2502.12853v1

# S2r: Teaching LLMs to Self-verify and Self-correct via Reinforcement Learning > Equal contribution. This work was done during Peisong, Cheng, Jiaqi and Bang were interning at Tencent.Corresponding authors. Abstract Recent studies have demonstrated the effectiveness of LLM test-time scaling. However, existing approaches to incentivize LLMs’ deep thinking abilities generally require large-scale data or significant training efforts. Meanwhile, it remains unclear how to improve the thinking abilities of less powerful base models. In this work, we introduce S 2 r, an efficient framework that enhances LLM reasoning by teaching models to self-verify and self-correct during inference. Specifically, we first initialize LLMs with iterative self-verification and self-correction behaviors through supervised fine-tuning on carefully curated data. The self-verification and self-correction skills are then further strengthened by both outcome-level and process-level reinforcement learning, with minimized resource requirements, enabling the model to adaptively refine its reasoning process during inference. Our results demonstrate that, with only 3.1k self-verifying and self-correcting behavior initialization samples, Qwen2.5-math-7B achieves an accuracy improvement from 51.0% to 81.6%, outperforming models trained on an equivalent amount of long-CoT distilled data. Extensive experiments and analysis based on three base models across both in-domain and out-of-domain benchmarks validate the effectiveness of S 2 r. Our code and data are available at https://github.com/NineAbyss/S2R. S 2 r: Teaching LLMs to Self-verify and Self-correct via Reinforcement Learning Ruotian Ma 1 thanks: Equal contribution. This work was done during Peisong, Cheng, Jiaqi and Bang were interning at Tencent., Peisong Wang 2 footnotemark: , Cheng Liu 1, Xingyan Liu 1, Jiaqi Chen 3, Bang Zhang 1, Xin Zhou 4, Nan Du 1 thanks: Corresponding authors. , Jia Li 5 To ensure a fair comparison, we report the Pass@1 (greedy) accuracy obtained without the process preference model of rStar, rather than the result obtained with increased test-time computation using 64 trajectories. 1 Tencent 2 Tsinghua University 3 The University of Hong Kong 4 Fudan University 5 The Hong Kong University of Science and Technology (Guangzhou) ruotianma@tencent.com, wps22@mails.tsinghua.edu.cn 1 Introduction Recent advancements in Large Language Models (LLMs) have demonstrated a paradigm shift from scaling up training-time efforts to test-time compute Snell et al. (2024a); Kumar et al. (2024); Qi et al. (2024); Yang et al. (2024). The effectiveness of scaling test-time compute is illustrated by OpenAI o1 OpenAI (2024), which shows strong reasoning abilities by performing deep and thorough thinking, incorporating essential skills like self-checking, self-verifying, self-correcting and self-exploring during the model’s reasoning process. This paradigm not only enhances reasoning in domains like mathematics and science but also offers new insights into improving the generalizability, helpfulness and safety of LLMs across various general tasks OpenAI (2024); Guo et al. (2025). <details> <summary>x1.png Details</summary>  ### Visual Description ## Scatter Plot: MATH500 ### Overview The image is a scatter plot comparing the accuracy (%) of different language models on the MATH500 dataset against their data size (log10). Each model is represented by a colored dot, with its name displayed next to it. The plot shows the relationship between model size and performance. ### Components/Axes * **Title:** MATH500 * **X-axis:** Data Size (log10), with ticks at 3, 4, 5, 6, 7, and 8. * **Y-axis:** Accuracy (%), with ticks at 76, 78, 80, 82, 84, and 86. * **Grid:** The plot has a light gray grid. * **Data Points (and their approximate coordinates):** * Qwen2.5-Math-7B-S²R-ORL (ours) (Green): Approximately (4, 84.5) * Qwen2.5-Math-7B-Instruct (Pink): Approximately (6.3, 83) * Eurus-2-7B-PRIME (Orange): Approximately (5.5, 79.5) * rStar-Math-7B (Blue): Approximately (7, 78.5) * Qwen2.5-7B-SimpleRL-Zero (Indigo): Approximately (3.8, 77.2) ### Detailed Analysis or ### Content Details The scatter plot displays the performance of five different language models. The x-axis represents the logarithm of the data size used to train the models, while the y-axis represents the accuracy achieved on the MATH500 dataset. * **Qwen2.5-Math-7B-S²R-ORL (ours):** Located at approximately (4, 84.5), this model has the highest accuracy among the models shown. * **Qwen2.5-Math-7B-Instruct:** Located at approximately (6.3, 83), this model has a relatively high accuracy. * **Eurus-2-7B-PRIME:** Located at approximately (5.5, 79.5), this model has a mid-range accuracy. * **rStar-Math-7B:** Located at approximately (7, 78.5), this model has a lower accuracy compared to the others. * **Qwen2.5-7B-SimpleRL-Zero:** Located at approximately (3.8, 77.2), this model has the lowest accuracy and smallest data size. ### Key Observations * The Qwen2.5-Math-7B-S²R-ORL model achieves the highest accuracy with a relatively smaller data size compared to other models. * The rStar-Math-7B model has a larger data size but lower accuracy compared to the Qwen2.5-Math-7B-Instruct model. * There is no clear linear correlation between data size and accuracy across all models. ### Interpretation The scatter plot suggests that model architecture and training methods play a significant role in achieving high accuracy, in addition to the size of the training data. The Qwen2.5-Math-7B-S²R-ORL model demonstrates that high accuracy can be achieved with a smaller data size, possibly due to a more efficient architecture or training process. The plot highlights the importance of factors beyond just data size in determining the performance of language models on the MATH500 dataset. </details> Figure 1: The data efficiency of S 2 r compared to competitive methods, with all models initialized from Qwen2.5-Math-7B. Recent studies have made various attempts to replicate the success of o1. These efforts include using large-scale Monte Carlo Tree Search (MCTS) to construct long-chain-of-thought (long-CoT) training data, or to scale test-time reasoning to improve the performance of current models Guan et al. (2025); Zhao et al. (2024); Snell et al. (2024b); constructing high-quality long-CoT data for effective behavior cloning with substantial human effort Qin et al. (2024); and exploring reinforcement learning to enhance LLM thinking abilities on large-scale training data and models Guo et al. (2025); Team et al. (2025); Cui et al. (2025); Yuan et al. (2024). Recently, DeepSeek R1 Guo et al. (2025) demonstrated that large-scale reinforcement learning can incentivize LLM’s deep thinking abilities, with the R1 series showcasing the promising potential of long-thought reasoning. However, these approaches generally requires significant resources to enhance LLMs’ thinking abilities, including large datasets, substantial training-time compute, and considerable human effort and time costs. Meanwhile, it remains unclear how to incentivize valid thinking in smaller or less powerful LLMs beyond distilling knowledge from more powerful models. In this work, we propose S 2 r, an efficient alternative to enhance the thinking abilities of LLMs, particularly for smaller or less powerful LLMs. Instead of having LLMs imitate the thinking process of larger, more powerful models, S 2 r focus on teaching LLMs to think deeply by iteratively adopting two critical thinking skills: self-verifying and self-correcting. By acquiring these two capabilities, LLMs can continuously reassess their solutions, identify mistakes during solution exploration, and refine previous solutions after self-checking. Such a paradigm also enables flexible allocation of test-time compute to different levels of problems. Our results show that, with only 3.1k training samples, Qwen2.5-math-7B significantly benefits from learning self-verifying and self-correcting behaviors, achieving a 51.0% to 81.6% accuracy improvement on the Math500 test set. This performance outperforms the same base model distilled from an equivalent amount of long-CoT data (accuracy 80.2%) from QwQ-32B-Preview Team (2024a). More importantly, S 2 r employs both outcome-level and process-level reinforcement learning (RL) to further enhance the LLMs’ self-verifying and self-correcting capabilities. Using only rule-based reward models, RL improves the validity of both the self-verification and self-correction process, allowing the models to perform more flexible and effective test-time scaling through a self-directed trial-and-error process. By comparing outcome-level and process-level RL for our task, we found that process-level supervision is particularly effective in boosting accuracy of the thinking skills at intermediate steps, which might benefit base models with limited reasoning abilities. In contrast, outcome-level supervision enables models explore more flexible trial-and-error paths towards the correct final answer, leading to consistent improvement in the reasoning abilities of more capable base models. Additionally, we further show the potential of offline reinforcement learning as a more efficient alternative to the online RL training. We conducted extensive experiments across 3 LLMs on 7 math reasoning benchmarks. Experimental results demonstrate that S 2 r outperforms competitive baselines in math reasoning, including recently-released advanced o1-like models Eurus-2-7B-PRIME Cui et al. (2025), rStar-Math-7B Guan et al. (2025) and Qwen2.5-7B-SimpleRL Zeng et al. (2025). We also found that S 2 r is generalizable to out-of-domain general tasks, such as MMLU-PRO, highlighting the validity of the learned self-verifying and self-correcting abilities. Additionally, we conducted a series of analytical experiments to better demonstrate the reasoning mechanisms of the obtained models, and provide insights into performing online and offline RL training for enhancing LLM reasoning. 2 Methodology The main idea behind teaching LLMs self-verification and self-correction abilities is to streamline deep thinking into a critical paradigm: self-directed trial-and-error with self-verification and self-correction. Specifically: (1) LLMs are allowed to explore any potential (though possibly incorrect) solutions, especially when tackling difficult problems; (2) during the process, self-verification is essential for detecting mistakes on-the-fly; (3) self-correction enables the model to fix detected mistakes. This paradigm forms an effective test-time scaling approach that is more accessible for less powerful base models and is generalizable across various tasks. In this section, we first formally define the problem (§ 2.1). Next, we present the two-stage training framework of S 2 r, as described in Figure 2: Stage 1: Behavior Initialization: We first construct dynamic self-verifying and self-correcting trial-and-error trajectories to initialize the desired behavior. Then, we apply supervised fine-tuning (SFT) to the initial policy models using these trajectories, resulting in behavior-initialized policy models (§ 2.2); Stage 2: Reinforcement Learning: Following behavior initialization, we employ reinforcement learning to further enhance the self-verifying and self-correcting capabilities of the policy models. We explore both outcome-level and process-level RL methods, as well as their offline versions (§ 2.3). <details> <summary>x2.png Details</summary>  ### Visual Description ## Data Construction and Reinforcement Learning Diagram ### Overview The image illustrates a multi-stage process involving data construction, behavior initialization, and reinforcement learning. It details how a model learns to solve problems through verification and trajectory construction based on difficulty levels. ### Components/Axes * **Stages:** The diagram is divided into three stages: * Stage 0: Data Construction * Stage 1: Behavior Initialization * Stage 2: Reinforcement Learning * **Verification Methods:** Two verification methods are presented: * "Problem-Solving" Verification (left side, outlined in orange) * "Confirmative" Verification (right side, outlined in green) * **Legend:** Located in the bottom-left corner: * Green circle: Correct Response * Red circle: Incorrect Response * **Difficulty Levels:** Three difficulty levels are shown: * Difficulty Level 1 * Difficulty Level 3 * Difficulty Level 5 * **Models:** * Initial Policy Model π₀ * SFT Model π_SFT ### Detailed Analysis or ### Content Details **Stage 0: Data Construction** * **Input Questions:** Input questions are fed into an "Initial Policy" block. * **Sample Responses:** The "Initial Policy" block outputs a set of responses, represented by green (correct) and red (incorrect) circles. * The first row of responses contains 3 red and 3 green circles. * The second row of responses contains 4 green and 2 red circles. * The third row of responses contains 5 green and 1 red circles. * **Sample K responses for each question:** Text below the responses. * **Difficulty Levels:** Trajectories are constructed based on difficulty distribution. * Difficulty Level 1: r = {s1, v1} * Difficulty Level 3: r = {s1, v1, s2, v2} * Difficulty Level 5: r = {s1, v1, s2, v2, s3, v3, s4, v4} * The variables s1, s2, s3, s4 are black. * The variables v1, v2, v3, v4 are teal. * The incorrect responses are red. **Stage 1: Behavior Initialization** * **Initial Policy Model:** An "Initial Policy Model π₀" block receives "Input question x". * **Supervised Fine-tuning:** The model undergoes supervised fine-tuning. * **Target Output:** The target output is defined as r = {s1, v1, s2, v2, s3, v3}. * The variables s1, s2, s3 are black. * The variables v1, v2, v3 are teal. * The incorrect responses are red. * **SFT Mask:** An SFT Mask m = {0, 1, 0, 1, 1, 1} is applied. **Stage 2: Reinforcement Learning** * **SFT Model:** An "SFT Model π_SFT" block receives "Input Questions". * **Outcome-level reward:** The model receives outcome-level rewards (checkmark for correct, X for incorrect). * **Process-level reward:** The model receives process-level rewards (checkmarks and X's). * The sequence of rewards is: checkmark, X, checkmark, checkmark, checkmark, X. * **Outputs:** The SFT Model outputs two sets of data: * {s1, v1, ..., sj, vj} * {s1, v1, s2, v2, s3, v3} **Verification Sections** * **"Problem-Solving" Verification:** * Problem: "27 increased by twice a number is 39. What is the number?" * Model's answer: 6 * Verification: "Let's denote the unknown number as x. The problem can be written as the equation: 27 + 2x = 39. To find x, we need to solve this equation step by step. Step 1:... Step 2: ... x = 6. The given answer is 6. Therefore, the answer is correct." * **"Confirmative" Verification:** * Problem: "27 increased by twice a number is 39. What is the number?" * Model's answer: 6 * Verification: "To verify the solution, we will substitute the given answer into the original statement to determine its validity. Given answer for the number: 6. We will now check the statement using the given answer: 27 increased by twice 6 should equal 39. Twice 6 is 12. 27 increased by 12 is 27 + 12, which equals 39. Therefore, the answer is correct." ### Key Observations * The diagram illustrates a learning process that starts with initial data construction, moves to behavior initialization, and culminates in reinforcement learning. * The difficulty levels increase the complexity of the trajectories. * The verification methods provide different approaches to validating the model's answers. ### Interpretation The diagram presents a structured approach to training a model to solve problems. It highlights the importance of data construction, initial behavior setup, and reinforcement learning. The use of different verification methods and difficulty levels suggests a comprehensive training strategy. The SFT Mask in Stage 1 likely serves to focus the model's learning on specific aspects of the target output. The overall process aims to improve the model's accuracy and problem-solving abilities through iterative learning and feedback. </details> Figure 2: Overview of S 2 r. 2.1 Problem Setup We formulate the desired LLM reasoning paradigm as a sequential decision-making process under a reinforcement learning framework. Given a problem $x$ , the language model policy $\pi$ is expected to generate a sequence of interleaved reasoning actions $y=(a_{1},a_{2},·s,a_{T})$ until reaching the termination action <end>. We represent the series of actions before an action $a_{t}∈ y$ as $y_{:a_{t}}$ , i.e., $y_{:a_{t}}=(a_{1},a_{2},·s,a_{t-i})$ , where $a_{t}$ is excluded. The number of tokens in $y$ is denoted as $|y|$ , and the total number of actions in $y$ is denoted as $|y|_{a}$ . We restrict the action space to three types: “ solve ”, “ verify ”, and “ <end> ”, where “ solve ” actions represent direct attempts to solve the problem, “ verify ” actions correspond to self-assessments of the preceding solution, and “ <end> ” actions signal the completion of the reasoning process. We denote the type of action $a_{i}$ as $Type(·)$ , where $Type(a_{i})∈\{\texttt{verify},\texttt{solve},\texttt{<end>}\}$ . We expect the policy to learn to explore new solutions by generating “ solve ” actions, to self-verify the correctness of preceding solutions with “ verify ” actions, and to correct the detected mistakes with new “ solve ” actions if necessary. Therefore, for each action $a_{i}$ , the type of the next action $a_{i+1}$ is determined by the following rules: $$ Type(a_{i+1})=\begin{cases}\texttt{verify},&Type(a_{i})=\texttt{solve}\\ \texttt{solve},&Type(a_{i})=\texttt{verify}\\ &\text{ and }\text{Parser}(a_{i})=\textsc{incorrect}\\ \texttt{<end>},&Type(a_{i})=\texttt{verify}\\ &\text{ and }\text{Parser}(a_{i})=\textsc{correct}\\ \end{cases} $$ Here, $Parser(a)∈\{\textsc{correct},\textsc{incorrect}\}$ (for any action $a$ where $Type(a)=\texttt{verify}$ ) is a function (e.g., a regex) that converts the model’s free-form verification text into binary judgments. For simplicity, we denote the $j$ -th solve action as $s_{j}$ and the $j$ -th verify action as $v_{j}$ . Then we have $y=(s_{1},v_{1},s_{2},v_{2},·s,s_{k},v_{k},\texttt{<end>})$ . 2.2 Initializing Self-verification and Self-correction Behaviors 2.2.1 Learning Valid Self-verification Learning to perform valid self-verification is the most crucial part in S 2 r, as models can make mistakes during trial-and-error, and recognizing intermediate mistakes is critical for reaching the correct answer. In this work, we explore two methods for constructing self-verification behavior. “Problem-Solving” Verification The most intuitive method for verification construction is to directly query existing models to generate verifications on the policy models’ responses, and then filter for valid verifications. By querying existing models using different prompts, we found that existing models tend to perform verification in a “Problem-Solving” manner, i.e., by re-solving the problem and checking whether the answer matches the given one. We refer to this kind of verification as “Problem-Solving” Verification. “Confirmative” Verification "Problem-solving" verification is intuitively not the ideal verification behavior we seek. Ideally, we expect the model to think outside the box and re-examine the solution from a new perspective, rather than thinking from the same problem-solving view for verification. We refer to this type of verification behavior as “Confirmative” Verification. Specifically, we construct “Confirmative” Verification by prompting existing LLMs to "verify the correctness of the answer without re-solving the problem", and filtering out invalid verifications using LLM-as-a-judge. The detail implementation can be found in Appendix § A.1. 2.2.2 Learning Self-correction Another critical part of S 2 r is enabling the model to learn self-correction. Inspired by Kumar et al. (2024) and Snell et al. (2024b), we initialize the self-correcting behavior by concatenating a series of incorrect solutions (each followed by a verification recognizing the mistakes) with a final correct solution. As demonstrated by Kumar et al. (2024), LLMs typically fail to learn valid self-correction behavior through SFT, but the validity of self-correction can be enhanced through reinforcement learning. Therefore, we only initialize the self-correcting behavior at this stage, leaving further enhancement of the self-correcting capabilities to the RL stage. 2.2.3 Constructing Dynamic Trial-and-Error Trajectory We next construct the complete trial-and-error trajectories for behavior initialization SFT, following three principles: - To ensure the diversity of the trajectories, we construct trajectories of various lengths. Specifically, we cover $k∈\{1,2,3,4\}$ for $y=(a_{1},·s,a_{2k})=(s_{1},v_{1},·s,s_{k},v_{k})$ in the trajectories. - To ensure that the LLMs learn to verify and correct their own errors, we construct the failed trials in each trajectory by sampling and filtering from the LLMs’ own responses. - As a plausible test-time scaling method allocates reasonable effort to varying levels of problems, it is important to ensure the trial-and-error trajectories align with the difficulty level of problems. Specifically, more difficult problems will require more trial-and-error iterations before reaching the correct answer. Thus, we determine the length of each trajectory based on the accuracy of the sampled responses for each base model. 2.2.4 Supervised Fine-tuning for Thinking Behavior Initialization Once the dynamic self-verifying and self-correcting training data $\mathcal{D}_{SFT}$ is ready, we optimize the policy $\pi$ for thinking behavior initialization by minimizing the following objective: $$ \mathcal{L}=-\mathbb{E}_{(x,y)\sim\mathcal{D}_{SFT}}\sum_{a_{t}\in y}\delta_{% mask}(a_{t})\log\pi(a_{t}\mid x,y_{:a_{t}}) \tag{1} $$ where the mask function $\delta_{mask}(a_{t})$ for action $a_{t}$ in $y=(a_{1},·s,a_{T})$ is defined as: $$ \delta_{mask}(a_{t})=\begin{cases}1,&\text{if }Type(a_{t})=\texttt{verify}\\ 1,&\text{if }Type(a_{t})=\texttt{solve}\text{ and }t=T-1\\ 1,&\text{if }Type(a_{t})=\texttt{<end>}\text{ and }t=T\\ 0,&\text{otherwise}\end{cases} $$ That is, we optimize the probability of all verifications and only the last correct solution $s_{N}$ by using masks during training. 2.3 Boosting Thinking Capabilities via Reinforcement Learning After Stage 1, we initialized the policy model $\pi$ with self-verification and self-correction behavior, obtaining $\pi_{SFT}$ . We then explore further enhancing these thinking capabilities of $\pi_{SFT}$ via reinforcement learning. Specifically, we explore two simple RL algorithms: the outcome-level REINFORCE Leave-One-Out (RLOO) algorithm and a proces-level group-based RL algorithm. 2.3.1 Outcome-level RLOO We first introduce the outcome-level REINFORCE Leave-One-Out (RLOO) algorithm Ahmadian et al. (2024); Kool et al. (2019) to further enhance the self-verification and self-correction capabilities of $\pi_{SFT}$ . Given a problem $x$ and the response $y=(s_{1},v_{1},...,s_{T},v_{T})$ , we define the reward function $R_{o}(x,y)$ based on the correctness of the last solution $s_{T}$ : $$ R_{o}(x,y)=\begin{cases}1,&V_{golden}(s_{T})=\texttt{correct}\\ -1,&otherwise\\ \end{cases} $$ Here $V_{golden}(·)∈\{\texttt{correct},\texttt{incorrect}\}$ represents ground-truth validation by matching the golden answer with the given solution. We calculate the advantage of each response $y$ using an estimated baseline and KL reward shaping as follows: $$ A(x,y)=R_{o}(x,y)-\hat{b}-\beta\log\frac{\pi_{\theta_{old}}(y|x)}{\pi_{ref}(y|% x)} \tag{2} $$ where $\beta$ is the KL divergence regularization coefficient, and $\pi_{\text{ref}}$ is the reference policy (in our case, $\pi_{SFT}$ ). $\hat{b}(x,y^{(m)})=\frac{1}{M-1}\sum_{\begin{subarray}{c}j=1,...,M\\ j≠ m\end{subarray}}.R_{o}(x,y^{(j)})$ is the baseline estimation of RLOO, which represents the leave-one-out mean of $M$ sampled outputs $\{y^{(1)},...y^{(M)}\}$ for each input $x$ , serving as a baseline estimation for each $y^{(m)}$ . Then, we optimize the policy $\pi_{\theta}$ by minimizing the following objective after each sampling episode based on $\pi_{\theta_{old}}$ : $$ \begin{split}\mathcal{L}(\theta)\ &=\ -\mathbb{E}_{\begin{subarray}{c}x\sim% \mathcal{D}\\ y\sim\pi_{\theta_{\text{old}}}(\cdot|x)\end{subarray}}\bigg{[}\min\big{(}r(% \theta)A(x,y),\\ &\text{clip}\big{(}r(\theta),1-\epsilon,1+\epsilon\big{)}A(x,y)\big{)}\bigg{]}% \end{split} \tag{3} $$ where $r(\theta)=\frac{\pi_{\theta}(y|x)}{\pi_{\theta_{\text{old}}}(y|x)}$ is the probability ratio. When implementing the above loss function, we treat $y$ as a complete trajectory sampled with an input problem $x$ , meaning we optimize the entire trajectory with outcome-level supervision. With this approach, we aim to incentivize the policy model to explore more dynamic self-verification and self-correcting trajectories on its own, which has been demonstrated as an effective practice in recent work Guo et al. (2025); Team et al. (2025). 