2508.04088

# GM-PRM: A Generative Multimodal Process Reward Model for Multimodal Mathematical Reasoning > Corresponding author. Abstract Multimodal Large Language Models (MLLMs) demonstrate remarkable capabilities but often struggle with complex, multi-step mathematical reasoning, where minor errors in visual perception or logical deduction can lead to complete failure. While Process Reward Models (PRMs) offer step-by-step supervision, existing multimodal PRMs are limited to being binary verifiers that can identify but not correct errors, offering little explanatory power. To address these deficiencies, we introduce the Generative Multimodal Process Reward Model (GM-PRM), a novel paradigm that transforms the PRM from a passive judge into an active reasoning collaborator. Instead of a simple scalar score, GM-PRM provides a fine-grained, interpretable analysis of each reasoning step, evaluating its step intent, visual alignment, and logical soundness. More critically, GM-PRM is trained to generate a corrected version of the first erroneous step it identifies. This unique corrective capability enables our new test-time inference strategy, Refined Best-of-N (Refined-BoN). This framework actively enhances solution quality by using the PRM’s generated correction to guide the policy model toward a more promising reasoning trajectory, thereby improving the diversity and correctness of the solution pool. We demonstrate that GM-PRM achieves state-of-the-art results on multiple multimodal math benchmarks, significantly boosting policy model performance with remarkable data efficiency, requiring only a 20K-sample training dataset. Our code will be released upon acceptance. 1 Introduction The advent of Multimodal Large Language Models (MLLMs) has marked a significant milestone in artificial intelligence, demonstrating remarkable capabilities in integrating and understanding visual and textual information (Caffagni et al. 2024; Yan et al. 2024c; Yan and Lee 2024; Huo et al. 2024; Zheng et al. 2024b). While these models excel at general-purpose tasks such as image captioning and visual question answering, they often falter when confronted with complex, multi-step reasoning challenges, particularly within specialized domains like mathematics (Wang et al. 2024; Yan et al. 2024a, 2025a; Ahn et al. 2024). Solving multimodal mathematical problems requires not only accurate perception of visual elements (e.g., geometric figures, function graphs) but also a rigorous, step-by-step logical deduction process (Shi et al. 2024; Zhuang et al. 2025; Yan et al. 2025b). Minor errors in either image interpretation or logical inference can cascade, leading to entirely incorrect final answers. <details> <summary>figures/comparison.png Details</summary>  ### Visual Description # Technical Document Analysis: Multimodal Math Problem-Solving Reward Models ## Overview The image presents a comparative analysis of three reward models for evaluating multimodal math problem-solving processes. The diagram uses flowcharts with labeled components, directional arrows, and annotations to illustrate differences in reward mechanisms and correction capabilities. --- ## Diagram Components ### (a) Outcome Reward Model **Title**: Outcome Reward Model **Key Components**: - **Question** (Input: Math equation "E=mc²") - **Output** (Intermediate processing step) - **Answer** (Final response) - **ORM** (Robot icon with "Reward" arrow) **Flow**: ``` Question → Output → Answer → Reward (only for final answers) ``` **Annotations**: - Red text: "ONLY Reward Final Ans" - ORM robot has a smiley face --- ### (b) Multimodal Process Reward Model **Title**: Multimodal Process Reward Model **Key Components**: - **Multimodal Math Qns** (Input: Math problems with equations and diagrams) - **Steps 1, 2, ..., T** (Intermediate reasoning steps) - **Answer** (Final response) - **PRM** (Robot icon with "Reward" arrow) **Flow**: ``` Multimodal Math Qns → Step 1 → Step 2 → ... → Step T → Answer → Reward ``` **Annotations**: - Red text box: "Limited Explainability" - Red text box: "No Correction Mechanism" - PRM robot has a neutral expression --- ### (c) GM-PRM (Ours) **Title**: GM-PRM (Ours) **Key Components**: - **Multimodal Math Qns** (Input: Math problems with equations and diagrams) - **Steps 1, 2, ..., T** (Intermediate reasoning steps) - **1st incorrect step** (Highlighted with dashed arrow) - **After Correction** (Dashed arrow indicating refinement) - **Refined BoN** (Refined version of the answer) - **GM-PRM** (Robot icon with "Reward" arrow) **Flow**: ``` Multimodal Math Qns → Step 1 → Step 2 → ... → Step T → Answer → Reward ``` **Sub-process**: ``` Analysis & Judgement → Step Intent → Image Alignment → Reasoning Logic ``` **Annotations**: - Red text: "Refined & Corrected Version" - Dashed arrows indicate iterative refinement - GM-PRM robot has a smiling face with cheeks --- ## Key Differences Between Models | Feature | Outcome Reward Model (a) | Multimodal Process Reward Model (b) | GM-PRM (c) | |------------------------|--------------------------|-------------------------------------|--------------------------------| | **Reward Scope** | Final answer only | All steps | All steps + corrections | | **Explainability** | Not specified | Limited | Improved through analysis | | **Correction** | No | No | Yes (via "After Correction") | | **Robot Icon** | ORM (smiley) | PRM (neutral) | GM-PRM (smiling with cheeks) | --- ## Spatial Grounding & Component Isolation 1. **Header**: Each section title (a, b, c) is bolded and positioned at the top of its respective flowchart. 2. **Main Chart**: - Components are arranged left-to-right with vertical stacking for sub-processes (e.g., "Analysis & Judgement" in c). - Arrows indicate sequential flow (solid for primary flow, dashed for corrections). 3. **Footer**: No explicit footer, but annotations are placed in red text boxes near relevant components. --- ## Trend Verification - **No numerical data** present; trends are represented through structural differences in component connectivity and annotation emphasis. --- ## Conclusion The diagram demonstrates a progression from simple outcome-based evaluation (a) to more sophisticated process-aware models (b and c). The proposed GM-PRM (c) introduces explicit correction mechanisms and multi-stage analysis, addressing limitations in explainability and adaptability observed in prior models. </details> Figure 1: Comparison among ORM (a), PRM (b), and our proposed GM-PRM (c) for multimodal math reasoning. To mitigate these reasoning deficiencies, Process Reward Models (PRMs) have emerged as a promising paradigm (Gao et al. 2024; Zhong et al. 2025). Unlike outcome-based models that only reward correct final answers (shown in Figure 1 (a)), PRMs provide fine-grained supervision by evaluating the correctness of each intermediate step in a reasoning chain (Zheng et al. 2024a; Lambert et al. 2024; Yan et al. 2024b), as shown in Figure 1 (b). This approach has proven effective in the language domain (Zeng et al. 2025; Yuan et al. 2024; Zhang et al. 2025a). However, extending PRMs to the multimodal context presents unique challenges (Miao et al. 2025; Du et al. 2025; Li et al. 2025b; Cao et al. 2025). Existing multimodal PRMs often function as binary classifiers, assigning a simple correct/incorrect label to each step, which offers limited explanatory power. Furthermore, they typically possess the ability to identify errors but lack the mechanism to correct them, leaving the reasoning process fundamentally broken. This limitation constrains their utility, especially within mechanisms like Best-of-N (BoN) sampling, which remain passive selection processes over a static set of potentially flawed solutions. In this work, we introduce a novel G enerative M ultimodal P rocess R eward M odel (GM-PRM) to address these limitations, as illustrated in Figure 1 (c). Our model transcends the role of a simple verifier and acts as an active reasoning collaborator. Instead of merely outputting a scalar score, our GM-PRM leverages its generative capabilities to produce a detailed, interpretable analysis for each reasoning step. This analysis deconstructs the step into three critical aspects: its fundamental step intent, the correctness of its image alignment, and the soundness of its reasoning logic. More importantly, our model is trained not only to identify errors but also to generate a refined, corrected version of the first incorrect step it encounters. This unique corrective capability enables us to propose a new test-time inference strategy: the Refined Best-of-N (Refined-BoN) process. This dynamic framework moves beyond passive selection by actively enhancing the quality of candidate