2506.09532v1

# Athena: Enhancing Multimodal Reasoning with Data-efficient Process Reward Models > Work done during the internship at AMD.Corresponding authors. Abstract We present Athena-PRM, a multimodal process reward model (PRM) designed to evaluate the reward score for each step in solving complex reasoning problems. Developing high-performance PRMs typically demands significant time and financial investment, primarily due to the necessity for step-level annotations of reasoning steps. Conventional automated labeling methods, such as Monte Carlo estimation, often produce noisy labels and incur substantial computational costs. To efficiently generate high-quality process-labeled data, we propose leveraging prediction consistency between weak and strong completers as a criterion for identifying reliable process labels. Remarkably, Athena-PRM demonstrates outstanding effectiveness across various scenarios and benchmarks with just 5,000 samples. Furthermore, we also develop two effective strategies to improve the performance of PRMs: ORM initialization and up-sampling for negative data. We validate our approach in three specific scenarios: verification for test time scaling, direct evaluation of reasoning step correctness, and reward ranked fine-tuning. Our Athena-PRM consistently achieves superior performance across multiple benchmarks and scenarios. Notably, when using Qwen2.5-VL-7B as the policy model, Athena-PRM enhances performance by 10.2 points on WeMath and 7.1 points on MathVista for test time scaling. Furthermore, Athena-PRM sets the state-of-the-art (SoTA) results in VisualProcessBench and outperforms the previous SoTA by 3.9 F1-score, showcasing its robust capability to accurately assess the correctness of the reasoning step. Additionally, utilizing Athena-PRM as the reward model, we develop Athena-7B with reward ranked fine-tuning and outperforms baseline with a significant margin on five benchmarks. 1 Introduction In recent years, Large Language Models (LLMs) [5, 1, 52, 30, 61] have achieved remarkable success in natural language processing. Building on this, Multimodal Large Language Models (MLLMs) [31, 26, 3, 8] have made significant strides in various vision-language tasks, such as visual question answering [2, 39] and chart understanding [38]. Despite their impressive performance, solving complex tasks involving mathematical and multi-step reasoning remains challenging. To enhance reasoning capabilities, several approaches have been explored, including fine-tuning on long chain-of-thought (CoT) data [59, 40], offline preference optimization [56, 43], and online reinforcement learning [47, 17, 21, 51]. Another promising avenue is test time scaling (TTS), which involves generating multiple responses from a policy model and selecting the most consistent answer [58] or the solution with the highest reward using reward models [9, 28, 53, 50, 46, 55]. Reward models for TTS mainly include two types: outcome reward models (ORMs) [9] and process reward models (PRMs) [28]. ORMs evaluate the reward score for a given question and its solution, whereas PRMs provide reward scores for each intermediate reasoning step, offering fine-grained feedback. PRMs typically deliver superior performance and better out-of-distribution generalization [28]. However, obtaining high-quality data with process labels poses significant challenges. PRM800K [28] involves collecting 800K labeled steps through human annotations, which is time-consuming and requires skilled annotators, particularly for complex multi-step reasoning and mathematical tasks. Math-Shepherd [55] proposes an automated labeling method using Monte Carlo (MC) estimation. It defines the quality of an intermediate step as its potential to reach the correct answer. MC estimation for step-level labels is typically performed by sampling numerous reasoning trajectories using a language model, referred to as the completer, to estimate the probability of arriving at the correct answer. However, this approach involves significant computational overhead. Besides, another shortcoming of MC-based estimation methods is that estimated step-level labels are inevitably noisy. We aim to address the above challenges: reducing computational costs and mitigating noisy step labels. We first find that the accuracy of step labels estimated by MC-based methods is influenced by the reasoning capability of the completer. A strong completer can arrive at the correct answer despite incorrect intermediate steps as shown in Figure 1, whereas a weak completer may struggle even with correct intermediate steps. The estimation of MC-based methods is biased toward the completers we use. However, the correctness of the intermediate step should not depend on the completer. Based on this insight, we propose using both weak and strong completers to estimate step labels, retaining only those steps where labels generated by both completers are consistent to remove the bias caused by completers. This approach improves the quality of the step label. Empirical results demonstrate that a small set of high-quality labels ( $≈$ 5K) achieves impressive performance compared to large-scale data ( $≈$ 300K) labeled by vanilla MC-based estimation [55]. In addition, our approach requires only 1/45 of the GPU hours for data synthesis and 1/60 of the GPU hours for reward model training, significantly lowering computational costs. <details> <summary>x1.png Details</summary>  ### Visual Description ## Image Analysis: Equations and Symbols ### Overview The image contains a green checkmark, a red "X" mark, and three text blocks in rounded rectangles on the right side of the image. The text blocks contain mathematical expressions and phrases. ### Components/Axes * **Symbols:** A green checkmark and a red "X" mark. * **Text Blocks:** Three rounded rectangles containing text. ### Detailed Analysis or ### Content Details * **Green Checkmark:** Located in the center-left of the image. * **Red "X" Mark:** Located to the right of the green checkmark. * **Top Text Block (Yellow):** Contains the following text: * `y² = 133.` * `> x - y, we` * `13 x 10, so` * **Middle Text Block (Light Blue):** Contains the text: * `two perfect` * **Bottom Text Block (Light Orange):** Contains the text: * `equations` * `there are` ### Key Observations * The image presents a combination of mathematical expressions and descriptive phrases. * The checkmark and "X" mark suggest a comparison or evaluation. ### Interpretation The image appears to be part of a mathematical problem or explanation. The equations and phrases in the text blocks likely relate to the concepts of perfect squares and equations. The checkmark and "X" mark could indicate whether a given statement or solution is correct or incorrect. The equation `y² = 133` suggests an attempt to find a perfect square, which is then evaluated. The phrase `> x - y, we` suggests a comparison between variables. The phrase `13 x 10, so` suggests a calculation. The phrases "two perfect" and "equations there are" suggest a discussion about the number of perfect squares or equations. </details> Figure 1: Illustration of different completers under the same question and solution steps. Even if given wrong intermediate steps, the strong completer stills reach the final answer while the weak completer fails. We omitted some intermediate steps in the figure for simplicity. After acquiring high-quality datasets, we explore two effective strategies for training PRMs: initialization from ORMs and up-sampling data with negative labels. PRMs are typically fine-tuned from pre-trained foundation models, such as LLaMA-3.1-8B-Instruct [16] in RLHF-workflow [13]. Previous studies indicate that ORMs possess some ability to assess intermediate step correctness through confidence variation [33] or parameterization [10, 63]. Inspired by this, we find that initializing PRMs from ORMs significantly boosts performance, as ORMs trained on large-scale response-level data serve as pre-training with weak supervision. And PRMs can be considered fine-tuning on high-quality, fine-grained step-level data. Additionally, we show that label imbalance exists in process-labeled solutions and we propose to up-sample data with negative step labels to address this. Building on these methodologies, we develop our outcome reward model Athena-ORM and process reward model