# Retrieval-Augmented Process Reward Model for Generalizable Mathematical Reasoning
> *Corresponding author
Abstract
While large language models (LLMs) have significantly advanced mathematical reasoning, Process Reward Models (PRMs) have been developed to evaluate the logical validity of reasoning steps. However, PRMs still struggle with out-of-distribution (OOD) challenges. This paper identifies key OOD issues, including step OOD—caused by differences in reasoning patterns across model types and sizes—and question OOD, which arises from dataset shifts between training data and real-world problems. To address these issues, we introduce Retrieval-Augmented Process Reward Model (RetrievalPRM), a novel framework designed to tackle these OOD issues. By utilizing a two-stage retrieval-enhanced mechanism, RetrievalPRM retrieves semantically similar questions and steps as a warmup, enhancing PRM’s ability to evaluate target steps and improving generalization and reasoning consistency across different models and problem types. Our extensive experiments demonstrate that RetrievalPRM outperforms existing baselines across multiple real-world datasets. Our open-source contributions include a retrieval-enhanced dataset, a tuning framework for PRM training, and the RetrievalPRM model, establishing a new standard for PRM performance.
Retrieval-Augmented Process Reward Model for Generalizable Mathematical Reasoning
Jiachen Zhu 1, Congmin Zheng 1, Jianghao Lin 1, Kounianhua Du 1 Ying Wen 1 ∗, Yong Yu 1, Jun Wang 2, Weinan Zhang 1 thanks: *Corresponding author 1 Shanghai Jiao Tong University, 2 University College London {gebro13,desp.zcm,chiangel,kounianhuadu,ying.wen,wnzhang}@sjtu.edu.cn, yyu@apex.sjtu.edu.cn jun.wang@cs.ucl.ac.uk
1 Introduction
While large language models (LLMs) have advanced mathematical reasoning OpenAI (2023); Dubey et al. (2024); Zhu et al. (2024); Shao et al. (2024); Yang et al. (2024b), they remain prone to critical flaws: explicit errors (e.g., miscalculations, logical inconsistencies) and implicit risks where correct answers mask flawed intermediate steps. Even when final results are accurate, LLMs often generate plausible-but-incorrect reasoning chains, eroding trust in their problem-solving processes Lightman et al. (2023). To address this, Process Reward Models (PRMs) Lightman et al. (2023); Wang et al. (2024b) have been developed to rigorously evaluate the logical validity of intermediate steps Cobbe et al. (2021), mirroring human pedagogical practices that prioritize reasoning quality over answer correctness.
<details>
<summary>x1.png Details</summary>
### Visual Description
## Scatter Plot: Dataset Distribution Analysis
### Overview
The image is a 2D scatter plot visualizing the distribution of three datasets (GSM8k, MATH, OlympiadBench) across two dimensions (x and y). Points are color-coded by dataset, with density shading indicating point concentration. The plot reveals distinct clustering patterns and spatial relationships between datasets.
### Components/Axes
- **Axes**:
- **X-axis**: "Dimension x" (range: -40 to 50)
- **Y-axis**: "Dimension y" (range: -20 to 25)
- **Legend**: Located in the top-right corner, associating:
- **Green**: GSM8k
- **Orange**: MATH
- **Blue**: OlympiadBench
- **Density Shading**: Semi-transparent gradients overlaying point clusters to highlight density.
### Detailed Analysis
1. **GSM8k (Green)**:
- **Positioning**: Dominates the right half of the plot (x: 20–40, y: 0–20).
- **Density**: Highest concentration near (x=30, y=10), with gradual tapering toward edges.
- **Outliers**: A few points extend leftward (x: 10–20, y: -5–5).
2. **MATH (Orange)**:
- **Positioning**: Concentrated in the lower-left quadrant (x: -30–0, y: -15–10).
- **Density**: Peaks near (x=-20, y=-5), with sparse points near the origin.
- **Overlap**: Minimal overlap with GSM8k; slight overlap with OlympiadBench near (x=0, y=0).
3. **OlympiadBench (Blue)**:
- **Positioning**: Broad distribution spanning x: -10–30 and y: -5–15.
- **Density**: Two main clusters:
- **Left Cluster**: x: -10–0, y: -5–5 (overlaps MATH).
- **Right Cluster**: x: 10–30, y: 5–15 (overlaps GSM8k).
- **Outliers**: Sparse points near (x=40, y=20) and (x=-20, y=-10).
### Key Observations
- **Clustering**: GSM8k and MATH form distinct clusters with minimal overlap, while OlympiadBench bridges the two.
- **Density Peaks**:
- GSM8k: (30, 10)
- MATH: (-20, -5)
- OlympiadBench: (-5, 0) and (25, 10)
- **Spatial Relationships**:
- OlympiadBench acts as a transitional dataset between GSM8k and MATH.
- No dataset dominates the entire plot; all occupy specific regions.
### Interpretation
The plot suggests that GSM8k and MATH represent distinct problem types or domains, as evidenced by their spatial separation. OlympiadBench, however, exhibits hybrid characteristics, potentially indicating a composite dataset or problems requiring skills from both GSM8k and MATH. The density shading confirms that GSM8k and MATH have more concentrated distributions, while OlympiadBench’s broader spread implies greater variability or interdisciplinary content. The overlap regions (e.g., near the origin) may highlight shared problem-solving strategies or ambiguous categorizations. This visualization could inform dataset curation, model training, or cross-domain analysis in computational problem-solving contexts.
</details>
Figure 1: The distribution differences across three datasets: GSM8K, MATH and Olympiad. We use sentence-bert to encode these questions and perform t-sne visualization.
Existing works Wang et al. (2024a); o1 Team (2024); Zheng et al. (2024) frame PRM as a binary classification problem. They train PRM on open-source base LLMs such as Qwen Yang et al. (2024b) or Llama Dubey et al. (2024) using human-annotated dataset Lightman et al. (2023) or automated process supervision method Wang et al. (2024b); Luo et al. (2024); Qin et al. (2024). Although these approaches show great performance and empirical success, they still face kinds of out-of-distribution challenges. We believe the out-of-distribution (OOD) problem can be viewed from the following perspectives:
<details>
<summary>x2.png Details</summary>
### Visual Description
## Diagram: AI Model Comparison for Mathematical Problem Solving
### Overview
The diagram compares three AI models (ChatGPT-4o, Qwen2.5-Math-72B-instruct, and Qwen2.5-Math-1.5B-instruct) in solving a mathematical problem involving five numbers added in pairs. It highlights differences in problem-solving approaches, computational steps, and final answers.
---
### Components/Axes
1. **Model Sections**:
- Three vertical rectangles represent the three models.
- Each section includes:
- **Process**: Step-by-step reasoning.
- **Chain Length**: Number of steps in the solution.
- **Solution Style**: Methodology (e.g., "Analyze, Calculate").
- **Answer**: Final numerical result.
2. **Comparison Section**:
- **Model Type**: Categorizes models as "Process" or "Math" types.
- **Model Size**: Indicates parameter scale (e.g., 72B vs. 1.5B).
---
### Detailed Analysis
#### Model 1: ChatGPT-4o
- **Process**:
- Step 1: Sum of all pairwise sums = 4S (where S = a+b+c+d+e).