solutions. When our GM-PRM identifies a flawed step within a generated solution, it intervenes by providing a corrected step, which is then used to guide the policy model in generating a new, more promising reasoning trajectory. This iterative refinement process significantly improves the diversity and correctness of the solution pool, leading to a substantial boost in the policy model’s problem-solving performance. Furthermore, we demonstrate that this powerful capability can be achieved with remarkable data efficiency, requiring a significantly smaller training dataset than previous approaches. Our primary contributions are as follows: - We develop a generative multimodal PRM called GM-PRM that provides fine-grained, interpretable feedback for mathematical reasoning. It analyzes each step’s purpose, image alignment, and logical validity, moving beyond simple binary classification to offer deeper insight into the model’s thought process. - We introduce a novel Refined-BoN framework that leverages the PRM’s generative power to actively correct errors at test time. It enhances the policy model’s ability to find correct solutions by iteratively improving flawed reasoning paths. - We demonstrate the effectiveness and data efficiency of GM-PRM, achieving state-of-the-art results on multiple multimodal math benchmarks. Our approach requires only a 20K sample dataset, highlighting the quality of data curation and the power of generative supervision strategy. 2 Related Work Process Reward Models (PRMs) PRMs have been proposed to evaluate the fine-grained step level for model reasoning. During the implementation process, annotating and obtaining a high-quality training dataset incurs a high cost. PRM800K (Lightman et al. 2023) is the first process supervision dataset completely annotated by humans. To mitigate annotation costs, Math-Shepherd (Wang et al. 2023) proposes Monte Carlo (MC) estimation, while OmegaPRM (Luo et al. 2024) leverages Monte Carlo Tree Search (MCTS) to automatically evaluate each reasoning step, both utilizing the generation capabilities of Large Language Models (LLMs). Subsequent research has enhanced the effectiveness of PRMs through various methods, including VersaPRM (Zeng et al. 2025), Implicit PRM (Yuan et al. 2024), OpenPRM (Zhang et al. 2025a), PQM (Li and Li 2024), PAV (Setlur et al. 2024), and others. In addition, GenRM (Zhao et al. 2025) utilizes the generation ability of reward models to analyze each reasoning step and obtain the score of each step by taking the probability of the special evaluation token. Furthermore, GenPRM (Zhao et al. 2025), ThinkPRM (Khalifa et al. 2025), R-PRM (She et al. 2025) extend the method of using model generation analysis to evaluate steps to PRMs. There are also many studies on benchmarks of PRMs such as ProcessBench (Zheng et al. 2024a), PRMBench (Song et al. 2025), and Socratic-PRMBench (Li et al. 2025a). Multimodal PRMs After achieving certain results and progress in the research of language modality in PRMs, research on PRMs has also begun to shift towards multimodal tasks. M-STAR (Liu et al. 2024a) proposes and implements multimodal PRM on multimodal problems. URSA (Luo et al. 2025) constructs a dataset by inserting errors and utilizes it to train a multimodal PRM. VisualPRM (Wang et al. 2025b) not only uses MC estimation to construct a multimodal VisualPRM400K training dataset, but also proposes a benchmark for multimodal PRMs called VisualProcessBench, which is entirely annotated by humans. Athena-PRM (Wang et al. 2025a) proposes using prediction consistency between strong and weak completers to enhance the quality of automatically annotated data based on MC estimation and improving multimodal PRM by ORM initialization and negative data up-sampling. Moreover, PRM-BAS (Hu et al. 2025), MM-PRM (Du et al. 2025) and DreamPRM (Cao et al. 2025) also improve the capability of multimodal PRMs. Although several studies have explored multimodal PRMs, applying them to multimodal tasks effectively remains certain challenges, such as insufficient interpretability of the labels assigned to each reasoning step and the inability to correct identified erroneous steps. In our work, we introduce a generative multimodal PRM, GM-PRM to solve the above problems. <details> <summary>figures/Case.png Details</summary>  ### Visual Description # Technical Document Extraction: Coordinate Translation Problem ## Visual Part Analysis ### Diagram Components - **Coordinate Grid**: - X-axis: 0 to 10 (horizontal) - Y-axis: 0 to 8 (vertical) - Grid spacing: 1 unit increments - **Triangle ABC**: - Vertex A: (3, 4) - Vertex B: (0, 0) - Vertex C: (4, 1) - Triangle orientation: Rightward-pointing ### Spatial Grounding - Legend placement: Not explicitly labeled (triangle components inferred from labels) - Color verification: No color coding used (black lines on white background) ## Textual Part Analysis ### Problem Statement "As shown in the diagram, triangle ABC is first translated 4 units to the right and then 3 units upward. The position of vertex A after the translation, represented as a coordinate pair, is ( )." ### Answer Options 1. A. (5, 4) 2. B. (8, 5) 3. C. (7, 7) 4. D. No correct answer ### Original Solution Analysis 1. **Step 1**: Incorrect premise - Assumes vertex A starts at (0,0) instead of (3,4) 2. **Step 2**: Flawed analysis - Uses incorrect initial coordinates 3. **Step 3**: Invalid translation application - Adds 4 to x and 3 to y of (0,0) → (4,3) (not an option) ### Refined Solution Analysis 1. **Step 2**: Correct initial coordinates - Vertex A at (3,4) 2. **Step 3**: Proper translation application: - Right translation: 3 + 4 = 7 (x-coordinate) - Upward translation: 4 + 3 = 7 (y-coordinate) - Result: (7,7) 3. **Step 6**: Final answer selection - Option C (7,7) ### GM-PRM Analysis - **Step Intent Analysis**: Correctly identifies initial coordinates as (3,4) - **Image Alignment Analysis**: Detects discrepancy between diagram (A at 3,4) and original solution's (0,0) - **Reasoning Logic Analysis**: Validates translation math (4 right + 3 up) ## Key Trends & Data Points 1. **Translation Vector**: (4,3) applied to all vertices 2. **Vertex Movement**: - A: (3,4) → (7,7) - B: (0,0) → (4,3) - C: (4,1) → (8,4) 3. **Answer Validation**: - Correct mathematical result: (7,7) - Available options: C matches calculated result ## Component Isolation ### Header - Title: "Question (Visual Part)" / "Question (Textual Part)" - Diagram/text separation: Clear visual boundary ### Main Chart - Grid structure: 10x8 coordinate plane - Triangle positioning: Bottom-left quadrant - Translation direction: Rightward + upward ### Footer - Solution steps: Numbered 1-6 - Checkmarks: Green ✓ for correct steps, red ✗ for errors ## Trend Verification 1. **Original Solution Trend**: Incorrect → (4,3) (not an option) 2. **Refined Solution Trend**: Correct → (7,7) (Option C) 3. **Answer Selection**: Final answer C validated through coordinate math ## Data Table Reconstruction | Step | Analysis Type | Correctness | Notes | |------|------------------------|-------------|--------------------------------| | 1 | Step Intent Analysis | ✓ | Identifies initial coordinates | | 2 | Image Alignment Analysis| ✓ | Detects diagram vs text error | | 3 | Reasoning Logic Analysis| ✓ | Validates translation math | ## Final Answer Determination - Mathematical result: (7,7) - Available options: C (7,7) - Correct answer: C - Original solution error: Assumed wrong initial coordinates </details> Figure 2: The illustration of a representative example before and after applying GM-PRM. In particular, GM-PRM first judges the steps of the original solution generated by the policy model. Subsequently, GM-PRM finds that the second step is incorrect and refines the second step to generate the correct version. The correct steps are input to the policy model to generate the refined solution, and finally the correct answer is obtained. 3 Methodology In this section, we first describe how to utilize PRMs and generative PRMs combined with the BoN method to improve the performance of policy models for mathematical problems in Section 3.1. Then, we introduce our process to implement multimodal generative PRM, including data construction and model training in Section 3.2. Finally, we propose a novel Refined-BoN framework for PRMs to enhance its performance beyond traditional BoN method in Section 3.3. 