arxiv: 2506.14245 · v2 · submitted 2025-06-17 · 💻 cs.AI · cs.CL

Recognition: 2 theorem links

· Lean Theorem

Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs

Xumeng Wen , Zihan Liu , Shun Zheng , Shengyu Ye , Zhirong Wu , Yang Wang , Zhijian Xu , Xiao Liang

show 4 more authors

Junjie Li Ziming Miao Jiang Bian Mao Yang

Authors on Pith no claims yet

Pith reviewed 2026-05-13 12:40 UTC · model grok-4.3

classification 💻 cs.AI cs.CL

keywords Reinforcement LearningVerifiable RewardsChain-of-Thought ReasoningLLM ReasoningMathematical ReasoningCoding TasksPolicy OptimizationEvaluation Metrics

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The pith

RL with verifiable rewards extends reasoning boundaries in base LLMs by incentivizing correct step-by-step thinking even when only final answers are rewarded.

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

This paper examines the effect of reinforcement learning with verifiable rewards on the reasoning capabilities of large language models. It finds that RLVR extends the reasoning boundary on both mathematical and coding tasks, as shown through experiments and a new CoT-Pass@K metric that tracks both final answers and intermediate reasoning steps. A theoretical framework demonstrates how the method encourages correct reasoning processes despite rewards depending solely on answer correctness. Training dynamics reveal that correct reasoning is incentivized early, with evaluations confirming substantial gains in reasoning quality.

Core claim

RLVR extends the reasoning boundary for mathematical and coding tasks while encouraging correct reasoning even when rewards are based solely on answer correctness, as demonstrated by improvements captured in the CoT-Pass@K metric and explained through a theoretical incentive framework.

What carries the argument

The RLVR incentive mechanism under Group Relative Policy Optimization, which uses verifiable final-answer rewards to drive policy updates that favor accurate intermediate reasoning chains.

If this is right

RLVR improves reasoning quality early in the training process for both math and coding.
Correct intermediate reasoning emerges without explicit process-level rewards.
The approach works when rewards verify only final-answer correctness.
Reasoning boundaries extend measurably on standard benchmarks via the CoT-Pass@K evaluation.

Where Pith is reading between the lines

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

Similar dynamics may appear in other domains with verifiable outcomes such as theorem proving or structured planning.
The method could lower reliance on human-annotated reasoning traces for training.
Design of future reward functions might prioritize easy-to-verify signals over detailed process supervision.

Load-bearing premise

That observed gains in CoT-Pass@K reflect genuine improvements in reasoning ability rather than changes in sampling behavior or post-hoc metric design.

What would settle it

A controlled test fixing the number of samples and sampling parameters across RLVR-trained and baseline models, then checking whether CoT-Pass@K scores remain higher for the RLVR models.

read the original abstract

Recent advancements in long chain-of-thought (CoT) reasoning, particularly through the Group Relative Policy Optimization algorithm used by DeepSeek-R1, have led to significant interest in the potential of Reinforcement Learning with Verifiable Rewards (RLVR) for Large Language Models (LLMs). While RLVR promises to improve reasoning by allowing models to learn from free exploration, there remains debate over whether it truly enhances reasoning abilities or simply boosts sampling efficiency. This paper systematically investigates the impact of RLVR on LLM reasoning. We revisit Pass@K experiments and demonstrate that RLVR can extend the reasoning boundary for both mathematical and coding tasks. This is supported by our introduction of a novel evaluation metric, CoT-Pass@K, which captures reasoning success by accounting for both the final answer and intermediate reasoning steps. Furthermore, we present a theoretical framework explaining RLVR's incentive mechanism, demonstrating how it can encourage correct reasoning even when rewards are based solely on answer correctness. Our analysis of RLVR's training dynamics reveals that it incentivizes correct reasoning early in the process, with substantial improvements in reasoning quality confirmed through extensive evaluations. These findings provide strong evidence of RLVR's potential to enhance LLM reasoning, offering valuable insights into its mechanisms and performance improvements.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

