2504.10559v1

# Efficient Process Reward Model Training via Active Learning > Work done during the internship in Sea AI Lab. Email:k.duan@u.nus.eduCorresponding authors. Abstract Process Reward Models (PRMs) provide step-level supervision to large language models (LLMs), but scaling up training data annotation remains challenging for both humans and LLMs. To address this limitation, we propose an active learning approach, ActPRM, which proactively selects the most uncertain samples for training, substantially reducing labeling costs. During training, we use the PRM to estimate uncertainty after the forward pass, retaining only highly uncertain data. A capable yet costly reasoning model then labels this data. Then we compute the loss w.r.t. the labels and update the PRM’s weights. We compare ActPRM vs. vanilla fine-tuning, on a pool-based active learning setting, demonstrating that ActPRM reduce 50% annotation, but achieving the comparable or even better performance. Beyond annotation efficiency, we further advance the actively trained PRM by filtering over 1M+ math reasoning trajectories with ActPRM, retaining 60% of the data. A subsequent training on this selected dataset yields a new state-of-the-art (SOTA) PRM on ProcessBench (75.0%) and PRMBench (65.5%) compared with same sized models The code is available at https://github.com/sail-sg/ActivePRM. 1 Introduction <details> <summary>x1.png Details</summary>  ### Visual Description ## Chart: Model Performance vs. Annotation Cost ### Overview The image is a chart comparing the performance (Average F1 Score) of three different models (ActPRM, UniversalPRM, and Qwen2.5-Math-PRM-7B) against their estimated annotation cost (in generated tokens). The chart shows how the F1 score increases with the annotation cost for ActPRM, and then compares the F1 scores of the other two models at specific annotation cost levels relative to ActPRM. ### Components/Axes * **X-axis:** Est. Annotation Cost (Gen. Tokens). The scale is logarithmic, with markers at 2²⁸, 2³⁰, 2³², and 2³⁴. * **Y-axis:** Average F1 Score. The scale ranges from 0.68 to 0.76, with markers at 0.68, 0.70, 0.72, 0.74, and 0.76. * **Legend (bottom-right):** * Red Star: ActPRM * Teal Star: UniversalPRM * Dark Blue Star: Qwen2.5-Math-PRM-7B ### Detailed Analysis * **ActPRM (Red Line):** The red line shows the performance of the ActPRM model as the annotation cost increases. * At approximately 2²⁸ tokens, the F1 score is around 0.68. * The line slopes upward, reaching an F1 score of 0.750 at 2³⁰ tokens. * A dashed vertical line extends downward from the ActPRM data point at 2³⁰. * **UniversalPRM (Teal Star):** The UniversalPRM model has an F1 score of 0.743. * The annotation cost for UniversalPRM is 4.8x the cost of ActPRM at 2³⁰ tokens. * The x-axis value for UniversalPRM is approximately 2³². * **Qwen2.5-Math-PRM-7B (Dark Blue Star):** The Qwen2.5-Math-PRM-7B model has an F1 score of 0.735. * The annotation cost for Qwen2.5-Math-PRM-7B is 17.3x the cost of ActPRM at 2³⁰ tokens. * The x-axis value for Qwen2.5-Math-PRM-7B is approximately 2³⁴. ### Key Observations * The ActPRM model's performance increases significantly as the annotation cost increases from 2²⁸ to 2³⁰ tokens. * UniversalPRM and Qwen2.5-Math-PRM-7B achieve lower F1 scores than ActPRM at 2³⁰, but at significantly higher annotation costs. * The annotation cost increases exponentially for UniversalPRM and Qwen2.5-Math-PRM-7B compared to ActPRM. ### Interpretation The chart suggests that while ActPRM's performance initially increases with annotation cost, there are diminishing returns. UniversalPRM and Qwen2.5-Math-PRM-7B achieve lower F1 scores than ActPRM at 2³⁰, but at significantly higher annotation costs. This indicates that ActPRM is more efficient in terms of performance per annotation cost, at least up to the 2³⁰ token mark. The 4.8x and 17.3x cost multipliers highlight the trade-offs between model complexity, annotation effort, and performance. The data implies that a simpler model with targeted annotation might be more effective than a more complex model requiring extensive annotation. </details> Figure 1: Average F1 score on ProcessBench (Zheng et al., 2024) versus estimated annotation cost. ActPRM outperforms prior SOTA models while requiring merely 20% of the annotation costs. Recently, Large Language Models (LLMs) (DeepSeek-AI et al., 2025; Yang et al., 2024; OpenAI et al., 2024b) have shown remarkable advances in mathematical reasoning, yet they can make mistakes during chain-of-thought (CoT) reasoning despite correct final answers (Zheng et al., 2024). To address this challenge, process reward models (Lightman et al., 2023; Wang et al., 2024; Zhang et al., 2025) were proposed, aiming to identify process errors and provide finer-grained supervision of the training process. The main challenge in training Process Reward Models (PRMs) lies in obtaining fine-grained step-level annotations, which remain prohibitively expensive. Lightman et al. (2023) pioneered PRM training by employing human experts to label 75K questions at the step level. While their approach achieved high-quality results (reaching 57.5% on ProcessBench (Zheng et al., 2024)), it does not scale automatically due to the heavy reliance on manual annotation. To reduce human efforts, Monte Carlo (M.C.) Estimation Methods (Wang et al., 2024; Wei et al., 2024; Luo et al., 2024) were proposed. However, these approaches come with high computational costs (massive rollouts are required for accurate estimation) and struggle to accurately identify the first error step (Zheng et al., 2024). To address this challenge, Qwen2.5-Math-PRM (Zhang et al., 2025) proposed using LLM-as-Judge — leveraging LLMs to detect the first error step — and filter out unreliable M.C. labels. It significantly boosts the performance of PRM on both ProcessBench (Zheng et al., 2024) and PRMBench (Song et al., 2025). More recently, UniversalPRM (Tan et al., 2025) relies solely on LLM-as-Judge with ensemble prompting (via majority voting), achieving new SOTA performance on ProcessBench within the same model size. However, the annotation costs are still considerable. We estimate the labeling costs of Qwen2.5-Math-PRM (Zhang et al., 2025) and UniversalPRM (Tan et al., 2025) and illustrate them in Figure 1. It shows that Qwen-Math-PRM-7B and UniversalPRM consume over $2^{34}$ and $2^{32}$ generated tokens, respectively. Refer to Appendix C for estimation strategy. To reduce annotation costs, we propose ActPRM, which uses a trained ensemble PRM to identify and select uncertain data for annotation by a high-capability reasoning model. Our approach trains a PRM with ensemble heads for uncertainty estimation. For each reasoning step, we compute the mean $\mu$ and standard deviation $\sigma$ of ensemble predictions, identifying uncertain steps when prediction confidence is outside threshold $1-\delta_{pred}<\mu<\delta_{pred}$ or variation exceeds $\delta_{std}$ . We consider a CoT trajectory uncertain if any step up to and including the first predicted error meets these criteria. By annotating only uncertain data and training exclusively on this subset, we significantly reduce labeling costs while maintaining PRM performance. To validate the effectiveness and efficiency of ActPRM, we conducted comprehensive experiments in multiple settings: - Pool-based Evaluation (Section 5.1): Using 100K labeled samples, ActPRM achieved performance comparable to full-data tuning while reducing annotation costs by 50%. It consistently outperformed random selection under identical budget constraints. - One-shot Active Learning (Section 5.2): Starting with our pool-based model, we applied ActPRM to select uncertain samples from 1M+ unlabeled CoT trajectories from NuminaMath (Li et al., 2024). After annotation and fine-tuning, we achieved new SOTA performance of 75.0% on ProcessBench. As shown in Figure 1, ActPRM surpasses prior SOTA models with significantly lower costs—outperforming UniversalPRM (Tan et al., 2025) by 0.7% using only 20% of its annotation budget and exceeding Qwen2.5-Math-PRM-7B by 1.5% with just 6% of its annotation budget. Our contributions are summarized as follows: ❶ We propose an uncertainty-aware active learning approach ActPRM for PRM training that selectively annotates informative reasoning steps using ensemble-based uncertainty estimation, significantly reducing labeling costs while maintaining performance. ❷ ActPRM achieves state-of-the-art results (75.0% on ProcessBench, 65.5% on PRMBench) while requiring only 20% of the annotation budget compared to prior SOTA method UniversalPRM. ❸ We release all trained models, datasets, and code to ensure reproducibility and facilitate community adoption. 2 Preliminaries 2.1 Process Reward Models Problem Formulation. Given a math problem $q$ and a corresponding solution trajectory $s=[s_{1},s_{2},...,s_{n}]$ , where $s_{i}$ denotes $i$ -th step., we require a PRM to identify the correctness of each step until a wrong step is identified. We only label the steps up to and including the first error step following prior works (Lightman et al., 2023; Zheng et al., 2024), since the afterward steps are genuinely difficult to define their correctness. As a result, in practice the labels for a solution trajectory are either $[1,1,...,1]$ or $[1,1,...,0]$ . Then a PRM could be trained using the typical BCE loss: $$ \mathcal{L}_{BCE}(s,y|\theta)=-\frac{1}{|s|}\sum_{1}^{|s|}y_{i}\log(P_{\theta}% (s_{i}|s_{[:i]},q))+(1-y_{i})\log(1-P_{\theta}(s_{i}|s_{[:i]},q)), \tag{1} $$ where $P_{\theta}$ is the PRM parameterized by $\theta$ and $s_{[:i]}$ denotes the steps before $s_{i}$ . When using PRM for inference, we set a threshold $\delta$ (usually $0.5$ ) to identify the first step that has a correctness probability $P_{\theta}(s_{i}|s_{[:i]},q)$ less than $\delta$ . PRM Implementation Details. A typical PRM is built upon a pretrained generative LLM by replacing the causal language model head with a binary classification head that outputs the probability of correctness at corresponding token position. In practice, we solely need the prediction at the end of each reasoning step and thus a prediction mask is used to mask out the prediction at the other positions. 2.2 Uncertainty Estimation for Classification Aleatoric Uncertainty. As aforementioned, a typical PRM $P_{\theta}$ is trained as a binary classification task. The simplest way to measure the uncertainty for it is to use aleatoric uncertainty (Valdenegro-Toro & Saromo, 2022): For simplicity, we use $P_{\theta}(s_{i})$ as the aleatoric probability for the $i$ -th step in the solution trajectory, where the full representation is $p_{\theta}(s_{i}|s_{[:i]},q)$ as in Equation 1. $$ \mbox{Aleatoric Uncertainty}\propto P_{\theta}(s_{i})\log P_{\theta}(s_{i}). \tag{2} $$ Epistemic Uncertainty. In addition, ensemble of models is also a common way to estimate epistemic uncertainty (Valdenegro-Toro & Saromo, 2022) by quantifying the disagreement among ensemble models. For example, Liang et al. (2022); Gleave & Irving (2022) use an ensemble of reward models to estimate uncertainty in preference learning. for process reward modeling, we could leverage the standard deviation of the ensemble predictions as the uncertainty estimation: $$ \mbox{Epistemic Uncertainty}\propto\mbox{Var}(\{P_{\theta}(s_{i})\}), \tag{3} $$ where $\{P_{\theta}\}$ is a set of models. It is worth noting that employing an ensemble of heads built upon a shared backbone is a common strategy to mitigate computational costs. We empirically study the combination of aleatoric and epistemic uncertainty and find that they are complementary to each other. Experimental results are shown in Section 5.1. 3 Related Work Active Learning and Uncertainty Estimation. Active learning has been widely explored in the alignment of LLMs. Several studies adopt an online bandit formulation, leveraging uncertainty-aware reward models (RMs) for active exploration in response selection (Mehta et al., 2023; Dwaracherla et al., 2024; Liu et al., 2024; Melo et al., 2024; Gleave & Irving, 2022). For instance, Mehta et al. (2023) and Dwaracherla et al. (2024) use ensemble-based LLM heads to estimate epistemic uncertainty, prioritizing data most informative for preference learning. Similarly, Melo et al. (2024) propose an acquisition function combining both entropy (aleatoric uncertainty) and epistemic uncertainty. Our work builds on these approaches, empirically evaluating the role of both uncertainty types in the context of process reward modeling. Beyond active learning, ensemble methods—such as those in Coste et al. (2024) —have also proven effective in mitigating reward hacking (Amodei et al., 2016). Process Reward Models. Different from outcome rewards (e.g., verifiable rewards (DeepSeek-AI et al., 2025) for mathematical reasoning problems) that assign rewards based on the final outcome, process rewards are assigned based on the intermediate steps of the problem-solving process. For a question and a corresponding solution with several steps, a PRM provides finer-grained rewards for each step. Til current stage, process reward modeling contains two categories: $(i)$ Process Reward as Q-values and $(ii)$ Process Reward as Judger. The former one (Wang et al., 2024; Luo et al., 2024; Wei et al., 2024; Li & Li, 2024) regards the process reward as the Q-values of the steps that estimate the probability of the policy model to reach the final correct answer. Specifically, they leverage the policy model that generates the solution steps to do Monte Carlo Estimation for each step. The estimated Q-values are used as the process rewards. However, recent works (Zhang et al., 2025; Zheng et al., 2024) show that this kind of process reward modeling suffers from identifying the process errors because it highly depends on the policy model and has large bias with the ground truth distribution. In contrast, the latter one (Lightman et al., 2023; Zhang et al., 2025) regards the process reward model as a proxy for identifying the intermediate process errors and the corresponding trained model achieves better performance on several benchmarks (Zheng et al., 2024; Song et al., 2025). In this work, we follow the latter one and regard the process reward as a judge that tries to identify the first error steps in the solutions if any. In addition, there are other works related to PRM. For example, Yuan et al. (2024) tries to train a PRM with a fashion of outcome reward modeling (ORM). Cheng et al. (2025); Cui et al. (2025) proposed RL training frameworks that integrate PRM as finer-grained supervison. 