3.1 PRMs for Mathematical Problem In this section, we present the implementation methods of PRM and GM-PRM, and provide formal explanations of their usage through mathematical notation. Problem and Reasoning Steps Generation Let $Q$ denote a mathematical problem. Firstly, an LLM $\pi$ is involved in solving the mathematical problem $Q$ . To facilitate reasoning, the problem is combined with a prompt $P$ , which includes specific instructions guiding the generation of a step-by-step reasoning process and a final answer. This composite input is then fed into the LLM. When generating a response, $\pi$ generates a sequence of reasoning steps, denoted as $R=\{r_{1},r_{2},...,r_{T}\}$ , where $T$ represents the total number of reasoning steps to the given mathematical problem. The above process can be explained as follows: $$ R=\pi(Q\parallel P), \tag{1} $$ where $\parallel$ denotes the concatenation of the problem $Q$ and the prompt $P$ , and $\pi(·)$ represents the inference of LLM. PRM A single instance in a training dataset $\mathcal{D}$ to train a PRM comprises three components: (1) a problem statement, (2) a generated response consisting of multiple inference steps, and (3) a corresponding set of binary labels, each taking a value of either 0 or 1, indicating whether the associated reasoning step is incorrect or correct, respectively. During training, the PRM is optimized using cross-entropy loss and supervised to align its predictions with the ground-truth labels. After being trained, the PRM model is capable of processing new reasoning steps generated by the LLM in response to a given mathematical problem, which means that the PRM is able to assign a scalar score to each individual reasoning step, reflecting the model’s confidence in the correctness of each step: $$ f_{\text{PRM}}:(Q,R)\mapsto(s_{1},s_{2},\dots,s_{T}), \tag{2} $$ where $f_{\text{PRM}}:(·)$ represents the inference of PRM, $s_{i}∈[0,1]$ denotes the confidence score assigned to the $i$ -th reasoning step $r_{i}$ , and $T$ denotes the number of reasoning steps. For generative PRM, the binary labels in the training dataset are replaced with textual analyses and judgments, each formulated as a textual choice such as “incorrect” or “correct”. During inference, generative PRM also generates textual critiques and judgments for each step. GM-PRM By extending generative PRMs from the textual modality to a multimodal setting, we introduce GM-PRM. In this setting, mathematical problems are represented using both textual and visual information. The input to the policy model comprises the image of the problem, its textual description and task-specific instructions, which are processed jointly to generate reasoning steps. Similarly, during both training and inference, it is essential to provide GM-PRM with inputs from both visual and textual modalities, enabling it to perform cross-modal analysis when assigning correctness labels to each reasoning step: $$ f_{\text{{GM-PRM}}}:(Q,I,R)\mapsto(c_{1},j_{1},\dots,c_{T},j_{T}), \tag{3} $$ where $f_{\text{{GM-PRM}}}:(·)$ represents the inference of GM-PRM, $I$ denotes the image of the mathematical problem, $c_{i}$ denotes the critique of the $i$ -th reasoning step $r_{i}$ , and $j_{i}$ denotes the textual judgment assigned to the $i$ -th reasoning step $r_{i}$ . 3.2 Data Construction In this section, we present our methodology employed to construct the training data for GM-PRM. The process consists of three key stages: (1) the selection of appropriate types and quantities of question data from the VisualPRM400K dataset (Wang et al. 2025b); (2) the generation of textual analysis and judgment data using GPT-4o; and (3) the filtering of the generated data through MC estimation and LLM-as-a-judge techniques to ensure quality and reliability. Data Selection VisualPRM400K is a large-scale dataset containing approximately 400,000 multimodal process supervision samples. In our work, we select plane geometry- and function-related problems from VisualPRM400K to construct a specialized subset and supplement it with corresponding textual analysis for training GM-PRM. This targeted subset with textual critiques supports the effective training of GM-PRM, yielding strong performance on geometric and function-based mathematical reasoning tasks. Generation of Analysis and Judgment To obtain textual analyses and judgments, we employ GPT-4o to critique each reasoning step from 4 key aspects: step intent, image alignment, reasoning logic, and step refinement. The aspect of step intent indicates identifying the purpose of each reasoning step. This initial analysis establishes a foundation that allows GM-PRM to interpret and evaluate each reasoning step in context more effectively. Furthermore, this level of understanding facilitates subsequent error detection and correction tasks, thereby enhancing the overall effectiveness of GM-PRM. The second aspect is image alignment. When MLLMs are used for inference in solving multimodal problems, MLLMs often make errors in image alignment, such as misidentifying parallel relationships or incorrectly annotating angles, which leads to flawed solutions. To address this, we employ GPT-4o to produce textual analysis and judgments in image alignment for inference steps, to form the dataset for training GM-PRM. Reasoning logic is an indispensable presence in the step-by-step problem-solving process of MLLMs. However, the occurrences of logical inconsistencies and errors, such as miscalculations and incorrect inferences significantly impact the correctness of the reasoning steps and the final answers. Therefore, it is crucial for GM-PRM to be capable of identifying such logical flaws and making accurate judgments regarding the validity of the reasoning logic for each step. In our work, we employ GPT-4o to generate textual analysis and judgments of each step in reasoning logic to form the training dataset. The above process can be formulated as follows: $$ \mathcal{F}:(Q,I,R\parallel P)\mapsto\{SI_{i},IA_{i},RL_{i},FJ_{i}\}_{i=1}^{t}, \tag{4} $$ where $\mathcal{F}:(·)$ represents the inference of GPT-4o, $SI_{i}$ denotes the textual analysis of step intent for the $i$ -th reasoning step, $t$ denotes the number of the first incorrect step or the last step, $1≤slant t≤slant T$ , $IA_{i}=\{IAC_{i},IAJ_{i}\}$ denotes the analysis which contains critique $IAC_{i}$ and judgment $IAJ_{i}$ in image alignment of the $i$ -th reasoning step, $RL_{i}=\{RLC_{i},RLJ_{i}\}$ denotes the analysis which contains critique $RLC_{i}$ and judgment $RLJ_{i}$ in image alignment of the $i$ -th reasoning step, $FJ_{i}$ denotes the final judgment of the $i$ -th reasoning step. Building on aforementioned three aspects, we further aim for GM-PRM to correct the first identified erroneous step. The above information enables GM-PRM to generate corrected reasoning steps that are logically coherent, visually accurate, and semantically aligned with the original step intent. The resulting corrected steps can then be used to construct more diverse and accurate inference solutions and ultimately produce more reliable final answers. In our work, we employ GPT-4o to generate a corrected version of the first identified error step in a reasoning process if the first error step is detected to exist by GPT-4o: $$ \mathcal{F}:(Q,I,R\parallel P)\mapsto\begin{cases}RS,&\text{if incorrect step exists},\\ \emptyset,&\text{otherwise}.\end{cases} \tag{5} $$ where $RS$ denotes refined step of the first error step in a reasoning process. In summary, we design a structured prompt for GPT-4o to generate comprehensive analysis data across four dimensions based on the provided problems, associated images, and step-by-step solutions: $$ \mathcal{F}:(Q,I,R\parallel P)\mapsto\mathcal{D}, \tag{6} $$ where $\mathcal{D}$ denotes the generated training dataset: $$ \mathcal{D}=\{(\{SI_{i}^{k},IA_{i}^{k},RL_{i}^{k},FJ_{i}^{k}\}_{i=1}^{t},RS^{k})\}_{k=1}^{K}, \tag{7} $$ where $k∈\{1,2,...,K\}$ represents the $k$ -th sample in the dataset, and $K$ denotes the number of the training instances. Data Filtering The process of constructing training data using GPT-4o can be regarded as an implementation of LLM-as-a-judge methodology. Inspired by the combination of LLM-as-a-judge and MC estimation techniques (Zhang et al. 2025b), we employ the MC estimation technique proposed by Math-Shepherd (Wang et al. 2023) to effectively filter and curate the generated data. Monte Carlo estimation is a strategy for automated annotation that leverages LLMs or MLLMs to generate multiple subsequent solutions for each step. When applying MC estimation to evaluate a step $r_{i}$ , we use an LLM or an MLLM as a ‘completer’ to finalize multiple subsequent reasoning processes from this step: $$ f_{comleter}\mapsto\{(r_{i+1}^{j},\dots,r_{L_{j}}^{j},a^{j})\}_{j=1}^{m}, \tag{8} $$ where $a^{j}$ is the final answer of the $j$ -th finalized solution and $L_{j}$ is the total number of steps. Within MC estimation, one type of evaluation method is commonly applied: hard estimation. In hard estimation, a step $r_{i}$ is deemed correct if at least one subsequent solution reaches the correct final answer $a^{*}$ ; otherwise, it is considered incorrect: $$ l_{i}^{HE}=\begin{cases}1,&\exists a_{j},a_{j}=a^{*},\\ 0,&\text{otherwise}.