Athena-PRM. Leveraging Athena-PRM, we introduce Athena-7B with reward ranked fine-tuning. We validate our approach across three scenarios: 1) test time scaling [50]: where Athena-PRM ranks multiple outputs generated by policies under the Best-of-N evaluation; 2) direct evaluation of the correctness of reasoning steps using Athena-PRM; 3) reward ranked fine-tuning [12]: where Athena-PRM ranks outputs sampled from the current policy, utilizing the highest reward response for policy model fine-tuning. In the TTS scenario, we evaluate Athena-PRM on seven multimodal math and reasoning benchmarks with three different policy models ranging from 7B to 72B, demonstrating significant improvements in reasoning abilities. For instance, on the WeMath benchmark [44], Athena-PRM enhances the zero-shot baseline by 10.2 points using Qwen2.5-VL-7B [3] as the policy model. Furthermore, Athena-PRM excels in text-only benchmarks, achieving an 8.9 points improvement on a challenging math benchmark [18] with Mistral-8B. To assess its ability to judge intermediate reasoning step correctness, Athena-PRM is evaluated on the VisualProcessBench [57], showcasing strong performance and outperforming VisualPRM-8B [57], an open-source multimodal PRM, and proprietary models as judge. In the reward ranked fine-tuning scenario, our fine-tuned model Athena-7B, based on Qwen2.5-VL-7B [3], significantly enhances the policy model’s reasoning capabilities across seven math and reasoning benchmarks. Our contributions are summarized as follows: - We propose using prediction consistency between weak and strong completers to filter noisy process labels, enhancing the quality of the automated-labeled process data. The high-quality process labeled data shows surprising data efficiency for training PRMs. - We introduce two effective strategies to improve the performance of PRMs: ORM initialization and negative data up-sampling. - We develop our reward models Athena-ORM and Athena-PRM, and evaluate them under the Best-of-N setting on nine math and reasoning benchmarks across various model sizes and families, demonstrating the effectiveness of our approach. Moreover, Athena-PRM achieves state-of-the-art (SoTA) performance in assessing the correctness of intermediate steps directly on the VisualProcessBench [57]. - Leveraging Athena-PRM, we introduce Athena-7B, a MLLM with exceptional capabilities fine-tuned using reward ranked fine-tuning. Extensive experiments highlight the superiority of Athena-7B across diverse benchmarks. 2 Method In this section, we first introduce basic concepts, including ORMs and PRMs in Sec. 2.1. Next, we present our method for constructing a high-quality PRM training set, along with practical training strategies to improve PRM performance in Sec. 2.2. The data collection process for training is detailed in Sec. 2.3. Finally, we discuss three application scenarios of PRMs in Sec. 2.4. 2.1 Reward Models for Mathematical Problem ORMs. Given a mathematical problem $x$ with golden answer $y^{*}$ and its solution $a$ generated by policy $\pi$ , i.e., $a\sim\pi\left(·\mid x\right)$ , ORMs assign a reward $r\left(x,y,y^{*}\right)$ to the final answer $y$ of $a$ to reflect the correctness of the solution. To train ORMs, we take inputs as $(x,y)$ and predict correctness $\delta(y,y^{*})$ of prediction with following loss function: $$ \mathcal{L}_{ORM}=\delta\left(y,y^{*}\right)\cdot\log r\left(x,y,y^{*}\right)+% \left(1-\delta\left(y,y^{*}\right)\right)\cdot\log\left(1-r\left(x,y,y^{*}% \right)\right), \tag{1} $$ where $\delta\left(y,y^{*}\right)=1$ if $y=y^{*}$ , otherwise $\delta\left(y,y^{*}\right)=0$ . PRMs. Different from ORMs, PRMs aim to predict the correctness of each step in the solution. Specifically, we sample response $a\sim\pi\left(·\mid x\right)$ . A response $a$ usually consists of multiple reasoning steps separated by a delimiter (e.g. \n\n), a.k.a $a=(a_{1},a_{2},...,a_{K})$ when $K$ is the number of reasoning steps. To predict the correctness of each step, we train PRMs with the cross-entropy loss for each step in the solution. Formally, we get the following loss function: $$ \mathcal{L}_{PRM}=\sum_{i=1}^{K}\delta_{i}\cdot\log r\left(x,a_{i}\right)+% \left(1-\delta_{i}\right)\cdot\log\left(1-r\left(x,a_{i}\right)\right), \tag{2} $$ where $\delta_{i}=1$ if the step $a_{i}$ is correct, otherwise $\delta_{i}=0$ . 2.2 Automated Data Scaling of PRMs Compared with ORMs, PRMs usually achieve better performance and exhibit stronger out-of-distribution generalization [28]. However, collecting high-quality data with process labels is challenging. Human annotation provides accurate supervision signals [28], but it requires expensive human labeling and difficult to scale up. Math-Shepherd [55] proposes to use a Monte Carlo (MC) sampling method to estimate the correctness of each step without human supervision. Specifically, to estimate the correctness of step $a_{i}$ , we use a completer $\phi$ to finalize the reasoning process from step $a_{i}$ , and get the final answer $y$ . We repeat sampling $T$ times and get corresponding answers $\mathcal{Y}=\{y_{j}\}_{j=1}^{T}$ . There are two ways to estimate the correctness of $a_{i}$ : soft estimation and hard estimation. For soft estimation, we assume that the frequency of getting the correct answer could dictate the quality of a step: $$ \delta_{i}^{soft}=\frac{\sum_{j=1}^{T}\mathbb{I}(y_{j}=y^{*})}{T}. \tag{3} $$ For hard estimation, we assume that the correct answer is right when the step could reach the answer in $T$ samples: $$ \delta_{i}^{hard}=\begin{cases}1&\text{if }\exists y_{j}\in\mathcal{Y},y_{j}=y% ^{*},\\ 0&\text{otherwise.}\end{cases} \tag{4} $$ In this paper, we mainly discuss the hard estimation for PRMs because it allows us to use standard language modeling loss to train PRMs and eliminates the adjustments of the training pipeline. MC-based estimation methods provide an automated and scalable labeling strategy for intermediate reasoning steps. However, automatically labeled data often contain incorrect labels, and MC-based methods typically incur substantial computational costs due to extensive sampling requirements. For instance, when processing a solution $a$ with $K$ steps, the completer $\phi$ must generate $T× K$ solutions, which produces $T× K$ times computation cost compared with ORMs. To enhance the accuracy of the labeling process and reduce the computational burden of data synthesis, our central approach is to utilize a smaller set of high-quality data with process labels. We improve the quality of automatically labeled process labels by introducing consistency filtering between weak and strong completers. The rationale is straightforward: the results of hard estimation depend on the number of samples $T$ , the difficulty of the problem $x$ , and the ability of the completer $\phi$ . A strong completer can arrive at the correct answer even when provided with incorrect steps, as shown in Figure 1, whereas a weak completer struggles to find the final answer even when given correct steps. We aim to enhance the quality of process labels through prediction consistency between weak and strong completers. Specifically, we only use steps whose assigned labels are consistent between different completers. To validate our method, we conducted a simple experiment using the PRM800K dataset [28]. We employed Mistral-7B-Instruct [22] as the weak completer ( $\phi_{w}$ ) and Qwen2.5-72B-Instruct [61] as the strong completer ( $\phi_{s}$ ), setting the number of samples $T=8$ for MC-based estimation. We randomly sampled 50 queries and reported the accuracy of estimated process labels: | | Weak Completer $\phi_{w}$ | Strong Completer $\phi_{s}$ | Consistency Filter | | --- | --- | --- | --- | | Accuracy (%) | 78.2 | 83.4 | 94.1 | Our experiments show that label quality improves significantly after applying the consistency filter. Empirically, we demonstrate that a small number of high-quality labels is sufficient to achieve superior performance, thereby reducing computational costs compared to the vanilla MC-based estimation [55]. Results of more strategies and combination of completers are provided in Appendix A.1. Additional strategies for PRMs. We offer two strategies for enhancing the performance of PRMs: initialization from ORMs and negative data up-sampling. First, initializing PRMs with ORMs trained on large-scale sample-level annotated data improves performance. It is shown that initialization from ORMs will improve performance with different datasets, see verification in Table 5. Previous work indicates that ORMs possess a certain capacity to assess the correctness of intermediate steps [10, 33, 63]. We empirically show that PRMs benefit from a few high-quality examples when initialized with ORMs. Large-scale sample-level annotation data provides weaker supervision, with outcome supervision acting as a simplified form of process supervision. Training ORMs on large-scale sample-level annotations serves as a “pre-training”, while training PRMs from pre-trained ORMs acts as“fine-tuning” with high-quality process-level annotated data. Secondly, our findings reveal that label imbalance is prevalent in most process-labeled datasets, such as PRM800K [28] and our synthetic data, where correct steps are more common than errors. We provide a detailed distribution of process labels in Table 9. Empirically, slightly up-sampling data with negative labels improves performance with minimal additional computation. 