2.3.2 Process-level Group-based RL Process-level supervision has demonstrated effectiveness in math reasoning Lightman et al. (2023a); Wang et al. (2024c). Since the trajectory of S 2 r thinking is naturally divided into self-verification and self-correction processes, it is intuitive to adopt process-level supervision for RL training. Inspired by RLOO and process-level GRPO Shao et al. (2024), we designed a group-based process-level optimization method. Specifically, we regard each action $a$ in the output trajectory $y$ as a sub-process and define the action level reward function $R_{a}(a\mid x,y_{:a})$ based on the action type. For each “ solve ” action $s_{j}$ , we expect the policy to generate the correct solution; for each “ verify ” action $v_{j}$ , we expect the verification to align with the actual solution validity. The corresponding rewards are defined as follows: $$ R_{a}(s_{j}\mid x,y_{:s_{j}})=\begin{cases}1,&V_{golden}(s_{j})=\texttt{% correct}\\ -1,&otherwise\\ \end{cases} $$ $$ R_{a}(v_{j}\mid x,y_{:v_{j}})=\begin{cases}1,&Parser(v_{j})=V_{golden}(s_{j})% \\ -1,&otherwise\\ \end{cases} $$ To calculate the advantage of each action $a_{t}$ , we estimate the baseline as the average reward of the group of actions sharing the same reward context: $$ \mathbf{R}(a_{t}\mid x,y)=\left(R_{a}(a_{i}\mid x,y_{:a_{i}})\right)_{i=1}^{t-1} $$ which is defined as the reward sequence of the previous actions $y_{:a_{t}}$ of each action $a_{t}$ . We denote the set of actions sharing the same reward context $\mathbf{R}(a_{t}\mid x,y)$ as $\mathcal{G}(\mathbf{R}(a_{t}\mid x,y))$ . Then the baseline can be estimated as follows: $$ \begin{split}&\hat{b}(a_{t}\mid x,y)=\\ &\frac{1}{|\mathcal{G}(\mathbf{R}(a_{t}|x,y))|}\sum_{a\in\mathcal{G}(\mathbf{R% }(a_{t}|x,y))}R_{a}(a|x^{(a)},y^{(a)}_{:a})\end{split} \tag{4} $$ And the advantage of each action $a_{t}$ is: $$ \begin{split}A(a_{t}\mid x,y)=&R_{a}(a_{t}\mid x,y_{:a_{t}})-\hat{b}(a_{t}\mid x% ,y)\\ &-\beta\log\frac{\pi_{\theta_{old}}(a_{t}\mid x,y)}{\pi_{\text{ref}}(a_{t}\mid x% ,y)}\end{split} \tag{5} $$ The main idea of the group-based baseline estimation is that the actions sharing the same reward context are provided with similar amounts of information before the action is taken. For instance, all actions sharing a reward context consisting of one failed attempt and one successful verification (i.e., $\mathbf{R}(a_{t}|x,y)=(-1,1)$ ) are provided with the information about the problem, a failed attempt, and the reassessment on the failure. Given the same amount of information, it is reasonable to estimate a baseline using the average reward of these actions. Putting it all together, we minimize the following surrogate loss function to update the policy parameters $\theta$ , using trajectories collected from $\pi_{old}$ : $$ \begin{split}\mathcal{L}(\theta)\ &=\ -\mathbb{E}_{\begin{subarray}{c}x\sim% \mathcal{D}\\ y\sim\pi_{\theta_{\text{old}}}(\cdot|x)\end{subarray}}\bigg{[}\frac{1}{|y|_{a}% }\sum_{a\in y}\min\big{(}r_{a}(\theta)A(a|x,y_{:a}),\\ &\text{clip}\big{(}r_{a}(\theta),1-\epsilon,1+\epsilon\big{)}A(a|x,y_{:a})\big% {)}\bigg{]}\end{split} \tag{6} $$ where $r_{a}(\theta)=\frac{\pi_{\theta}(a|x,y_{:a})}{\pi_{\theta_{\text{old}}}(a|x,y_% {:a})}$ is the importance ratio. 2.4 More Efficient Training with Offline RL While online RL is known for its high resource requirements, offline RL, which does not require real-time sampling during training, offers a more efficient alternative for RL training. Additionally, offline sampling allows for more accurate baseline calculations with better trajectories grouping for each policy. As part of our exploration into more efficient RL training in S 2 r framework, we also experimented with offline RL to assess its potential in further enhancing the models’ thinking abilities. In Appendix § D.2, we include more details and formal definition for offline RL training. 3 Experiment To verify the effectiveness of the proposed method, we conducted extensive experiments across 3 different base policy models on various benchmarks. | Stage 1: Behavior Initialization | | | | --- | --- | --- | | Base Model | Source | # Training Data | | Llama-3.1-8B-Instruct | MATH | 4614 | | Qwen2-7B-Instruct | MATH | 4366 | | Qwen2.5-Math-7B | MATH | 3111 | | Stage 2: Reinforcement Learning | | | | Base Model | Source | # Training Data | | Llama-3.1-8B-Instruct | MATH+GSM8K | 9601 | | Qwen2-7B-Instruct | MATH+GSM8K | 9601 | | Qwen2.5-Math-7B | MATH+OpenMath2.0 | 10000 | Table 1: Training data statistics. Table 2: The performance of S 2 r and other strong baselines on the most challenging math benchmarks is presented. BI refers to the behavior-initialized models through supervised fine-tuning, ORL denotes models trained with outcome-level RL, and PRL refers to models trained with process-level RL. The highest results are highlighted in bold and the second-best results are marked with underline. For some baselines, we use the results from their original reports or from Guan et al. (2025), denoted by ∗. 3.1 Experiment Setup Base Models To evaluate the general applicability of our method across different LLMs, we conducted experiments using three distinct base models: Llama-3.1-8B-Instruct Dubey et al. (2024), Qwen2-7B-Instruct qwe (2024), and Qwen2.5-Math-7B Qwen (2024). Llama-3.1-8B-Instruct and Qwen2-7B-Instruct are versatile general-purpose models trained on diverse domains without a specialized focus on mathematical reasoning. In contrast, Qwen2.5-Math-7B is a state-of-the-art model specifically tailored for mathematical problem-solving and has been widely adopted in recent research on math reasoning Guan et al. (2025); Cui et al. (2025); Zeng et al. (2025). Training Data Setup For Stage 1: Behavior Initialization, we used the widely adopted MATH Hendrycks et al. (2021a) training set for dynamic trial-and-error data collection We use the MATH split from Lightman et al. (2023a), i.e., 12000 problems for training and 500 problems for testing.. For each base model, we sampled 5 responses per problem in the training data. After data filtering and sampling, we constructed a dynamic trial-and-error training set consisting of 3k-4k instances for each base model. Detailed statistics of the training set are shown in Table 1. For Stage 2: Reinforcement Learning, we used the MATH+GSM8K Cobbe et al. (2021a) training data for RL training on the policy $\pi_{SFT}$ initialized from Llama-3.1-8B-Instruct and Qwen2-7B-Instruct. Since Qwen2.5-math-7b already achieves high accuracy on the GSM8K training data after Stage 1, we additionally include training data randomly sampled from the OpenMath2 dataset Toshniwal et al. (2024). Following Cui et al. (2025), we filter out excessively easy or difficult problems based on each $\pi_{SFT}$ from Stage 1 to enhance the efficiency and stability of RL training, resulting in RL training sets consisting of approximately 10000 instances. Detailed statistics of the final training data can be found in Table 1. Additional details on training data construction can be found in in Appendix § A.1. Baselines We benchmark our proposed method against four categories of strong baselines: - Frontier LLMs includes cutting-edge proprietary models such as GPT-4o, the latest Claude, and OpenAI’s o1-preview and o1-mini. - Top-tier open-source reasoning models covers state-of-the-art open-source models known for their strong reasoning capabilities, including Mathstral-7B-v0.1 Team (2024b), NuminaMath-72B LI et al. (2024), LLaMA3.1-70B-Instruct Dubey et al. (2024), and Qwen2.5-Math-72B-Instruct Yang et al. (2024). - Enhanced models built on Qwen2.5-Math-7B: Given the recent popularity of Qwen2.5-Math-7B as a base policy model, we evaluate S 2 r against three competitive baselines that have demonstrated superior performance based on Qwen2.5-Math-7B: Eurus-2-7B-PRIME Cui et al. (2025), rStar-Math-7B Guan et al. (2025), and Qwen2.5-7B-SimpleRL Zeng et al. (2025). These models serve as direct and strong baseline for our Qwen2.5-Math-7B-based variants. - SFT with different CoT constructions: We also compare with training on competitive types of CoT reasoning, including the original CoT solution in the training datasets, and Long-CoT solutions distilled from QwQ-32B-Preview Team (2024a), a widely adopted open-source o1-like model Chen et al. (2024c); Guan et al. (2025); Zheng et al. (2024). Specifically, to ensure a fair comparison between behavior initialization with long-CoT and S 2 r, we use long-CoT data of the same size as our behavior initialization data. We provide more details on the baseline data construction in Appendix § A.2.3. More details on the baselines are included in Appendix § A.2. Evaluation Datasets We evaluate the proposed method on 7 diverse mathematical benchmarks. To ensure a comprehensive evaluation, in addition to the in-distribution GSM8K Cobbe et al. (2021b) and MATH500 Lightman et al. (2023a) test sets, we include challenging out-of-distribution benchmarks covering various difficulty levels and mathematical domains, including the AIME 2024 competition problems AI-MO (2024a), the AMC 2023 exam AI-MO (2024b), the advanced reasoning tasks from Olympiad Bench He et al. (2024), and college-level problem sets from College Math Tang et al. (2024a). Additionally, we assess performance on real-world standardized tests, the GaoKao (Chinese College Entrance Exam) En 2023 Liao et al. (2024). A detailed description of these datasets is provided in Appendix § B.1. Evaluation Metrics We report Pass@1 accuracy for all baselines. For inference, we employ vLLM Kwon et al. (2023) and develop evaluation scripts based on Qwen Math’s codebase. All evaluations are performed using greedy decoding. Details of the prompts used during inference are provided in Appendix § A.3. All implementation details, including hyperparameter settings, can be found in Appendix § B.2. 3.2 Main Results Table 2 shows the main results of S 2 r compared with baseline methods. We can observe that: (1) S 2 r consistently improves the reasoning abilities of models across all base models. Notably, on Qwen2.5-Math-7B, the proposed method improves the base model by 32.2% on MATH500 and by 34.3% on GSM8K. (2) Generally, S 2 r outperforms the baseline methods derived from the same base models across most benchmarks. Specifically, on Qwen2.5-Math-7B, S 2 r surpasses several recently proposed competitive baselines, such as Eurus-2-7B-PRIME, rStar-Math-7B and Qwen2.5-7B-SimpleRL. While Eurus-2-7B-PRIME and rStar-Math-7B rely on larger training datasets (Figure 1) and require more data construction and reward modeling efforts, S 2 r only needs linear sampling efforts for data construction, 10k RL training data and rule-based reward modeling. These results highlight the efficiency of S 2 r. (3) With the same scale of SFT data, S 2 r also outperforms the long-CoT models distilled from QwQ-32B-Preview, demonstrating that learning to self-verify and self-correct is an effective alternative to long-CoT for test-time scaling in smaller LLMs. Comparing process-level and outcome-level RL, we find that outcome-level RL generally outperforms process-level RL across the three models. This is likely because outcome-level RL allows models to explore trajectories without emphasizing intermediate accuracy, which may benefit enhancing long-thought reasoning in stronger base models like Qwen2.5-Math-7B. In contrast, process-level RL, which provides guidance for each intermediate verification and correction step, may be effective for models with lower initial capabilities, such as Qwen2-7B-Instruct. As shown in Figure 3, process-level RL can notably enhance the verification and correction abilities of Qwen2-7B- S 2 r -BI. | Model | FOLIO | CRUX- Eval | Strategy- QA | MMLUPro- STEM | | --- | --- | --- | --- | --- | | Qwen2.5-Math-72B-Instruct | 69.5 | 68.6 | 94.3 | 66.0 | | Llama-3.1-70B-Instruct ∗ | 65.0 | 59.6 | 88.8 | 61.7 | | OpenMath2-Llama3.1-70B ∗ | 68.5 | 35.1 | 95.6 | 55.0 | | QwQ-32B-Preview ∗ | 84.2 | 65.2 | 88.2 | 71.9 | | Eurus-2-7B-PRIME | 56.7 | 50.0 | 79.0 | 53.7 | | Qwen2.5-Math-7B-Instruct | 61.6 | 28.0 | 81.2 | 44.7 | | Qwen2.5-Math-7B | 37.9 | 40.8 | 61.1 | 46.0 | | Qwen2.5-Math-7B- S 2 r -BI (ours) | 58.1 | 48.0 | 88.7 | 49.8 | | Qwen2.5-Math-7B- S 2 r -ORL (ours) | 61.6 | 50.9 | 90.8 | 50.0 | Table 3: Performance of the proposed method and the baseline methods on 4 cross-domain tasks. The results with ∗ are reported by Shen et al. (2025). 