\end{cases} \tag{9} $$ In our data construction process, we employ hard estimation to label the correctness of individual reasoning steps. By integrating LLM-as-a-judge technique and MC estimation, we compare the labels acquired by MC estimation and judgments generated by GPT-4o. Data samples that receive consistent evaluations from both methods are selected as our final training dataset. By integrating these two methods, we aim to further enhance the reliability and quality of the training data, ensuring better performance of GM-PRM. 3.3 Refined-BoN Process When applying Test-time Scaling (TTS) for LLMs and MLLMs, a widely adopted method is Best-of-N (BoN) approach. In the BoN process, a policy model is employed to generate N candidate solutions, which are then evaluated by reward models or self-consistency to select the optimal solution. However, during the BoN process, policy models are under identical prompting conditions when generating multiple solutions, which leads to the problem that the solutions often lack diversity and may exhibit limited correctness. In our work, we propose a novel Refined-BoN framework utilizing TTS techniques to enhance the diversity and accuracy of generated solutions, thereby improving the reasoning capabilities of policy models. Refined-BoN Method As shown in Figure 2, in Refined-BoN process, we first employ an MLLM as the policy model to generate $N/2$ initial solutions to a multimodal problem, and then these solutions are evaluated step-by-step by GM-PRM. For the subsequent $N/2$ solutions, the policy model generates them under varying conditions, informed by the evaluation of the preceding $N/2$ solutions: If GM-PRM identifies an incorrect reasoning step within a solution, it stops evaluating and refines the first erroneous step by generating a corrected version. This corrected step, along with all previously validated correct steps, is then input back into the policy model to continue the solution generation process. Conversely, if GM-PRM determines that all steps in a particular solution are correct, we employ the policy model to generate a new solution using the same prompt. Through this regeneration mechanism, we obtain the additional $N/2$ solutions. Subsequently, we employ GM-PRM to evaluate the subsequent $N/2$ solutions. Solution Selection After applying the Refined-BoN process, we obtain $N$ solutions for each problem, each accompanied by step-level correctness judgments. Moreover, we divide all the solutions into two categories: one where GM-PRM judges that it contains incorrect steps, and the other where GM-PRM judges that all its steps are correct. Furthermore, we take the probability of GM-PRM generating the associated “Correct” and “Incorrect” tokens as the score of each step. Among the $N$ generated solutions, if there exist solutions in which all reasoning steps are judged correct, we calculate the average of the scores of all steps in these solutions as the overall score of the solution, and select the solution with the highest average score as the optimal solution. For N solutions to a problem, if GM-PRM determines that all $N$ solutions contain incorrect steps, we calculate the average score of all steps in a solution as the overall score of the solution, and select the solution with the highest overall score as the final answer. 4 Experiments | MiniCPM-V2.6-8B + GM-PRM (Ours) Improvements | 44.3 51.0 +6.7 | 16.0 18.1 +2.1 | 18.9 24.4 +5.5 | 22.6 25.7 +3.1 | 38.6 51.0 +12.4 | 28.1 34.0 +5.9 | | --- | --- | --- | --- | --- | --- | --- | | Llama-3.2-11B-Vision | 44.5 | 14.3 | 16.5 | 28.4 | 46.1 | 30.0 | | + GM-PRM (Ours) | 49.5 | 18.2 | 18.8 | 32.7 | 53.4 | 34.5 | | Improvements | +5.0 | +3.9 | +2.3 | +4.3 | +7.3 | +4.5 | | Qwen2.5-VL-7B | 63.2 | 25.1 | 32.8 | 35.0 | 60.6 | 43.3 | | + GM-PRM (Ours) | 65.0 | 28.2 | 37.4 | 39.2 | 69.0 | 47.8 | | Improvements | +1.8 | +3.1 | +4.6 | +4.2 | +8.4 | +4.5 | | InternVL3-8B | 50.6 | 20.3 | 25.0 | 27.0 | 50.9 | 34.8 | | + GM-PRM (Ours) | 55.7 | 22.2 | 31.7 | 33.4 | 59.2 | 40.4 | | Improvements | +5.1 | +1.9 | +6.7 | +6.4 | +8.3 | +5.6 | | InternVL3-38B | 68.3 | 34.9 | 37.8 | 40.1 | 66.4 | 49.5 | | + GM-PRM (Ours) | 69.9 | 37.0 | 39.1 | 43.1 | 72.9 | 52.4 | | Improvements | +1.6 | +2.1 | +1.3 | +3.0 | +6.5 | +2.9 | | InternVL3-78B | 68.0 | 34.6 | 36.0 | 38.1 | 65.7 | 48.5 | | + GM-PRM (Ours) | 70.7 | 37.1 | 40.6 | 39.9 | 72.2 | 52.1 | | Improvements | +2.7 | +2.5 | +4.6 | +1.8 | +6.5 | +3.6 | Table 1: Percentage accuracy scores (%) of multiple MLLMs across five datasets. For each MLLM, the first row shows the baseline, the second shows the final result with GM-PRM, and the third shows the improvement. Only positive improvements are underlined. The best results are highlighted in bold. All values are reported after rounding to three decimal places. In this section, we introduce our experimental setup to assess GM-PRM under the Refined-BoN process on five multimodal mathematical benchmarks in Section 4.1. In addition, we present the results of our experiments and three conclusions analyzed from the results in Section 4.2. Finally, we show the ablation studies in Section 4.3. 4.1 Experimental Setup Benchmarks. We evaluate GM-PRM across five datasets, including MathVista (Lu et al. 2023), MathVision (Wang et al. 2024), MathVerse (Zhang et al. 2024), DynaMath (Zou et al. 2024) and WeMath (Qiao et al. 2024). The datasets contain diverse problem types, such as plane geometry, functions, puzzle tests, etc. We use Vision-Only subset of MathVerse dataset and Plane-Geometry subset of DynaMath. Settings. We employ GM-PRM as the critic model for Refined-BoN evaluation and set N to 8 by default. For MLLMs, we select six models as the policy models to generate step-by-step reasoning processes. When reasoning, we set the temperature of the policy models to 0.7 and top-p to 0.9. For comparison, we use the average accuracy of N sets of answers generated by policy models as baselines. Training Details. To train GM-PRM, we use Qwen2.5-VL-7B-Instruct as our base model and perform supervised fine-tuning (SFT) with all parameters trainable except for the frozen Vision Transformer (ViT) encoder. During the training process, we utilize bfloat16 mixed-precision and DeepSpeed with zero3 technology and set the training consists of 2 epochs. For batch size, the batch size on each training device is set to 2, and through gradient accumulation, the effective batch size is extended to 16. Moreover, we use two A800 GPUs to train GM-PRM, and the AdamW optimizer is used with an initial learning rate of $1× 10^{-5}$ . The learning rate schedule involves a linear warm-up with the warm-up ratio equal to 0.05 followed by linear decay. 4.2 Main Results As shown in Table 1, integrating GM-PRM with the Refined-BoN process consistently improves performance across five benchmark datasets for six different MLLMs. On average, our method yields notable accuracy gains, with improvements of +5.9 for MiniCPM-V2.6-8B, +4.5 for Llama-3.2-11B-Vision, +4.5 for Qwen2.5-VL-7B, and +5.6 for InternVL3-8B. A closer look at dataset-level results reveals that the improvements are not uniform. The WeMath benchmark shows the most significant enhancement, with MiniCPM-V2.6-8B improving by +12.4 points, highlighting the ability of our method to strengthen mathematical reasoning on challenging problems. Similarly, MathVerse and DynaMath exhibit consistent gains of +4.5–6.7 points across multiple models, suggesting that our approach particularly benefits datasets requiring complex symbolic manipulation and multi-step reasoning. In contrast, MathVision improvements are more modest (+1.9–3.9), indicating that the visual reasoning component may already be relatively strong in baseline models. GM-PRM combined with the Refined-BoN process demonstrates strong generalization across diverse multimodal mathematical problems, with particularly remarkable gains in plane geometry tasks. As illustrated in Figure 3, even after excluding plane geometry and function problems, policy models still achieve notable improvements across the datasets. This indicates that although GM-PRM is primarily trained on a dataset composed of plane geometry and function problems, it generalizes effectively to other types of multimodal mathematical problems. Moreover, as shown by the averaged results in Figure 3, the improvements achieved by GM-PRM with Refined-BoN on plane geometry problems consistently exceed those on the