2.3 Data Collection To construct a diverse and high-quality dataset, we first gather data from various public multimodal and text-only datasets. For multimodal datasets, we collect from MathV360k [48], UniGeo [7], Geometry3k [34], CLEVR-Math [29], ScienceQA [35], GeomVerse [24], GeoQA-plus [6], and DocVQA [39]. For text-only datasets, we utilize GSM8K [9], MATH [18], and NuminaMath-1.5. https://huggingface.co/datasets/AI-MO/NuminaMath-1.5 If the original dataset includes different splits, we use only the training portion to prevent test set contamination. We remove all judgment questions and convert multiple-choice questions to open-ended ones to deter policy models from “guessing” answers. Then, for each query, we sample 8 responses from the policy model. To ensure dataset diversity, we employ multiple models as policies, including InternVL2.5-8B/78B [8], Qwen2.5-VL-7/72B-Instruct [3], LLaVA-OneVision-7B/72B [26] for multimodal datasets, and Qwen2.5-7/72B-Instruct [61] for text-only datasets. Next, we apply carefully designed filtering rules to all responses. We exclude responses with too many or too few tokens and eliminate those exhibiting repetitive patterns. We employ $n$ -gram deduplication to remove similar responses and enhance diversity. Ultimately, we obtain approximately 600K queries with corresponding responses. We parse answers from responses and assign correctness labels of 0 or 1 to each query-response pair. We refer to this dataset as Athena-600K and employ it to train our outcome reward model Athena-ORM. For training PRMs, we generate process labels using the proposed method introduced in Sec. 2.2. We use Qwen2.5-VL-7B [3] as the weak completer ( $\phi_{w}$ ) and Qwen2.5-VL-72B [3] as the strong completer ( $\phi_{s}$ ). Finally, we collect approximately 5K samples for training our process reward model Athena-PRM. We also compare with PRMs trained using approximately 300K samples estimated by the vanilla MC-based method outlined in [55]. We denote these datasets as Athena-5K and MC-300K, respectively, and compare their performance and computational cost in Sec. 3.4. 2.4 Application Following reward model training, we assess the performance of ORMs and PRMs in three scenarios: test time scaling [50], direct judgment, and reward ranked fine-tuning [12]. Verification for test time scaling. We adopt a Best-of-N evaluation paradigm. Specifically, given a problem $x$ in the test set, we sample $N$ solutions from the policy $\pi$ . All solutions are scored using a reward model, and we choose the solution with the highest score. For ORMs, we directly use the outputs from ORMs as the reward of solutions. For PRMs, the minimum reward across all steps is used to rank all solutions. We test different choices, including the reward from the last step, the minimum reward across all steps, or the product of the rewards from all steps, as the reward for solutions in Appendix A.3. Direct judgment for reasoning steps. Besides verification at test time under the Best-of-N setting, Athena-PRM can also be used to directly identify erroneous steps in the mathematical reasoning process. Given a solution $a$ with $K$ steps as input, Athena-PRM outputs the correctness score $\{\delta_{i}\}_{i=1}^{K}$ for each step $\{a_{i}\}_{i=1}^{K}$ in a single forward pass. Response ranking for reward ranked fine-tuning. We explore the use of PRMs for data synthesis in reward ranked fine-tuning [12, 64, 49, 66]. Using the current policy $\pi$ , we generate $M=8$ solutions for an input problem $x$ . Subsequently, we filter out solutions with incorrect answers and apply deduplication to remove highly similar responses, enhancing the diversity of the remaining solutions. We retain queries where 2 to 6 out of 8 responses are correct, excluding those with too few or too many correct answers, as such cases are either too easy or too challenging for the current policy $\pi$ . Following this filtering step, we use Athena-PRM to score all solutions for each query and select the solution with the highest reward as the corresponding label. The policy $\pi$ is fine-tuned using the synthetic data generated as described above. 3 Experiments Table 1: Results on seven multimodal reasoning benchmarks under Best-of-N (N=8) evaluation. $\dagger$ denotes that the results are from [57]. We mark our results and highlight the best result. | | WeMath | MathVista | MathVision | MathVerse | DynaMath | MMMU | LogicVista | | --- | --- | --- | --- | --- | --- | --- | --- | | Qwen2.5-VL-7B [3] | 36.2 | 68.1 | 25.4 | 41.1 | 21.8 | 58.0 | 47.9 | | Self-consistency [58] | 44.7 | 71.6 | 28.6 | 43.7 | 22.9 | 60.1 | 49.5 | | $\text{VisualPRM-8B}^{\dagger}$ [57] | 39.8 | 70.3 | 31.3 | 44.3 | 23.0 | 58.6 | 48.3 | | Athena-ORM | 45.1 | 72.8 | 29.8 | 44.1 | 23.1 | 62.7 | 51.3 | | Athena-PRM | 46.4 | 75.2 | 32.5 | 46.3 | 23.4 | 63.8 | 53.0 | | InternVL2.5-8B [8] | 23.5 | 64.5 | 17.0 | 22.8 | 9.4 | 56.2 | 36.0 | | Self-consistency [58] | 28.4 | 66.1 | 21.1 | 24.7 | 13.8 | 57.8 | 40.2 | | $\text{VisualPRM-8B}^{\dagger}$ [57] | 36.5 | 68.5 | 25.7 | 35.8 | 18.0 | 60.2 | 43.8 | | Athena-ORM | 28.6 | 66.9 | 22.0 | 25.8 | 15.2 | 59.1 | 40.8 | | Athena-PRM | 30.1 | 71.4 | 23.4 | 26.1 | 18.7 | 60.3 | 44.4 | | Qwen2.5-VL-72B [3] | 49.1 | 74.2 | 39.3 | 47.3 | 35.9 | 70.2 | 55.7 | | Self-consistency [58] | 54.8 | 77.0 | 43.1 | 50.8 | 37.6 | 71.1 | 59.6 | | Athena-ORM | 55.6 | 77.8 | 43.0 | 51.2 | 39.6 | 72.3 | 60.1 | | Athena-PRM | 58.7 | 79.1 | 44.8 | 54.6 | 42.5 | 75.8 | 60.9 | Table 2: Results on text-only math benchmarks under Best-of-N (N=8) evaluation. $\dagger$ denotes that the results are from [57]. We mark our results and highlight the best result. (a) Qwen2.5-7B | | GSM8K | MATH | | --- | --- | --- | | Qwen2.5-7B [61] | 91.6 | 75.5 | | Self-consistency [58] | 93.4 | 79.9 | | $\text{VisualPRM-8B}^{\dagger}$ [57] | 94.5 | 81.6 | | Athena-ORM | 94.0 | 81.1 | | Athena-PRM | 94.8 | 82.4 | (b) Ministral-8B | | GSM8K | MATH | | --- | --- | --- | | Ministral-8B | 85.6 | 54.5 | | Self-consistency [58] | 90.7 | 62.0 | | Athena-ORM | 91.4 | 63.2 | | Athena-PRM | 93.1 | 65.4 | (c) Qwen2.5-72B | | GSM8K | MATH | | --- | --- | --- | | Qwen2.5-72B [61] | 95.8 | 83.1 | | Self-consistency [58] | 96.0 | 86.0 | | $\text{VisualPRM-8B}^{\dagger}$ [57] | 96.5 | 85.2 | | Athena-ORM | 96.0 | 86.2 | | Athena-PRM | 97.0 | 87.4 | To comprehensively evaluate the proposed method, we assess Athena-ORM and Athena-PRM under the Best-of-N setting on seven multimodal benchmarks and two text-only benchmarks in Sec. 3.1. Furthermore, we evaluate Athena-PRM on VisualProcessBench, comparing its performance with MLLMs as judges and a recent open-source multimodal PRM, VisualPRM-8B [57], as described in Sec. 3.2. We assess the capability of Athena-7B on multimodal math and reasoning tasks in Sec. 3.3. To validate the effectiveness of our designs for Athena-PRM, an in-depth analysis is provided in Sec. 3.4. 