- Step 2: Total pairwise sums = 1066 → S = 266.5.
- **Chain Length**: 6 steps.
- **Solution Style**: Analyze, Calculate.
- **Answer**: 56.
#### Model 2: Qwen2.5-Math-72B-instruct
- **Process**:
- Step 1: Smallest sum = a+b=101; largest sum = d+e=111.
- Step 2: Derived equations (e.g., c-b=1) to solve for variables.
- **Chain Length**: 10 steps.
- **Solution Style**: Solve Equations.
- **Answer**: 57.
#### Model 3: Qwen2.5-Math-1.5B-instruct
- **Process**:
- Step 1: Enumerated all possible pairs to identify the largest sum.
- Step 2: Listed pairs (e.g., (101,101), (101,102)) to deduce the largest number.
- **Chain Length**: 3 steps.
- **Solution Style**: Enumerate.
- **Answer**: 56.
#### Comparison Section
- **Model Type**:
- ChatGPT-4o: "Process".
- Qwen2.5-Math-72B-instruct: "Math".
- Qwen2.5-Math-1.5B-instruct: "Math".
- **Model Size**:
- Qwen2.5-Math-72B-instruct: 72B parameters.
- Qwen2.5-Math-1.5B-instruct: 1.5B parameters.
---
### Key Observations
1. **Chain Length Variance**:
- Qwen2.5-Math-72B-instruct uses the most steps (10), while Qwen2.5-Math-1.5B-instruct uses the fewest (3).
2. **Solution Methodology**:
- ChatGPT-4o relies on algebraic summation.
- Qwen2.5-Math-72B-instruct uses equation-solving.
- Qwen2.5-Math-1.5B-instruct employs enumeration.
3. **Answer Consistency**:
- Two models (ChatGPT-4o and Qwen2.5-Math-1.5B-instruct) arrive at the same answer (56), while Qwen2.5-Math-72B-instruct achieves 57.
---
### Interpretation
1. **Efficiency vs. Accuracy**:
- The 72B model (Qwen2.5-Math-72B-instruct) achieves a slightly higher answer (57) but requires significantly more steps (10) compared to the 1.5B model (3 steps for 56). This suggests larger models may prioritize thoroughness over efficiency.
- The 1.5B model’s enumeration approach is faster but risks missing edge cases, yet it matches the answer of the larger model in this instance.
2. **Model Type Implications**:
- "Math"-specialized models (Qwen variants) outperform the general "Process" model (ChatGPT-4o) in both steps and accuracy for this problem.
- The 72B model’s equation-solving method aligns with the problem’s structured nature, while the 1.5B model’s enumeration works for simpler cases.
3. **Scalability Trade-offs**:
- The 72B model’s increased parameter count does not guarantee proportional performance gains, as the 1.5B model achieves near-identical results with fewer resources.
---
### Conclusion
The diagram illustrates how model architecture, size, and problem-solving strategies influence outcomes. While larger models (72B) may offer nuanced solutions, smaller models (1.5B) can achieve comparable results through optimized methods like enumeration. This highlights the importance of aligning model capabilities with problem complexity.
</details>
Figure 2: Processes and problem-solving ideas for the same question vary from different models with the perspectives of model types and model sizes. GPT tends to analyze and calculate, while Qwen-72B tends to solve equations. Qwen-1.5B is small and relatively weak. It can only enumerate, and its thinking chain is short, so its answers are also very wrong.
Firstly, Step OOD may occur because of different processes generated by different models. Due to the high cost of manual annotation, there are very few accurately labeled PRM expert datasets, such as PRM800K and ProcessBench, with processes generated by GPT OpenAI (2023) and Qwen Yang et al. (2024b), respectively. However, different model types (e.g., GPT, Qwen, Llama Dubey et al. (2024)) approach problem-solving differently. As is shown in Figure 2, when facing the same question, GPT-4o tends to analyze and calculate, while Qwen-72B tends to solve questions directly. They have different solution styles. Therefore, using process data generated by one model to train a PRM and then applying it to guide another model leads to an OOD issue. Moreover, models of different sizes also exhibit different reasoning processes. Larger models, like exceptional students, tend to have clearer and more accurate reasoning steps, while smaller models tend to have very short reasoning chains, as shown in Figure 2.
Secondly, Question OOD emerges because of dataset shift. Current PRM datasets contain only a limited number of problems. For example, Math Shepherd and PRM800K cover problems from the GSM8K and MATH datasets, with GSM8K being at the elementary to middle school level and MATH at the high school to university level. However, real-world problems are far more diverse, such as those in the Olympic math competition dataset He et al. (2024), leading to OOD issues in other datasets. As shown in the Figure 1, we used Sentence-BERT Reimers (2019) to encode all the problems from the three datasets and visualized the distribution with t-SNE. It is evident that the distributions differ, and since both Olympic and MATH problems are typically from high school-level exams, they are semantically closer to each other than to GSM8K.
To address this issue, we propose a new framework, Retrieval Augmented Process Reward Model (RetrievalPRM), which leverages a Two-stage Retrieval-enhanced Mechanism to help PRMs solve the OOD problem. we retrieve relevant questions and steps in these two stages to address the issues of question OOD and step OOD, respectively. Specifically, when predicting a step for a given question, we select semantically similar questions based on their embeddings, placing them at the beginning of the entire prompt. Additionally, we select more fine-grained, similar steps and use them as references when predicting the correctness of the step. These retrieved questions and steps serve as a kind of warm-up for PRM, acting as example problems for reference. They not only help stimulate PRM’s potential by warming up but also allow the system to handle more difficult problems by identifying similarities, thus alleviating OOD issues.
Our main contributions are summarized as follows:
- To the best of our knowledge, we are the first to highlight the key OOD problems in Process Reward Models (PRMs), particularly the question OOD and step OOD, which arise due to differences in reasoning patterns across model types (e.g., GPT, Qwen), model sizes (1.5B, 72B) and varying problem difficulties in real-world datasets.
- We introduce the Retrieval-Augmented Process Reward Model (RetrievalPRM) framework, which utilizes a Two-stage Retrieval-enhanced Mechanism to address OOD issues by incorporating both Question-level Retrieval and Step-level Retrieval, thereby enhancing PRM’s ability to generalize across diverse problem-solving scenarios.
- We build a Retrieval-enhanced dataset for training PRM using RetrievalPRM framework. We have made our code publicly available. https://anonymous.4open.science/r/RetrievalPRM-1C77 Our dataset https://huggingface.co/datasets/gebro13/RetrievalPRM_ Dataset and model https://huggingface.co/gebro13/RetrievalPRM are open-sourced.
- Extensive experiments on the ProcessBench Zheng et al. (2024) on four public real-world datasets demonstrate that RetrievalPRM outperforms strong baselines and that the Out-of-distribution issue has been alleviated due to our retrieval approach.
2 Preliminary
In this section, we formulate the whole problem and introduce PRM as a binary classification model.