3.3 Generalizing to Cross-domain Tasks Despite training on math reasoning tasks, we found that the learned self-verifying and self-correcting capability can also generalize to out-of-distribution general domains. In Table 3, we evaluate the SFT model and the outcome-level RL model based on Qwen2.5-Math-7B on four cross-domain tasks: FOLIO Han et al. (2022) on logical reasoning, CRUXEval Gu et al. (2024) on code reasoning, StrategyQA Geva et al. (2021) on multi-hop reasoning and MMLUPro-STEM on multi-task complex understanding Wang et al. (2024d); Shen et al. (2025), with details of these datasets provided in Appendix § B.1. The results show that after learning to self-verify and self-correct, the proposed method effectively boosts the base model’s performance across all tasks and achieves comparative results to the baseline models. These findings indicate that the learned self-verifying and self-correcting capabilities are general thinking skills, which can also benefit reasoning in general domains. Additionally, we expect that the performance in specific domains can be further improved by applying S 2 r training on domain data with minimal reward model requirements (e.g., rule-based or LLM-as-a-judge). For better illustration, we show cases on how the trained models perform self-verifying and self-correcting on general tasks in Appendix § E. 3.4 Analyzing Self-verification and Self-correction Abilities In this section, we conduct analytical experiments on the models’ self-verification and self-correction capabilities from various perspectives. 3.4.1 Problem-solving v.s. Confirmative Verification We first compare the Problem-solving and Confirmative Verification methods described in § 2.2.1. In Table 4, we present the verification results of different methods on the Math500 test set. We report the overall verification accuracy, as well as the initial verification accuracy when the initial answer is correct ( $V_{golden}(s_{0})=\texttt{correct}$ ) and incorrect ( $V_{golden}(s_{0})=\texttt{incorrect}$ ), respectively. | Base Model | Methods | Overall Verification Acc. | Initial Verification Acc. | | | --- | --- | --- | --- | --- | | $V_{golden}(s_{0})$ $=\texttt{correct}$ | $V_{golden}(s_{0})$ $=\texttt{incorrect}$ | | | | | Llama3.1-8B-Instruct | Problem-solving | 80.10 | 87.28 | 66.96 | | Confirmative | 65.67 | 77.27 | 78.22 | | | Qwen2-7B-Instruct | Problem-solving | 73.28 | 90.24 | 67.37 | | Confirmative | 58.31 | 76.16 | 70.05 | | | Qwen2.5-Math-7B | Problem-solving | 77.25 | 91.21 | 56.67 | | Confirmative | 61.58 | 82.80 | 68.04 | | Table 4: Comparison of problem-solving and confirmative verification. We observe from the table that: (1) Generally, problem-solving verification achieves superior overall accuracy compared to confirmative verification. This result is intuitive, as existing models are trained for problem-solving, and recent studies have highlighted the difficulty of existing LLMs in performing reverse thinking Berglund et al. (2023); Chen et al. (2024b). During data collection, we also found that existing models tend to verify through problem-solving, even when prompted to verify without re-solving (see Table 6 in Appendix § A.1). (2) In practice, accuracy alone does not fully reflect the validity of a method. For example, when answer accuracy is sufficiently high, predicting all answers as correct will naturally lead to high verification accuracy, but this is not a desired behavior. By further examining the initial verification accuracy for both correct and incorrect answers, we found that problem-solving verification exhibits a notable bias toward predicting answers as correct, while the predictions from confirmative verification are more balanced. We deduce that this bias arises might be because problem-solving verification is more heavily influenced by the preceding solution, aligning with previous studies showing that LLMs struggle to identify their own errors Huang et al. (2023); Tyen et al. (2023). In contrast, confirmative verification performs verification from different perspectives, making it less influenced by the LLMs’ preceding solution. In all experiments, we used confirmative verification for behavior initialization. 3.4.2 Boosting Self-verifying and Self-correcting with RL In this experiment, we investigate the effect of RL training on the models’ self-verifying and self-correcting capabilities. We assess self-verification using the following metrics: (1) Verification Accuracy: The overall accuracy of verification predictions, as described in § 3.4.1. (2) Error Recall: The recall of verification when the preceding answers are incorrect. (3) Correct Precision: The precision of verification when it predicts the answers as correct. Both Error Recall and Correct Precision directly affect the final answer accuracy: if verification fails to detect an incorrect answer, or if it incorrectly predicts an answer as correct, the final answer will be wrong. For self-correction, we use the following metrics: (1) Incorrect to Correct Rate: the rate at which the model successfully corrects an incorrect initial answer to a correct final answer. (2) Correct to Incorrect Rate: the rate at which the model incorrectly changes a correct initial answer to an incorrect final answer. We provide the formal definitions of the metrics used in Appendix § C. <details> <summary>x3.png Details</summary>  ### Visual Description ## Bar Chart: Evaluation on Verification and Correction (Base Model: Qwen2-7B-Instruct) ### Overview The image presents two bar charts comparing the performance of different models (SFT, SFT + Process-level RL, and SFT + Outcome-level RL) on self-verification and self-correction metrics. The left chart focuses on self-verification, showing Verification Accuracy, Error Recall, and Correct Precision. The right chart focuses on self-correction, showing Incorrect to Correct and Correct to Incorrect ratios. ### Components/Axes **Overall Title:** Evaluation on Verification and Correction (Base Model: Qwen2-7B-Instruct) **Left Chart:** * **Title:** Self-verification Metrics * **Y-axis:** Value (%) * Scale: 50 to 100, incrementing by 10. * **X-axis:** * Verification Accuracy * Error Recall * Correct Precision * **Legend:** Located in the top-left corner. * SFT (Gray) * SFT + Process-level RL (Teal) * SFT + Outcome-level RL (Salmon) **Right Chart:** * **Title:** Self-correction Metrics * **Y-axis:** Value (%) * Scale: 0 to 25, incrementing by 5. * **X-axis:** * Incorrect to Correct * Correct to Incorrect * **Legend:** (Same as left chart, located in the top-left corner of the left chart) * SFT (Gray) * SFT + Process-level RL (Teal) * SFT + Outcome-level RL (Salmon) ### Detailed Analysis **Left Chart (Self-verification Metrics):** * **Verification Accuracy:** * SFT (Gray): 58.31% * SFT + Process-level RL (Teal): 67.86% * SFT + Outcome-level RL (Salmon): 63.93% * Trend: SFT + Process-level RL performs best, followed by SFT + Outcome-level RL, and then SFT. * **Error Recall:** * SFT (Gray): 81.91% * SFT + Process-level RL (Teal): 86.67% * SFT + Outcome-level RL (Salmon): 87.34% * Trend: SFT + Outcome-level RL performs best, closely followed by SFT + Process-level RL, and then SFT. * **Correct Precision:** * SFT (Gray): 65.58% * SFT + Process-level RL (Teal): 73.59% * SFT + Outcome-level RL (Salmon): 69.80% * Trend: SFT + Process-level RL performs best, followed by SFT + Outcome-level RL, and then SFT. **Right Chart (Self-correction Metrics):** * **Incorrect to Correct:** * SFT (Gray): 20.00% * SFT + Process-level RL (Teal): 22.17% * SFT + Outcome-level RL (Salmon): 19.55% * Trend: SFT + Process-level RL performs best, followed by SFT, and then SFT + Outcome-level RL. * **Correct to Incorrect:** * SFT (Gray): 8.42% * SFT + Process-level RL (Teal): 5.39% * SFT + Outcome-level RL (Salmon): 3.93% * Trend: SFT performs worst, followed by SFT + Process-level RL, and then SFT + Outcome-level RL. ### Key Observations * For self-verification metrics, SFT + Process-level RL and SFT + Outcome-level RL generally outperform the base SFT model. * For self-correction metrics, SFT + Process-level RL shows the highest rate of correcting incorrect answers. * SFT + Outcome-level RL has the lowest rate of correct answers becoming incorrect. ### Interpretation The charts suggest that incorporating reinforcement learning (RL), particularly process-level RL, enhances the performance of the Qwen2-7B-Instruct model in both self-verification and self-correction tasks. Process-level RL seems to be more effective at improving the model's ability to correct its mistakes, while outcome-level RL excels at maintaining the correctness of already correct answers. The base SFT model consistently underperforms compared to the RL-enhanced models, indicating the value of RL in improving model reliability and accuracy. </details> (a) <details> <summary>x4.png Details</summary>  ### Visual Description ## Bar Chart: Evaluation on Verification and Correction (Base Model: Qwen2.5-Math-7B) ### Overview The image presents two bar charts comparing the performance of different models (SFT, SFT + Process-level RL, and SFT + Outcome-level RL) on self-verification and self-correction metrics. The left chart focuses on self-verification metrics (Verification Accuracy, Error Recall, Correct Precision), while the right chart focuses on self-correction metrics (Incorrect to Correct, Correct to Incorrect). ### Components/Axes **Left Chart (Self-verification Metrics):** * **Title:** Self-verification Metrics * **X-axis:** Categorical axis with three categories: "Verification Accuracy", "Error Recall", and "Correct Precision". * **Y-axis:** "Value (%)", ranging from 50 to 90 in increments of 10. * **Legend:** Located in the top-left corner. * Gray: SFT * Teal: SFT + Process-level RL * Peach: SFT + Outcome-level RL **Right Chart (Self-correction Metrics):** * **Title:** Self-correction Metrics * **X-axis:** Categorical axis with two categories: "Incorrect to Correct" and "Correct to Incorrect". * **Y-axis:** "Value (%)", ranging from 0 to 14 in increments of 2. * **Legend:** (Same as left chart) * Gray: SFT * Teal: SFT + Process-level RL * Peach: SFT + Outcome-level RL ### Detailed Analysis **Left Chart (Self-verification Metrics):** * **Verification Accuracy:** * SFT (Gray): 61.58% * SFT + Process-level RL (Teal): 74.61% * SFT + Outcome-level RL (Peach): 66.49% * Trend: SFT + Process-level RL performs the best, followed by SFT + Outcome-level RL, and then SFT. * **Error Recall:** * SFT (Gray): 66.83% * SFT + Process-level RL (Teal): 64.75% * SFT + Outcome-level RL (Peach): 70.11% * Trend: SFT + Outcome-level RL performs the best, followed by SFT, and then SFT + Process-level RL. * **Correct Precision:** * SFT (Gray): 84.94% * SFT + Process-level RL (Teal): 90.28% * SFT + Outcome-level RL (Peach): 87.85% * Trend: SFT + Process-level RL performs the best, followed by SFT + Outcome-level RL, and then SFT. **Right Chart (Self-correction Metrics):** * **Incorrect to Correct:** * SFT (Gray): 6.52% * SFT + Process-level RL (Teal): 12.22% * SFT + Outcome-level RL (Peach): 13.64% * Trend: SFT + Outcome-level RL performs the best, followed by SFT + Process-level RL, and then SFT. * **Correct to Incorrect:** * SFT (Gray): 1.96% * SFT + Process-level RL (Teal): 1.46% * SFT + Outcome-level RL (Peach): 0.97% * Trend: SFT performs the worst, followed by SFT + Process-level RL, and then SFT + Outcome-level RL. ### Key Observations * SFT + Process-level RL generally performs better than SFT and SFT + Outcome-level RL in self-verification metrics, especially in Verification Accuracy and Correct Precision. * SFT + Outcome-level RL shows the best performance in Error Recall and Incorrect to Correct metrics. * All models have a low percentage of "Correct to Incorrect" transitions. ### Interpretation The charts suggest that incorporating reinforcement learning (RL) into the SFT model, either at the process level or outcome level, generally improves performance in both self-verification and self-correction tasks. The specific type of RL (process-level vs. outcome-level) seems to have a varying impact depending on the metric being evaluated. SFT + Process-level RL excels in accuracy and precision, while SFT + Outcome-level RL is better at recalling errors and correcting incorrect answers. The low "Correct to Incorrect" values across all models indicate a relatively stable performance in maintaining correct answers. The base SFT model consistently underperforms compared to the RL-enhanced models, highlighting the benefits of incorporating RL techniques for these