overall dataset, function problems, and other categories, underscoring the exceptional effectiveness of our method in tackling plane geometry tasks. <details> <summary>figures/Question_types.png Details</summary>  ### Visual Description # Technical Document Extraction: Bar Chart Analysis ## 1. Chart Type and Structure - **Chart Type**: Grouped bar chart - **Purpose**: Compares accuracy improvements across three AI models - **Key Components**: - **X-axis**: Model names (categorical) - **Y-axis**: Improvements of accuracy (%) (numerical) - **Legend**: Four categories with color coding ## 2. Axis Labels and Markers - **X-axis Title**: Not explicitly labeled (categories listed directly) - **X-axis Categories**: - MiniCPM-V2.6-8B - InternVL3-8B - InternVL3-78B - **Y-axis Title**: "Improvements of accuracy (%)" - **Y-axis Scale**: 0% to 10% in 2% increments ## 3. Legend Analysis - **Position**: Top-right corner - **Color-Category Mapping**: - 🔴 Red: Overall - 🔵 Blue: Plane Geometry - 🟢 Green: Functions - 🟠 Orange: Others ## 4. Data Extraction and Trends ### Model: MiniCPM-V2.6-8B - **Overall (🔴)**: ~6.0% - **Plane Geometry (🔵)**: ~9.0% (highest value) - **Functions (🟢)**: ~5.5% - **Others (🟠)**: ~1.2% ### Model: InternVL3-8B - **Overall (🔴)**: ~5.5% - **Plane Geometry (🔵)**: ~6.3% - **Functions (🟢)**: ~0.5% (lowest value) - **Others (🟠)**: ~3.2% ### Model: InternVL3-78B - **Overall (🔴)**: ~3.5% - **Plane Geometry (🔵)**: ~5.8% - **Functions (🟢)**: ~5.5% - **Others (🟠)**: ~0.8% ## 5. Spatial Grounding - **Legend Coordinates**: Top-right quadrant (standard position for bar charts) - **Bar Alignment**: All bars aligned vertically under respective model categories ## 6. Trend Verification - **Plane Geometry (🔵)**: - MiniCPM-V2.6-8B shows steepest improvement (~9%) - InternVL3-78B demonstrates ~5.8% improvement - Trend: Decreasing from MiniCPM to InternVL3-78B - **Functions (🟢)**: - MiniCPM-V2.6-8B: ~5.5% - InternVL3-8B: Sharp drop to ~0.5% - InternVL3-78B: Recovers to ~5.5% - Trend: U-shaped pattern across models - **Others (🟠)**: - Peaks at InternVL3-8B (~3.2%) - Lowest in InternVL3-78B (~0.8%) - Trend: Inverse relationship with Plane Geometry improvements ## 7. Data Table Reconstruction | Model | Overall (%) | Plane Geometry (%) | Functions (%) | Others (%) | |---------------------|-------------|--------------------|---------------|------------| | MiniCPM-V2.6-8B | 6.0 | 9.0 | 5.5 | 1.2 | | InternVL3-8B | 5.5 | 6.3 | 0.5 | 3.2 | | InternVL3-78B | 3.5 | 5.8 | 5.5 | 0.8 | ## 8. Critical Observations 1. **Plane Geometry Dominance**: - MiniCPM-V2.6-8B shows highest Plane Geometry improvement (9%) - All models show Plane Geometry as second-highest category after Overall 2. **Functions Paradox**: - InternVL3-8B shows near-zero improvement in Functions (~0.5%) - InternVL3-78B matches MiniCPM-V2.6-8B in Functions (~5.5%) 3. **Others Category**: - InternVL3-8B has highest "Others" improvement (3.2%) - Suggests potential misclassification or model-specific edge cases ## 9. Language Analysis - **Primary Language**: English (all labels and axis titles) - **Secondary Language**: None detected ## 10. Data Validation - All legend colors match bar colors exactly - Y-axis values align with bar heights - X-axis categories match model names in legend </details> Figure 3: Improvements of the average percentage accuracy (%) of multiple MLLMs across different question types in MathVista, MathVision and MathVerse datasets. The Refined-BoN process yields disproportionately larger gains for models with lower baseline performance. As shown in Table 1, InternVL3-38B starts with the highest initial average accuracy among all policy models (49.5%) and achieves a modest improvement of +2.9 points (+5.9%). In contrast, Qwen2.5-VL-7B, which has the highest baseline accuracy (43.3%) among models with fewer than 12 billion parameters, improves by +4.5 points (+10.4%), surpassing the relative gains of InternVL3-38B. Notably, MiniCPM-V-2.6-8B demonstrates the most significant relative improvement, achieving +5.9 points (+21.0%), despite its lower initial score. These results suggest that models with weaker baseline performance benefit more from the refinement mechanism of GM-PRM with Refined-BoN, likely because the process effectively corrects errors in reasoning steps, leaving greater room for improvement. <details> <summary>figures/MiniCPM-BoN.png Details</summary>  ### Visual Description # Technical Document Extraction: Line Chart Analysis ## Chart Type - **Line Chart** comparing two methods: GM-PRM and Self-Consistency. ## Axes Labels - **X-Axis**: `# Solutions per Problem` (values: 1, 4, 6, 8) - **Y-Axis**: `Accuracy (%)` (range: 38% to 53%) ## Legend - **Location**: Right side of the chart. - **Labels**: - **Blue Line**: GM-PRM - **Orange Line**: Self-Consistency ## Data Points ### GM-PRM (Blue Line) - **Trend**: Steep upward slope. - **Values**: - At `# Solutions = 1`: 38% - At `# Solutions = 4`: 46% - At `# Solutions = 6`: 48% - At `# Solutions = 8`: 51% ### Self-Consistency (Orange Line) - **Trend**: Gradual upward slope. - **Values**: - At `# Solutions = 1`: 38% - At `# Solutions = 4`: 43% - At `# Solutions = 6`: 45% - At `# Solutions = 8`: 46% ## Key Observations 1. **Initial Overlap**: Both methods start at 38% accuracy when `# Solutions = 1`. 2. **Divergence**: GM-PRM outperforms Self-Consistency across all solution counts. 3. **Slope Comparison**: - GM-PRM exhibits a steeper increase in accuracy as `# Solutions` increases. - Self-Consistency shows a more gradual improvement. 4. **Final Accuracy**: - At `# Solutions = 8`, GM-PRM achieves 51% accuracy vs. 46% for Self-Consistency. ## Spatial Grounding - **Legend Position**: Right-aligned, outside the main chart area. - **Data Point Alignment**: All points match legend colors (blue for GM-PRM, orange for Self-Consistency). ## Trend Verification - **GM-PRM**: Consistent upward trajectory with increasing slope. - **Self-Consistency**: Linear growth with diminishing returns. ## Component Isolation - **Header**: No title present. - **Main Chart**: Grid lines for reference; no annotations. - **Footer**: No additional text or metadata. ## Conclusion The chart demonstrates that GM-PRM consistently outperforms Self-Consistency in accuracy as the number of solutions per problem increases. The steeper slope of GM-PRM suggests greater efficiency in leveraging additional solutions for performance gains. </details> (a) MiniCPM-V-2.6-8B <details> <summary>figures/QwenVL-BoN.png Details</summary>  ### Visual Description # Technical Document Extraction: Line Chart Analysis ## 1. Labels and Axis Titles - **Y-Axis**: "Accuracy (%)" with tick markers at 60, 62, 64, 66, 68, 70. - **X-Axis**: "# Solutions per Problem" with tick markers at 1, 4, 6, 8. ## 2. Legend - **Location**: Upper right corner of the chart. - **Entries**: - **Blue Line**: Labeled "GM-PRM". - **Orange Line**: Labeled "Self-Consistency". ## 3. Data Points ### GM-PRM (Blue Line) - **Trend**: Steep upward slope from x=1 to x=4, followed by a gradual increase. - **Points**: - [1, 60.5] - [4, 66.0] - [6, 67.8] - [8, 68.8] ### Self-Consistency (Orange Line) - **Trend**: Gradual upward slope with slight acceleration between x=4 and x=6. - **Points**: - [1, 60.5] - [4, 64.2] - [6, 65.6] - [8, 66.4] ## 4. Visual Trends - **GM-PRM**: - Starts at 60.5% accuracy for 1 solution. - Sharp increase to 66.0% at 4 solutions. - Continues rising to 68.8% at 8 solutions. - **Self-Consistency**: - Starts at 60.5% accuracy for 1 solution. - Slower rise to 64.2% at 4 solutions. - Accelerates slightly to 66.4% at 8 solutions. ## 5. Spatial Grounding - **Legend Position**: Upper right corner. - **Line Colors**: - Blue (GM-PRM) matches legend entry. - Orange (Self-Consistency) matches legend entry. ## 6. Component Isolation - **Main Chart**: Line plot with grid lines for reference. - **No Header/Footer**: Chart focuses solely on data visualization. ## 7. Key Observations - Both methods start with identical accuracy (60.5%) at 1 solution. - GM-PRM outperforms Self-Consistency across all solution counts. - GM-PRM achieves a 8.3% accuracy gain (60.5% → 68.8%) from 1 to 8 solutions. - Self-Consistency gains 5.9% accuracy (60.5% → 66.4%) over the same range. ## 8. Data Table Reconstruction | # Solutions | GM-PRM Accuracy (%) | Self-Consistency Accuracy (%) | |-------------|----------------------|-------------------------------| | 1 | 60.5 | 60.5 | | 4 | 66.0 | 64.2 | | 6 | 67.8 | 65.6 | | 8 | 68.8 | 66.4 | ## 9. Conclusion The chart demonstrates that GM-PRM consistently achieves higher accuracy than Self-Consistency as the number of solutions per problem increases. The performance gap widens significantly between 1 and 4 solutions, then narrows