2.1 Problem Formulation
We denote the Math dataset as $\mathcal{D}=\{(q_{i},\mathbf{s}_{i},\mathbf{y}_{i})\}_{i=1}^{N}$ , where $N$ is the number of data instances. The input $q_{i}$ is the $i^{th}$ Math question. $\mathbf{s}_{i}=\{s^{1}_{i},s^{2}_{i},...,s^{n_{i}}_{i}\}$ are the solution steps, where $n_{i}$ is the step number of solution $\mathbf{s}_{i}$ . $\mathbf{y}_{i}=\{y^{1}_{i},y^{2}_{i},...,y^{n_{i}}_{i}\}$ and the label $y^{j}_{i}$ indicates the correctness from the $1^{st}$ step to the $j^{th}$ step.
$$
y^{j}_{i}=\begin{cases}1,~{}(s^{1}_{i},\ldots,s^{j}_{i})~{}\text{is correct %
for}~{}q_{i};\\
0,~{}\text{otherwise.}\end{cases} \tag{1}
$$
2.2 ORM vs. PRM
Outcome-supervised Reward Models are introduced (ORM) by Cobbe et al. (2021), where verifiers are trained for judging the final correctness of generated solutions. ORM only predicts the final label $\hat{y}^{n_{i}}_{i}$ , which can be formulated as
$$
\forall{i},\hat{y}^{n_{i}}_{i}=\text{ORM}(q_{i},s^{1}_{i},\ldots,s^{n_{i}}_{i}). \tag{2}
$$
Building on this, the concept of process reward models (PRM) is introduced as a more granular and transparent approach. Not only does PRM evaluate the final solutions but it also assesses intermediate processes, where $\hat{y}^{j}_{i}$ represents the predicted label for the $j^{th}$ step by PRM.
$$
\forall{i,j},\hat{y}^{j}_{i}=\text{PRM}(q_{i},s^{1}_{i},\ldots,s^{j}_{i}). \tag{3}
$$
2.3 Large Language Model for PRM scoring
When directly adopting LLMs as the PRM for scoring, we need to convert the data $(q_{i},\mathbf{s}_{i},\mathbf{y}_{i})$ with a hard prompt template. The whole template example is illustrated in Appendix B.2.
The textual input consists of the question $q_{i}$ and steps $\mathbf{s}_{i}$ , followed by a binary question about the correctness of these steps.
To obtain the floating-point correctness estimation $\hat{y}_{i}^{j}∈[0,1]$ instead of discrete word tokens ’+’ or ’-’, we apply bidimensional softmax over the corresponding logits of the binary key answer tokens (ie., + & -) from LLMs to accomplish the correctness estimation during evaluation:
$$
\hat{y}_{i}^{j}=\frac{\exp(l_{i,\text{+}})}{\exp(l_{i,\text{+}})+\exp(l_{i,%
\text{-}})}\in(0,1). \tag{4}
$$
where $l_{i,\text{+}}$ and $l_{i,\text{-}}$ are the logits of token + and - in the $i^{th}$ instance, respectively.
It is important to note that the estimated PRM scoring $\hat{y}_{i}^{j}$ is used solely for evaluation on the testing set. If training is involved, we maintain the standard instruction tuning and causal language modeling paradigm for LLMs. In this way, we don’t need to replace the language model head with binary classification head which is the last layer of LLM.
3 Methodology
In this section, we introduce our proposed RetrievalPRM framework in detail.
<details>
<summary>x3.png Details</summary>
### Visual Description
## Diagram: Comparison of PRM Frameworks
### Overview
The image compares three problem-solving frameworks: **Traditional PRM**, **Two-stage Retrieval-enhanced Mechanism**, and **Retrieval PRM Framework**. Each framework is structured with system prompts, target questions, solution steps, and feedback mechanisms.
### Components/Axes
1. **Traditional PRM**
- **System Prompt**: "I want you to act as a math teacher..."
- **Target Question**: "How many seconds are in 5.5 minutes?"
- **Solution Steps**:
- Step 1: 5.5 minutes = 5 minutes + 0.5 minutes.
- Step 2: 5 minutes = 300 seconds.
- Step 3: 0.5 minutes = 30 seconds.
- **Target Step**: "Is that step correct?" with feedback (Yes: 0.9, No: 0.1).
2. **Two-stage Retrieval-enhanced Mechanism**
- **Question Pool**: Contains reference questions (e.g., "What is the equivalent number of seconds in 7.8 minutes?").
- **Reference Questions**:
- Reference Question 1: Process involves converting 7.8 minutes to seconds (7.8 × 60 = 468 seconds).
- Reference Question 2: Placeholder for additional examples.
- **New Step Pool**: Stores validated steps (e.g., "0.3 hours = 18 minutes").
3. **Retrieval PRM Framework**
- **Reference Questions**:
- Reference Question 1: "How many seconds are in 5.5 minutes?"
- Reference Question 2: Placeholder for additional examples.
- **Reference Steps**:
- Reference Step 1: "0.3 hours = 18 minutes" (correct).
- Reference Step 2: Placeholder for additional steps.
- **Feedback**: Emojis (😊 for correct, 😠for incorrect) and probabilities (Yes: 0.2, No: 0.8).
### Detailed Analysis
- **Traditional PRM**:
- The target question is solved via direct calculation (5.5 minutes = 330 seconds).
- Feedback shows high confidence in correctness (Yes: 0.9).
- **Two-stage Retrieval-enhanced Mechanism**:
- Uses a database of reference questions to guide problem-solving.
- Example process: 7.8 minutes × 60 = 468 seconds.
- **Retrieval PRM Framework**:
- Integrates reference steps (e.g., "0.3 hours = 18 minutes") to validate target steps.
- Feedback uses emojis and probabilities to indicate correctness.
### Key Observations
1. **Feedback Mechanisms**:
- Traditional PRM uses binary feedback (Yes/No) with probabilities.
- Retrieval PRM Framework employs emojis for intuitive feedback.
2. **Reference Utilization**:
- Two-stage and Retrieval frameworks leverage reference questions/steps to enhance accuracy.
3. **Step Validation**:
- The Retrieval PRM Framework explicitly cross-references steps (e.g., "Is the target step correct?").
### Interpretation
- **Traditional PRM** relies on direct computation without external references, which may limit adaptability.
- **Retrieval-enhanced frameworks** improve robustness by integrating reference data, reducing errors in complex conversions (e.g., 7.8 minutes → 468 seconds).
- The use of emojis in the Retrieval PRM Framework suggests a user-friendly approach to feedback, potentially improving interpretability.
- The Two-stage mechanism’s question pool and new step pool indicate a modular design for scalable problem-solving.
## Additional Notes
- **Language**: All text is in English.
- **Missing Data**: No numerical trends or charts are present; the diagram focuses on structural and procedural comparisons.
- **Spatial Grounding**:
- **Traditional PRM**: Left section with linear flow (System Prompt → Target Question → Solution Steps → Feedback).
- **Two-stage Retrieval**: Central section with bidirectional flow (Question Pool ↔ Reference Questions ↔ New Step Pool).
- **Retrieval PRM**: Right section with reference-question-driven feedback loop.
</details>
Figure 3: The model structure of our proposed RetrievalPRM framework and its difference with traditional PRM. We design a Two-stage Retrieval Module to retrieve reference questions and steps in each stage.