3.1 Best-of-N Evaluation Benchmarks. We evaluate Athena-ORM and Athena-PRM across multiple multimodal mathematical and logical reasoning benchmarks, including MathVista [36], MathVision [54], MathVerse [68], WeMath [44], DynaMath [70], LogicVista [60], and MMMU [65]. To validate the effectiveness of Athena-ORM and Athena-PRM in the text-only scenario, we conduct experiments on the GSM8K [9] and MATH [18] under the BoN settings. Further details can be found in Appendix A.2. Settings. We employ Athena-ORM and Athena-PRM as the reward model for BoN evaluation, sampling $N=8$ solutions for every problem by default. For decoding, we set the temperature to 0.8 and use nucleus sampling [20] with top- $p$ set to 0.9, and report the average of five runtimes. For multimodal benchmarks, we choose Qwen2.5-VL-7B/72B [3] and InternVL2.5-8B [8] as policy models. All models used in this paper are “instruct” version. For simplicity, we ignore “instruct” in the text. For text-only datasets, we choose Qwen2.5-7/72B [61] and Ministral-8B https://huggingface.co/mistralai/Ministral-8B-Instruct-2410 as policy models. Zero-shot and self-consistency [58] serve as our baselines, alongside a recent public multimodal process reward model, VisualPRM-8B [57]. Training details. To train ORMs and PRMs, we fine-tune Qwen2.5-VL-7B on our dataset as introduced in Sec. 2.3. We use the AdamW optimizer [32] with the weight decay as 0 and set the learning rate as $1e^{-6}$ with cosine decay. We train all models with one epoch and set the global batch size to 64. We use DeepSpeed with zero2 [45] and flash-attention-2 [11] to improve training efficiency. To train ORMs, a new special token <step> is added at the end of solutions for ORMs training. This makes a consistent formulation for ORMs and PRMs. For PRMs, <step> is added to separate all reasoning steps and assign labels to corresponding tokens. We use “ $+$ ” and “ $-$ ” as labels to denote the correctness of each step. We use $8×$ AMD-MI250 GPUs for training and data synthesis. Results. Table 1 demonstrates that Athena-ORM and Athena-PRM generally enhance the reasoning performance of MLLMs across policy models of different sizes and benchmarks. Notably, Athena-PRM achieves a +10.2 points improvement on the WeMath dataset with Qwen2.5-VL-7B [3] as the policy model, compared to the zero-shot baseline. ORMs also exhibit general improvement across all benchmarks and policy models, although some gains are limited, such as +0.4 points on MathVerse with Qwen2.5-VL-72B [3]. In contrast, PRMs consistently show substantial performance gains over ORMs, illustrating the benefits of fine-grained rewards in test-time scaling. Table 2 highlights that Athena-PRM enhances reasoning abilities for text-only inputs in both the Qwen2.5 series and Ministral-8B models. Specifically, Athena-PRM achieves +9.9 points improvement for Ministral-8B on the challenging MATH [18] benchmark, demonstrating its effectiveness in text-only scenarios. Table 3: Results on VisualProcessBench. We report the F1-score on different dataset and macro F1 in overall. We highlight the best result across all models. | | MMMU | MathVision | MathVerse | DynaMath | WeMath | Overall | | --- | --- | --- | --- | --- | --- | --- | | Random Guessing | 50.0 | 50.0 | 50.0 | 50.0 | 50.0 | 50.0 | | Proprietary Models as Judge | | | | | | | | GPT-4o-mini [41] | 53.6 | 58.9 | 57.1 | 56.7 | 58.5 | 57.9 | | GPT-4o [41] | 56.3 | 60.2 | 59.7 | 59.0 | 63.3 | 60.3 | | Gemini-2.0-Flash [15] | 58.5 | 60.1 | 62.8 | 66.7 | 58.7 | 62.3 | | Open-source Models as Judge | | | | | | | | MiniCPM-V2.6-8B [62] | 44.9 | 50.9 | 58.9 | 46.7 | 57.4 | 50.4 | | LLaVA-OV-7B [26] | 45.7 | 43.0 | 42.2 | 44.7 | 52.5 | 44.4 | | LLaVA-OV-72B [26] | 46.1 | 48.4 | 53.0 | 57.0 | 57.3 | 52.3 | | Qwen2.5-VL-7B [3] | 53.1 | 51.8 | 47.8 | 51.3 | 54.2 | 51.0 | | Qwen2.5-VL-72B [3] | 59.2 | 59.0 | 59.7 | 62.9 | 62.3 | 60.5 | | InternVL2.5-8B [8] | 47.1 | 45.5 | 47.8 | 50.3 | 50.8 | 48.0 | | InternVL2.5-26B [8] | 48.8 | 47.4 | 49.2 | 50.4 | 51.4 | 49.2 | | InternVL2.5-38B [8] | 51.5 | 48.4 | 50.9 | 51.8 | 52.5 | 50.8 | | InternVL2.5-78B [8] | 52.0 | 51.7 | 53.7 | 50.8 | 52.5 | 52.6 | | Multimodal PRMs | | | | | | | | VisualPRM-8B [57] | 58.5 | 62.1 | 61.0 | 62.7 | 61.8 | 62.0 | | Athena-PRM | 74.1 | 61.2 | 60.7 | 72.7 | 73.8 | 65.9 | 3.2 Evaluation on Direct Judgment of Steps Setup. In addition to Best-of-N evaluation, we assess Athena-PRM on VisualProcessBench [57], comparing it with general MLLMs and VisualPRM-8B [57]. VisualProcessBench [57] aims to evaluate the ability to directly judge the correctness of each step. For MLLMs, we prompt the model to analyze and judge the correctness of each step. We report the F1-score for every subset and the micro-F1 score overall. Results. The results of VisualProcessBench are presented in Table 3. It is evident that current general open-source MLLMs struggle to judge the correctness of reasoning steps, with most $\scriptstyle\sim$ 7B models performing no better than the random guessing baseline. Conversely, multimodal process reward models achieve superior performance, rivaling proprietary models. Notably, our process reward model Athena-PRM, with only 7B parameters, achieves a +3.9 points improvement compared to VisualPRM-8B [57], even outperforming proprietary models as judges. 3.3 Evaluation on the Fine-tuned Model Table 4: Comparison of multimodal reasoning and mathematical performance across different models. $\Delta$ denotes improvements compared to Qwen2.5-VL-7B [61]. We highlight improvements at least +0.5 points with green. | | MMMU | WeMath | MathVista | MathVision | MathVerse | DynaMath | LogicVista | | --- | --- | --- | --- | --- | --- | --- | --- | | Proprietary Models | | | | | | | | | GPT-4o [41] | 72.9 | 50.6 | 71.6 | 43.8 | 49.9 | 48.5 | 64.4 | | Claude-3.7-Sonnet [1] | 71.0 | 49.3 | 66.8 | 41.9 | 46.7 | 39.7 | 58.2 | | Gemini-2.0-Pro [15] | 72.6 | 56.5 | 71.3 | 48.1 | 67.3 | 43.3 | 53.2 | | Open-source Models (>70B) | | | | | | | | | InternVL2.5-78B-MPO [56] | 68.2 | 37.6 | 76.6 | 36.2 | 43.7 | 21.2 | 50.8 | | InternVL2.5-78B [8] | 70.1 | 39.8 | 70.6 | 32.2 | 39.2 | 19.2 | 49.0 | | Qwen2.5-VL-72B [3] | 68.2 | 49.1 | 74.2 | 39.3 | 47.3 | 35.9 | 55.7 | | Open-source Models (~7B) | | | | | | | | | InternVL2.5-8B [8] | 56.2 | 20.2 | 58.3 | 20.0 | 20.4 | 9.2 | 33.6 | | MiniCPM-o-2.6-8B [62] | 50.9 | 25.2 | 73.3 | 21.7 | 35.0 | 10.4 | 36.0 | | Qwen2.5-VL-7B [3] | 58.0 | 36.2 | 68.1 | 25.4 | 41.1 | 21.8 | 47.9 | | Athena-7B | 61.1 | 43.0 | 71.4 | 25.7 | 45.7 | 21.9 | 51.3 | | $\Delta$ | +3.1 | +6.8 | +3.3 | +0.3 | +4.6 | +0.1 | +3.4 | Setup. We select Qwen2.5-VL-7B [3] as the initial policy model, obtaining training data as outlined in Sec. 2.4. We use AdamW [32] optimizer and set the learning rate as $1e^{-6}$ with cosine decay. The batch size is set to 128, and the number of epochs is set to one. We denote our fine-tuned model as Athena-7B and evaluate it on the same benchmarks as Sec. 3.1 in the zero-shot setting. For evaluation, we use VLMEvalKit [14] to evaluate Athena-7B and report the results of all baselines from the open leaderboard. https://huggingface.co/spaces/opencompass/Open_LMM_Reasoning_Leaderboard Results. As shown in Table 4, Athena-7B consistently improves upon Qwen2.5-VL-7B [3] across all benchmarks, with significant enhancements of at least $+$ 0.5 points on five benchmarks. Athena-7B outperforms large open-source models (e.g. InternVL2.5-78B [8]) on five benchmarks. On the MathVista [36], Athena-7B even achieves the comparable performance with proprietary models such as Gemini-2.0-Pro [15], Claude-3.7-Sonnet [1], and GPT-4o [41]. 