slightly at higher solution counts. </details> (b) Qwen2.5-VL-7B <details> <summary>figures/InternVL3-78B-BoN.png Details</summary>  ### Visual Description # Technical Document Extraction: Line Chart Analysis ## 1. Chart Type & Purpose - **Chart Type**: Line chart comparing two optimization methods - **Purpose**: Visualize accuracy trends as the number of solutions per problem increases ## 2. Axis Labels & Scale - **X-axis**: - Title: "# Solutions per Problem" - Range: 1 to 8 (integer increments) - Ticks: 1, 4, 6, 8 - **Y-axis**: - Title: "Accuracy (%)" - Range: 65% to 73% (1% increments) - Ticks: 65, 67, 69, 71, 73 ## 3. Legend & Color Coding - **Legend Position**: Right side of chart - **Color Assignments**: - Blue (`#0000FF`): GM-PRM - Orange (`#FFA500`): Self-Consistency ## 4. Data Series Analysis ### GM-PRM (Blue Line) - **Trend**: Steep upward slope initially, then gradual increase - **Data Points**: - [1, 65.5] - [4, 71.0] - [6, 71.5] - [8, 72.5] ### Self-Consistency (Orange Line) - **Trend**: Gradual upward slope throughout - **Data Points**: - [1, 65.5] - [4, 67.5] - [6, 67.8] - [8, 67.8] ## 5. Key Observations 1. **Initial Performance**: Both methods start at identical accuracy (65.5%) with 1 solution 2. **Scaling Behavior**: - GM-PRM shows 5.5% accuracy gain at 4 solutions - Self-Consistency gains only 2% at same point 3. **Divergence**: Accuracy gap widens to 4.7% by 8 solutions 4. **Plateau**: Self-Consistency stabilizes at 67.8% after 6 solutions ## 6. Spatial Grounding Confirmation - Legend colors match line colors exactly - All data points align with legend assignments - Axis markers correspond to grid lines ## 7. Trend Verification - **GM-PRM**: - Initial slope: ~16.7% per solution (1→4) - Later slope: ~0.83% per solution (4→8) - **Self-Consistency**: - Consistent slope: ~0.5% per solution throughout ## 8. Data Table Reconstruction | # Solutions | GM-PRM Accuracy | Self-Consistency Accuracy | |-------------|------------------|---------------------------| | 1 | 65.5% | 65.5% | | 4 | 71.0% | 67.5% | | 6 | 71.5% | 67.8% | | 8 | 72.5% | 67.8% | ## 9. Language Analysis - **Primary Language**: English (all labels, titles, and annotations) - **No Secondary Languages Detected** ## 10. Critical Findings 1. GM-PRM demonstrates significantly better scalability 2. Self-Consistency shows diminishing returns after 6 solutions 3. Method selection should consider problem complexity: - GM-PRM preferred for problems requiring >4 solutions - Self-Consistency may suffice for simpler problems </details> (c) InternVL3-78B Figure 4: The results of changing the value of N in the Refined-BoN process on the WeMath across different policies. As N increases, the effectiveness of GM-PRM in enhancing accuracy improves and surpasses that of Self-Consistency. 4.3 Hyperparameter & Ablation Study Number of solution samples N in Refined BoN. Following Test-time Scaling technique, we vary the number of N in the Refined-BoN process to evaluate the performance of GM-PRM in comparison to the Self-Consistency baseline. Figure 4 depicts WeMath accuracy as the number of sampled solutions per problem ( $N$ ) increases from 1 to 8. Across all three backbones—MiniCPM‑V‑2.6‑8B, Qwen2.5‑VL‑7B, and InternVL3‑78B—both GM-PRM and the Self‑Consistency (SC) baseline benefit from a larger sampling budget, yet GM-PRM exhibits a noticeably steeper growth curve. Under the widely adopted Best‑of‑8 setting, GM-PRM delivers gains of 4.9 and 3.5 over SC on MiniCPM‑V‑2.6‑8B and Qwen2.5‑VL‑7B, respectively. Even for the 78B‑parameter InternVL3, GM-PRM maintains a substantial 4.1 margin. These results indicate that the proposed refinement strategy not only scales to larger models but also converts additional candidate solutions into accuracy more effectively than Self‑Consistency, thereby underscoring the robustness and versatility of GM-PRM. Furthermore, for MiniCPM-V-2.6-8B, GM-PRM surpasses the self-consistency baseline by 2.1, 2.2, and 4.9 points under the Best-of-4, Best-of-6, and Best-of-8 settings, respectively, indicating a steadily increasing performance gap between GM-PRM and self-consistency as N increases. Methods for aggregating step scores. For PRMs, the method used to aggregate step scores into an overall solution score plays a critical role. In this part, we compare several different aggregation strategies, including averaging step scores, selecting the maximum step score, and selecting the minimum step score. Since step-by-step solutions that contain steps judged incorrect are often not evaluated or scored for all steps, this experiment focuses exclusively on solutions where all steps are judged correct. <details> <summary>figures/Aggregation.png Details</summary>  ### Visual Description # Technical Document Analysis of Accuracy Comparison Chart ## Language Declaration - **Primary Language**: English - **Secondary Languages**: None detected --- ## Chart Structure and Components ### Header - **Title**: Not explicitly labeled (implied by context) - **Legend**: Located in the **top-right corner** - **Labels**: - `Original` (gray) - `Average` (blue) - `Min` (dark teal) - `Max` (light teal) ### Main Chart - **Type**: Grouped bar chart - **X-Axis**: - **Labels**: - `MiniCPM-V-2.6-8B` - `Qwen2.5-VL-7B` - `InternVL3-8B` - **Position**: Bottom of chart - **Y-Axis**: - **Title**: `Accuracy (%)` - **Range**: 0% to 60% (increments of 10%) - **Position**: Left side of chart ### Data Points and Trends #### 1. **MiniCPM-V-2.6-8B** - **Original**: ~28% (gray bar) - **Average**: ~34% (blue bar) - **Min**: ~35% (dark teal bar) - **Max**: ~30% (light teal bar) - **Trend**: - Average > Min > Original > Max #### 2. **Qwen2.5-VL-7B** - **Original**: ~43% (gray bar) - **Average**: ~48% (blue bar) - **Min**: ~47% (dark teal bar) - **Max**: ~43% (light teal bar) - **Trend**: - Average > Min > Original = Max #### 3. **InternVL3-8B** - **Original**: ~35% (gray bar) - **Average**: ~40% (blue bar) - **Min**: ~40% (dark teal bar) - **Max**: ~37% (light teal bar) - **Trend**: - Average = Min > Original > Max --- ## Data Table Reconstruction | Model | Original (%) | Average (%) | Min (%) | Max (%) | |---------------------|--------------|-------------|---------|---------| | MiniCPM-V-2.6-8B | 28 | 34 | 35 | 30 | | Qwen2.5-VL-7B | 43 | 48 | 47 | 43 | | InternVL3-8B | 35 | 40 | 40 | 37 | --- ## Spatial Grounding - **Legend**: Top-right corner (aligned with chart title area) - **X-Axis Labels**: Centered below each group of bars - **Y-Axis Labels**: Left-aligned, vertical ticks at 0%, 10%, ..., 60% --- ## Component Isolation 1. **Header**: Legend and implied title 2. **Main Chart**: Three grouped bars per model, with distinct colors for metrics 3. **Footer**: Y-axis percentage labels and gridlines --- ## Key Observations 1. **Accuracy Trends**: - `Average` consistently outperforms `Original` across all models. - `Min` and `Max` metrics show variability: - `Min` often matches or exceeds `Average` (e.g., Qwen2.5-VL-7B). - `Max` frequently underperforms `Original` (e.g., MiniCPM-V-2.6-8B). 2. **Model Performance**: - `Qwen2.5-VL-7B` achieves the highest `Average` accuracy (~48%). - `MiniCPM-V-2.6-8B` has the lowest `Original` accuracy (~28%). --- ## Validation Checks - **Color Consistency**: - All `Original` bars are gray. - `Average` bars are blue. - `Min` bars are dark teal. - `Max` bars are light teal. - **Trend Verification**: - Numerical values align with visual bar heights (e.g., Qwen2.5-VL-7B’s `Average` bar is the tallest). --- ## Conclusion The chart compares accuracy metrics (`Original`, `Average`, `Min`, `Max`) across three language models. `Qwen2.5-VL-7B` demonstrates the highest overall performance, while `MiniCPM-V-2.6-8B` lags in `Original` accuracy. The `Average` metric consistently outperforms `Original`, suggesting potential for optimization in model configurations. </details> Figure 5: Average percentage accuracy (%) of MLLMs via different aggregation methods across five datasets. The results are illustrated in the Figure 5. Across all policy models and datasets, we find that both averaging the step scores and selecting the minimum score significantly outperform the strategy of selecting the maximum score. This suggests that either the average or the minimum score provides a more accurate reflection of the overall quality of a solution than the maximum score. Between the minimum and average aggregation methods, we observe that averaging performs slightly better. This improvement may stem from the fact that the average score takes into account all problem-solving steps, providing a more comprehensive evaluation, whereas the minimum score reflects only the step with the lowest score and thus offers a less holistic assessment. Refined-BoN vs. BoN. The Refined-BoN process aims to enhance the diversity of N candidate solutions by refining the steps judged incorrect and integrating the refined steps with the steps judged correct into the prompt for the policy models. In this part, we use the Pass@k metric to evaluate the diversity and accuracy of policy models in generating multiple solutions to the given problems. The results are summarized in the Table 2. Overall, the Refined-BoN process improves Pass@8 scores compared to the standard BoN process across multiple policy models and five benchmark datasets. Specifically, it increases the average Pass@8 values of MiniCPM-V-2.6-8B, Llama-3.2-11B-Vision, and InternVL3-8B by 0.9, 1.3, and 0.9 points, respectively, across the five datasets, demonstrating the effectiveness of the Refined-BoN approach in enhancing the diversity and correctness of the generated solutions. | MiniCPM-V-2.6-8B Llama-3.2-11B-Vision InternVL3-8B | 62.5 62.7 65.3 | 63.4 64.0 66.2 | +0.9 +1.3 +0.9 | | --- | --- | --- | --- | Table 2: Average percentage Pass@8 scores of BoN and Refined-BoN across five datasets for different models. 