4 Efficient Process Reward Labeling via Active Learning Labeling the process rewards for a large-scale dataset is very expensive as it either requires human experts to annotate the correctness of each step for each solution as in the previous work (Lightman et al., 2023) or leverages highly capable generative models to imitate human experts (Zhang et al., 2025). Even though the latter one is automated, it is still resource-consuming since the test time scales up with the difficulty of math problems. Algorithm 1 PRM Active Learning with Cold Start. 1: // The difference with full data tuning is colored. 2: Ensemble PRM $P_{\theta}$ , dataset $\mathcal{D}=\{(q,s)\}$ , uncertainty thresholds $\delta_{pred}$ and $\delta_{std}$ , generative LLM $M$ , batch size $B$ , learning rate $\eta$ 3: for $\mathcal{B}⊂\mathcal{D}$ do 4: $P_{\theta}(\mathcal{B})←\text{Forward}(\mathcal{B})$ 5: $\widetilde{\mathcal{B}}=\{\}$ 6: for $(q,s)∈\mathcal{B}$ do 7: if $\mathcal{U}^{\text{alea}}_{\theta}(s)\lor\mathcal{U}^{\text{epis}}_{\theta}(s)$ then $\triangleright$ Equation 5 8: $\widetilde{\mathcal{B}}←\widetilde{\mathcal{B}}\cup\{(q,s)\}$ 9: end if 10: end for 11: $Y_{\widetilde{\mathcal{B}}}←\text{labeling}(\widetilde{\mathcal{B}})$ $\triangleright$ Labeling using generative LLM 12: $\mathcal{L}←\frac{1}{|\widetilde{\mathcal{B}}|}\sum_{(s,y)∈(% \widetilde{\mathcal{B}},Y_{\widetilde{\mathcal{B}}})}{\color[rgb]{% 0.65234375,0.125,0.08984375}\mathcal{L}(s,y)}$ $\triangleright$ Equation 4 13: $∇_{\theta}\mathcal{L}←\text{Backward}(\mathcal{L})$ 14: $\theta←\theta-\eta∇_{\theta}\mathcal{L}$ 15: end for 16: $P_{\theta}$ To mitigate this issue, we propose to leverage active learning to make the PRM proactively select the data that is most informative to train on. Specifically, we train a PRM with ensemble heads to enable uncertainty estimation following Liang et al. (2022); Gleave & Irving (2022). As shown in Algorithm 1, We forward the data candidates to the ensemble PRM (line 3) and estimate the prediction uncertainty for each data point (line 5-6). Then we omit the data that the ensemble PRM is confident about (line 7) and only label the other retained data with a generative reasoning LLM (line 10). Then, we only backpropagate from the loss of labeled data (line 11). By doing so, we could considerably reduce the labeling cost while maintaining the PRM performance. Now we introduce our two key differences with the original finetuning: ensemble PRM training and uncertain data selection. Ensemble PRM Training. In this work, we use ensemble of PRMs to estimate the epistemic uncertainty following Gleave & Irving (2022); Liang et al. (2022). Specifically, we use a shared LLM backbone and build multiple binary classification heads on top of it. In our training, the diversity of ensemble models is ensured by two ways: $(i)$ the random initialization of the head layers and $(ii)$ a diversity regularization term (Dwaracherla et al., 2024): $\mathcal{L}_{\mbox{div}}=\lambda·\frac{1}{n}\sum_{i=1}^{n}||\phi^{i}-\phi_% {\text{init}}^{i}||_{2},$ where $\{\phi^{i}\}$ are the parameters of the ensemble heads and n is the number of ensemble heads. It is a $L2$ term penalizing the distance between the ensemble head parameters and their initial parameters. Our training objective for the ensemble PRM is therefore formulated as follows $$ \mathcal{L}(y,s)=\frac{1}{n}\sum_{i=1}^{n}\left(\mathcal{L}_{BCE}(y,s|\theta,% \phi^{i})+\lambda||\phi^{i}-\phi_{\text{init}}^{i}||_{2}\right), \tag{4} $$ where $\theta$ denotes the backbone parameters and $\mathcal{L}_{B}CE$ is from Equation 1, that computes the loss for a certain head. Uncertain Data Selection. Considering a batch of data candidates $\mathcal{D}=\{(q,s)\}$ , we first forward the data to the ensemble PRM $P_{\theta}$ to get the ensemble predictions $P_{\theta}(\mathcal{D})∈\mathbb{R}^{n×|\mathcal{D}|×|s|}$ . for each data $(q,s)∈\mathcal{D}$ , we could determine the hard-value aleatoric and epistemic uncertainty with pre-set thresholds. Briefly, the aleatoric (or epistemic) uncertainty is defined as 1 if uncertainty occurs at any step prior to the first predicted error step; otherwise, it is 0. A formal definition is as follows: $$ \begin{split}\mathcal{U}^{\text{alea}}_{\theta}(s)=\bigvee_{i=0}^{\mathcal{E}(% s)}\left(\max\left(\mu(P_{\theta}(s_{i})),1-\mu(P_{\theta}(s_{i}))\right)<% \delta_{pred}\right)\,\,;\,\,\mathcal{U}^{\text{epis}}_{\theta}(s)=\bigvee_{i=% 0}^{\mathcal{E}(s)}\left(\sigma(P_{\theta}(s_{i}))>\delta_{std}\right),\end{split} \tag{5} $$ where $\mu(·)$ and $\sigma(·)$ are the mean and standard deviation of the ensemble predictions among ensemble heads and $\lor$ denotes the logical ‘OR’ operation. Moreover, the $\mathcal{E}(s)$ denotes the first error step in the solution trajectory $s$ , defined as $\mathcal{E}(s)=\min\{j\mid\mu(s_{j})<\delta\}$ , where $\delta$ is the threshold for the correctness, typically set to $0.5$ . This is because we only care about the correctness of the steps before the first error step since it is genuinely difficult to define the correctness of the steps afterwards. By following the uncertainty estimation strategy, we retain the data in $\mathcal{D}$ that satisfies either $\mathcal{U}^{\text{alea}}_{\theta}$ or $\mathcal{U}^{\text{epis}}_{\theta}$ as $\widetilde{\mathcal{D}}$ . Then we could leverage expensive generative LLMs as judger (Zheng et al., 2024) to label the retained data in $\widetilde{\mathcal{D}}$ . 5 Experiments In Section 5.1, we first validate ActPRM in a pool-based active learning setting using 100K labeled samples, including ablation studies on our uncertainty estimation strategy. Based on the optimal configuration, we then scale up to 1M unlabeled samples in Section 5.2, further proving our pipeline’s efficiency and effectiveness. 