3.4 Analysis Test time scaling. To evaluate the scalability of Athena-ORM and Athena-PRM, we compare their performance using different numbers of samples ranging from 4 to 64 on the MathVista [36]. As shown in Figure 2, reward ranking for solutions at test time improves reasoning performance across different policy models and number of samples. Specifically, ORMs and PRMs improve the self-consistency [58] baseline by 0.8 and 2.1 points with Qwen2.5-VL-72B [3] using eight samples, respectively. As the sample size increases, the performance improves steadily. However, the performance gain of ORMs is markedly lower than that of PRMs. For instance, Athena-PRM achieves 82.4% with Qwen2.5-VL-72B [3] using 64 samples, while Athena-ORM achieves similar performance with self-consistency [58] baseline. These findings underscore the advancements and scalability of PRMs in test-time scaling scenarios. <details> <summary>x2.png Details</summary>  ### Visual Description ## Line Chart: MathVista Accuracy vs. Number of Solutions per Problem ### Overview The image is a line chart comparing the MathVista accuracy (%) of four different methods (PRM, ORM, Self-consistency, and Zero-shot) against the number of solutions per problem. The x-axis represents the number of solutions per problem, ranging from 4 to 64. The y-axis represents the MathVista accuracy in percentage, ranging from 68% to 76%. ### Components/Axes * **X-axis:** "# Solutions per problem" with markers at 4, 8, 16, 32, and 64. * **Y-axis:** "MathVista Accuracy (%)" with markers at 68, 70, 72, 74, and 76. * **Legend:** Located in the center-right of the chart, it identifies the four methods: * PRM (light teal line with diamond markers) * ORM (light orange line with triangle markers) * Self-consistency (light red line with square markers) * Zero-shot (light blue dashed line with 'x' markers) ### Detailed Analysis * **PRM (light teal line with diamond markers):** The line starts at approximately 72.5% accuracy with 4 solutions, increases to approximately 75.2% at 8 solutions, decreases slightly to approximately 74.8% at 16 solutions, and then increases to approximately 75.8% at 32 solutions, and finally reaches approximately 76.2% at 64 solutions. * **ORM (light orange line with triangle markers):** The line starts at approximately 69.8% accuracy with 4 solutions, increases to approximately 72.8% at 8 solutions, remains relatively stable at approximately 72.8% at 16 solutions, increases to approximately 73.4% at 32 solutions, and reaches approximately 73.8% at 64 solutions. * **Self-consistency (light red line with square markers):** The line starts at approximately 69.3% accuracy with 4 solutions, increases to approximately 71.5% at 8 solutions, increases to approximately 71.9% at 16 solutions, increases to approximately 72.2% at 32 solutions, and reaches approximately 73.2% at 64 solutions. * **Zero-shot (light blue dashed line with 'x' markers):** The line remains relatively constant at approximately 68.1% accuracy across all numbers of solutions per problem. ### Key Observations * PRM consistently outperforms the other methods, achieving the highest accuracy across all numbers of solutions per problem. * Zero-shot has the lowest accuracy and remains constant regardless of the number of solutions. * ORM and Self-consistency show similar trends, with accuracy increasing as the number of solutions increases. ### Interpretation The chart demonstrates the impact of the number of solutions per problem on the MathVista accuracy of different methods. PRM appears to be the most effective method, while Zero-shot is the least effective. The increasing accuracy of ORM and Self-consistency with more solutions suggests that these methods benefit from having more data to work with. The relatively flat performance of Zero-shot indicates that it is not influenced by the number of solutions provided. </details> (a) Qwen2.5-VL-7B <details> <summary>x3.png Details</summary>  ### Visual Description ## Line Chart: MathVista Accuracy vs. Number of Solutions per Problem ### Overview The image is a line chart comparing MathVista Accuracy (in percentage) against the number of solutions per problem. There are four data series represented by different colored lines: a teal line with diamond markers, a peach line with triangle markers, a coral line with square markers, and a light blue dashed line with 'x' markers. The chart shows how accuracy changes as the number of solutions increases. ### Components/Axes * **Y-axis (Vertical):** "MathVista Accuracy (%)". The scale ranges from 65.0% to 75.0% with tick marks at 65.0, 67.5, 70.0, 72.5, and 75.0. * **X-axis (Horizontal):** "# Solutions per problem". The scale has values 4, 8, 16, 32, and 64. * **Data Series:** Four distinct lines, each with unique markers and colors. ### Detailed Analysis * **Teal Line (Diamond Markers):** This line represents the highest accuracy among the four. * At 4 solutions, the accuracy is approximately 68%. * At 8 solutions, the accuracy is approximately 71.5%. * At 16 solutions, the accuracy is approximately 73.5%. * At 32 solutions, the accuracy is approximately 74%. * At 64 solutions, the accuracy is approximately 75%. * Trend: The teal line shows an upward trend, indicating increasing accuracy with more solutions, but the rate of increase slows down as the number of solutions increases. * **Peach Line (Triangle Markers):** This line shows a moderate level of accuracy. * At 4 solutions, the accuracy is approximately 65%. * At 8 solutions, the accuracy is approximately 67%. * At 16 solutions, the accuracy is approximately 68%. * At 32 solutions, the accuracy is approximately 69.5%. * At 64 solutions, the accuracy is approximately 69.5%. * Trend: The peach line shows an upward trend initially, but plateaus after 32 solutions. * **Coral Line (Square Markers):** This line shows a lower level of accuracy compared to the teal and peach lines. * At 4 solutions, the accuracy is approximately 64%. * At 8 solutions, the accuracy is approximately 66%. * At 16 solutions, the accuracy is approximately 68%. * At 32 solutions, the accuracy is approximately 68.5%. * At 64 solutions, the accuracy is approximately 68%. * Trend: The coral line shows an upward trend initially, but plateaus and slightly decreases after 32 solutions. * **Light Blue Dashed Line (X Markers):** This line represents the lowest and most stable accuracy. * The accuracy remains constant at approximately 65% across all solution counts. * Trend: The light blue line is relatively flat, indicating no significant change in accuracy with an increasing number of solutions. ### Key Observations * The teal line (diamond markers) consistently outperforms the other lines in terms of accuracy. * The light blue dashed line (X markers) shows no improvement in accuracy as the number of solutions increases. * The peach and coral lines show improvement up to 32 solutions, after which they plateau or slightly decline. ### Interpretation The chart suggests that increasing the number of solutions per problem generally improves MathVista accuracy, but the extent of improvement varies depending on the specific approach represented by each line. The teal line represents the most effective approach, as it consistently achieves the highest accuracy with increasing solutions. The light blue line likely represents a baseline or control method that does not benefit from additional solutions. The peach and coral lines show diminishing returns after a certain point, suggesting that their effectiveness plateaus beyond 32 solutions per problem. </details> (b) InternVL2.5-8B <details> <summary>x4.png Details</summary>  ### Visual Description ## Line Chart: MathVista Accuracy vs. Number of Solutions per Problem ### Overview The image is a line chart illustrating the relationship between MathVista Accuracy (in percentage) and the number of solutions per problem. There are four distinct data series represented by different colored lines with different markers. The x-axis represents the number of solutions per problem, while the y-axis represents the MathVista accuracy in percentage. ### Components/Axes * **X-axis:** "# Solutions per problem" with markers at 4, 8, 16, 32, and 64. * **Y-axis:** "MathVista Accuracy (%)" with markers at 74, 76, 78, 80, and 82. * **Data Series:** Four distinct data series are plotted on the chart, each represented by a different color and marker. The legend is missing, so the exact meaning of each series is unknown. ### Detailed Analysis * **Teal Line (Diamond Markers):** This line shows an upward trend. * At 4 solutions, the accuracy is approximately 76%. * At 8 solutions, the accuracy is approximately 79%. * At 16 solutions, the accuracy is approximately 80%. * At 32 solutions, the accuracy is approximately 81%. * At 64 solutions, the accuracy is approximately 82.5%. * **Orange Line (Triangle Markers):** This line shows an upward trend that plateaus. * At 4 solutions, the accuracy is approximately 75.5%. * At 8 solutions, the accuracy is approximately 78%. * At 16 solutions, the accuracy is approximately 79%. * At 32 solutions, the accuracy is approximately 79.2%. * At 64 solutions, the accuracy is approximately 79.5%. * **Red Line (Square Markers):** This line shows an upward trend that plateaus. * At 4 solutions, the accuracy is approximately 75%. * At 8 solutions, the accuracy is approximately 77%. * At 16 solutions, the accuracy is approximately 78.5%. * At 32 solutions, the