3.1 Overview of RetrievalPRM
The RetrievalPRM is developed to address the problem of out-of-distribution (OOD) scenarios in mathematical problem-solving, specifically focusing on both question OOD and step OOD. According to Figure 3, traditional PRM models are constrained by predefined solution steps and are unable to handle unseen questions or steps effectively, especially when the problem context shifts or the solution process deviates from previously seen examples. RetrievalPRM overcomes this challenge by incorporating a Two-stage Retrieval-enhanced Mechanism that dynamically fetches relevant questions and steps from a large pool of questions and their solutions. These retrieved questions and steps serve as a kind of warm-up for PRM, acting as example problems for reference. They not only help stimulate PRM’s potential by warming up but also allow the system to handle more difficult problems by identifying similarities.
3.2 Two-stage Retrieval-enhanced Mechanism
The core of RetrievalPRM is the Two-stage Retrieval-enhanced Mechanism, which consists of two key phases: Question-level Retrieval and Step-level Retrieval.
3.2.1 Question-level Retrieval
The first stage of retrieval tackles the question OOD issue. As is shown in Figure 3, the retrieval pool is the question database $\mathbb{D}_{q}=\{q_{i}\}_{i=1}^{N}$ . During retrieval process, we treat:
- Query: the target question $q_{t}$ .
- Key: all $q_{i}$ in the retrieval pool.
- Value: all the $(q_{i},\mathbf{s}_{i})$ pair in the retrieval pool.
We calculate their similarities $<q_{i},q_{t}>$ to match the most similar n questions. Specifically, all questions will first pass through a Sentence-BERT model to encode questions and obtain their semantic representations.
$$
\{e_{q_{i}}\}_{i=1}^{N}=\text{SentenceBERT}(\{q_{i}\}_{i=1}^{N}) \tag{5}
$$
where $e_{q_{i}}∈\mathbb{R}^{D}$ is the embedding vector of the question $q_{i}$ .
And then all the embeddings undergo Principle Component Analysis (PCA) Kurita (2021) for dimensionality reduction to extract the most important dimensions.
$$
\{e^{\prime}_{q_{i}}\}_{i=1}^{N}=\text{PCA}(\{e_{q_{i}}\}_{i=1}^{N}) \tag{6}
$$
where $e^{\prime}_{q_{i}}∈\mathbb{R}^{d}$ is the embedding after dimension reduction.
Finally, we compute the cosine similarity between the target question and the entire question pool, selecting the top- k most similar questions and inputting them into the text.
$$
\displaystyle\langle q_{i},q_{t}\rangle \displaystyle=\frac{e^{\prime}_{q_{t}}\cdot e^{\prime}_{q_{i}}}{|e^{\prime}_{q%
_{t}}|\cdot|e^{\prime}_{q_{i}}|}. \tag{7}
$$
Now we sort the vector $\{\langle q_{i},q_{t}\rangle\}_{i=1}^{N}$ of similarity and choose top- k $(q_{i},\mathbf{s}_{i})$ pairs as reference questions $q_{r}$ and put them in RetrievalPRM’s input together with the target question. Furthermore, we store all the solutions $\{\mathbf{s}_{i}\}_{i=1}^{m}$ of top- m ( $m>k$ ) questions in a new database to conduct a further step-level retrieval.
3.2.2 Step-level Retrieval
We place step-level retrieval in the second stage of the two-stage retrieval process, rather than as a separate module, for two key reasons:
Firstly, for a solution to be meaningful, both the question and the steps must be similar. For example, two different types of questions might both use the letter "p" to represent an unknown variable, but in some problems, "p" represents a prime number, while in others, it represents probability. This results in steps that may appear similar but have entirely different meanings, rendering the retrieved steps potentially unhelpful.
Secondly, since there are many possible solutions to a question, this leads to a large number of steps. If the majority of these steps are irrelevant, the time spent calculating similarities becomes inefficient. By placing step-level retrieval in the second stage, we can save both time and computational resources.
Therefore, after retrieving the top- m most similar questions, we inject all their solutions into a new steps database $\mathbb{D}_{s}$ . Then, we use the target step as the query to retrieve reference steps from this new database. The similarity for retrieval is still calculated using Sentence-BERT, PCA, and cosine similarity, as mentioned in 3.2.1.
3.3 Retrieval-based System Prompt
In RetrievalPRM, The system prompt serves as the instruction set for the model, framing the problem and directing it to evaluate each step of the solution. Besides the traditional system prompt for PRM, the Retrieval-based System Prompt (RetSP) is extended with additional instructions, as shown in the red sentence in Figure 3, which encourages the model to leverage knowledge from reference questions. For example, we inform PRM that step labels "+" and "-" represent correct and incorrect steps, respectively. At the same time, to avoid noise, we specify that if the reference question or step contains no relevant or helpful information, it should not be considered. These retrieval-based system prompts give PRM a more flexible thinking process, enabling it to actively decide whether to use retrieval-based knowledge.
We define reference questions of $q_{i}$ as $\mathbf{q}_{i}^{r}$ and reference steps as $\mathbf{s}_{i}^{r}$ . The whole input $\mathbf{x}_{i}^{j}$ of predicting the $j_{th}$ step of $q_{i}$ in RetrievalPRM can be formulated as:
$$
\displaystyle\mathbf{x}^{j}_{i}=(RetSP,\mathbf{q}^{r}_{i}, \displaystyle q_{i},s^{1}_{i},\ldots,s^{j-1}_{i},\mathbf{s}^{r}_{i},s^{j}_{i},%
y^{j}_{i}), \displaystyle\hat{y}^{j}_{i}= \displaystyle\text{PRM}(\mathbf{x}^{j}_{i}) \tag{8}
$$
where $s^{j}_{i}$ is the $j_{th}$ step of solution $\mathbf{s}_{i}$ .
According to the input template above, it is worth noting that when predicting step n, we assume that steps 1 through n-1 are correct Luo et al. (2024); Zheng et al. (2024). At this point, the most important task for PRM is to predict step n, so PRM can only access the reference steps for step n and cannot see the reference steps for steps $1\sim n-1$ .
4 Experiments
Table 1: The performance of different models on ProcessBench. The best result is given in bold, and the second-best value is underlined. See Table 3 in Appendix D for breakdown of evaluation results.