tasks. </details> (b) Figure 3: Evaluation on verification and correction. In Figure 3, we present the results of the behavior-initialized model (SFT) and different RL models obtained from Qwen2.5-Math-7B. We observe that: (1) Both RL methods effectively enhance self-verification accuracy. The process-level RL shows larger improvement on accuracy, while the outcome-level RL consistently improves Error Recall and Correct Precision. This might be because process-level supervision indiscriminately promotes verification accuracy in intermediate steps, while outcome-level supervision allows the policy model to explore freely in intermediate steps and only boosts the final answer accuracy, thus mainly enhancing Error Recall and Correct Precision (which directly relate to final answer accuracy). (2) Both RL methods can successfully enhance the models’ self-correction capability. Notably, the model’s ability to correct incorrect answers is significantly improved after RL training. The rate of model mistakenly altering correct answers is also notably reduced. This comparison demonstrates that S 2 r can substantially enhance the validity of models’ self-correction ability. <details> <summary>x5.png Details</summary>  ### Visual Description ## Bar Chart: Accuracy and Trial Numbers across Difficulty Level ### Overview The image is a bar chart comparing the accuracy and trial numbers of a model (Llama3.1-8B-Instruct) across different difficulty levels. The chart displays two accuracy metrics (SFT Accuracy and SFT+RL Accuracy) and two trial number metrics (SFT Trials and SFT+RL Trials) for each difficulty level (Level 1 to Level 5). ### Components/Axes * **Title:** Accuracy and Trial Numbers across Difficulty Level (Base Model: Llama3.1-8B-Instruct) * **X-axis:** Difficulty Level (Levels 1, 2, 3, 4, 5) * **Left Y-axis:** Accuracy (ranging from 0.2 to 1.0, with gridlines at intervals of 0.1) * **Right Y-axis:** Trial Numbers (ranging from 0 to 6, with gridlines at intervals of 1) * **Legend:** Located at the top-right of the chart. * SFT Accuracy (light green) * SFT+RL Accuracy (dark green) * SFT Trials (light red) * SFT+RL Trials (dark red) ### Detailed Analysis The chart presents four data series across five difficulty levels. **Accuracy Data Series:** * **SFT Accuracy (light green):** * Level 1: 0.814 * Level 2: 0.733 * Level 3: 0.610 * Level 4: 0.367 * Level 5: 0.239 * Trend: Decreasing accuracy as difficulty level increases. * **SFT+RL Accuracy (dark green):** * Level 1: 0.930 * Level 2: 0.722 * Level 3: 0.638 * Level 4: 0.445 * Level 5: 0.276 * Trend: Decreasing accuracy as difficulty level increases. **Trial Numbers Data Series:** * **SFT Trials (light red):** * Level 1: 3.279 * Level 2: 3.367 * Level 3: 3.924 * Level 4: 5.117 * Level 5: 4.104 * Trend: Increasing trial numbers from Level 1 to Level 4, then a decrease at Level 5. * **SFT+RL Trials (dark red):** * Level 1: 2.209 * Level 2: 2.844 * Level 3: 4.219 * Level 4: 4.234 * Level 5: 5.254 * Trend: Increasing trial numbers as difficulty level increases. ### Key Observations * Accuracy decreases as the difficulty level increases for both SFT and SFT+RL models. * Trial numbers generally increase with difficulty level, especially for the SFT+RL model. * The SFT+RL model consistently shows higher accuracy than the SFT model at Level 1, but the difference diminishes at higher difficulty levels. * The SFT+RL model requires fewer trials than the SFT model at Level 1 and Level 2, but more trials at Level 3, Level 4, and Level 5. ### Interpretation The data suggests that the Llama3.1-8B-Instruct model's performance degrades as the difficulty level increases. Reinforcement learning (RL) initially improves accuracy (Level 1), but this advantage diminishes with increasing difficulty. The increasing trial numbers for SFT+RL at higher difficulty levels may indicate that the model requires more attempts to achieve a comparable level of performance to the SFT model, suggesting that the RL component struggles to maintain its effectiveness as the task complexity grows. The model may benefit from further training or adjustments to the RL component to improve its robustness across varying difficulty levels. </details> (a) <details> <summary>x6.png Details</summary>  ### Visual Description ## Bar Chart: Accuracy and Trial Numbers across Difficulty Level (Base Model: Qwen2.5-Math-7B) ### Overview The image is a bar chart comparing the accuracy and trial numbers of two models, SFT and SFT+RL, across five difficulty levels. The chart displays accuracy on the left y-axis and trial numbers on the right y-axis. The x-axis represents the difficulty levels. ### Components/Axes * **Title:** Accuracy and Trial Numbers across Difficulty Level (Base Model: Qwen2.5-Math-7B) * **X-axis:** Difficulty Level, with categories: Level 1, Level 2, Level 3, Level 4, Level 5 * **Left Y-axis:** Accuracy, ranging from 0.60 to 1.00 in increments of 0.05. * **Right Y-axis:** Trial Numbers, ranging from 0.0 to 2.5 in increments of 0.5. * **Legend:** Located at the top-right of the chart. * SFT Accuracy (light green) * SFT+RL Accuracy (dark green) * SFT Trials (light red) * SFT+RL Trials (dark red) ### Detailed Analysis * **Level 1:** * SFT Accuracy: 0.930 * SFT+RL Accuracy: 0.930 * SFT Trials: 1.116 * SFT+RL Trials: 1.047 * **Level 2:** * SFT Accuracy: 0.944 * SFT+RL Accuracy: 0.944 * SFT Trials: 1.311 * SFT+RL Trials: 1.244 * **Level 3:** * SFT Accuracy: 0.943 * SFT+RL Accuracy: 0.962 * SFT Trials: 1.771 * SFT+RL Trials: 1.790 * **Level 4:** * SFT Accuracy: 0.773 * SFT+RL Accuracy: 0.836 * SFT Trials: 1.828 * SFT+RL Trials: 1.883 * **Level 5:** * SFT Accuracy: 0.619 * SFT+RL Accuracy: 0.649 * SFT Trials: 2.254 * SFT+RL Trials: 2.149 ### Key Observations * **Accuracy:** Both SFT and SFT+RL models show high accuracy at Levels 1, 2, and 3, with accuracy decreasing significantly at Levels 4 and 5. SFT+RL accuracy is slightly higher than SFT accuracy at levels 3, 4, and 5. * **Trial Numbers:** The number of trials generally increases with difficulty level. Levels 4 and 5 have the highest trial numbers for both SFT and SFT+RL. ### Interpretation The chart suggests that the base model (Qwen2.5-Math-7B) performs well on easier difficulty levels (1-3) but struggles with more difficult levels (4-5), as indicated by the lower accuracy and higher number of trials required. The addition of Reinforcement Learning (RL) in the SFT+RL model seems to provide a slight improvement in accuracy, particularly at higher difficulty levels, but does not drastically change the overall trend. The increase in trial numbers at higher difficulty levels indicates that the model requires more attempts to achieve a correct answer, reflecting the increased complexity of the problems. </details> (b) Figure 4: The accuracy and average trial number of different models across difficulty levels. Evaluated on MATH500 test set. 3.4.3 Improvement across Difficulty Levels To further illustrate the effect of S 2 r training, Figure 4 shows the answer accuracy and average number of trials (i.e., the average value of " $K$ " across all $y=(s_{1},v_{1},·s,s_{K},v_{K})$ under each difficulty level) for the SFT and SFT+RL models. We observe that: (1) By learning to self-verify and self-correct during reasoning, the models learn to dynamically allocate test-time effort. For easier problems, the models can reach a confident answer with fewer trials, while for more difficult problems, they require more trials to achieve a confident answer. (2) RL further improves test-time effort allocation, particularly for less capable model (e.g., Llama3.1-8B-Instruct). (3) After RL training, the answer accuracy for more difficult problems is notably improved, demonstrating the effectiveness of the self-verifying and self-correcting paradigm in enhancing the models’ reasoning abilities. | | Datasets | xx Average xx | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Model | MATH 500 | AIME 2024 | AMC 2023 | College Math | Olympiad Bench | GSM8K | GaokaoEn 2023 | | | General Model: Qwen2-7B-Instruct | | | | | | | | | | Qwen2-7B-Instruct | 51.2 | 3.3 | 30.0 | 18.2 | 19.1 | 86.4 | 39.0 | 35.3 | | Qwen2-7B- S 2 r -BI (ours) | 61.2 | 3.3 | 27.5 | 41.1 | 27.1 | 87.4 | 49.1 | 42.4 | | Qwen2-7B- S 2 r -PRL (ours) | 65.4 | 6.7 | 35.0 | 36.7 | 27.0 | 89.0 | 49.9 | 44.2 | | Qwen2-7B- S 2 r -ORL (ours) | 64.8 | 3.3 | 42.5 | 34.7 | 26.2 | 86.4 | 50.9 | 44.1 | | Qwen2-7B–Instruct- S 2 r -PRL-offline (ours) | 61.6 | 10.0 | 32.5 | 40.2 | 26.5 | 87.6 | 50.4 | 44.1 | | Qwen2-7B-Instruct- S 2 r -ORL-offline (ours) | 61.0 | 6.7 | 37.5 | 40.5 | 27.3 | 87.4 | 49.6 | 44.3 | | Math-Specialized Model: Qwen2.5-Math-7B | | | | | | | | | | Qwen2.5-Math-7B | 51.0 | 16.7 | 45.0 | 21.5 | 16.7 | 58.3 | 39.7 | 35.6 | | Qwen2.5-Math-7B- S 2 r -BI (ours) | 81.6 | 23.3 | 60.0 | 43.9 | 44.4 | 91.9 | 70.1 | 59.3 | | Qwen2.5-Math-7B- S 2 r -PRL (ours) | 83.4 | 26.7 | 70.0 | 43.8 | 46.4 | 93.2 | 70.4 | 62.0 | | Qwen2.5-Math-7B- S 2 r -ORL (ours) | 84.4 | 23.3 | 77.5 | 43.8 | 44.9 | 92.9 | 70.1 | 62.4 | | Qwen2.5-Math-7B- S 2 r -PRL-offline (ours) | 83.4 | 23.3 | 62.5 | 50.0 | 46.7 | 92.9 | 72.2 | 61.6 | | Qwen2.5-Math-7B- S 2 r -ORL-offline (ours) | 82.0 | 20.0 | 67.5 | 49.8 | 45.8 | 92.6 | 70.4 | 61.2 | Table 5: Comparison of S 2 r using online and offline RL training. 3.5 Exploring Offline RL As described in § 2.4, we explore offline RL as a more efficient alternative to online RL training, given the effectiveness of offline RL has been demonstrated in recent studies Baheti et al. (2023); Cheng et al. (2025); Wang et al. (2024b). Table 5 presents the results of offline RL with process-level and outcome-level supervision, compared to online RL. We can observe that: (1) Different from online RL, process-level supervision outperforms outcome-level supervision in offline RL training. This interesting phenomenon may be due to: a) Outcome-level RL, which excels at allowing models to freely explore dynamic trajectories, is more suitable for on-the-fly sampling during online parameter updating. b) In contrast, process-level RL, which requires accurate baseline estimation for intermediate steps, benefits from offline trajectory sampling, which can provide more accurate baseline estimates with larger scale data sampling. (2) Offline RL consistently improves performance over the behavior-initialized models across most benchmarks and achieves comparable results to online RL. These results highlight the potential of offline RL as a more efficient alternative for enhancing LLMs’ deep reasoning. 4 Related Work 4.1 Scaling Test-time Compute Scaling test-time compute recently garners wide attention in LLM reasoning Snell et al. (2024b); Wu et al. (2024); Brown et al. (2024). Existing studies have explored various methods for scaling up test-time compute, including: (1) Aggregation-based methods that samples multiple responses for each question and obtains the final answer with self-consistency Wang et al. (2023) or by selecting best-of-N answer using a verifier or reward model Wang et al. (2024c); Zhang et al. (2024b); Lightman et al. (2023b); Havrilla et al. (2024b); (2) Search-based methods that apply search algorithms such as Monte Carlo Tree Search Tian et al. (2024); Wang et al. (2024a); Zhang et al. (2024a); Qi et al. (2024), beam search Snell et al. (2024b), or other effective algorithms Feng et al. (2023); Yao et al. (2023) to search for correct trajectories; (3) Iterative-refine-based methods that iteratively improve test performance through self-refinement Madaan et al. (2024a); Shinn et al. (2024); Chen et al. (2024a, 2025). Recently, there has been a growing focus on training LLMs to perform test-time search on their own, typically by conducting longer and deeper thinking OpenAI (2024); Guo et al. (2025). These test-time scaling efforts not only directly benefit LLM reasoning, but can also be integrated back into training time, enabling iterative improvement for LLM reasoning Qin et al. (2024); Feng et al. (2023); Snell et al. (2024b); Luong et al. (2024). In this work, we also present an efficient framework for training LLMs to perform effective test-time scaling through self-verification and self-correction iterations. This approach is achieved without extensive efforts, and the performance of S 2 r can also be consistently promoted via iterative training. 