accuracy is approximately 78.5%. * At 64 solutions, the accuracy is approximately 79%. * **Light Blue Dashed Line (X Markers):** This line remains relatively constant. * The accuracy hovers around 74% for all values of solutions per problem. ### Key Observations * The teal line (diamond markers) consistently outperforms the other lines in terms of MathVista accuracy. * The light blue dashed line (X markers) shows almost no change in accuracy as the number of solutions per problem increases. * The orange and red lines show similar trends, with accuracy increasing initially and then plateauing. ### Interpretation The chart suggests that increasing the number of solutions per problem generally improves MathVista accuracy, but the extent of improvement varies depending on the specific data series (likely representing different models or algorithms). The teal line demonstrates the most significant improvement with increasing solutions, while the light blue line indicates a method that is unaffected by the number of solutions. The plateauing of the orange and red lines suggests diminishing returns for those methods beyond a certain number of solutions. Without a legend, it's impossible to definitively state what each line represents, but the data implies that the teal line represents the most effective approach among the four shown. </details> (c) Qwen2.5-VL-72B Figure 2: Best-of-N results on the MathVista [36] across different policies. The number of solutions we sample is from 4 to 64. Table 5: Ablation study of our design choices on the MathVista [54] and WeMath [44] with Qwen2.5-VL-7B [3] as the policy model. | Method | Data | ORM init | Up-sample | MathVista | WeMath | | --- | --- | --- | --- | --- | --- | | Zero-shot | - | - | - | 68.1 | 36.2 | | Self-consistency | - | - | - | 71.6 | 44.7 | | Athena-ORM | Athena-600K | - | - | 72.8 | 45.1 | | Athena-PRM | Random 5K | ✘ | ✘ | 73.1 | 45.2 | | Athena-5K | ✘ | ✘ | 74.1 | 45.8 | | | Random 5K | ✔ | ✘ | 73.6 | 45.4 | | | Athena-5K | ✔ | ✘ | 74.8 | 46.2 | | | Athena-5K | ✔ | ✔ | 75.2 | 46.4 | | | MC-300K | ✔ | ✔ | 75.5 | 46.4 | | The effectiveness of our design choices. Table 5 presents results under various design choices. Our findings indicate that PRMs trained with selectively chosen 5K samples significantly outperform those trained with randomly selected samples, achieving scores 74.1 vs. 73.1 and 74.8 vs. 73.6 when initialized from ORMs. Moreover, a large-scale dataset of 300K samples labeled using vanilla MC-based methods yields similar performance to our selected 5K samples. With these selected samples, ORM initialization enhances performance by 0.7 points (74.8 vs. 74.1), and up-sampling negative labels further boosts performance by 0.4 points on the MathVista [36]. Computation cost analysis. Vanilla MC-based estimation [55] incurs substantial computational demands to estimate the correctness of intermediate steps. Our prediction consistency filter strategy significantly reduces this burden, requiring only 5K samples for training, thereby lowering the computational cost by a factor of 60. For data synthesis, we employ vLLM primarily [25] to accelerate MLLM inference. Compared to the MC-300K dataset, our Athena-5K dataset reduces GPU hours by approximately 45-fold while achieving similar performance. 4 Related Work Test time scaling. While scaling data and model parameters during training has been extensively studied [23, 19], scaling computation at test time has garnered interest more recently [50, 46]. TTS enables language models to leverage additional computational resources when faced with challenging questions. Our study primarily focuses on parallel scaling [67], wherein language models generate multiple outputs simultaneously and aggregate them into a final answer. This aggregation can involve selecting the most common outcome [58] or using reward models [50] to rank solutions, selecting the one with the highest reward, as demonstrated in our Best-of-N settings. Process reward models. Reward models are crucial in reinforcement learning [17, 47, 55, 42, 4, 13] and test time scaling [46, 28, 55, 9, 50]. Outcome reward models [9] assign a scalar value to question-response pairs, whereas process reward models evaluate each step, typically achieving superior performance and generalization [28, 37, 46, 27]. While better performance and generalization are achieved, collecting data to train PRMs is challenging. PRM800K [28] is the first open-source process supervision dataset annotated by humans, but scaling it is challenging due to the time and skill required for annotating reasoning steps. Math-Shepherd [55] proposes an automated method for estimating intermediate step correctness using Monte Carlo estimation, though it generates some incorrect labels and demands extensive computational resources. Our work explores strategies to reduce computational costs while improving process label accuracy. 5 Conclusion In this paper, we introduce Athena-PRM, trained on 5K high-quality data with process labels. 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Appendix A More Experiments and Details A.1 Choices of Completers Here we verify the effect of combining different completers on the accuracy of estimated labels. Results are listed in Table 6. From Table 6, it is noted that the consistency filter for two completers usually improves the accuracy of estimated labels. For example, combining two weak or strong completers improves accuracy by almost two points in general. Furthermore, the accuracy of estimated labels is improved by combining one weak and one strong completer, achieving the most competitive accuracy, e.g., 95.2% with weak completer $\phi_{w}^{2}$ and strong completer $\phi_{s}^{1}$ . This shows that the proposed consistency filter strategy is effective across all combinations of different completers and achieves the best performance when we combine a weak completer and a strong completer. Table 6: The accuracy of estimated process labels with Different completers combination. In Two Completers, we use consistency filter introduced in Sec. 2.2. $\phi_{w}^{1}$ , $\phi_{w}^{2}$ , $\phi_{s}^{1}$ , and $\phi_{w}^{2}$ denote Mistral-7B-Instruct, Qwen2-7B-Instruct, Qwen2.5-72B-Instruct and Llama-3-70B-Instruct, respectively. | | Weak Completer | Strong Completer | Accuracy(%) | | | | --- | --- | --- | --- | --- | --- | | $\phi_{w}^{1}$ | $\phi_{w}^{2}$ | $\phi_{s}^{1}$ | $\phi_{s}^{2}$ | | | | Single Completer | ✔ | | | | 78.2 | | ✔ | | | 80.1 | | | | ✔ | | 83.4 | | | | | ✔ | 84.7 | | | | | | Two Completers | ✔ | ✔ | | | 82.3 | | ✔ | ✔ | 85.6 | | | | | ✔ | | ✔ | | 94.1 | | | ✔ | | | ✔ | 93.8 | | | ✔ | ✔ | | 95.2 | | | | ✔ | | ✔ | 94.7 | | | A.2 Benchmarks We provide more details about used benchmarks in Table 7. Table 7: Details of all benchmarks used and corresponding metrics. | | Split | # Samples | Metric | | --- | --- | --- | --- | | MathVista [36] | Testmini | 1000 | Accuracy | | MathVision [54] | Full | 3040 | Accuracy | | MathVerse [68] | Vision-Only | 788 | Accuracy | | DynaMath [70] | Full | 5050 | Accuracy | | WeMath [44] | Testmini | 1740 | Score (Strict) | | LogicVista [60] | Full | 448 | Accuracy | | MMMU [65] | Validation | 900 | Accuracy | | GSM8K [9] | Test | 1319 | Accuracy | | MATH [18] | Test | 5000 | Accuracy | | VisualProcessBench [57] | - | 2866 | F1-score | A.3 Reward Choice In this paper, we consider three reward choices of Athena-PRM for aggregating step scores into a final score [69], including PRM-minium, PRM-last, and PRM-product. For all reasoning steps $\{a_{i}\}_{i=1}^{K}$ , its corresponding reward for each step is $\{r_{i}\}_{i=1}^{K}$ , the reward $r$ for each solution with different scoring methods are calculated as follows: 1) PRM-minium uses the minium reward across all steps, i.e., $r=\min\{r_{i}\}_{i=1}^{K}$ ; 2) PRM-last uses the reward of last step, i.e., $r=r_{K}$ ; 3) PRM-product uses the reward product of all steps, i.e. $r=\prod_{i=1}^{K}r_{i}$ . We list results in Table 8. It is shown that different choices of selecting a reward for PRMs could produce different performance. In all other experiments, we use the minimum reward across all steps as the reward for the solution unless otherwise specified. Table 8: Best-of-8 performance comparison on the MathVista [36] with Qwen2.5-VL-7B [3] with different reward choices: last step reward, minium reward