RLVR extends reasoning boundaries on math and coding per their CoT-Pass@K, but the gains could still come from sampling shifts rather than deeper capability.

read the letter

The main point is that this paper finds RLVR can push performance on harder math and coding instances, with their new CoT-Pass@K metric showing gains in both final answers and intermediate reasoning steps. They also sketch a theory for why answer-only rewards create an implicit push toward correct chains during policy gradient updates. The experiments track training dynamics and show the incentive appearing early, which directly engages the debate about whether RLVR improves reasoning or just sampling efficiency. That part is concrete and moves past earlier RLVR results like DeepSeek-R1 by trying to measure reasoning quality itself. The soft spot is that the theory stays high-level and does not clearly derive a positive gradient on reasoning tokens after controlling for length or format biases. Without that or an ablation that holds the output distribution fixed, the lifts remain compatible with a pure sampling-efficiency story where the model simply learns to emit longer or more structured chains. This work is aimed at people doing RL post-training for LLM reasoning. The metric and dynamics analysis are worth attention even if the central claim needs tighter controls. It deserves peer review because the question is live and they bring new tools to it, though revisions should address the alternative explanations.

Referee Report

2 major / 2 minor

Summary. The paper claims that Reinforcement Learning with Verifiable Rewards (RLVR) extends the reasoning boundary of base LLMs on mathematical and coding tasks. It introduces the CoT-Pass@K metric to evaluate both final-answer correctness and intermediate reasoning steps, reports empirical gains under RLVR training, and presents a theoretical framework arguing that policy-gradient dynamics implicitly incentivize correct reasoning even when the reward depends only on the final answer. Training-dynamics analysis is said to show that correct reasoning is incentivized early in RLVR.

Significance. If the central theoretical derivation holds and the empirical gains survive controls for sampling distribution shifts, the result would be significant: it would supply a mechanistic account of why RLVR (as used in systems like DeepSeek-R1) can improve reasoning quality rather than merely altering output length or format, with direct implications for reward design in long-CoT training.

major comments (2)

[Theoretical framework] Theoretical framework section: the central claim that RLVR produces a strictly positive expected gradient on reasoning tokens conditional on answer correctness (independent of length or format biases) is load-bearing yet only described at high level. An explicit derivation or inequality showing this incentive is required; absent it, observed CoT-Pass@K lifts remain compatible with a pure sampling-efficiency account.
[Experiments] Experimental results (Pass@K and CoT-Pass@K tables): no ablation is reported that holds the sampling distribution fixed (e.g., temperature, length penalty, or decoding constraints) while measuring reasoning improvement. Without such a control, it is unclear whether gains reflect extended reasoning capability or post-RL changes in output distribution.

minor comments (2)

[Evaluation metrics] Notation for CoT-Pass@K should be defined formally (e.g., as an indicator over both answer and step correctness) rather than described only in prose.
[Training dynamics] Figure captions for training-dynamics plots should state the exact reward threshold and number of samples per prompt used to compute the curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments help clarify how to strengthen the presentation of the theoretical framework and the empirical controls. We address each major comment below and commit to revisions that directly respond to the concerns.

read point-by-point responses

Referee: Theoretical framework section: the central claim that RLVR produces a strictly positive expected gradient on reasoning tokens conditional on answer correctness (independent of length or format biases) is load-bearing yet only described at high level. An explicit derivation or inequality showing this incentive is required; absent it, observed CoT-Pass@K lifts remain compatible with a pure sampling-efficiency account.