5.1 Pool-Based Active Learning 5.1.1 Experimental Settings To evaluate our active learning strategy’s effectiveness, we first conduct experiments in an offline setting where ActPRM iteratively selects the most informative examples from a large unlabeled pool as detailed in Algorithm 1. We establish a strong baseline by comparing against full data tuning, where a model is trained on the complete dataset labeled by a single annotator. It is worth noting that as our data is randomly shuffled, the performance of full data tuning at intermediate training steps is essentially equivalent to the performance of random selection with the corresponding budget. Evaluation Benchmark. We utilize ProcessBench (Zheng et al., 2024) to evaluate the effectiveness of PRMs. The test data in ProcessBench contains intermediate step errors and requires the PRM to identify the first error step. ProcessBench contains four subsets, and we report the average F1 score following the original work. Models. We train ActPRM based on Qwen2.5-Math-7B-Instruct. Training Dataset. For dataset construction, we randomly select 100K data from Numinamath (Li et al., 2024) dataset after decontamination against the ProcessBench (Zheng et al., 2024) and PRMBench (Song et al., 2025). We leverage Qwen-2.5-Math-7B-Instruct to generate CoT reasoning trajectories for the selected data and further use QwQ-32B as a judge to annotate the process correctness for all trajectories following Zhang et al. (2025). For completeness, we provide the prompt template in Appendix A. 5.1.2 Experimental Results a) <details> <summary>x2.png Details</summary>  ### Visual Description ## Chart: Model Performance vs. Budget ### Overview The image is a line chart comparing the performance of two models, "ActPRM" and "Full Data Tuning," as a function of "Budget." Performance is measured by "Average F1 Score." The chart also includes horizontal and vertical dashed lines indicating specific values for F1 score and Budget, respectively. ### Components/Axes * **X-axis:** "Budget," ranging from 0.0 to 1.0 in increments of 0.2. * **Y-axis:** "Average F1 Score," ranging from 0.45 to 0.70 in increments of 0.05. * **Legend (bottom-right):** * "ActPRM" (red line with red data points) * "Full Data Tuning" (blue line with white-filled blue data points) * **Horizontal dashed line:** Indicates F1 = 0.673. * **Vertical dashed line:** Indicates Budget = 0.5. * **Text labels:** "F1 = 0.673" (near the horizontal line) and "Budget = 0.5" (near the vertical line). ### Detailed Analysis * **ActPRM (red):** * Trend: The red line generally slopes upward, indicating increasing F1 score with increasing budget. The slope decreases as the budget approaches 1.0. * Data points: * Budget 0.0: F1 score approximately 0.50 * Budget 0.2: F1 score approximately 0.62 * Budget 0.4: F1 score approximately 0.65 * Budget 0.6: F1 score approximately 0.68 * Budget 0.8: F1 score approximately 0.67 * Budget 1.0: F1 score approximately 0.67 * **Full Data Tuning (blue):** * Trend: The blue line also generally slopes upward, but its increase is less steep than the red line, especially at lower budget values. * Data points: * Budget 0.0: F1 score approximately 0.48 * Budget 0.2: F1 score approximately 0.59 * Budget 0.4: F1 score approximately 0.64 * Budget 0.6: F1 score approximately 0.64 * Budget 0.8: F1 score approximately 0.66 * Budget 1.0: F1 score approximately 0.67 ### Key Observations * At lower budget values (e.g., 0.2), ActPRM outperforms Full Data Tuning. * As the budget increases, the performance of both models converges. * The F1 score of ActPRM plateaus around 0.67 for budgets greater than 0.6. * The F1 score of Full Data Tuning appears to be still increasing slightly at a budget of 1.0. * The horizontal dashed line intersects the ActPRM line near a budget of 0.6 and the Full Data Tuning line near a budget of 1.0. ### Interpretation The chart suggests that ActPRM is more efficient at lower budget levels, achieving a higher F1 score compared to Full Data Tuning with the same budget. However, as the budget increases, the performance gap between the two models diminishes. The horizontal line at F1 = 0.673 indicates a target performance level. ActPRM reaches this level at a lower budget than Full Data Tuning, suggesting it's a more cost-effective option if the goal is to achieve an F1 score of approximately 0.67. The vertical line at Budget = 0.5 could represent a budget constraint, and the chart shows the relative performance of each model under that constraint. </details> ( ( b) <details> <summary>x3.png Details</summary>  ### Visual Description ## Line Chart: Average F1 Score vs. Budget for Different Models ### Overview The image is a line chart comparing the average F1 score of three different models as a function of budget. The chart displays how the performance of each model changes with increasing budget allocation. The models are distinguished by their standard deviation (std) and p-value (p). ### Components/Axes * **X-axis (Horizontal):** "Budget". The scale ranges from approximately 0 to 0.6, with tick marks at 0.2, 0.4, and 0.6. * **Y-axis (Vertical):** "Average F1 Score". The scale ranges from 0.45 to 0.70, with tick marks at 0.45, 0.50, 0.55, 0.60, 0.65, and 0.70. * **Legend (Bottom-Right):** A box labeled "Model" that identifies the three models represented by different colored lines: * Red line with circle markers: "(std: 0.005, p: 0.95)" * Blue line with triangle markers: "(p: 0.95)" * Teal line with square markers: "(std: 0.005)" ### Detailed Analysis * **Red Line (std: 0.005, p: 0.95):** This line generally slopes upward, indicating an increase in the average F1 score as the budget increases. * At a budget of approximately 0.1, the F1 score is around 0.51. * At a budget of approximately 0.2, the F1 score is around 0.60. * At a budget of approximately 0.4, the F1 score is around 0.65. * At a budget of approximately 0.6, the F1 score is around 0.68. * **Blue Line (p: 0.95):** This line also slopes upward, showing an improvement in the average F1 score with increasing budget. * At a budget of approximately 0.1, the F1 score is around 0.47. * At a budget of approximately 0.2, the F1 score is around 0.56. * At a budget of approximately 0.4, the F1 score is around 0.62. * At a budget of approximately 0.6, the F1 score is around 0.65. * **Teal Line (std: 0.005):** This line shows an upward trend, similar to the other two lines. * At a budget of approximately 0.1, the F1 score is around 0.48. * At a budget of approximately 0.2, the F1 score is around 0.59. * At a budget of approximately 0.4, the F1 score is around 0.64. * At a budget of approximately 0.6, the F1 score is around 0.65. ### Key Observations * The red line (std: 0.005, p: 0.95) consistently outperforms the other two models across the range of budgets. * The blue line (p: 0.95) generally has the lowest average F1 score compared to the other two models. * All three models show diminishing returns as the budget increases, with the rate of improvement slowing down at higher budget levels. * The teal line (std: 0.005) and the blue line (p: 0.95) converge to similar F1 scores at higher budget levels. ### Interpretation The chart demonstrates the relationship between budget allocation and model performance, as measured by the average F1 score. The model with (std: 0.005, p: 0.95) appears to be the most effective, achieving the highest F1 scores