| Model | GSM8k | MATH | OlympiadBench | OmniMATH | Avg.F1 | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| ArithACC | F1 | ArithACC | F1 | ArithACC | F1 | ArithACC | F1 | | | |
| Open-source PRM | RetrievalPRM-7B(Ours) | 76.0 | 74.6 | 70.6 | 71.1 | 59.1 | 60.2 | 55.2 | 57.33 | 65.8 |
| Qwen2.5-Math-7B-PRM800K | 73.5 | 68.2 | 65.1 | 62.6 | 53.2 | 50.7 | 43.4 | 44.3 | 56.5 | |
| Skywork-PRM-7B | 71.6 | 70.8 | 54.5 | 53.6 | 25.6 | 22.9 | 23.7 | 21.0 | 42.1 | |
| Skywork-PRM-1.5B | 59.9 | 59.0 | 49.1 | 48.0 | 20.5 | 19.3 | 19.7 | 19.2 | 36.4 | |
| Math-Shepherd-PRM-7B | 58.3 | 47.9 | 45.1 | 29.5 | 39.7 | 24.8 | 34.8 | 23.8 | 31.5 | |
| RLHFlow-PRM-Mistral-8B | 62.3 | 50.4 | 42.1 | 33.4 | 22.3 | 13.8 | 19.1 | 15.8 | 28.4 | |
| RLHFlow-PRM-Deepseek-8B | 56.9 | 38.8 | 45.1 | 33.8 | 26.5 | 16.9 | 23.2 | 16.9 | 26.6 | |
| Language Models as Critic | QwQ-32B-Preview | 87.9 | 88.0 | 78.5 | 78.7 | 59.2 | 57.8 | 61.1 | 61.3 | 71.5 |
| GPT-4o | 80.2 | 79.2 | 63.4 | 63.6 | 50.1 | 51.4 | 50.1 | 53.5 | 61.9 | |
| Qwen2.5-72B-Instruct | 77.9 | 76.2 | 65.4 | 61.8 | 59.8 | 54.6 | 55.1 | 52.2 | 61.2 | |
| Llama-3.3-70B-Instruct | 83.7 | 82.9 | 63.7 | 59.4 | 54.3 | 46.7 | 51.0 | 43.0 | 58.0 | |
| Qwen2.5-Coder-32B-Instruct | 72.0 | 68.9 | 64.5 | 60.1 | 57.0 | 48.9 | 52.5 | 46.3 | 56.1 | |
| Llama-3.1-70B-Instruct | 75.3 | 74.9 | 52.6 | 48.2 | 50.0 | 46.7 | 43.2 | 41.0 | 52.7 | |
| Qwen2.5-14B-Instruct | 72.3 | 69.3 | 59.2 | 53.3 | 50.2 | 45.0 | 43.5 | 41.3 | 52.2 | |
| Qwen2-72B-Instruct | 67.8 | 67.6 | 52.3 | 49.2 | 43.3 | 42.1 | 39.3 | 40.2 | 49.8 | |
| Qwen2.5-32B-Instruct | 70.6 | 65.6 | 61.9 | 53.1 | 53.5 | 40.0 | 47.7 | 38.3 | 49.3 | |
| Qwen2.5-Math-72B-Instruct | 70.3 | 65.8 | 59.6 | 52.1 | 56.1 | 32.5 | 55.1 | 31.7 | 45.5 | |
| Qwen2.5-Coder-14B-Instruct | 61.9 | 50.1 | 54.2 | 39.9 | 51.4 | 34.0 | 55.6 | 27.3 | 37.8 | |
| Qwen2.5-7B-Instruct | 37.8 | 36.5 | 36.9 | 36.6 | 29.9 | 29.7 | 27.3 | 27.4 | 32.6 | |
| Meta-Llama-3-70B-Instruct | 62.4 | 52.2 | 48.3 | 22.8 | 46.2 | 21.2 | 44.8 | 20.0 | 29.1 | |
| Qwen2.5-Math-7B-Instruct | 54.4 | 26.8 | 50.3 | 25.7 | 43.1 | 14.2 | 41.6 | 12.7 | 19.9 | |
| Qwen2-7B-Instruct | 25.1 | 8.4 | 20.4 | 19.0 | 16.1 | 14.7 | 13.8 | 12.1 | 13.6 | |
| Meta-Llama-3-8B-Instruct | 27.1 | 13.1 | 17.3 | 13.8 | 14.2 | 4.8 | 19.7 | 12.6 | 11.1 | |
| Qwen2.5-Coder-7B-Instruct | 49.1 | 14.3 | 46.3 | 6.5 | 47.2 | 4.1 | 48.9 | 1.8 | 6.7 | |
| Llama-3.1-8B-Instruct | 27.3 | 10.9 | 20.5 | 5.1 | 16.0 | 2.8 | 15.0 | 1.6 | 5.1 | |
Table 2: The performance of different variants of RetrievalPRM on ProcessBench. We remove different components of RetrievalPRM to evaluate the contribution of each part to the model. The best result is given in bold, and the second-best value is underlined. See Table 4 in Appendix D for breakdown of evaluation results.
| Retrieval Components | GSM8k | MATH | OlympiadBench | OmniMATH | Avg.F1 | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Question-level | Step-level | ArithACC | F1 | ArithACC | F1 | ArithACC | F1 | ArithACC | F1 | |
| ✓ | ✓ | 76.0 | 74.6 | 70.6 | 71.1 | 59.1 | 60.2 | 55.2 | 57.3 | 65.8 |
| ✓ | $×$ | 77.8 | 74.9 | 70.7 | 71.2 | 58.4 | 59.8 | 50.5 | 54.4 | 65.0 |
| $×$ | ✓ | 73.8 | 67.5 | 69.5 | 69.2 | 58.2 | 58.9 | 52.2 | 56.3 | 63.0 |
| $×$ | $×$ | 71.0 | 65.6 | 67.3 | 67.5 | 54.3 | 55.8 | 47.2 | 50.9 | 59.9 |
<details>
<summary>x4.png Details</summary>
### Visual Description
## Bar Chart: F1 Score Comparison Across Datasets and Retrieval Question Numbers
### Overview
The chart compares F1 scores for four datasets (GSM8k, MATH, OlympiadBench, OmniMATH) across four retrieval question thresholds (Top-0 to Top-3). A dashed black line represents the average F1 score across all datasets. Bars are color-coded per dataset, with numerical values labeled on top of each bar.
### Components/Axes
- **X-axis**: "Number of Retrieval Questions" with categories: Top-0, Top-1, Top-2, Top-3.
- **Y-axis**: "F1 Score" scaled from 50 to 80 in increments of 5.
- **Legend**: Located in the top-right corner, mapping colors to datasets:
- Gray: GSM8k
- Blue: MATH
- Teal: OlympiadBench
- Orange: OmniMATH
- Black circle: Average F1 (dashed line).
- **Axis Markers**: Y-axis gridlines at every 5-point interval; X-axis labels centered below categories.
### Detailed Analysis
#### Dataset Performance by Retrieval Threshold
- **Top-0**:
- GSM8k: 65.60
- MATH: 67.50
- OlympiadBench: 55.80
- OmniMATH: 50.90
- Average F1: 60.00 (dashed line)
- **Top-1**:
- GSM8k: 70.50
- MATH: 72.60
- OlympiadBench: 60.80
- OmniMATH: 56.90
- Average F1: 65.20
- **Top-2**:
- GSM8k: 74.90
- MATH: 71.20
- OlympiadBench: 59.80
- OmniMATH: 54.40
- Average F1: 65.00
- **Top-3**:
- GSM8k: 72.30
- MATH: 71.60
- OlympiadBench: 57.30
- OmniMATH: 56.70
- Average F1: 64.50
#### Trends
- **GSM8k**: Shows a consistent upward trend (65.60 → 74.90) until Top-2, then a slight decline at Top-3.
- **MATH**: Peaks at Top-1 (72.60), then declines steadily (71.20 → 71.60).
- **OlympiadBench**: Gradual improvement (55.80 → 57.30) across all thresholds.
- **OmniMATH**: Slight improvement (50.90 → 56.70) but remains the lowest performer.
- **Average F1**: Peaks at Top-1 (65.20), then declines slightly (65.00 → 64.50).
### Key Observations
1. **MATH** achieves the highest F1 score at Top-1 (72.60) but underperforms relative to GSM8k at Top-2 and Top-3.