Authors: We agree that the current high-level description leaves the central incentive claim open to alternative interpretations. In the revised manuscript we will insert a self-contained derivation (new subsection 3.2 plus supporting inequalities in the appendix) that starts from the policy-gradient objective, conditions on answer correctness, and shows that the expected gradient on reasoning tokens is strictly positive under the verifiable-reward setting, independent of length or format biases. The derivation will explicitly separate the reasoning-token contribution from any sampling-efficiency effects. revision: yes
Referee: Experimental results (Pass@K and CoT-Pass@K tables): no ablation is reported that holds the sampling distribution fixed (e.g., temperature, length penalty, or decoding constraints) while measuring reasoning improvement. Without such a control, it is unclear whether gains reflect extended reasoning capability or post-RL changes in output distribution.

Authors: We acknowledge that the reported tables do not contain an explicit ablation that freezes the sampling distribution. In the revision we will add a controlled experiment (new Table X and accompanying text) that evaluates both the base and RLVR models under identical decoding settings: fixed temperature, identical maximum length, and no length penalty. This will isolate changes in reasoning quality (measured by CoT-Pass@K) from post-training shifts in output distribution. revision: yes

Circularity Check

0 steps flagged

No significant circularity; theoretical claims do not reduce to inputs by construction

full rationale

The paper's central theoretical framework is presented at a high level to explain how RLVR can incentivize correct intermediate reasoning from answer-only rewards. No specific equations, fitted parameters, or self-citations are quoted that would make any prediction equivalent to its inputs by definition. The CoT-Pass@K metric is introduced as a new evaluation tool separate from the reward definition, and empirical results on math/coding tasks are reported as independent support. The derivation chain remains self-contained against external benchmarks; no load-bearing step collapses to a renaming, ansatz smuggling, or self-referential fit.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5555 in / 1000 out tokens · 34627 ms · 2026-05-13T12:40:22.879832+00:00 · methodology

discussion (0)

Forward citations

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 23 Pith papers · 2 internal anchors

[1]

Chain-of-thought reasoning in the wild is not always faithful.arXiv preprint arXiv: 2503.08679, 2025

[Released 23-05-2025]. Iv´an Arcuschin, Jett Janiak, Robert Krzyzanowski, Senthooran Rajamanoharan, Neel Nanda, and Arthur Conmy. Chain-of-thought reasoning in the wild is not always faithful.arXiv preprint arXiv:2503.08679, 2025. Shiyi Cao, Sumanth Hegde, Dacheng Li, Tyler Griggs, Shu Liu, Eric Tang, Jiayi Pan, Xingyao Wang, Akshay Malik, Graham Neubig, ...

work page arXiv 2025
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DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning

[Released 19-02-2025]. Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et al. DeepSeek-R1: Incentivizing reasoning capability in LLMs via reinforcement learning.arXiv preprint arXiv:2501.12948, 2025. Jujie He, Jiacai Liu, Chris Yuhao Liu, Rui Yan, Chaojie Wang, Peng Cheng, Xiaoyu Zhang...

work page internal anchor Pith review Pith/arXiv arXiv 2025
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Qwen2.5 Technical Report

[Released 10-06-2025]. Team OpenAI. Learning to reason with LLMs.https://openai.com/index/ learning-to-reason-with-llms/, 2024. [Released 12-09-2024]. Team OpenAI. Introducing OpenAI o3 and o4-mini.https://openai.com/index/ introducing-o3-and-o4-mini/, 2025. [Released 16-04-2025]. Team Qwen. Qwen2.5 technical report.arXiv preprint arXiv:2412.15115, 2024. ...

work page internal anchor Pith review Pith/arXiv arXiv 2025
[4]

**Analyze the solution step-by-step**, paying close attention to: - Computational accu- racy - Logical consistency - Conceptual understanding - Whether the reasoning is valid and complete

work page
[5]

Clas- sify them into the following categories (if applicable): - **Calculation Error**: Mistakes in arithmetic, algebraic manipulation, or numerical computation

