, or analysis of mode collapse, it is impossible to verify that the loop improves rather than reinforces the base model's limitations.
- [Evaluation] Evaluation section: only aggregate benchmark scores are given for the final models. The absence of ablation tables isolating the iterative component, error bars across multiple seeds, or comparisons against single-round synthesis with matched data volume leaves the source of any observed gains (self-improvement vs. scale vs. base model) unidentified.
minor comments (2)
- [Abstract] The abstract states that ten datasets are used but does not enumerate them; a compact table or appendix reference would aid reproducibility.
- [Inference stage] Inference-stage RM guidance is mentioned but the precise algorithm (best-of-N, process supervision, etc.) and hyper-parameters are not specified.
Simulated Author's Rebuttal
We thank the referee for the constructive feedback on our technical report. The points raised are important for substantiating the self-improvement claims, and we respond to each below. Revisions have been made to the manuscript to provide greater transparency on the post-training pipeline and evaluation.
read point-by-point responses
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Referee: [Post-training phase] Post-training phase (abstract and §3): the central claim that iterative RM-SFT evolution produces net gains in data quality rests on the untested assumption that repeated sampling plus RM filtering avoids distributional shift or reward hacking. No per-iteration quality metrics, error rates on generated traces, or control runs that hold total tokens fixed while disabling iteration are reported.
Authors: We agree that explicit per-iteration metrics and control experiments would provide stronger support for our claims. In the revised manuscript, we have added per-iteration quality metrics in Section 3, including the evolution of average RM scores and the percentage of samples passing the reward threshold. We also include a qualitative error analysis on generated mathematical traces. For the control runs with matched token counts, we note this as a limitation due to computational constraints and have added a discussion in the limitations section. A partial comparison to non-iterative data synthesis is provided using available resources. revision: partial
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Referee: [Abstract and post-training] Abstract and post-training description: the RM is first trained on samples from Qwen2-Math-Instruct and then used to filter the next SFT round, creating an explicit circular dependency; without reported RM accuracy on held-out human data, agreement statistics, or analysis of mode collapse, it is impossible to verify that the loop improves rather than reinforces the base model's limitations.
Authors: The design intentionally uses iterative updates to the RM to leverage improving model capabilities and reduce the risk of reinforcing initial limitations. We have revised the post-training section to include the RM accuracy on held-out human-annotated data, along with inter-rater agreement statistics between the RM and human evaluators. Additionally, we have added an analysis of response diversity across iterations to address potential mode collapse. These revisions allow for better verification that the process leads to net improvements. revision: yes
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Referee: [Evaluation] Evaluation section: only aggregate benchmark scores are given for the final models. The absence of ablation tables isolating the iterative component, error bars across multiple seeds, or comparisons against single-round synthesis with matched data volume leaves the source of any observed gains (self-improvement vs. scale vs. base model) unidentified.
Authors: We concur that ablations are necessary to attribute the gains. The revised evaluation section now features an ablation study isolating the iterative self-improvement component, with comparisons to single-round SFT/RL using similar data volumes. We have also included error bars representing standard deviation over multiple evaluation runs on the main benchmarks. While full multi-seed training was not feasible, these additions help identify the contributions of self-improvement. revision: partial
- Full control experiments that hold total tokens fixed while disabling iteration, due to high computational costs.
Circularity Check
No significant circularity in claimed derivation chain
full rationale
The paper describes an empirical iterative pipeline (pre-training data generation from Qwen2-Math-Instruct, RM training via sampling from the same model, iterative SFT data evolution, RM updates, and final RL) but presents no equations, first-principles derivations, or uniqueness theorems whose outputs reduce to inputs by construction. The self-improvement process is a standard training loop whose net gains are claimed via end-to-end benchmarks rather than tautological redefinitions or fitted parameters renamed as predictions. Self-citations to prior Qwen models exist but are not load-bearing for the central claim, which remains an independently verifiable empirical procedure without self-referential reduction.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Reward model trained on model-generated samples accurately ranks mathematical correctness
- ad hoc to paper Iterative self-sampling does not introduce compounding distributional shift or reward hacking
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Reference graph
Works this paper leans on
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[1]
For s(P (0)) to have the same parity as s(0), b0 must be 0
**Case 1: k ≡ 0 mod 2 ** P (0) ≡ b0 mod 2 . For s(P (0)) to have the same parity as s(0), b0 must be 0. Therefore, a0 must be even
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[2]
For s(P (1)) to have the same parity as s(1), 1 + bn−1 + · · · + b1 + b0 must be odd
**Case 2: k ≡ 1 mod 2 ** P (1) ≡ 1n + bn−1 · 1n−1 + · · · + b1 · 1 + b0 ≡ 1 + bn−1 + · · · + b1 + b0 mod 2 . For s(P (1)) to have the same parity as s(1), 1 + bn−1 + · · · + b1 + b0 must be odd. Since 1 is odd, the sum bn−1 + · · · + b1 + b0 must be even. From the above cases, we have: - a0 must be even. - The sum bn−1 + · · · + b1 + b0 must be even. Howe...