2. **GSM8k** demonstrates the strongest overall improvement with increased retrieval questions, reaching 74.90 at Top-2.
3. **OlympiadBench** and **OmniMATH** lag significantly behind other datasets, with OmniMATH showing the lowest scores across all thresholds.
4. The average F1 score plateaus after Top-1, suggesting diminishing returns for additional retrieval questions beyond this point.
### Interpretation
The data suggests that increasing the number of retrieval questions improves performance for most datasets, with **GSM8k** benefiting the most. However, the average F1 score plateaus after Top-1, indicating that further retrieval questions may not yield proportional gains. **MATH**'s decline after Top-1 could imply over-reliance on retrieval at higher thresholds, while **OlympiadBench** and **OmniMATH** require targeted improvements to close the performance gap. The trend highlights a trade-off between retrieval depth and practical utility, as gains beyond Top-1 become marginal for the average case.
</details>
Figure 4: We show the F1 scores of Retrieval-PRM on four datasets and their average, as the number of retrieval questions varies. Specifically, Top-0 means no retrieval questions.
In this section, we present the experimental settings and results. Our implementation code of RetrievalPRM is publicly available.
4.1 Experiment Setup
4.1.1 Datasets
Datasets are categorized into two kinds: Math reasoning datasets, and prm training datasets.
Math Reasoning Datasets
We conduct experiments on four public and widely used datasets in mathematical reasoning tasks: GSM8K Cobbe et al. (2021) which contains math problems from elementary to middle school, MATH Hendrycks et al. (2021) which contains math problems from basic to university level, OlympiadBench He et al. (2024) which involves questions from the Mathematical Olympiad, Omni-MATH Gao et al. (2024b) which covers multi-domain high-difficulty problems. Further details are provided in Appendix C.
Except for GSM8K, which focuses on grade school math problems, the other three datasets feature problems of competition or Olympiad-level difficulty.
PRM training datasets
We conduct experiments on two publicly available datasets for PRM:
PRM800K Lightman et al. (2023): Based on the MATH dataset, it contains 800,000 manually annotated step-level correctness labels for training the Process Reward Model. It relies on expensive manual annotations.
Math-Shepherd Wang et al. (2024b): It generates 400,000 machine-annotated step-level labels (covering MATH and GSM8K datasets) by automatically building process supervision data, without manual annotation.
4.1.2 Evaluation Metrics
We evaluate our model in a public PRM benchmark ProcessBench Zheng et al. (2024). The aim is to judge whether PRM can find the first wrong step. It divides data into two parts: samples with incorrect and correct final answers and then conducts harmonic mean on the accuracy of these two parts to get the final F1-score. Moreover, we think since the sample number of each part isn’t balanced, We add an additional metric: weighted arithmetic mean of these two parts, which is shown in Table 1 as ArithACC.
4.1.3 Baselines
Following Zheng et al. (2024), we divide all baselines into two parts:
(1) Open-source PRM, including Skywork o1 Team (2024), Qwen2.5-PRM Zheng et al. (2024), Math-Shepherd Wang et al. (2024b) and RLHFlow Xiong et al. (2024). These models are binary classification PRMs.
(2) Language Models as Critic, including Llama Dubey et al. (2024), Qwen2 Yang et al. (2024b), Qwen2.5 Team (2024), Qwen2.5-MATH Yang et al. (2024a), Qwen2.5-Coder Hui et al. (2024), GPT-4o OpenAI et al. (2024). These models are promoted to judge the steps with the help of majority voting.
Further details of these baselines are provided in Appendix A due to article length limitations.
4.1.4 Implementation Details
Details like base models, hyperparameters, prompts, and training sizes are provided in Appendix B due to the article length limitations.
4.2 Overall Performance
We evaluate RetrievalPRM against existing baselines on ProcessBench, and the results are presented in Table 1. The findings are as follows:
- RetrievalPRM-7B surpasses all open-source PRM baselines, achieving the highest performance. Notably, the most significant improvement is observed on OmniMATH, the most challenging dataset, with performance gains increasing as dataset difficulty rises. This phenomenon may stem from the fact that most baseline PRMs are trained on human- or machine-annotated datasets such as PRM800K or Math-Shepherd, which primarily focus on GSM8K or MATH and exhibit OOD issues when applied to more complex datasets. In contrast, our RetrievalPRM effectively mitigates the OOD problem through its retrieval-based approach, demonstrating the efficacy of our Two-stage Retrieval-enhanced Mechanism.
- When comparing models of different scales, RetrievalPRM outperforms all evaluated language models, including Qwen2.5-72B-Instruct and Llama3.3-70B-Instruct, with the sole exception of QwQ-32B-Preview. Remarkably, RetrievalPRM achieves this with a model size of just 7B. This highlights that PRMs, being both lightweight and task-specific, maintain strong competitiveness and potential compared to LLMs as critics.
4.3 Ablation Study
We analyze two main components in the Two-stage Retrieval-enhanced Mechanism: Question-level Retrieval and Step-level Retrieval —through the following ablations:
RetrievalPRM (Ours): The complete version of our proposed method.
RetrievalPRM (w/o Step-level Retrieval): This variant retains only the Question-level Retrieval, removing Step-level Retrieval during both training and inference.
RetrievalPRM (w/o Question-level Retrieval): This variant retains only the Step-level Retrieval, removing Question-level Retrieval during both training and inference.
RetrievalPRM (w/o Question-level and Step-level Retrieval): In this variant, both Question-level and Step-level Retrieval are removed during training and inference.
The performance of these variants is presented in Table 2, from which we can draw the following observations:
- The performance of RetrievalPRM (w/o Step-level Retrieval) remains almost identical to that of RetrievalPRM on GSM8K and MATH but exhibits a slight decline on OlympiadBench and OmniMATH. This can be attributed to the fact that Step-level Retrieval information is partially absorbed by Question-level Retrieval. As a result, Question-level Retrieval alone may be sufficiently effective for relatively easy datasets, as the reference steps it provides contain adequate knowledge for step prediction. However, for more challenging datasets, Step-level Retrieval becomes significantly more crucial, as it offers finer-grained guidance essential for handling complex problem-solving processes.
- RetrievalPRM (w/o Question-level Retrieval) shows lower performance, as it relies solely on Step-level Retrieval. The model lacks knowledge of reference questions, which is useful to alleviate question OOD, restricting its overall performance.
- RetrievalPRM (w/o both Retrieval) performs the worst, which is expected, demonstrating the effectiveness of both question-level and Step-level Retrieval.
4.4 Hyperparameter Study
Figure 4 illustrates the impact of the number of retrieval questions on the model’s performance. The findings are as follows:
Compared to Top-0, where no retrieval questions are used, models that incorporate retrieval questions show improved performance, highlighting the importance of Question-level Retrieval. It inspires us that Reference questions are important for PRM to get warmup, no matter how many reference questions there are.
The performance of Top-3 exhibits a slight decline, potentially due to two factors: (1) An excessive number of reference questions may lead to an overly long input prompt, making it difficult for PRMs to comprehend or extract key information effectively. (2) A limited retrieval pool might result in later reference questions being less relevant than earlier ones, increasing the likelihood of misjudgments in the model’s predictions.