across all steps and product of all rewards. Default settings are marked in gray. | Method | Accuracy(%) | | --- | --- | | Zero-shot | 68.1 | | Self-consistency | 71.6 | | ORM | 72.8 | | PRM-product | 74.5 | | PRM-last | 75.1 | | PRM-minium | 75.2 | A.4 Label Distribution We present the distribution of process labels in Table 9. It is shown that label imbalance exists in PRM800K and Athena-300K. We try different up-sample rates (e.g. $× 2$ ) for samples with error steps and find that slightly up-sampling negative data is beneficial for training PRMs, as shown in Table 10. We find that increasing the up-sample rate will not improve performance. We set the up-sample rate to 2 for other experiments. Table 9: Distribution of labels across different datasets. Good, neutral, and bad denote different process labels. | Dataset | Good | Neutral | Bad | | --- | --- | --- | --- | | PRM800K [28] | 76% | 12% | 12% | | Athena-300K | 81% | - | 19% | | Athena-5K | 79% | - | 21% | Table 10: Results on MathVista under Best-of-8 setting with different up-sampling rate. We use Athena-5K as training dataset and Qwen-2.5-VL-7B as the policy model. Default settings are marked in gray. | Method | Up-sample rate | Accuracy(%) | | --- | --- | --- | | Zero-shot | - | 68.1 | | Self-consistency | - | 71.6 | | Athena-ORM | - | 72.8 | | Athena-PRM | ✘ | 74.8 | | Athena-PRM | $× 2$ | 75.2 | | Athena-PRM | $× 5$ | 75.2 | | Athena-PRM | $× 10$ | 74.9 | Appendix B Limitations and Social Impacts Limitations. Athena-PRM achieves impressive performance across various scenarios and benchmarks. However, Athena-PRM is specifically designed to enhance the reasoning capabilities of LLMs and MLLMs. Its application may be limited to solving problems with clear answers, such as multi-step reasoning and math problems. Effectively assigning rewards for intermediate steps in open-ended text generation tasks remains an open problem, and we leave this for future research. In addition, how to define a step for complex multi-step reasoning is still underexplored. Existing PRMs typically split solutions into multiple steps using predefined symbols (e.g., “\n\n”) or fixed numbers of tokens or lines. This approach may result in steps that lack sufficient information. To prevent steps from containing too little information, we design a simple strategy that merges steps with too few tokens or without any mathematical symbols. However, better splitting strategies without human-designed rules still need to be explored. Social Impacts. The paper introduces an effective method for generating high-quality, process-labeled data for reasoning problems. Our approach enhances the efficiency and effectiveness of training PRMs. We hope that our research will have a broader impact on improving the reasoning abilities of foundation models. Since our method focuses on general improvements in the reasoning capabilities of foundation models, it does not directly produce negative societal impacts. Appendix C Case Study We provide some examples from VisualProcessBench and MathVista to help readers understand how PRMs work, see Figure 3, 4, 5, 6. <details> <summary>extracted/6532028/fig/example_fig/images_MathVision_MINI_15.png Details</summary>  ### Visual Description ## Route Finding Puzzle ### Overview The image presents a series of simple matching tasks followed by a route-finding puzzle on a grid. The puzzle involves finding a path for an ant to reach its home, guided by arrows on the grid. The image concludes with a multiple-choice question asking to identify the last animal on the path. ### Components/Axes * **Matching Tasks:** The image starts with a series of paired images. The first image in each pair shows an animal or object, and the second image is blank. * **Grid:** A 7x7 grid with alternating white and gray squares. * **Path:** A path marked with arrows, starting from an ant and leading through the grid. * **Animals/Objects on the Grid:** Various animals and objects are placed on the grid, including a bee, ladybug, butterfly, house, snail, caterpillar, and frog. * **Multiple Choice Options:** Five images of animals labeled (A) through (E) are presented at the bottom of the image. ### Detailed Analysis 1. **Matching Tasks:** * Image 1: Ant * Image 2: House * Image 3: Ladybug * Image 4: Ant * Image 5: House 2. **Grid and Path:** * The path starts at the bottom-left of the grid, where an ant is located next to a house. * The path proceeds as follows: * Right two squares * Up three squares * Right two squares * Up one square * Right one square 3. **Animals/Objects Encountered on the Path:** * Starting Point: Ant near a house * Along the path: Caterpillar, Snail, Ladybug, Butterfly 4. **Multiple Choice Options:** * (A): Butterfly * (B): Bee * (C): Caterpillar * (D): Snail * (E): Frog ### Key Observations * The path is clearly marked with arrows, making it easy to follow. * The animals and objects are placed strategically on the grid to create a simple puzzle. * The multiple-choice options include animals that are both on and off the path, requiring careful observation. ### Interpretation The image presents a simple puzzle designed to test visual tracking and pattern recognition skills. The initial matching tasks serve as a warm-up, while the grid-based pathfinding requires more focused attention. The final multiple-choice question assesses the ability to recall the sequence of elements encountered along the path. The correct answer is (A) Butterfly, as it is the last animal on the path. </details> | Hint: Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end. Question: When the ant <image1> walks from home <image2> along the arrows $→ 3,\uparrow 3,→ 3,\uparrow 1$ , he gets to the ladybird <image3>. Which animal does the ant <image1> get to when he walks from home <image2> along the following arrows: $→ 2,\downarrow 2,→ 3,\uparrow 3,→ 2,\uparrow 2$ ? <image6> <image7> Choices:(A) A (B) B (C) C (D) D (E) E (Answer: A) | | | | --- | --- | --- | | Step | Reward | Label | | To determine which animal the ant reaches, we will follow the given movements step-by-step on the grid: | 0.95 | ✔ | | 1. Start at the home position (‘<image2>‘), which is at the bottom-left corner of the grid. | 0.99 | ✔ | | 2. Follow the arrows: - $→ 2$ : Move 2 squares to the right. - $\downarrow 2$ : Move 2 squares down. - $→ 3$ : Move 3 squares to the right. - $\uparrow 3$ : Move 3 squares up. - $→ 2$ : Move 2 squares to the right. - $\uparrow 2$ : Move 2 squares up. | 0.99 | ✔ | | 3. Trace the movement: - Start at the bottom-left corner. - Move 2 squares right: You are now in the third column of the first row. - Move 2 squares down: You are now in the third column of the third row. - Move 3 squares right: You are now in the sixth column of the third row. - Move 3 squares up: You are now in the sixth column of the top row. - Move 2 squares right: You are now in the eighth column of the top row. - Move 2 squares up: You are now in the eighth column of the top row (still at the butterfly position). | 0.12 | ✘ | | The ant reaches the butterfly, which corresponds to option A. | 0.87 | ✔ | | ### Final Answer: A | 0.75 | ✔ | Figure 3: A case study from VisualProcessBench [57]. <details> <summary>extracted/6532028/fig/example_fig/images_MathVerse_MINI_Vision_Only_35.png Details</summary>  ### Visual Description ## Geometry Diagram: Circle and Tangent ### Overview The image presents a geometry problem involving a circle with center O, a tangent line at point C, and an angle A formed by lines intersecting the circle. The problem asks for the degree of angle A, given that the angle between the tangent line and line segment CB is 50 degrees. Multiple choice answers are provided. ### Components/Axes * **Circle:** Labeled with center "O". * **Points:** Labeled A, B, C, and D. Point C lies on the circle and is the point of tangency. * **Tangent Line:** A line that touches the circle at point C. * **Lines:** Line segments connect points A to C, A to B, C to B, and C to D (tangent line). * **Angle:** The angle between the tangent line CD and the line segment CB is labeled as 50°. * **Text:** * "As shown in the figure, passing point C to draw the tangent of circle O. then the degree of angle A is ()" * "Choices: A:20° B:25° C:.40° D:50°" ### Detailed Analysis The diagram shows a circle with center O. Point C lies on the circle, and line CD is tangent to the