across different budget levels. The diminishing returns suggest that there is a point beyond which increasing the budget yields only marginal improvements in model performance. The convergence of the teal and blue lines at higher budgets indicates that the impact of 'p' becomes less significant as the budget increases, while the standard deviation remains a more critical factor. </details> c) <details> <summary>x4.png Details</summary>  ### Visual Description ## Bar Chart: Average F1 Score vs. Number of Heads ### Overview The image is a bar chart showing the relationship between the average F1 score and the number of heads. The chart displays red bars representing the average F1 score for different numbers of heads (2, 4, 8, 16, 32, and 64). Error bars are included on each bar to indicate the variability or uncertainty in the F1 score. ### Components/Axes * **X-axis:** "#Heads" with values 2, 4, 8, 16, 32, and 64. * **Y-axis:** "Average F1 Score" with values ranging from 0.600 to 0.650, with a gridline at 0.625. * **Bars:** Red bars representing the average F1 score for each number of heads. * **Error Bars:** Black vertical lines extending above and below each bar, indicating the standard deviation or confidence interval. ### Detailed Analysis The chart shows how the average F1 score changes as the number of heads increases. * **#Heads = 2:** Average F1 Score is approximately 0.600, with an error bar extending from approximately 0.585 to 0.615. * **#Heads = 4:** Average F1 Score is approximately 0.624, with an error bar extending from approximately 0.618 to 0.630. * **#Heads = 8:** Average F1 Score is approximately 0.635, with an error bar extending from approximately 0.628 to 0.642. * **#Heads = 16:** Average F1 Score is approximately 0.638, with an error bar extending from approximately 0.632 to 0.644. * **#Heads = 32:** Average F1 Score is approximately 0.653, with an error bar extending from approximately 0.647 to 0.659. * **#Heads = 64:** Average F1 Score is approximately 0.652, with an error bar extending from approximately 0.646 to 0.658. ### Key Observations * The average F1 score generally increases as the number of heads increases from 2 to 32. * The average F1 score plateaus or slightly decreases when the number of heads increases from 32 to 64. * The error bars suggest that the variability in F1 score is relatively consistent across different numbers of heads. ### Interpretation The data suggests that increasing the number of heads initially improves the average F1 score, indicating better model performance. However, there appears to be a point of diminishing returns, as increasing the number of heads beyond 32 does not lead to a significant improvement and may even slightly decrease the F1 score. This could be due to overfitting or increased complexity without a corresponding increase in useful information. The error bars provide an indication of the uncertainty in these average scores, which should be considered when drawing conclusions. </details> ( Figure 2: (a) Comparison of the average F1 score on ProcessBench between ActPRM and random selection, plotted against the normalized budget positively correlated the number of labeled data instances consumed. The semi-transparent points represent all results in grid searching w.r.t. $\delta_{pred}$ and $\delta_{std}$ . For the highlighted ActPRM curve in the figure, $\delta_{pred}=0.95$ and $\delta_{std}=0.005$ . (b) Ablation: uncertainty estimation strategies. (c) Ablation: number of ensemble PRM heads. ActPRM achieves comparable performance while reducing annotation costs by 50%. We compare ActPRM with full data tuning across a normalized budget, as illustrated in Figure 2 (a). The results demonstrate that ActPRM achieves an average F1 score of $0.673$ on ProcessBench, matching baseline performance while using only half the annotation budget. Furthermore, ActPRM consistently outperforms random selection under the same budget constraints. Notably, at 50% budget, ActPRM surpasses random selection by a significant margin of 3.3%. at the end of pool-based active training, ActPRM achieves a better performance of 0.680 on ProcessBench while consuming solely 62.5% budget. ActPRM Consistently Outperforms Random Selection Under Diverse $\delta_{pred}$ and $\delta_{std}$ . As shown in Figure 2 (a), the semi-transparent blue points represent all results of a grid searching over $\delta_{pred}∈\{0.9,0.95,0.97\}$ and $\delta_{std}∈\{0.01,0.005,0.002,0.001\}$ . One can see that most blue points are above the baseline (gray line) with the same budget, further demonstrating the effectiveness and robustness of ActPRM. Ablation Study on Uncertainty Estimation Strategies. We conduct an ablation study on uncertainty estimation strategies, i.e. using epistemic and aleatoric uncertainty. We selected the best setting ( $\delta_{std}=0.005,\delta_{pred}=0.95$ ) searched by a grid search as in Figure 2 and ablates epistemic and aleatoric uncertainty by setting $\delta_{std}=∈f$ and $\delta_{pred}=0.5$ , respectively. As shown illustrated in Figure 2 (b), solely use either epistemic or aleatoric uncertainty under-performs using both, indicating that epistemic and aleatoric uncertainty are complementary to each other. Ablation Study on Number of Heads for Ensemble PRM. The number of heads for ensemble PRM controls how accurate our estimated epistemic uncertainty is. To find the trade-off between good estimation and computational overhead, we conduct an ablation study regarding it and show the results in Figure 2 (c), where we only consider epistemic uncertainty by setting $\delta_{std}=0.005,\delta_{pred}=0.5$ and report the averaged results with 3 runs. We empirically find that the performance continually grows with the number of heads and converges at about 32. 5.2 Achieving New SOTA Performance on ProcessBench (75.0%) with Solely 6% Annotation Cost. <details> <summary>x5.png Details</summary>  ### Visual Description ## Bar Chart: Estimated Annotation Cost Comparison ### Overview The image is a bar chart comparing the estimated annotation cost (in generated tokens) for different methods: "Ours", "Ensemble Prompting", "Math Shepherd", and "Consensus Filtering". The y-axis represents the estimated annotation cost, scaled in scientific notation. The chart uses red bars to represent the cost for each method, with numerical multipliers displayed above each bar indicating the relative cost compared to the "Ours" method. ### Components/Axes * **X-axis:** Categorical axis representing different methods: "Ours", "Ensemble Prompting", "Math Shepherd", and "Consensus Filtering". * **Y-axis:** Numerical axis representing "Est. Annotation Cost (Gen. Token)". The scale ranges from 0.0e+00 to 2.0e+10, with increments at 5.0e+09, 1.0e+10, and 1.5e+10. * **Bars:** Red bars representing the estimated annotation cost for each method. * **Multipliers:** Numerical values above each bar indicating the relative cost compared to