4.2 Self-verification and Self-correction Enabling LLMs to perform effective self-verification and self-correction is a promising solution for achieving robust reasoning for LLMs Madaan et al. (2024b); Shinn et al. (2023); Paul et al. (2023); Lightman et al. (2023a), and these abilities are also critical for performing deep reasoning. Previous studies have shown that direct prompting of LLMs for self-verification or self-correction is suboptimal in most scenarios Huang et al. (2023); Tyen et al. (2023); Ma et al. (2024); Zhang et al. (2024c). As a result, recent studies have explored various approaches to enhance these capabilities during post-training Saunders et al. (2022); Rosset et al. (2024); Kumar et al. (2024). These methods highlight the potential of using human-annotated or LLM-generated data to equip LLMs with self-verification or self-correction capabilities Zhang et al. (2024d); Jiang et al. (2024), while also indicating that behavior imitation via supervised fine-tuning alone is insufficient for achieving valid self-verification or self-correction Kumar et al. (2024); Qu et al. (2025); Kamoi et al. (2024). In this work, we propose effective methods to enhance LLMs’ self-verification and self-correction abilities through principled imitation data construction and RL training, and demonstrate the effectiveness of our approach with in-depth analysis. 4.3 RL for LLM Reasoning Reinforcement learning has proven effective in enhancing LLM performance across various tasks Ziegler et al. (2019); Stiennon et al. (2020); Bai et al. (2022); Ouyang et al. (2022); Setlur et al. (2025). In LLM reasoning, previous studies typically employ RL in an actor-critic framework Lightman et al. (2024); Tajwar et al. (2024); Havrilla et al. (2024a), and research on developing accurate reward models for RL training has been a long-standing focus, particularly in reward modeling for Process-level RL Lightman et al. (2024); Setlur et al. (2024, 2025); Luo et al. (2024). Recently, several studies have demonstrate that simplified reward modeling and advantage estimation Ahmadian et al. (2024); Shao et al. (2024); Team et al. (2025); Guo et al. (2025) in RL training can also effectively enhance LLM reasoning. Recent advances in improving LLMs’ deep thinking Guo et al. (2025); Team et al. (2025) further highlight the effectiveness of utilizing unhackable rewards Gao et al. (2023); Everitt et al. (2021) to consistently enhance LLM reasoning. In this work, we also show that simplified advantage estimation and RL framework enable effective improvements on LLM reasoning. Additionally, we conducted an analysis on process-level RL, outcome-level RL and offline RL, providing insights for future work in RL for LLM reasoning. 5 Conclusion In this work, we propose S 2 r, an efficient framework for enhancing LLM reasoning by teaching LLMs to iteratively self-verify and self-correct during reasoning. We introduce a principled approach for behavior initialization, and explore both outcome-level and process-level RL to further strengthen the models’ thinking abilities. Experimental results across three different base models on seven math reasoning benchmarks demonstrate that S 2 r significantly enhances LLM reasoning with minimal resource requirements. Since self-verification and self-correction are two crucial abilities for LLMs’ deep reasoning, S 2 r offers an interpretable framework for understanding how SFT and RL enhance LLMs’ deep reasoning. It also offers insights into the selection of RL strategies for enhancing LLMs’ long-CoT reasoning. 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(2019) Daniel M Ziegler, Nisan Stiennon, Jeffrey Wu, Tom B Brown, Alec Radford, Dario Amodei, Paul Christiano, and Geoffrey Irving. 2019. Fine-tuning language models from human preferences. arXiv preprint arXiv:1909.08593. Appendix A Implementation Details A.1 Verification Processing and SFT Data Construction Given the responses sampled from the original LLM policy, we prompt frontier LLMs for initial verifications. In order to construct more valid verification, we force the LLMs to “verify without re-solving the problem” and filter out invalid verifications during data processing. We found that despite being instructed to "verify without re-solving the problem", most existing LLMs still biased to solve the problem again, as shown in Table 6. Finally, we collected the verification data by querying gpt-4-preview-1106 https://openai.com/api/ , which shows strong instruction-following ability to "verify without re-solving the problem" and can perform plausible verification such as adopting reverse thinking, inductive reasoning and other methods. For these collected prompts, we refine the remaining verifications using gpt-4o to improve fluency and clarity. During this refinement, we instruct gpt-4o to append a conclusion at the end of each verification based on its stance—for example: “Therefore, the answer is correct/incorrect/cannot verify.” Finally, we discard any verifications where the judgment does not align with the actual correctness of the answer. The prompts we used during the whole process are provided in Appendix § A.3. With the refined and filtered verifications, we construct the SFT data as follows. For each problem, we determine the number of answer attempts required to eventually obtain a correct answer based on the accuracy from the initial sampling. The lower the accuracy, the more rounds of responses are generated. In our implementation, we categorize all problems into four difficulty levels and construct answer sequences with 1, 2, 3, or 4 rounds, according to descending accuracy. Then, after an incorrect answer, we append “Wait, let me recheck my solution” along with the corresponding verification. If that answer is not the final attempt, we further append “Let me try again.” We ensure that the last answer in the sequence is correct. Additionally, we ensure that the answers in each round for a given problem are distinct. Figure 5 is an example of SFT data constructed with 4 rounds of responses. A.2 Baseline Details A.2.1 Baseline Implementations In Table 2, the reported results for Frontier LLMs and Top-tier Open-source Reasoning LLMs are sourced from the original reports and Guan et al. (2025). We evaluate Llama-3.1-8B-Instruct Dubey et al. (2024), Qwen2-7B-Instruct qwe (2024), Qwen2.5-Math-7B, Qwen2.5-Math-7B-Instruct and Qwen2.5-Math-72B-Instruct Yang et al. (2024) using the same process described in Section § 3.1. For Eurus-7B-PRIME Cui et al. (2025), rStar-Math-7B Guan et al. (2025), and Qwen2.5-7B-SimpleRL Zeng et al. (2025), we report results directly from the original papers. In Table 3, the results for Llama-3.1-70B-Instruct and QwQ-32B-Preview are taken from Shen et al. (2025). For the remaining baselines, we follow the official evaluation protocol of the dataset project https://github.com/Yale-LILY/FOLIO https://github.com/facebookresearch/cruxeval https://github.com/eladsegal/strategyqa https://github.com/TIGER-AI-Lab/MMLU-Pro . A.2.2 Baseline License In this work, we utilize the Llama-3.1-8B-Instruct model, whose license can be reviewed at https://huggingface.co/meta-llama/Llama-3.1-8B-Instruct/blob/main/LICENSE. In addition, the models Qwen2-7B-Instruct, Qwen2.5-Math-7B, Eurus-2-7B-PRIME, and project vLLM are distributed under the Apache License 2.0. We gratefully acknowledge the contributions of the open-source community and strictly adhere to the terms of the respective licenses. A.2.3 Baseline SFT Data Construction Original Solution SFT Data In this setting, we use the solution from the original dataset as sft data. To ensure a fair comparison, we maintain the same training data volume as our behavior initialization approaches. Long CoT SFT Data We also introduce a baseline by fine-tuning on Long CoT responses generated by QwQ-32B-Preview Team (2024a). Specifically, we instruct QwQ to generate responses to given problems and filter out those with incorrect answers. The remaining high-quality responses are then used for supervised fine-tuning. Importantly, we ensure that the total training data volume remains consistent with that used in our behavior initialization approach. The prompt we use for QwQ is provided in Appendix § A.3. A.3 Prompts The prompts we use in all experiments are as follows: Sampling Responses During Training/Inference ⬇ Please reason step by step, and put your final answer within \ boxed {}. Problem: {problem} Verification Refinement ⬇ You are a math teacher. I will give you a math problem and an answer. Verify the answer ’ s correctness without step - by - step solving. Use alternative verification methods. Question: {problem} Answer: {answer} Verification: Verification Collection ⬇ Refine this verification text to read as a natural self - check within a solution. Maintain logical flow and professionalism. Key Requirements: 1. Avoid phrases like " without solving step - by - step " or " as a math teacher ". 2. Treat the answer as your own prior solution. 3. Conclude with EXACTLY one of: Therefore, the answer is correct. Therefore, the answer is incorrect. Therefore, the answer cannot be verified. Original text: {verification} Appendix B Detailed Experiment Settings | Without Asking for Confirmative Verification | | | --- | --- | | Model | Confirmative out of 100 | | GPT-4o | 26 | | GPT-4-Preview-1106 | 32 | | QwQ-32B-preview | 37 | | Llama-3.1-70B-Instruct | 28 | | Asking for Confirmative Verification | | | Model | Confirmative out of 100 | | GPT-4o | 44 | | GPT-4-Preview-1106 | 61 | | QwQ-32B-preview | 58 | | Llama-3.1-70B-Instruct | 50 | Table 6: B.1 Datasets Details of each test dataset we used as benchmark are as follows: B.1.1 In-domain Datasets MATH500 Lightman et al. (2023b) offers a streamlined slice of the broader MATH Hendrycks et al. (2021b) dataset, comprising 500 test problems selected through uniform sampling. Despite its smaller scope, it maintains a distribution of topics and difficulty levels that mirrors the larger MATH corpus. GSM8K Cobbe et al. (2021a) features around 8,500 grade-school math word problems. The dataset focuses on simple arithmetic through early algebra and includes 1,319 distinct tasks in its test set. OlympiadBench He et al. (2024) collects 8,476 advanced math and physics questions drawn from Olympiad contexts, with some originating from the Chinese college entrance exam. We use the subset of 674 text-only competition questions, providing open-ended math challenges. AMC2023 AI-MO (2024b) and AIME AI-MO (2024a) each supply a set of challenging exam-style problems: 40 questions from AMC 2023 and 30 from AIME 2024, all in text-only format. CollegeMath Tang et al. (2024b) is a dataset targeting advanced college-level mathematics, drawn from nine textbooks spanning seven major fields—algebra, pre-calculus, calculus, vector calculus, probability, linear algebra, and differential equations. The final collection comprises 1,281 training examples and 2,818 test examples. Gaokao2023en Liao et al. (2024) is a dataset consisting of 385 mathematics problems sourced from the 2023 Chinese higher education entrance examination, which have been professionally translated into English. B.1.2 Cross-domain Datasets FOLIO Han et al. (2022) is meticulously annotated to assess intricate logical reasoning in natural language. It pairs 1,430 conclusions with 487 sets of premises—each verified using first-order logic (FOL)—and contains 203 unique problems in its test portion. CRUXEval Gu et al. (2024) tests code comprehension and reasoning through 800 concise Python functions (spanning 3–13 lines). Each function is accompanied by one or more input-output examples. The goal is to predict the correct outputs given the function body and a specific input. The test partition encompasses all 800 problems. StrategyQA Geva et al. (2021) targets multi-hop reasoning questions where the necessary intermediate steps are not explicit. Each of its 2,780 items includes a strategic query, a breakdown of the reasoning steps, and supporting evidence drawn from Wikipedia. MMLUProSTEM is extracted from MMLU-Pro Wang et al. (2024d). Following Satori Shen et al. (2025), we conduct evaluations on six STEM subsets—physics, chemistry, computer science, engineering, biology, and economics. B.2 Hyperparameters Setting | Model | Learning Rate | Batch Size | KL Coefficient | Max Length | Training Epochs | | --- | --- | --- | --- | --- | --- | | Llama-3.1-8B-Instruct | 5e-6 | 32 | 0.1 | 8000 | 3 | | Qwen2-7B-Instruct | 5e-6 | 32 | 0.1 | 6000 | 3 | | Qwen2.5-Math-7B | 5e-6 | 32 | 0.01 | 8000 | 3 | Table 7: Model Training Hyperparameter Settings (SFT) | Model | Learning Rate | Training Batch Size | Forward Batch Size | KL Coefficient | Max Length | Sampling Temperature | Clip