circle at point C. The angle between the tangent line CD and the line segment CB is given as 50°. The problem asks to find the measure of angle A. The multiple-choice options provided are: * A: 20° * B: 25° * C: 40° * D: 50° ### Key Observations * The angle between the tangent and the chord (angle BCD) is 50 degrees. * Angle A subtends the same arc as angle BCD. * The inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. * The angle formed by a tangent and a chord is half the measure of the intercepted arc. ### Interpretation The problem is a geometry question that requires knowledge of circle theorems, specifically the relationship between the angle formed by a tangent and a chord, and the inscribed angle subtending the same arc. Since the angle between the tangent CD and the chord CB is 50°, the inscribed angle A, which subtends the same arc BC, should also be 50°. However, the relationship between the tangent-chord angle and the inscribed angle is that they are equal if they intercept the same arc. Therefore, angle A should be 50°. </details> | According to the question shown in the image, please directly answer the question and provide the correct option letter, e.g., A, B, C, D. (Answer: A) | | | | --- | --- | --- | | Step | Reward | Label | | To find the degree of angle A, we can use the property that the angle subtended by an arc at the center of the circle is twice the angle subtended at any point on the circle. | 0.82 | ✔ | | 1. Identify the key points: - The arcs are AC and BC. - Angle A is subtended by these arcs at point O (the center of the circle). | 0.85 | ✔ | | 2. Understand the given information: - CD is a tangent at point C. - The angle between the tangent and the chord is equal to the angle subtended by the chord at the circumference. Therefore, angle CDB = 50°. | 0.85 | ✔ | | 3. Angle properties: - The angle A is subtended by arc BC at the center. - Since CD is a tangent, angle A = 50° (as the inscribed angle is half the central angle). | 0.23 | ✘ | | Thus, the degree of angle A is 20°. | 0.67 | ✔ | | Final answer: A | 0.96 | ✔ | Figure 4: A case study from VisualProcessBench [57]. <details> <summary>extracted/6532028/fig/example_fig/mathvista_559.jpg Details</summary>  ### Visual Description ## Watch: Omega Speedmaster ### Overview The image is a close-up shot of an Omega Speedmaster watch with a stainless steel band. The watch features a black dial with digital display elements, gold-colored hands, and a black bezel with blue markings. ### Components/Axes * **Bezel:** Black with blue markings from 05 to 55 in increments of 5. * **Dial:** Black with digital display showing "6:09". The Omega logo is present. The word "Speedmaster" is printed on the dial. The word "Chronometer" is printed on the dial. * **Hands:** Gold-colored hour and minute hands, and a red-tipped seconds hand. * **Band:** Stainless steel. ### Detailed Analysis or Content Details * The digital display shows the time as "6:09". * The bezel has markings at intervals of 5, ranging from "05" to "55". * The Omega logo is positioned above the center of the dial. * The word "Speedmaster" is printed below the Omega logo. * The word "Chronometer" is printed above the digital display. ### Key Observations * The watch is in a well-lit environment, highlighting its features. * The digital display is clearly visible. * The gold-colored hands contrast well with the black dial. ### Interpretation The image showcases the design and features of an Omega Speedmaster watch. The combination of analog hands and a digital display provides both classic and modern functionality. The blue markings on the bezel and the red tip on the seconds hand add subtle color accents to the overall design. The presence of "Speedmaster" and "Chronometer" on the dial indicates the watch's heritage and precision. </details> | Col1 | Col2 | Col3 | | --- | --- | --- | | Hint: Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end. Question: what is the highest number shown on the black outer part of the watch? (Answer: 55) | | | | Step | Reward | Label | | To determine the highest number shown on the black outer part of the watch, let’s analyze the image step by step | 0.90 | ✔ | | 1. The black outer part of the watch appears to be a bezel with numbers printed on it. | 0.87 | ✔ | | 2. The numbers on the bezel are arranged in a circular pattern. | 0.96 | ✔ | | 3. Observing the numbers, they range from 0 to 60, increasing in increments of 5. | 0.83 | ✔ | | 4. The highest number visible on the bezel is 60. | 0.34 | ✘ | | Therefore, the highest number shown on the black outer part of the watch is 60. | 0.15 | ✘ | | Answer: $\boxed{60}$ | 0.04 | ✘ | Figure 5: A case study from MathVista [36] with Qwen2.5-VL-72B [3]. <details> <summary>extracted/6532028/fig/example_fig/mathvista_605.jpg Details</summary>  ### Visual Description ## Image Analysis: Geometric Shapes ### Overview The image depicts a collection of various geometric shapes arranged on a surface. The shapes include spheres, cubes, and cylinders, each with distinct colors and reflective properties. The scene is lit to create shadows and highlights, enhancing the three-dimensional appearance of the objects. ### Components/Axes * **Shapes:** * Gold Sphere: Located on the left side of the image. * Silver Cylinder: Located to the left of the gold sphere. * Green Cube: Located in the center-top of the image. * Teal Cylinder: Located in the center-right of the image. * Gold Cube: Located on the right side of the image. * Teal Sphere: Located in the center-bottom of the image. * Gray Sphere: Located on top of the teal cylinder. * **Surface:** A light-colored, possibly white or gray, surface on which the shapes are arranged. * **Lighting:** The scene is lit to create shadows and highlights on the shapes. ### Detailed Analysis * **Gold Sphere:** A reflective, golden-colored sphere positioned on the left side of the image. * **Silver Cylinder:** A small, silver-colored cylinder located to the left of the gold sphere. * **Green Cube:** A green cube positioned in the center-top of the image. * **Teal Cylinder:** A teal-colored cylinder located in the center-right of the image. * **Gold Cube:** A reflective, golden-colored cube positioned on the right side of the image. * **Teal Sphere:** A small, teal-colored sphere located in the center-bottom of the image. * **Gray Sphere:** A small, gray-colored sphere located on top of the teal cylinder. ### Key Observations * The shapes are arranged in a seemingly random manner, with no apparent pattern or order. * The lighting creates shadows and highlights on the shapes, enhancing their three-dimensional appearance. * The different colors and reflective properties of the shapes add visual interest to the scene. ### Interpretation The image appears to be a simple arrangement of geometric shapes, possibly for artistic or illustrative purposes. The different colors and reflective properties of the shapes, combined with the lighting, create a visually appealing scene. The arrangement of the shapes does not seem to convey any specific meaning or message, but rather serves as a study in form, color, and light. </details> | Hint: Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end. Question: Subtract all large green matte cubes. Subtract all big green blocks. How many objects are left? (Answer: 6) | | | | --- | --- | --- | | Step | Reward | Label | | To solve this problem, let’s identify and count the objects in the image: | 0.95 | ✔ | | 1. There is one large green matte cube. | 0.99 | ✔ | | 2. There are no other big green blocks since the only green block is already counted. | 0.90 | ✔ | | Now, we subtract these objects as per the instructions | 0.92 | ✔ | | - Subtract all large green matte cubes: This removes the large green matte cube from the count. | 0.99 | ✔ | | - Subtract all big green blocks: Since there are no other big green blocks, nothing additional is removed. | 0.99 | ✔ | | After removing the large green matte cube, the remaining objects are: - A small gray metallic cylinder. - A large gold metallic sphere. - A small gray matte sphere. - A large cyan matte cylinder. - A small cyan matte sphere. - A large gold metallic cube. | 0.84 | ✔ | | Counting the remaining objects, we have 6 objects left. | 0.92 | ✔ | | Answer: $\boxed{6}$ | 0.99 | ✔ | Figure 6: A case study from MathVista [36] with Qwen2.5-VL-72B [3].