5 Conclusion In this work, we introduced GM-PRM, a novel paradigm that transforms the reward model from a passive judge into an active reasoning collaborator for multimodal mathematics. By providing fine-grained, interpretable analysis and, more critically, generating corrections for erroneous steps, GM-PRM moves beyond simple binary verification. 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The development of MLLMs has been driven by extensive efforts, including enhancements across model structure and data curation. In terms of model structure, multiple studies (Bai et al. 2025; Liu et al. 2023; Yao et al. 2024) achieve notable performance through a method that utilizing connectors to align the embeddings of vision from Vision Foundation Models (VFMs) (Chen et al. 2024) with the latent space of LLMs (Bai et al. 2023; Touvron et al. 2023a, b). Alternatively, another line of research (Grattafiori et al. 2024; Tian et al. 2024) enhances pre-trained LLMs by adding supplementary layers to integrate visual features, which reduces the number of visual tokens but incurs additional training costs. Regarding dataset curation, recent research has achieved substantial advancements. Specifically, MultimodalC4 (Zhu et al. 2023) extends C4 corpus containing only text with images and constructs a corpus that supports pre-training for MLLMs. Furthermore, OmniCorpus (Li et al. 2024) delivers a large-scale yet noisy multimodal dataset suitable for pre-training, and MMInstruct (Liu et al. 2024b) presents an open-source collection of high-quality data designed for instruction tuning. The majority of research efforts have been concentrated on the training processes of MLLMs, leaving significant room for exploration in Test-Time Scaling (TTS) technique. In our work, we investigate the potential of enhancing the performance of MLLMs by incorporating Process Reward Model (PRM) into the TTS framework. A.2 Benchmark We provide more details about the Refined-BoN test benchmarks in Table 3: | DynaMath | Plane Geometry | 770 | | --- | --- | --- | | MathVerse | Vision-Only | 788 | | MathVista | Testmini | 1000 | | WeMath | Testmini | 1740 | | MathVision | Full | 3040 | Table 3: More details about the Refined-BoN test benchmarks. A.3 Dataset To ensure a balanced distribution of process labels, we carefully construct the training dataset. The final dataset used to train GM-PRM contains 19,614 samples in total, comprising 9,061 solutions that contain incorrect steps—as jointly identified by GPT-4o and Monte Carlo (MC) estimation—and 10,553 solutions in which all steps are judged to be correct. A.4 Prompt In this section, we introduce the prompts used to construct the training dataset and generate the reasoning processes and final answers. The prompt we guide the policy models to generate reasoning processes and final answers of multi-choice problems is represented in Figure 6. <details> <summary>figures/Prompt_multi_choice.png Details</summary>  ### Visual Description # Technical Document: Text Extraction from Image ## Overview The image contains a structured textual prompt designed for generating reasoning for multi-choice problems. It includes formatting instructions, a problem placeholder, and a question placeholder. No charts, diagrams, or data tables are present. --- ## Textual Content ### Header - **Text**: "Prompt for generating reasoning of multi-choice problems:" - **Formatting**: Bold, followed by a dashed line separator. ### Main Instructions - **Text**: "You are an expert in solving multimodal mathematical problems. I will provide a mathematical problem along with its corresponding image. According to the problem and the image, please first conduct step-by-step reasoning, and after your reasoning, please provide the correct option letter (e.g., A, B, C, D, E) using the format: 'Final answer: ...'" - **Key Elements**: - Role definition: "expert in solving multimodal mathematical problems." - Input requirements: "mathematical problem + corresponding image." - Output format: "Final answer: ..." with option letters (A–E). ### Problem Placeholder - **Label**: "Problem:" (in red). - **Content**: `<Question>` (in blue, acting as a placeholder for the actual question). --- ## Structural Analysis 1. **Header Section**: - Purpose: Introduce the prompt's objective. - Visual Cue: Bold text and dashed line for emphasis. 2. **Main Body**: - Role: Define the user's expected behavior (step-by-step reasoning). - Output Specification: Strict format for final answers (e.g., "Final answer: A"). 3. **Footer Section**: - Placeholders: - "Problem:" (red) to denote the input problem. - `<Question>` (blue) to indicate where the question will be inserted. --- ## Observations - **Language**: All text is in English. No non-English content detected. - **Formatting**: - Color coding: "Problem:" in red, `<Question>` in blue. - Placeholders: `<Question>` suggests dynamic insertion of content. - **No Data Visualization**: No charts, diagrams, or numerical data present. --- ## Conclusion The image provides a template for generating reasoning for multi-choice problems, emphasizing structured input (problem + image) and output (step-by-step reasoning + final answer). No additional data or visual elements are included. </details> Figure 6: Prompt for policy models to generate reasoning and final answers of multi-choice problem. The prompt we guide the policy models to generate reasoning processes and final answers of free-form problems is represented in Figure 7. <details> <summary>figures/Prompt_free_form.png Details</summary>  ### Visual Description # Technical Document Extraction: Image Analysis ## Image Description The image contains a textual prompt for generating reasoning for free-form mathematical problems. It is structured as follows: --- ### Header **Prompt for generating reasoning of free-form problems:** *(Bold text, followed by a dashed line separator)* --- ### Main Content **Textual Instructions:** "You are an expert in solving multimodal mathematical problems. I will provide a mathematical problem along with its corresponding image. According to the problem and the image, please first conduct step-by-step reasoning, and after your reasoning, please provide your final answer using the format: 'Final answer: ...'" --- ### Footer **Problem Section:** - **Label:** "Problem:" (in red) - **Placeholder:** `<Question>` (in blue) --- ## Key Observations 1. **No Charts/Diagrams:** The image contains no visual data, charts, or diagrams. It is purely textual. 2. **Language:** All text is in English. 3. **Formatting:** - Bold headers for emphasis. - Placeholders (`<Question>`) for dynamic content insertion. - Color-coded labels ("Problem:" in red, `<Question>` in blue). ## Transcribed Text </details> Figure 7: Prompt for policy models to generate reasoning and final answers of free-form problem. The prompt we use to employ GPT-4o to generate the training dataset is shown in Figure 8. <details> <summary>figures/Prompt.png Details</summary>  ### Visual Description # Prompt for GPT-4o to Construct Training Dataset --- ## Introduction You are an expert in solving multimodal mathematical problems. You will be given: 1. A multimodal mathematical problem and its corresponding image. 2. A multiple-step solution (each step on a new line). --- ## **Task** (Red Text) The tasks you need to do are: 1. Analyze the purpose of each step and what specific actions were taken in each step. 2. Analyze each step's correctness in terms of: - **Image alignment**: Whether the information and reasoning used in the step are consistent with the content of the provided image. - **Reasoning logic**: Whether the reasoning is logically sound, calculations are correct, and information used matches that from previous steps and the question. 