Range | Training Steps | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Llama-3.1 | 5e-7 | 64 | 256 | 0.05 | 8000 | 0.7 | 0.2 | 500 | | Qwen2-7B-Instruct | 5e-7 | 64 | 256 | 0.05 | 6000 | 0.7 | 0.2 | 500 | | Qwen2.5-Math-7B | 5e-7 | 64 | 256 | 0.01 | 8000 | 0.7 | 0.2 | 500 | Table 8: Model Training Hyperparameter Settings (RL) During behavior initialization with SFT, we use a batch size of 32 and adopt a learning rate of 5e-6. We set the maximum sequence length 8000 to accommodate long responses and verifications. To balance stability and convergence during training, we add a KL punishment to the training loss, and the KL coefficient is set to 0.1. During reinforcement learning, for each training batch, we use a training batch size of 64, and sample $n$ responses for each question in a batch, resulting a forward batch size of $64n$ . For each forward batch, we update the model for $n$ step with the training batch size 64. Specifically, for both process-level and outcome-level RL, we adopt $n=4$ (i.e., for RLOO, the sample number is also $4$ ). More hyperparameters of the RL training are presented in Table 8. We use the BF16 model precision in all experiments. Main hyperparameters used in the experiments are illustrated in Table 7 and 8. B.3 Experiment Environment All experiments are implemented using the PyTorch framework on 32 NVIDIA H20 (96GB) GPUs or 32 NVIDIA A100Pro (40GB) GPUs. Our training code is built upon Hugging Face TRL https://github.com/huggingface/trl. For inference, we use a single NVIDIA A100 (40GB) GPU with vLLM-0.5.4 https://github.com/vllm-project/vllm. We utilize transformers version 4.39.3 for fine-tuning Qwen2-7B-Instruct and Qwen2.5-Math-7B, version 4.44.0 for fine-tuning Llama-3.1-8B, and version 4.46.3 for reinforcement learning. We use PyTorch 2.1.1 across our training pipeline. Our evaluation code is built upon Qwen Math’s evaluation codebase https://github.com/QwenLM/Qwen2.5-Math. Appendix C Metrics Definition We include the formal definition of metrics we use for analyzing self-verification and self-correction behaviors of the post-trained models as follows. C.1 Notations We first present the main notations used in our formulation in Table 9. | Variable | Description | | --- | --- | | $\pi$ | The policy | | $x$ | Problem instance | | $y$ | Series of predefined actions: $y=\{a_{1},a_{2},...,a_{n}\}$ | | $a_{i}$ | The $i$ -th action in the response $y$ , and let | | $Type(a_{i})∈\{\texttt{verify},\texttt{solve},\texttt{<end>}\}$ | | | $s_{j}$ | $j^{th}$ attempt to solve the problem | | $v_{j}$ | $j^{th}$ self-verification for the $j^{th}$ attempt | | $Parser(·)$ | $Parser(v_{j})∈\{\texttt{correct},\texttt{incorrect}\}$ | | The text parser to get the self-verification result | | | indicating the correctness of action $s_{j}$ | | | $V_{golden}(·)$ | $V_{golden}(a_{i})∈\{\texttt{correct},\texttt{incorrect}\}$ | | $R(·)$ | The rule based reward function | | $R(·)∈\{-1,1\}$ | | | $R(s_{j})=\begin{cases}1,&V_{golden}(s_{j})=\texttt{correct}\\ -1,&otherwise\\ \end{cases}$ | | | $R(v_{j})=\begin{cases}1,&Parser(v_{j})=V_{golden}(s_{j})\\ -1,&otherwise\\ \end{cases}$ | | | <end> | End of action series | | $\mathbb{I}(·)$ | The indicator function, $\mathbb{I}(·)∈\{0,1\}$ . $\mathbb{I}(·)=1$ if the condition inside holds true, and $\mathbb{I}(·)=0$ otherwise. | Table 9: Variable Lookup Table C.2 Self-Verification Metrics C.2.1 Verification Accuracy (VA) Verification Accuracy measures how often the verification prediction matches the ground-truth correctness ( $N$ is the total number of verifications in the responses to the test set): C.2.2 Error Recall (ER) Error Recall measures the recall of detecting incorrect answers (i.e., the fraction of actually incorrect answers that are successfully identified as incorrect): where $|y|_{a}$ is the total number of actions in $y$ and $\frac{|y|_{a}}{2}$ is the total number of attempts to solve the problem ( $y=\{a_{1},a_{2},·s,a_{|y|_{a}}\}=\{s_{1},v_{1},·s,s_{\frac{|y|_{a}}{2}% },v_{\frac{|y|_{a}}{2}}\}$ ). C.2.3 Correct Precision (CP) Correct Precision measures the precision when the verification model predicts an answer to be correct (i.e., among all “correct” predictions, how many are truly correct): C.3 Self-Correction Metrics C.3.1 Incorrect to Correct Rate (ICR) The rate at which the model successfully corrects an initially incorrect answer ( $R(s_{1})=-1$ ) into a correct final answer ( $R(s_{T_{y}})=1$ ), where $T_{y}=|y|_{a}/2$ is the total number of attempts to solve the problem in each $y$ . Formally: $$ \text{ICR}=\frac{\sum_{y}\mathbb{I}\bigl{(}R(s_{1})=-1\bigr{)}\,\mathbb{I}% \bigl{(}R(s_{T_{y}})=1\bigr{)}}{\sum_{y}\mathbb{I}\bigl{(}R(s_{1})=-1\bigr{)}}. \tag{10} $$ C.3.2 Correct to Incorrect Rate (CIR) The rate at which the model incorrectly alters an initially correct answer ( $R(s_{1})=1$ ) into an incorrect final answer ( $R(s_{T_{y}})=-1$ ), where $T_{y}=|y|_{a}/2$ is the total number of attempts to solve the problem in each $y$ . Formally: $$ \text{CIR}=\frac{\sum_{y}\mathbb{I}\bigl{(}R(s_{1})=1\bigr{)}\,\mathbb{I}\bigl% {(}R(s_{T_{y}})=-1\bigr{)}}{\sum_{y}\mathbb{I}\bigl{(}R(s_{1})=1\bigr{)}}. \tag{11} $$ Appendix D Offline RL Training Details In this section, we provide additional details on the offline reinforcement learning training process, including formal definition, ablation studies, and implementation details. D.1 Accuracy-Grouped Baseline Definition To fully leverage the advantages of offline RL, which does not require real-time sampling, we explore more appropriate baseline selection by further grouping trajectories based on problem difficulty. Intuitively, for two trajectories $y^{(1)}$ and $y^{(2)}$ sampled under questions of different difficulty levels, and their corresponding actions $a^{(1)}_{t}$ and $a^{(2)}_{t}$ at the same position, even if they share identical reward contexts, their expected returns (baselines) should differ, i.e., the expected return is typically lower for more challenging problems. We measure a problem’s difficulty by estimating how often it is solved correctly under the current sampling policy. Concretely, we sample multiple trajectories in parallel for each problem. The fraction of these trajectories that yield a correct final answer serves as the problem’s accuracy. We then discretize this accuracy into separate bins, effectively grouping the problems according to their estimated difficulty. All trajectories belonging to problems within the same accuracy bin form a common subset. Compared to using direct reward contexts alone, this accuracy-based grouping offers a more robust estimate of expected returns, problems in the same bin share similar success rates. Moreover, unlike a pre-defined difficulty grouping, these bins adjust dynamically as the model’s capabilities evolve. Building on this approach, we propose two accuracy-based baseline estimation methods for offline RL as follows. D.1.1 Accuracy-Grouped Baseline With Position Group Within each accuracy bin, we further split actions based on their position in the trajectory. Concretely, we consider all actions occurring at the same step index across trajectories in the same bin to be comparable, and we compute their average return to serve as the baseline. Thus, when we look up the baseline for a particular action at a given step in a trajectory, we use the average return of all actions taken at that same step index in all trajectories belonging to the same accuracy bin. D.1.2 Accuracy-Grouped Baseline With Reward Context We also propose combining accuracy-based grouping with reward-context grouping. The underlying assumption is that even if two actions share the same immediate reward context, their expected returns can differ if they originate from different difficulty bins. Generally, problems that are harder to solve exhibit lower expected returns. Consequently, we first bin the trajectories by accuracy, then further group them by common reward context. Within each sub-group, we average the returns of all relevant actions to obtain the baseline. D.2 Offline RL Implementation Details In each iteration of offline RL training, we generate multiple trajectories (e.g., eight) per prompt in parallel. We then apply prompt filtering, rejection sampling, accuracy-based baseline estimation, advantage computation, and policy updates. Implementation details follow. D.2.1 Prompt Filtering | Accuracy Range | Retained Questions | MATH500 | AIME2024 | AMC2023 | College Math | Olympiad Bench | GSM8K | GaokaoEn2023 | Average | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | $[0.1-0.7]$ | 1805 | 83.4 | 23.3 | 62.5 | 50.0 | 46.7 | 92.9 | 72.2 | 61.6 | | $[0.2-0.8]$ | 2516 | 82.6 | 23.3 | 70.0 | 49.8 | 45.3 | 92.4 | 70.1 | 61.9 | | $[0.3-0.9]$ | 4448 | 81.6 | 23.3 | 70.0 | 49.4 | 44.7 | 92.0 | 68.1 | 61.3 | | $[0-1]$ | Full | 80.6 | 26.7 | 67.5 | 50.0 | 43.0 | 91.4 | 67.0 | 60.9 | Table 10: Comparison of question filtering accuracy selection. As we sample multiple trajectories for each prompt, we compute the accuracy of each prompt. We retain prompts whose accuracy falls within a predefined range. Our ablation study on Qwen2.5-Math-7B shown in Table 10 confirms that filtering improves performance. The most stable results are obtained with an accuracy range of $[0.1,0.7]$ , suggesting that including moderately difficult samples enhances the model’s reasoning capabilities. D.2.2 Rejection Sampling We discard any trajectory that does not follow the alternation pattern of solution and verification: $y=(s_{1},v_{1},...,s_{k},v_{k})$ . Additionally, we remove malformed trajectories such as $y=(s_{1},s_{2},v_{1})$ . To mitigate reward hacking due to excessively long outputs, we eliminate trajectories where $R(s_{t})=1$ and $R(v_{t})=1$ at timestep $t$ , but further actions are taken at $t+1$ . Moreover, we discard trajectories containing more than 20 actions, as excessive action sequences can introduce instability and deviate from expected solution structures. D.2.3 Loss Function | | Datasets | xx Average xx | | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Baseline Method | MATH500 | AIME2024 | AMC2023 | College Math | Olympiad Bench | GSM8K | GaokaoEn2023 | | | Based on reward context | 82.4 | 26.7 | 65.0 | 50.1 | 46.1 | 92.9 | 71.2 | 62.1 | | Based on accuracy group with position | 83.4 | 23.3 | 62.5 | 50.0 | 46.7 | 92.9 | 72.2 | 61.6 | | Based on accuracy group with reward context | 82.4 | 23.3 | 67.5 | 49.3 | 45.8 | 93.3 | 71.2 | 61.8 | Table 11: The performance of different baselines To determine the best offline baseline method, we conducted ablation studies on Qwen2.5-Math-7B shown in Table 11. We found that using the accuracy-grouped baseline with an additional division by position provides the most stable results. When computing advantages, we subtract both the baseline and a scaled relative policy term like Equation 5. Notably, we fix $\pi_{\text{ref}}$ as the reference policy instead of being updated at each iteration. D.2.4 Training Hyperparameter Settings We use a batch size of 64, a maximum learning rate of $5× 10^{-7}$ , and a KL penalty coefficient of 0.1. The maximum training sequence length is set to 8192. We apply a warm-up phase of 5 steps and a clipping range parameter of 0.2. We use BF16 model precision in all experiments. Appendix E Demo Cases <details> <summary>x7.png Details</summary>  ### Visual Description ## Math Problem and Solution ### Overview The image presents a math problem involving the distribution of candy among four people (Lizzy, Megan, Oscar, and Patrick) and a proposed solution. The problem asks which of several statements about the number of candies Patrick has at the end are true. The solution analyzes each statement and determines its validity. ### Components/Axes * **Problem Statement:** Defines the initial conditions, the candy distribution process, and the question to be answered. * **Statements (a) to (f):** A list of six statements about the divisibility of *P* (the number of candies Patrick has at the end). * **Response:** A step-by-step analysis of each statement, providing justification for whether it is true or false. ### Detailed Analysis or ### Content Details **Problem Statement:** * Lizzy, Megan, Oscar, and Patrick each start with *x* pieces of candy, where *x* is a positive integer. * Patrick is the only one who likes candy. * Lizzy gives all her candy to Megan. * Megan gives all her candy to Oscar. * Oscar gives all his candy to Patrick. * *P* is the number of pieces of candy Patrick has in the end. * Question: How many of the following