the "Ours" method. ### Detailed Analysis * **Ours:** The bar for "Ours" starts at approximately 0.0e+00 and extends to an estimated value of approximately 0.3e+09. * **Ensemble Prompting:** The bar for "Ensemble Prompting" extends to approximately 4.8x the cost of "Ours". * **Math Shepherd:** The bar for "Math Shepherd" extends to approximately 15.9x the cost of "Ours". * **Consensus Filtering:** The bar for "Consensus Filtering" extends to approximately 17.3x the cost of "Ours". ### Key Observations * The "Ours" method has the lowest estimated annotation cost. * "Ensemble Prompting" has a significantly higher cost than "Ours", but lower than "Math Shepherd" and "Consensus Filtering". * "Math Shepherd" and "Consensus Filtering" have the highest estimated annotation costs, with "Consensus Filtering" being slightly higher. ### Interpretation The bar chart demonstrates that the "Ours" method is the most cost-effective in terms of estimated annotation cost (generated tokens) compared to the other methods. "Ensemble Prompting" offers a moderate increase in cost, while "Math Shepherd" and "Consensus Filtering" incur substantially higher costs. This suggests that the "Ours" method is a more efficient approach for annotation, potentially due to optimized token generation or reduced annotation requirements. The multipliers above each bar provide a clear comparison of the relative cost increase for each method compared to the baseline "Ours" method. </details> Figure 3: Estimated annotation costs (generated tokens) comparison between ActPRM and popular methods, including Ensemble Prompting (Tan et al., 2025), MathShepherd (Wang et al., 2024) and Consensus Filtering (Zhang et al., 2025). Obtaining high-quality process supervision labels is costly. To demonstrate the efficiency of ActPRM, we evaluate it in a one-shot active learning setting. Starting with the model trained in Section 5.1, we select the most uncertain samples from over 1M+ unlabeled examples and annotate them using a powerful reasoning model. Figure 3 compares our estimated labeling costs with those of other real-world datasets for training PRMs, including MathShepherd (Wang et al., 2024), Consensus Filtering (Zhang et al., 2025), and Ensemble Prompting (Tan et al., 2025). Since the training data for Consensus Filtering is not publicly available, we estimate costs based on our data statistics. We introduce our estimation strategy in Appendix C. Training a Qwen2.5-Math-7B-Instruct on our data, Ensemble Prompting data, MathShepherd data, and Consensus Filtering data yields ActPRM, UniversalPRM, Qwen2.5-Math-7B-Math-Shepherd, and Qwen2.5-Math-PRM-7B in Table 1. We evaluated the performance of models trained on this labeled data on both ProcessBench and PRMBench benchmarks. 5.2.1 Experimental Settings Data Filtering with ActPRM. We used Qwen2.5-Math-7B-Instruct and Qwen2.5-Math-72B-Instruct to collect over 1 million (1,061,763) Chain-of-Thought (COT) trajectories from the Numinamath problem set (Li et al., 2024), after decontamination against the test benchmarks. ActPRM was then applied to filter out high-confidence ( $\delta_{pred}>0.95$ and $\delta_{std}<0.005$ following Section 5.1) data instances that were unnecessary for training, retaining the remaining data for labeling and training. This process resulted in a final dataset of 563,030 PRM data points labeled by QwQ-32B, reducing annotation costs by 47.0%. Models. Obtaining the dataset, we continually train our ActPRM in Section 5.1 on our filtered dataset. In addition, we empirically find that our retrained data is generally useful to other PRMs. Specifically, we also continually train Qwen2.5-Math-PRM-7B (the previous SOTA model on ProcessBench) on our constructed data. The resultant model is named ActPRM-X X stands for extended version.. Benchmarks. We use ProcessBench (Zheng et al., 2024) and PRMBench (Song et al., 2025) to evaluate the effectiveness of our trained model. Different from ProcessBench that collects intermediate errors from real-world generative models, PRMBench heuristically builds intermediate errors by manipulating correct steps. Baselines. We compare with the following PRMs: ❶ Qwen2.5-Math-PRM-7B (Zhang et al., 2025): This model uses consensus filtering for labeling. It labels 860K data twice using two methods (LLM-as-judge [Zheng et al., 2024] and Mathshepherd [Wang et al., 2024]) and filters out 40% of the data where the labels disagree. ❷ Pure-PRM-7B (Cheng et al., 2025): A Qwen2.5-Math-based PRM trained on PRM800K using a two-stage strategy: warming up the PRM head and then fine-tuning the entire model. ❸: EurusPRM-Stage2 (Cui et al., 2025): A PRM resulting from the Implicit PRM approach (Yuan et al., 2024), which derives process rewards from an ORM. ❹ Universal-PRM (Tan et al., 2025): A Qwen2.5-Math-based model trained with data augmentation techniques like ensemble prompting and reverse verification. ❺ Math-Shepherd-PRM-7B (Wang et al., 2024): a PRM trained on process labels that estimates hard Q-values for the policy model. ❻ Qwen2.5-Math-7B-Math-Shepherd (Zhang et al., 2025): a PRM trained on 860K data labeled using MathShepherd. ❼ Ensemble-PRM-PRM800K (ours): a model with ensemble heads trained by ourselves on PRM800K without active learning. 5.2.2 Experimental Results | Models | GSM8K | MATH | Olympiad Bench | OmniMath | Average F1 | | --- | --- | --- | --- | --- | --- | | LLM-as-judge | | | | | | | o1-Mini ⋄ | 0.932 | 0.889 | 0.872 | 0.824 | 0.879 | | Deepseek-R1-Distill-32B | 0.817 | 0.739 | 0.659 | 0.585 | 0.700 | | QwQ-32B | 0.871 | 0.834 | 0.787 | 0.771 | 0.816 | | Process Reward Models (72B) | | | | | | | Qwen2.5-Math-PRM-72B ⋄ | 0.873 | 0.806 | 0.743 | 0.711 | 0.783 | | Process Reward Models (7B+) | | | | | | | Math-Shepherd-PRM-7B ⋄ | 0.479 | 0.295 | 0.248 | 0.238 | 0.315 | | Qwen2.5-Math-7B-Math-Shepherd ⋄ | 0.625 | 0.316 | 0.137 | 0.077 | 0.289 | | EurusPRM-Stage2 ⋄ | 0.473 | 0.357 | 0.212 | 0.209 | 0.313 | | Qwen2.5-Math-7B-PRM800K ⋄ | 0.683 | 0.626 | 0.507 | 0.443 | 0.565 | | Ensemble-PRM-PRM800K (ours) | 0.705 | 0.630 | 0.472 | 0.433 | 0.560 | | PURE-PRM-7B | 0.690 | 0.665 | 0.484 | 0.459 | 0.575 | | Qwen2.5-Math-PRM-7B ⋄ | 0.824 | 0.776 | 0.675 | 0.663 | 0.735 | | Universal-PRM | 0.858 | 0.777 | 0.676 | 0.664 | 0.743 | | \hdashline ActPRM (ours) | 0.816 | 0.798 | 0.714 | 0.670 | 0.750 | | ActPRM-X (ours) | 0.827 | 0.820 | 0.720 | 0.673 | 0.760 | Table 1: Performance comparison on ProcessBench. We report the results in the same calculation method with ProcessBench. ⋄ denotes the results are from Qwen PRM’s report (Zhang et al., 2025). | # | Models | Simlicity | Soundness | Sensitivity | Average | | --- | --- | --- | --- | --- | --- | | LLM-as-judge | | | | | | | 1 | Gemini-2.0-thinking-exp-1219 | 0.662 | 0.718 | 0.753 | 0.688 | | 1 | o1-mini | 0.646 | 0.721 | 0.755 | 0.688 | | 4 | GPT-4o | 0.597 | 0.709 | 0.758 | 0.668 | | 6 | Gemini-2.0-flash-exp | 0.627 | 0.673 | 0.754 | 0.660 | | Process Reward