5 Related Works
5.1 Process Reward Models
Process reward models (PRMs) have demonstrated significant advantages over traditional outcome reward models (ORMs) Cobbe et al. (2021) in enhancing process-level reasoning accuracy and improving long-process reasoning abilities in model training Lightman et al. (2023); Uesato et al. (2022). A growing number of PRMs have been proposed for application in process-level reinforcement learning with human feedback (RLHF) Wang et al. (2024b); Qin et al. (2024); Xia et al. (2025); o1 Team (2024). For instance, Lightman et al. (2023) made a substantial contribution by releasing a large set of human-annotated data at the process level, opening up new research opportunities for multi-step reasoning.
Additionally, Wang et al. (2024b) introduces an automatic, self-supervised pipeline for generating process-level labels and training PRMs, enabling efficient data generation. Xia et al. (2025) employs PRMs as automatic evaluators to assess the accuracy of multi-step reasoning in language models (LMs). With the surge in PRM-focused research and data curation, numerous PRMs o1 Team (2024); Xiong et al. (2024); Sun et al. (2024); Gao et al. (2024a); Wang et al. (2024a) have been proposed. Additionally, several studies focus on leveraging natural language feedback from large language models (LLMs) as rewards, which are termed critic models McAleese et al. (2024); Zhang et al. (2024); Gao et al. (2024a).
However, most existing PRMs trained on math datasets such as GSM8K and MATH inevitably encounter Out-of-distribution issues, which can be divided into two categories: question OOD, where PRMs trained on simpler or medium-difficulty datasets lack understanding of questions from more challenging datasets, and step OOD, where different base models and model sizes in LLMs lead to different step distributions for the same question. This is reflected in differences in chain length, problem-solving approaches, and methods. To address these issues, we propose the RetrievalPRM framework to tackle the OOD problems encountered in the current PRM field, achieving promising results.
5.2 Retrieval-Augmented Generation
Retrieval-augmented generation (RAG) enhances language models by dynamically integrating external knowledge, pioneered by Lewis et al. (2021) through their joint retrieval-generation architecture. Subsequent advances refined this paradigm. Guu et al. (2020) introduced REALM to co-train retrieval and generation modules via masked language modeling, while Izacard and Grave (2021) proposed Fusion-in-Decoder (FiD) to process multi-document contexts efficiently. Research further optimized retrieval precision through dense passage embeddings Karpukhin et al. (2020) and scaled retrieval to web-level corpora Borgeaud et al. (2022).
6 Conclusion
In this paper, we have addressed the significant out-of-distribution (OOD) challenges faced by Process Reward Models (PRMs), particularly step OOD and question OOD. By introducing the Retrieval Augmented Process Reward Model (RetrievalPRM), we propose an effective solution that leverages a Two-stage Retrieval-enhanced Mechanism to improve the generalization of PRMs across diverse models and problems. Extensive experiments on multiple real-world datasets have shown that RetrievalPRM consistently outperforms existing methods, highlighting its effectiveness in tackling OOD issues.
7 Limitation
RetrievalPRM has two main limitations. Firstly, the retrieval pool is only constructed from PRM800K and Math-Shepherd at present, which is relatively small and limits the diversity and breadth of the mathematical problems. Second, using Sentence-BERT to embed questions and steps struggles to capture the full complexity of mathematical problems as semantic similarity doesn’t mean knowledge similarity in Math problems. As a result, the naive cosine similarity calculated through embeddings may fail to accurately reflect the true similarity between two questions.
References
- Borgeaud et al. (2022) Sebastian Borgeaud, Arthur Mensch, Jordan Hoffmann, Trevor Cai, Eliza Rutherford, Katie Millican, George van den Driessche, Jean-Baptiste Lespiau, Bogdan Damoc, Aidan Clark, Diego de Las Casas, Aurelia Guy, Jacob Menick, Roman Ring, Tom Hennigan, Saffron Huang, Loren Maggiore, Chris Jones, Albin Cassirer, Andy Brock, Michela Paganini, Geoffrey Irving, Oriol Vinyals, Simon Osindero, Karen Simonyan, Jack W. Rae, Erich Elsen, and Laurent Sifre. 2022. Improving language models by retrieving from trillions of tokens. Preprint, arXiv:2112.04426.
- Cobbe et al. (2021) Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. 2021. Training verifiers to solve math word problems. CoRR, abs/2110.14168.
- Dubey et al. (2024) Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Amy Yang, Angela Fan, et al. 2024. The llama 3 herd of models. arXiv preprint arXiv:2407.21783.
- Gao et al. (2024a) Bofei Gao, Zefan Cai, Runxin Xu, Peiyi Wang, Ce Zheng, Runji Lin, Keming Lu, Dayiheng Liu, Chang Zhou, Wen Xiao, Junjie Hu, Tianyu Liu, and Baobao Chang. 2024a. Llm critics help catch bugs in mathematics: Towards a better mathematical verifier with natural language feedback. Preprint, arXiv:2406.14024.
- Gao et al. (2024b) Bofei Gao, Feifan Song, Zhe Yang, Zefan Cai, Yibo Miao, Qingxiu Dong, Lei Li, Chenghao Ma, Liang Chen, Runxin Xu, Zhengyang Tang, Benyou Wang, Daoguang Zan, Shanghaoran Quan, Ge Zhang, Lei Sha, Yichang Zhang, Xuancheng Ren, Tianyu Liu, and Baobao Chang. 2024b. Omni-math: A universal olympiad level mathematic benchmark for large language models. Preprint, arXiv:2410.07985.
- Guu et al. (2020) Kelvin Guu, Kenton Lee, Zora Tung, Panupong Pasupat, and Ming-Wei Chang. 2020. Realm: Retrieval-augmented language model pre-training. Preprint, arXiv:2002.08909.
- He et al. (2024) Chaoqun He, Renjie Luo, Yuzhuo Bai, Shengding Hu, Zhen Leng Thai, Junhao Shen, Jinyi Hu, Xu Han, Yujie Huang, Yuxiang Zhang, Jie Liu, Lei Qi, Zhiyuan Liu, and Maosong Sun. 2024. Olympiadbench: A challenging benchmark for promoting agi with olympiad-level bilingual multimodal scientific problems. Preprint, arXiv:2402.14008.
- Hendrycks et al. (2021) Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. 2021. Measuring mathematical problem solving with the math dataset. arXiv preprint arXiv:2103.03874.
- Hui et al. (2024) Binyuan Hui, Jian Yang, Zeyu Cui, Jiaxi Yang, Dayiheng Liu, Lei Zhang, Tianyu Liu, Jiajun Zhang, Bowen Yu, Keming Lu, Kai Dang, Yang Fan, Yichang Zhang, An Yang, Rui Men, Fei Huang, Bo Zheng, Yibo Miao, Shanghaoran Quan, Yunlong Feng, Xingzhang Ren, Xuancheng Ren, Jingren Zhou, and Junyang Lin. 2024. Qwen2.5-coder technical report. Preprint, arXiv:2409.12186.
- Izacard and Grave (2021) Gautier Izacard and Edouard Grave. 2021. Leveraging passage retrieval with generative models for open domain question answering. Preprint, arXiv:2007.01282.