work page 2022
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[3]
Let’s assume f (x) = kx for some integer k
f (4x) = f (3x) + f (x) SUBSTITUTION 3: y = 1 IN THE FIRST EQUATION f (3 · 1) = f (1) + f (2 · 1) f (3) = f (1) + f (2) SUBSTITUTION 4: x = 1 IN THE SECOND EQUATION f (4 · 1) = f (3 · 1) + f (1) f (4) = f (3) + f (1) Step 3: Using the Equations to Find a Pattern We have: f (3) = f (1) + f (2) f (4) = f (3) + f (1) Substitute f (3) from the first equation ...
work page 2023
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[4]
**Understanding the Game Rules:** - Initially, there are 5000 stones. - Joe makes the first move (odd-numbered move), Penny makes the second move (even-numbered move), and so on. - On the k-th move, a player can remove between 1 and k stones. - The player who removes the last stone wins
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[5]
**Key Insight:** - The game is symmetric in the sense that the number of stones a player can remove on their turn increases linearly with the move number. - The total number of moves in the game is equal to the number of stones initially in the pile, which is 5000
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[6]
- Joe makes the odd-numbered moves: 1st, 3rd, 5th, ..., 4999th
**Total Number of Moves:** - Since there are 5000 stones, there will be 5000 moves in total. - Joe makes the odd-numbered moves: 1st, 3rd, 5th, ..., 4999th. - Penny makes the even-numbered moves: 2nd, 4th, 6th, ..., 5000th
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[7]
- Therefore, Penny will make the last move
**Determining the Winner:** - The last move (5000th move) is an even-numbered move. - Therefore, Penny will make the last move
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[8]
- Hence, Penny wins the game if both players play perfectly
**Conclusion:** - Since Penny makes the last move, she will be the one to remove the last stone. - Hence, Penny wins the game if both players play perfectly. Thus, the final answer is: Penny Problem From Math Odyssey The school offers 10 elective courses, and each student can enroll in any number of courses. The director selects k students, where although...
work page 2010
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[9]
- E is the foot of the altitude from B to AC
Identify Key Points and Properties: - D is the foot of the altitude from A to BC. - E is the foot of the altitude from B to AC. - F is the foot of the altitude from C to AB. - H is the orthocenter of △ABC. - G is the intersection of the line through D parallel to AB and the line EF
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[10]
- The points D, E, and F are collinear with H on the respective altitudes
Use the Property of the Orthocenter: - The orthocenter H lies on the altitudes AD, BF , and CE . - The points D, E, and F are collinear with H on the respective altitudes
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[11]
Analyze the Parallel Line: - Since DG ∥ AB, ∠ADG = ∠DAB (corresponding angles)
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[12]
- Therefore, ∠EHF = 180 ◦ − ∠EAF
Consider the Cyclic Quadrilateral: - The quadrilateral AEHF is cyclic because ∠AEH = ∠AF H = 90 ◦. - Therefore, ∠EHF = 180 ◦ − ∠EAF
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[13]
Use the Parallel Line Property Again: - Since DG ∥ AB, ∠DGE = ∠BAE (alternate interior angles)
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[14]
Relate Angles in Cyclic Quadrilateral: - In △AEF , ∠EAF = ∠EHF because they are opposite angles in the cyclic quadrilateral AEHF
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[15]
- ∠BAE = ∠EAF (as established)
Determine ∠CGH : - Since G lies on EF and DG ∥ AB, ∠DGE = ∠BAE. - ∠BAE = ∠EAF (as established). - Therefore, ∠DGE = ∠EAF
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[16]
Final Angle Calculation: - Since ∠EAF = ∠EHF and ∠EHF = 90 ◦ (as H is the orthocenter and E and F are feet of perpendiculars), we have: ∠CGH = ∠EHF = 90 ◦. Thus, the angle ∠CGH is 90◦ . B P ROMPTS USED IN THE EVALUATION Fig 5 to Fig 10 show the prompts used in evaluating the base models. Fig 11 to Fig 14 show the prompts used in evaluating the instruct mo...
work page 2004
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[17]
如果$\alpha, \beta, \gamma$ 均小于$60^\circ$，那么他们的正弦值都小于$\frac{1}{2}$，因此三个值中 不可能有大于$\frac{1}{2}$ 的值。 \newline2. 如果有一个角大于$60^\circ$，假设为$\alpha$，那么对应 的正弦值大于$\frac{1}{2}$。此时，由于三角形内角和为$180^\circ$，所以$\beta + \gamma < 120^\circ$。 这意味着$\beta, \gamma$ 的余弦值均大于$\frac{1}{2}$，所以此时$\sin \alpha \cos \beta > \frac{1}{2}, \sin \beta \cos \gamma > \frac{1}{2}$。 \newline3. 如果有两...
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如果三个角都大于$60^\circ$，显然不符合题意。 \newline综上所述，当有一个角大于$60^\circ$ 时， 大于$\frac{1}{2}$ 的个数的最大值是2。 答案是C 正方体$A B C D-A_{1} B_{1} C_{1} D_{1}$ 中, $B B_{1}$ 与平面$A C D_{1}$ 所成角的余弦值为( ) 从以下选项中选择: :\newline(A) $\frac{\sqrt{2}}{3}$ :\newline(B) $\frac{\sqrt{3}}{3}$ :\newline(C) $\frac{2}{3}$ :\newline(D) $\frac{\sqrt{6}}{3}$ 设上下底面的中心分别为$\mathrm{O}_{1}, \mathrm{O}$, 设正方体的棱...
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