Models (72B) | | | | | | | 3 | Qwen-2.5-Math-PRM-72B | 0.546 | 0.739 | 0.770 | 0.682 | | Process Reward Models (7B+) | | | | | | | 7 | Qwen2.5-Math-PRM-7B | 0.521 | 0.710 | 0.755 | 0.655 | | 9 | Pure-PRM-7B | 0.522 | 0.702 | 0.758 | 0.653 | | \hdashline 7 | ActPRM (ours) | 0.536 | 0.713 | 0.752 | 0.655 | | 5 | ActPRM-X (ours) | 0.545 | 0.727 | 0.756 | 0.667 | Table 2: Performance comparison on PRMBench. All results of the other models are from the official leaderboard. # denotes the ranking. ActPRM and ActPRM-X achieve new SOTA performance on ProcessBench compared with same size models. The evaluation results on ProcessBench are shown in Table 1. ActPRM achieves an average F1 score of $0.750$ , outperforming Qwen2.5-Math-PRM-7B by a margin of 1.5%. Furthermore, ActPRM-X training based on Qwen2.5-Math-PRM-7B achieves a new SOTA performance on ProcessBench with an average F1 of 0.760, outperforming the second-place model (Universal-PRM) with a margin of 1.7% and improve the performance of Qwen2.5-Math-PRM-7B by a significant margin of 2.5%. QwQ-32B (our PRM label annotator) outperforms all PRMs on ProcessBench. As shown in Table 1, QwQ-32B outperforms all PRMs including 72B models. It indicates the reliability of utilizing QwQ-32B as a PRM label annotator as it provides a high empirical upperbound for the training PRMs. ActPRM-X achieves new SOTA performance on PRMBench, on-par with GPT-4o. We further test our models on PRMBench and show the results in Table 2. As on the leaderboard, ActPRM achieves the best performance within 7B PRMs and ActPRM-X achieves new SOTA performance (0.667), outperforming the other models by a large margin of at least $1.2\%$ and on-par with GPT-4o (OpenAI et al., 2024a). <details> <summary>x6.png Details</summary>  ### Visual Description ## Line Charts: Performance Comparison of ActPRM Selected vs. Randomly Selected Data ### Overview The image contains two line charts comparing the performance of "ActPRM Selected" data against "Randomly Selected" data. The left chart displays the "Average F1 Score" over "Step" iterations, while the right chart shows the "Training Loss" over "Step" iterations. Both charts share a common x-axis ("Step") and a legend that distinguishes between the two data selection methods. ### Components/Axes **Left Chart:** * **Title:** Average F1 Score vs. Step * **X-axis:** * Label: "Step" * Scale: 500, 1000, 1500, 2000 * **Y-axis:** * Label: "Average F1 Score" * Scale: 0.67, 0.68, 0.69, 0.70, 0.71, 0.72, 0.73 * **Legend:** Located in the bottom-right of the left chart. * "ActPRM Selected" - Represented by a red line. * "Randomly Selected" - Represented by a blue line. **Right Chart:** * **Title:** Training Loss vs. Step * **X-axis:** * Label: "Step" * Scale: 0, 500, 1000, 1500, 2000 * **Y-axis:** * Label: "Training Loss" * Scale: 0.10, 0.15 * **Legend:** Located in the top-right of the right chart. * "ActPRM Selected" - Represented by a red line. * "Randomly Selected" - Represented by a blue line. ### Detailed Analysis **Left Chart (Average F1 Score):** * **ActPRM Selected (Red Line):** The line slopes upward, indicating an increasing F1 score as the step increases. * Step 500: F1 Score ≈ 0.68 * Step 1000: F1 Score ≈ 0.71 * Step 1500: F1 Score ≈ 0.725 * Step 2000: F1 Score ≈ 0.73 * **Randomly Selected (Blue Line):** The line initially slopes upward, then plateaus and slightly decreases. * Step 500: F1 Score ≈ 0.67 * Step 1000: F1 Score ≈ 0.69 * Step 1500: F1 Score ≈ 0.705 * Step 2000: F1 Score ≈ 0.70 **Right Chart (Training Loss):** * **ActPRM Selected (Red Line):** The line generally slopes downward with significant fluctuations, indicating a decreasing training loss over time. The line is consistently above the "Randomly Selected" line. * The training loss starts around 0.16 and decreases to approximately 0.135 by step 2000. * **Randomly Selected (Blue Line):** The line slopes downward with fluctuations, indicating a decreasing training loss over time. The line is consistently below the "ActPRM Selected" line. * The training loss starts around 0.10 and decreases to approximately 0.075 by step 2000. ### Key Observations * In the left chart, the "ActPRM Selected" method consistently outperforms the "Randomly Selected" method in terms of Average F1 Score. * In the right chart, the "Randomly Selected" method consistently achieves a lower Training Loss compared to the "ActPRM Selected" method. * Both methods show a decreasing trend in Training Loss as the number of steps increases. * The "ActPRM Selected" method shows a more pronounced increase in F1 score compared to the "Randomly Selected" method. ### Interpretation The data suggests that while the "ActPRM Selected" method leads to a higher Average F1 Score (better performance), it also results in a higher Training Loss compared to the "Randomly Selected" method. This could indicate that the "ActPRM Selected" method is more effective at learning relevant features, but requires more computational resources or is more prone to overfitting. The "Randomly Selected" method, on the other hand, might be more stable and less resource-intensive, but at the cost of lower overall performance. The relationship between the two charts highlights a potential trade-off between performance (F1 score) and efficiency (training loss). Depending on the specific application and constraints, one method might be preferred over the other. The trends observed suggest that both methods benefit from increased training steps, as evidenced by the decreasing training loss. </details> Figure 4: ProcessBench performance (left) and training loss (right): ActPRM v.s. random data selection on 1M NuminaMath Rollouts. 5.2.3 Comparative Experiment with Random Selection A potential concern is that while ActPRM achieves state-of-the-art (SOTA) performance on several benchmarks, this success might be attributed solely to the high quality of our collected data pool, rather than the method itself. To address this, we conducted a comparative study with random selection. Specifically, we randomly selected 256K data points from our retained dataset as the experimental group. For the control group, we randomly selected the same number of data points from the entire data pool (over 1M) and used the same annotator to label any unlabeled data (i.e., data not in the retained set). We then continually train ActPRM checkpoint, as in Sec 5.1, on both datasets. The results, including performance on ProcessBench and training loss, are shown in Figure 4. ActPRM outperforms random selection of the same amount of data. As illustrated in Figure 4 (left), the model trained on data selected by ActPRM consistently achieves significantly better results than the model trained on randomly selected data. To further validate this, we compare their training losses in Figure 4 (right). The model trained on ActPRM -selected data exhibits a consistently higher training loss, with a margin of 0.05, suggesting that the data selected by ActPRM is more challenging and informative, thereby enhancing the learning process. 