statements are true? (Assume we do not know exactly what *x* is.) **Statements:** * (a) 2 can be a divisor of *P*. * (b) 2 must be a divisor of *P*. * (c) 3 can be a divisor of *P*. * (d) 3 must be a divisor of *P*. * (e) 4 can be a divisor of *P*. * (f) 4 must be a divisor of *P*. **Response:** The response analyzes each statement as follows: * **(a) 2 can be a divisor of P:** This is true because if *x* is even, then 4*x* is even and divisible by 2. * **(b) 2 must be a divisor of P:** This is true because 4*x* is always divisible by 2, regardless of *x*. * **(c) 3 can be a divisor of P:** This is true if *x* is a multiple of 3, then 4*x* will also be a multiple of 3. * **(d) 3 must be a divisor of P:** This is false because if *x* is not a multiple of 3, then 4*x* will not be a multiple of 3. * **(e) 4 can be a divisor of P:** This is true as 4*x* is divisible by 4. * **(f) 4 must be a divisor of P:** This is true because 4*x* is always divisible by 4, regardless of *x*. **Conclusion:** * Five statements are true: (a), (b), (c), (e), and (f). * The only false statement is (d). * Therefore, the answer is correct. Throughout the response, the phrase "Wait, let me recheck my solution" and "Let me try again" appear, suggesting an iterative problem-solving process. ### Key Observations * The problem involves basic divisibility rules. * The key to solving the problem is recognizing that *P* = 4*x*. * The response provides clear and concise justifications for each statement. ### Interpretation The problem demonstrates the application of divisibility rules in a simple scenario. The solution highlights the importance of considering all possible cases when determining whether a statement is always true or only sometimes true. The iterative nature of the response, indicated by the repeated phrases "Wait, let me recheck my solution" and "Let me try again," suggests a process of refinement and verification, which is a common practice in problem-solving. The final answer correctly identifies the true and false statements based on the divisibility of 4*x*. </details> Figure 5: SFT data example. <details> <summary>x8.png Details</summary>  ### Visual Description ## Math Problem: Finding a Vector with Smallest Magnitude ### Overview The image presents a math problem focused on finding a vector **v** that satisfies a cross-product equation and has the smallest magnitude. The problem is solved twice, with the first attempt resulting in an incorrect answer and the second attempt resulting in a correct answer. The solution involves setting up a system of equations, solving for the components of **v**, and minimizing the magnitude of **v**. ### Components/Axes The problem involves vectors, cross products, systems of equations, and magnitude minimization. ### Detailed Analysis or ### Content Details **Problem Statement:** * There are an infinite number of vectors **v** such that (1, 2, -5) x **v** = (90, 30, 30). * Find the vector **v** that has the smallest magnitude. **First Attempt (Incorrect):** 1. **Setup:** Let **v** = (x, y, z). The cross product (1, 2, -5) x (x, y, z) is calculated as (2z + 5y, -5x + z, 2x - y). 2. **System of Equations:** * 2z + 5y = 90 * -5x + z = 30 * 2x - y = 30 3. **Solving for y:** y = 2x - 30 4. **Substitution:** Substituting y = 2x - 30 into the first equation: 2z + 5(2x - 30) = 90 => 2z + 10x = 240 => z + 5x = 120 => z = 120 - 5x 5. **Substitution:** Substituting z = 120 - 5x into the second equation: -5x + (120 - 5x) = 30 => -10x = -90 => x = 9 6. **Finding y and z:** * y = 2(9) - 30 = -12 * z = 120 - 5(9) = 75 7. **Result:** **v** = (9, -12, 75) 8. **General Solution:** **v** = (9, -12, 75) + t(1, 2, -5) for some scalar t. 9. **Magnitude Minimization:** The magnitude of **v** is minimized when t = 0, which corresponds to the vector (9, -12, 75). 10. **Recheck:** The solution is rechecked by verifying if **a** and **v** are orthogonal to **c** by taking the dot product. * **a** · **c** = (1)(90) + (2)(30) + (-5)(30) = 90 + 60 - 150 = 0 * **v** · **c** = (9)(90) + (-12)(30) + (75)(30) = 810 - 360 + 2250 = 2700 11. **Conclusion:** Since **v** · **c** ≠ 0, the answer is incorrect. **Second Attempt (Correct):** 1. **Setup:** Let **v** = (x, y, z). The cross product (1, 2, -5) x (x, y, z) is calculated using the determinant method, resulting in the vector (2z + 5y, -(z + 5x), y - 2x). 2. **System of Equations:** * 2z + 5y = 90 * -z - 5x = 30 * y - 2x = 30 3. **Solving for z:** z = -5x - 30 4. **Solving for y:** y = 2x + 30 5. **Substitution:** Substituting y = 2x + 30 and z = -5x - 30 into the first equation: 2(-5x - 30) + 5(2x + 30) = 90 => -10x - 60 + 10x + 150 = 90 => 90 = 90. 6. **General Solution:** **v** = (x, 2x + 30, -5x - 30) 7. **Magnitude Minimization:** The magnitude of **v** is ||**v**|| = sqrt(x² + (2x + 30)² + (-5x - 30)²). 8. **Simplification:** x² + (2x + 30)² + (-5x - 30)² = 30x² + 420x + 1800 9. **Completing the Square:** 30x² + 420x + 1800 = 30((x + 7)² + 11) = 30(x + 7)² + 330 10. **Minimum Value:** The minimum value occurs when x = -7. 11. **Result:** Substituting x = -7 into the expression for **v**: * x = -7 * y = 2(-7) + 30 = 16 * z = -5(-7) - 30 = 5 * **v** = (-7, 16, 5) 12. **Recheck:** The solution is rechecked by verifying if **a** and **v** are orthogonal to **c** by taking the dot product. * **a** · **c** = (1)(90) + (2)(30) + (-5)(30) = 90 + 60 - 150 = 0 * **v** · **c** = (-7)(90) + (16)(30) + (5)(30) = -630 + 480 + 150 = 0 13. **Conclusion:** Since both dot products are zero, the answer is correct. ### Key Observations * The first attempt fails because the dot product of the calculated vector **v** and the vector **c** (90, 30, 30) is not zero, indicating that **v** is not orthogonal to **c**. * The second attempt correctly finds a vector **v** that is orthogonal to both **a** and **c**, satisfying the conditions of the problem. * The magnitude minimization involves completing the square to find the value of x that minimizes the expression. ### Interpretation The problem demonstrates the process of finding a vector that satisfies a cross-product equation and has the smallest magnitude. The initial incorrect attempt highlights the importance of verifying the solution by checking orthogonality. The correct solution involves a more rigorous approach, including the use of the determinant method for calculating the cross product and completing the square for magnitude minimization. The problem emphasizes the application of vector algebra and calculus concepts to solve geometric problems. </details> Figure 6: MATH500 Case. <details> <summary>x9.png Details</summary>  ### Visual Description ## Logical Reasoning Problem ### Overview The image presents a logical reasoning problem with two facts, a question, and a series of attempts to answer the question. The problem asks whether chlorine can be bought at a dollar store, given that chlorine, when added to water, creates household bleach, and household bleach is available at most dollar stores. The image shows three attempts to solve the problem, with the first two resulting in the answer "False" and the third resulting in the answer "True". Each attempt includes an explanation of the reasoning process. ### Components/Axes * **Problem Statement:** Defines the facts and the question to be answered. * **Facts:** * Fact 1: Chlorine, when added to water, creates household bleach. * Fact 2: Household bleach is available at most dollar stores. * **Question:** Can you buy chlorine at a dollar store? * **Response:** The attempt to answer the question, including the reasoning and the final answer ("True" or "False"). * **Recheck:** A section where the provided answer is verified by examining the logical structure of the statements and the inferred conclusion. ### Detailed Analysis or ### Content Details **Attempt 1:** * **Answer:** False * **Reasoning:** Fact 2 states that household bleach is available at dollar stores. Fact 1 does not provide any direct information about the availability of chlorine itself at dollar stores. Since Fact 1 does not provide information about the direct sale of chlorine, we cannot conclude that chlorine is available at dollar stores based solely on the given facts. * **Recheck:** The answer "False" implies that it is not possible to buy chlorine at a dollar store. Given that the statements do not provide information about the direct sale of chlorine, but rather the creation of bleach from chlorine, the answer "False" is not necessarily incorrect. It is possible that chlorine is not sold at dollar stores, but it could be used to create bleach that is sold there. Therefore, the answer is incorrect. **Attempt 2:** * **Answer:** False * **Reasoning:** Fact 2 states that household bleach is a product that can be found at dollar stores. Fact 1 only states that chlorine, when added to water, creates household bleach. It does not provide any information about whether chlorine itself is sold at dollar stores. Since the availability of chlorine at dollar stores is not mentioned in the facts provided, we cannot definitively conclude that chlorine is available at dollar stores based solely on the given information. * **Recheck:** The answer "False" implies that it is not possible to buy chlorine at a dollar store. Given that the statements do not provide information about the direct sale of chlorine, but rather the creation of bleach from chlorine, the answer "False" is not necessarily incorrect. It is possible that chlorine is not sold at dollar stores, but it could be used to create bleach that is sold there. Therefore, the answer is incorrect. **Attempt 3:** * **Answer:** True * **Reasoning:** Fact 2 states that household bleach is available at most dollar stores. Since household bleach is made by adding chlorine to water, it implies that the process of making household bleach involves chlorine. Given that household bleach is available at dollar stores and it is made from chlorine, it is reasonable to infer that chlorine is also available at dollar stores, as it is a necessary component for making household bleach. Therefore, based on the given facts, it is reasonable to conclude that you can buy chlorine at a dollar store. * **Recheck:** This inference supports the conclusion that chlorine can be bought at a dollar store, as it is a necessary component for producing household bleach. Therefore, the answer is correct. ### Key Observations * The initial attempts focus on the lack of direct information about chlorine's availability. * The successful attempt infers chlorine's availability based on its role as a necessary component of household bleach. * The "Recheck" sections analyze the logical validity of the answers. ### Interpretation The problem demonstrates the importance of inference in logical reasoning. While the given facts do not directly state that chlorine is available at dollar stores, it can be reasonably inferred based on the fact that household bleach, which requires chlorine, is available. The repeated attempts highlight the process of refining one's reasoning to arrive at a more accurate conclusion. The problem also illustrates the difference between explicit information and implicit knowledge that can be derived from the given facts. </details> Figure 7: StrategyQA Case. To intuitively demonstrate the effectiveness of our proposed method, we present the model’s inference examples after RL on the MATH500 and StrategyQA datasets in the Figure 6 and Figure 7. Appendix F Other Discussion F.1 Discussion on Potential Risk We have carefully considered potential risks associated with our work and found no significant concerns. Our approach, focused on enhancing LLM reasoning through self-verification and self-correction, does not introduce malicious or harmful effects, privacy issues, or security threats. Additionally, it does not contribute to biases, fairness concerns, or environmental impact. We believe our work is safe for responsible use in research. F.2 Use of AI Assistant In this work, we utilized an AI assistant solely for the purpose of refining and polishing the language of the manuscript. The AI assistant was employed to improve clarity, flow, and overall readability, ensuring the text adhered to academic writing standards. It was not involved in any data analysis, experimentation, or formulation of ideas. All research design, methodology, results, and conclusions were developed independently by the authors. The use of the AI assistant was limited to language enhancement and did not influence the content or scientific integrity of the work.