**Identify any issues or errors in the reasoning**, even if the final answer is correct. Clas- sify them into the following categories (if applicable): - **Calculation Error**: Mistakes in arithmetic, algebraic manipulation, or numerical computation. - **Logical Error**: Invalid reasoning, flawed logic, or incorrect inference. - **Conceptual Error**: Misu...

work page
[6]

unideal case

**Provide a final judgment** on whether the solution is logically sound and free of errors in reasoning. Please format your response as follows: — **Issues Identified:** - [Issue 1]: [Classification] - [Brief explanation] - [Issue 2]: [Classification] - [Brief expla- nation] - ... Let’s think step by step and output your final judgment within\boxed{} \box...

work page 2025
[7]

Choosing 2 lines from the 12 sides: There are 12 choose 2 ways to do this, which is equal to 66. DeepSeek-R1-0528-Qwen3-8B verify: the method of counting two pairs of parallel lines assumes that any such choice forms a rectangle, which is not necessarily true as the lines may not intersect properly to form a rectangle

work page
[8]

Choosing 2 lines from the 12 diagonals that connect every other vertex: There are 12 choose 2 ways to do this, which is equal to 66

work page
[9]

Choosing 2 lines from the 12 diagonals that connect every third vertex: There are 12 choose 2 ways to do this, which is equal to 66

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[10]

Choosing 1 line from the sides and 1 line from the diagonals that connect every other vertex: There are 12 * 12 ways to do this, which is equal to 144

work page
[11]

Choosing 1 line from the sides and 1 line from the diagonals that connect every third vertex: There are 12 * 12 ways to do this, which is equal to 144

work page
[12]

Choosing 1 line from the diagonals that connect every other vertex and 1 line from the diagonals that connect every third vertex: There are 12 * 12 ways to do this, which is equal to 144. DeepSeek-R1-0528-Qwen3-8B verify: **Logical Error**: The solution only con- siders three types of lines (sides, diagonals connecting every other vertex, and diag- onals ...

work page
[13]

This distance can be written in the form m√n p , wherem,n, andpare positive integers,mandpare relatively prime, andn is not divisible by the square of any prime

There exists a pointIinside the tetrahedron such that the distances fromIto each of the faces of the tetrahedron are all equal. This distance can be written in the form m√n p , wherem,n, andpare positive integers,mandpare relatively prime, andn is not divisible by the square of any prime. Findm+n+p. Qwen2.5-32B Answer To find the distance from the pointIt...

work page 2025
[14]

So, the area of triangleAF Mis 1 2 ·AF·height= 1 2 ·13· 72 17 = 468 17 . DeepSeek-R1-0528-Qwen3-8B verify: - - **Conceptual Error**: The solution in- correctly assumes that the height of triangle ADE is derived from the area of quadri- lateral DEGF using DE + EG, but EG is not defined in the problem. Next, let’s find the area of triangleN CE. SinceNis the...

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[15]

DeepSeek-R1-0528-Qwen3-8B verify: The area calculation for triangle MNE uses DE + EG as a base, which is not a valid base unless DE and EG are collinear

So, the area of triangleM N Eis 1 2 ·DE+EG·height= 1 2 ·68· 72 17 = 144. DeepSeek-R1-0528-Qwen3-8B verify: The area calculation for triangle MNE uses DE + EG as a base, which is not a valid base unless DE and EG are collinear. Now, we can find the area of the heptagonAF N BCEMby adding the areas of the triangles AF M,N CE,M N E, and the quadrilateralDEGF:...

work page 2023
[16]

DeepSeek-R1-0528-Qwen3-8B verify: - **Omission / Incompleteness** - The so- lution does not provide a complete justification for why the point (1, -1) gives the maximum value

Therefore, the maximum value of the expression is 1 4 . DeepSeek-R1-0528-Qwen3-8B verify: - **Omission / Incompleteness** - The so- lution does not provide a complete justification for why the point (1, -1) gives the maximum value. It only evaluates one specific point and claims it is the maximum without comparing it to other points or demonstrating that ...

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