6 Conclusion and Future Work In this work, we address the high annotation costs associated with training Process Reward Models (PRMs) by proposing ActPRM, an uncertainty-aware active learning framework that selectively annotates the most informative reasoning steps. By leveraging an ensemble PRM to estimate uncertainty and strategically labeling only uncertain data, ActPRM significantly reduces annotation costs while maintaining competitive performance. Extensive experiments demonstrate that ActPRM achieves a new state-of-the-art (75.0% on ProcessBench) with merely at most 20% of the labeling budget required by prior methods. Our results highlight the potential of efficient data selection for scalable PRM training, and we commit to releasing all models, datasets, and code to foster further research in this direction. To further enhance PRM’s performance, several promising directions can be explored. First, leveraging larger base models and more advanced LLM judges (e.g., O1-mini) could yield significant improvements. Second, implementing the framework in an online setting would ultimately enable PRM to iteratively refine its performance through active learning. 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(2025) Zhenru Zhang, Chujie Zheng, Yangzhen Wu, Beichen Zhang, Runji Lin, Bowen Yu, Dayiheng Liu, Jingren Zhou, and Junyang Lin. The Lessons of Developing Process Reward Models in Mathematical Reasoning, 2025. - Zheng et al. (2024) Chujie Zheng, Zhenru Zhang, Beichen Zhang, Runji Lin, Keming Lu, Bowen Yu, Dayiheng Liu, Jingren Zhou, and Junyang Lin. ProcessBench: Identifying Process Errors in Mathematical Reasoning, 2024. Appendix A LLM-as-Judger Prompt Template For LLM-as-Judger, we follow the prompt in Zhang et al. (2025). ⬇ I will provide a math problem along with a solution. They will be formatted as follows: [Math Problem] < math_problem > ...(math problem)... </ math_problem > [Solution] < paragraph_1 > ...(paragraph 1 of solution)... </ paragraph_1 > ... < paragraph_n > ...(paragraph n of solution)... </ paragraph_n > Your task is to review each paragraph of the solution in sequence, analyzing, verifying, and critiquing the reasoning in detail. You need to provide the analyses and the conclusion in the following format: < analysis_1 > ...(analysis of paragraph 1)... </ analysis_1 > ... < analysis_n > ...(analysis of paragraph n)... </ analysis_n > < conclusion > Correct / Incorrect </ conclusion > * When you analyze each paragraph, you should use proper verification, recalculation, or reflection to indicate whether it is logically and mathematically valid. Please elaborate on the analysis process carefully. * If an error is detected in any paragraph, you should describe the nature and cause of the error in detail, and suggest how to correct the error or the correct approach. Once a paragraph is found to contain any error, stop further analysis of subsequent paragraphs (as they may depend on the identified error) and directly provide the conclusion of " Incorrect." For instance, given a solution of five paragraphs, if an error is found in the third paragraph, you should reply in the following format: < analysis_1 > ...(analysis of paragraph 1)... </ analysis_1 > < analysis_2 > ...(analysis of paragraph 2)... </ analysis_3 > < analysis_3 > ...(analysis of paragraph 3; since an error is found here, also provide detailed critique and correction guideline)... </ analysis_3 > < conclusion > Incorrect </ conclusion > Note that the analyses of paragraphs 4 and 5 should be skipped as the paragraph 3 has been found to contain an error. * Respond with your analyses and conclusion directly. -------------------------------------------------- The following is the math problem and the solution for your task: [Math Problem] {tagged_problem} [Solution] {tagged_response} Appendix B More Experiment Results B.1 Problem diversity is important for Training PRMs PRM800K (Lightman et al., 2023) is a widely used and human-annotated dataset for PRM training, which contains 800K step-level labels across 75K tree-of-thoughts solutions to 12K MATH (Hendrycks et al., 2021). Our empirical results show that models trained on our dataset (100K samples from 100K diverse questions) consistently and significantly outperform those trained on PRM800K https://huggingface.co/datasets/HuggingFaceH4/prm800k-trl-dedup (369K samples from only 12K questions) on ProcessBench. These findings suggest that problem diversity plays a more crucial role in PRM training than the number of step-level annotations. | | # Problem set | # CoT Trajectories | ProcessBench F1 score | | --- | --- | --- | --- | | PRM800K | 7,500 | 460,000 | 0.575 | | NuminaMath (Random Selected) | 100,000 | 100,000 | 0.673 | Table 3: Comparison between PRM800K and 100K data collected from NuminaMath labeled by Qwen-QwQ. Appendix C Annotation Cost Estimation We estimate the labeling cost based on the statistics of our 1M data collected from NuminaMath Li et al. (2024) using Qwen2.5-Math-7B-Instruct and Qwen2.5-Math-72B-Instruct. We introduce the statistics in Table 4. | | Value | Source | | --- | --- | --- | | # Reasoning Steps ( $S$ ) | 8.845 | Qwen Models’ rollouts | | # Tokens per Rollout ( $R$ ) | 625.098 | Qwen Models’ rollouts | | # Tokens per Critic Response from Judge ( $C$ ) | 1,919.860 | Qwen-QwQ’s responses as LLM-as-Judge | Table 4: Statistics of 1M NuminaMath CoT Trajectories collected by Qwen2.5-Math Models. In addition to the statistics, we also use $N$ to denote the data number of the dataset and show this statistic of each model’s training dataset in Table 5 | Dataset | # Labeled Data | | --- | --- | | ActPRM | 624,000 (labeled in two stages) | | Qwen2.5-Math-PRM-Math-shepherd | 860,000 | | Qwen2.5-Math-PRM | 860,000 | | UniversalPRM | 690,000 | Table 5: Data number of datasets. Using the statistics, we compute the estimated labeling cost for ActPRM, Qwen2.5-Math-PRM-Math-shepherd (Zhang et al., 2025), Qwen2.5-Math-PRM (Zhang et al., 2025), UniversalPRM Tan et al. (2025) as follows: - Qwen2.5-Math-PRM-Math-shepherd: $N× S× 8× R/2$ , where $8$ is the number of rollouts per step set in Zhang et al. (2025). We divided by two since the number of tokens for rollouts varies based on the position of reasoning step. For latter reasoning step, it requires less reasoning tokens. As a result, the expectation of tokens per rollout should be half of the number of tokens of the complete rollout. - Qwen2.5-Math-PRM: $N× S× 8× R/2+N*C$ . It used consensus filtering for each data, the cost is both from MathShepherd ( $S× 8× R/2$ ) and LLM-as-Judge (C). - UniversalPRM: $N× C× 4+N× S$ , where 4 is the number of ensemble prompts from the original paper (Tan et al., 2025) and another $N× S$ is for its semantic-based step seperation. - ActPRM: $N× C$ . We solely use Qwen-QwQ as Judge and do not include any other operations.