3. When outputting judgments, choose one output from "Correct" or "Incorrect". 4. For the first incorrect step, correct it based on your analysis of its error, and output the corrected step at the end of your output. --- ## **Question** (Red Text) The multimodal mathematical problem is as follows: <Question> --- ## **Solution Steps** (Red Text) The multiple-step solution is as follows: <Solution Steps> --- ## **Output Format** (Red Text) You must output your content in the following format: ### #### Step 1 #### - **Step intent analysis**: [Describe what the step aims to do and the specific actions] - **Image alignment analysis**: [Analyze the consistency of image alignment] - **Judgment of image alignment**: [Correct/Incorrect] - **Reasoning logic analysis**: [Analyze the rationality of logic, correctness of calculations, and consistency with prior step] - **Judgment of reasoning logic**: [Correct/Incorrect] - **Final judgment of the current step**: [Correct/Incorrect] ### #### Step 2 #### ... ### Corrected step of the first incorrect step in solution: - **Step n**: [Assume that the first incorrect step is step n, and fill in the corrected step n in the square bracket] --- ## Notes - **Language**: English (no other languages present). - **Structure**: Text-based prompt with no charts, diagrams, or data tables. - **Placeholders**: `<Question>`, `<Solution Steps>`, and `Step n` are placeholders for user input. - **No numerical data or visual trends** to extract. </details> Figure 8: Prompt for GPT-4o to construct training dataset. A.5 More Cases We collect 2 more cases of GM-PRM and the Refined-BoN process illustrated in Figure 9 and 10. <details> <summary>figures/More_Cases_1.png Details</summary>  ### Visual Description # Technical Document Extraction: Area Remaining Analysis ## Visual Part ### Diagram Description - **Question**: "How much area is remaining?" - **Diagram Components**: - **Main Rectangle**: - Divided into three vertical sections. - **Left Section**: Labeled `6000` (area). - **Middle Section**: Labeled `60` (area). - **Right Section**: Contains a white square (unlabeled area). - **Axes**: - **X-axis**: Labeled `1000` (top), `10` (middle), `6` (bottom). - **Y-axis**: Labeled `6` (left). - **Total Area**: Annotated as `6096` square units at the bottom. ### Key Observations - The diagram visually represents a subtraction problem where two allocated areas (`6000` and `60`) are subtracted from a total area (`6096`). - The white square in the right section implies the remaining area to be calculated. --- ## Textual Part ### Question - **Text**: "How much area is remaining?" ### Original Solution 1. **Step 1**: "To determine the remaining area, we start with the total area and subtract the areas that have been allocated." 2. **Step 2**: - **Total Area**: `6096` square units. 3. **Step 3**: - **Allocated Areas**: - First: `6000` square units. - Second: `60` square units. 4. **Step 4**: - **Subtraction**: - First: `6096 - 6000 = 96`. - Second: `96 - 60 = 6`. - **Error**: A red `X` marks this step, indicating incorrect logic. ### GM-PRM Analysis & Judgement - **Step Intent Analysis**: Subtract allocated areas from total area to find remaining area. - **Image Alignment Analysis**: The diagram supports subtraction of `6000` and `60` from `6096`. - **Reasoning Logic Analysis**: - Correct calculations: `6096 - 6000 = 96` and `96 - 60 = 36`. - **Error**: Original solution incorrectly states `6` instead of `36`. ### Refined Solution 1. **Step 3**: - **Subtract Allocated Areas from Total Area**: - First: `6096 - 6000 = 96`. - Second: `96 - 60 = 36`. 2. **Step 5**: - **Final Answer**: "The remaining area is `36` square units." - **Checkmark**: Green checkmark confirms correctness. --- ## Key Data Points | Component | Value | Notes | |--------------------|-------------|--------------------------------| | Total Area | 6096 | Base value for subtraction | | Allocated Areas | 6000, 60 | Subtracted from total area | | Remaining Area | 36 | Correct result (Refined Solution) | | Error in Original | 6 | Incorrect final result | ## Diagram-Specific Details - **Spatial Grounding**: - Legend: Not explicitly present; labels are directly embedded in the diagram. - Axis Labels: - X-axis: `[1000, 10, 6]` (top to bottom). - Y-axis: `[6]` (left). - **Trend Verification**: - Subtraction logic: Total area (`6096`) decreases by allocated areas (`6000`, `60`), resulting in `36`. ## Conclusion The image combines a visual diagram and textual steps to solve an area subtraction problem. The **Original Solution** contains an error in the final calculation (`6` instead of `36`), which is corrected in the **Refined Solution**. The GM-PRM analysis highlights the discrepancy, emphasizing the importance of accurate arithmetic in multi-step problems. </details> Figure 9: The illustration of a representative example before and after applying GM-PRM. In particular, GM-PRM first judges the steps of the original solution generated by the policy model. Subsequently, GM-PRM finds that the forth step is incorrect regarding reasoning logic and refines the forth step to generate the correct version. The correct steps are input to the policy model to generate the refined solution, and finally the correct answer is obtained. <details> <summary>figures/More_Cases_2.png Details</summary>  ### Visual Description # Technical Document Extraction: Graph and Textual Analysis ## Visual Component Analysis ### Graph Structure - **Axes Labels**: - X-axis: Rainfall (cm) - Y-axis: Number of bats/mosquitoes - **Legend**: - Position: Top-right quadrant - Colors: - Blue: Mosquitoes (y = 5x - x²) - Red: Bats (y = 3x - 0.5x²) - **Key Data Points**: - Intersection Points: (0,0) and (4,0) - Peak Values: - Mosquitoes: ~2.5 cm rainfall (y ≈ 6.25) - Bats: ~3 cm rainfall (y ≈ 4.5) ### Equation Analysis 1. **Mosquitoes Equation**: y = 5x - x² - Parabolic curve opening downward - Vertex at (2.5, 6.25) - X-intercepts at 0 and 5 cm 2. **Bats Equation**: y = 3x - 0.5x² - Parabolic curve opening downward - Vertex at (3, 4.5) - X-intercepts at 0 and 6 cm ### Spatial Grounding - **Legend Position**: [x=0.8, y=0.9] (normalized coordinates) - **Intersection Points**: - Confirmed via visual inspection at (0,0) and (4,0) - Algebraic verification required for precision ## Textual Component Analysis ### Question Text > "In both equations x represents rainfall (in centimeters). When there is 0cm of rainfall, the number of mosquitoes is the same as the number of bats. What is another rainfall amount where the number of mosquitoes is the same as the number of bats? Round your answer to the nearest half centimeter." ### Solution Analysis #### Original Solution 1. **Step 1**: Correct (identify intersection goal) 2. **Step 2**: Correct (equation identification) 3. **Step 3**: Correct (set equations equal) 4. **Step 4**: ❌ **Error** - Incorrect simplification: `[5x - x² = 3x - 0.5x²]` → `[5x - x² - 3x + 0.5x² = 0]` → `[2x - 0.5x² = 0]` → `[x(2 - 0.5x) = 0]` → Solutions: x=0, x=4 - **Mistake**: Final answer incorrectly stated as 4 cm despite correct math #### Refined Solution 1. **Step 4 Correction**: - Proper simplification: `[5x - x² = 3x - 0.5x²]` → `[5x - x² - 3x + 0.5x² = 0]` → `[2x - 0.5x² = 0]` → `[x(2 - 0.5x) = 0]` → Solutions: x=0, x=4 2. **Step 5**: Correct (solve for x) 3. **Step 8**: Final Answer: **4 cm** ### GM-PRM Analysis - **Step Intent Analysis**: - Step 3 simplification prepares equation for solving - **Image Alignment Analysis**: - Simplification process matches visual intersection points - **Reasoning Logic Analysis**: - Original solution contained algebraic errors in term rearrangement ## Cross-Reference Verification 1. **Legend Validation**: - Blue (Mosquitoes) matches y=5x-x² curve - Red (Bats) matches y=3x-0.5x² curve 2. **Trend Verification**: - Both curves show downward parabolas - Mosquitoes peak higher (6.25 vs 4.5) - Bats have wider spread (0-6 cm vs 0-5 cm) ## Final Answer The rainfall amount where mosquitoes equal bats is **4 cm** (nearest half centimeter). </details> Figure 10: The illustration of a representative example before and after applying GM-PRM. In particular, GM-PRM first judges the steps of the original solution generated by the policy model. Subsequently, GM-PRM finds that the forth step is incorrect regarding reasoning logic and refines the forth step to generate the correct version. The correct steps are input to the policy model to generate the refined solution, and finally the correct answer is obtained.