- Karpukhin et al. (2020) Vladimir Karpukhin, Barlas OÄŸuz, Sewon Min, Patrick Lewis, Ledell Wu, Sergey Edunov, Danqi Chen, and Wen tau Yih. 2020. Dense passage retrieval for open-domain question answering. Preprint, arXiv:2004.04906.
- Kurita (2021) Takio Kurita. 2021. Principal component analysis (pca). In Computer vision: a reference guide, pages 1013–1016. Springer.
- Lewis et al. (2021) Patrick Lewis, Ethan Perez, Aleksandra Piktus, Fabio Petroni, Vladimir Karpukhin, Naman Goyal, Heinrich Küttler, Mike Lewis, Wen tau Yih, Tim Rocktäschel, Sebastian Riedel, and Douwe Kiela. 2021. Retrieval-augmented generation for knowledge-intensive nlp tasks. Preprint, arXiv:2005.11401.
- Lightman et al. (2023) Hunter Lightman, Vineet Kosaraju, Yura Burda, Harri Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. 2023. Let’s verify step by step. arXiv preprint arXiv:2305.20050.
- Luo et al. (2024) Liangchen Luo, Yinxiao Liu, Rosanne Liu, Samrat Phatale, Meiqi Guo, Harsh Lara, Yunxuan Li, Lei Shu, Yun Zhu, Lei Meng, Jiao Sun, and Abhinav Rastogi. 2024. Improve mathematical reasoning in language models by automated process supervision. Preprint, arXiv:2406.06592.
- McAleese et al. (2024) Nat McAleese, Rai Michael Pokorny, Juan Felipe Ceron Uribe, Evgenia Nitishinskaya, Maja Trebacz, and Jan Leike. 2024. Llm critics help catch llm bugs. Preprint, arXiv:2407.00215.
- o1 Team (2024) Skywork o1 Team. 2024. Skywork-o1 open series. https://huggingface.co/Skywork.
- OpenAI et al. (2024) OpenAI, :, Aaron Hurst, Adam Lerer, Adam P. Goucher, Adam Perelman, Aditya Ramesh, Aidan Clark, AJ Ostrow, and Akila Welihinda et al. 2024. Gpt-4o system card. Preprint, arXiv:2410.21276.
- OpenAI (2023) R OpenAI. 2023. Gpt-4 technical report. arxiv 2303.08774. View in Article, 2(5).
- Qin et al. (2024) Yiwei Qin, Xuefeng Li, Haoyang Zou, Yixiu Liu, Shijie Xia, Zhen Huang, Yixin Ye, Weizhe Yuan, Hector Liu, Yuanzhi Li, and Pengfei Liu. 2024. O1 replication journey: A strategic progress report – part 1. Preprint, arXiv:2410.18982.
- Reimers (2019) N Reimers. 2019. Sentence-bert: Sentence embeddings using siamese bert-networks. arXiv preprint arXiv:1908.10084.
- Shao et al. (2024) Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, YK Li, Y Wu, et al. 2024. Deepseekmath: Pushing the limits of mathematical reasoning in open language models. arXiv preprint arXiv:2402.03300.
- Sun et al. (2024) Zhiqing Sun, Longhui Yu, Yikang Shen, Weiyang Liu, Yiming Yang, Sean Welleck, and Chuang Gan. 2024. Easy-to-hard generalization: Scalable alignment beyond human supervision. Preprint, arXiv:2403.09472.
- Team (2024) Qwen Team. 2024. Qwen2.5: A party of foundation models.
- Uesato et al. (2022) Jonathan Uesato, Nate Kushman, Ramana Kumar, Francis Song, Noah Siegel, Lisa Wang, Antonia Creswell, Geoffrey Irving, and Irina Higgins. 2022. Solving math word problems with process- and outcome-based feedback. Preprint, arXiv:2211.14275.
- Wang et al. (2024a) Jun Wang, Meng Fang, Ziyu Wan, Muning Wen, Jiachen Zhu, Anjie Liu, Ziqin Gong, Yan Song, Lei Chen, Lionel M. Ni, Linyi Yang, Ying Wen, and Weinan Zhang. 2024a. Openr: An open source framework for advanced reasoning with large language models. Preprint, arXiv:2410.09671.
- Wang et al. (2024b) Peiyi Wang, Lei Li, Zhihong Shao, Runxin Xu, Damai Dai, Yifei Li, Deli Chen, Yu Wu, and Zhifang Sui. 2024b. Math-shepherd: Verify and reinforce llms step-by-step without human annotations. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 9426–9439.
- Xia et al. (2025) Shijie Xia, Xuefeng Li, Yixin Liu, Tongshuang Wu, and Pengfei Liu. 2025. Evaluating mathematical reasoning beyond accuracy. Preprint, arXiv:2404.05692.
- Xiong et al. (2024) Wei Xiong, Hanning Zhang, Nan Jiang, and Tong Zhang. 2024. An implementation of generative prm. https://github.com/RLHFlow/RLHF-Reward-Modeling.
- Yang et al. (2024a) An Yang, Beichen Zhang, Binyuan Hui, Bofei Gao, Bowen Yu, Chengpeng Li, Dayiheng Liu, Jianhong Tu, Jingren Zhou, Junyang Lin, Keming Lu, Mingfeng Xue, Runji Lin, Tianyu Liu, Xingzhang Ren, and Zhenru Zhang. 2024a. Qwen2.5-math technical report: Toward mathematical expert model via self-improvement. Preprint, arXiv:2409.12122.
- Yang et al. (2024b) An Yang, Beichen Zhang, Binyuan Hui, Bofei Gao, Bowen Yu, Chengpeng Li, Dayiheng Liu, Jianhong Tu, Jingren Zhou, Junyang Lin, et al. 2024b. Qwen2. 5-math technical report: Toward mathematical expert model via self-improvement. arXiv preprint arXiv:2409.12122.
- Zhang et al. (2024) Lunjun Zhang, Arian Hosseini, Hritik Bansal, Mehran Kazemi, Aviral Kumar, and Rishabh Agarwal. 2024. Generative verifiers: Reward modeling as next-token prediction. Preprint, arXiv:2408.15240.
- Zheng et al. (2024) Chujie Zheng, Zhenru Zhang, Beichen Zhang, Runji Lin, Keming Lu, Bowen Yu, Dayiheng Liu, Jingren Zhou, and Junyang Lin. 2024. Processbench: Identifying process errors in mathematical reasoning. arXiv preprint arXiv:2412.06559.
- Zhu et al. (2024) Qihao Zhu, Daya Guo, Zhihong Shao, Dejian Yang, Peiyi Wang, Runxin Xu, Y Wu, Yukun Li, Huazuo Gao, Shirong Ma, et al. 2024. Deepseek-coder-v2: Breaking the barrier of closed-source models in code intelligence. arXiv preprint arXiv:2406.11931.
Appendix A Baselines
A.1 Open-source PRM
<details>
<summary>x5.png Details</summary>
### Visual Description
## Textual Analysis: Math Problem Evaluation System
### Overview
The image depicts a structured system for evaluating mathematical problem-solving steps. It presents a question ("How many seconds are in 5.5 minutes?"), a three-step solution process, and an output indicating correctness. The system requires the evaluator to judge the validity of the solution steps using only "+" (correct) or "-" (incorrect) responses.
### Components/Axes
- **System Definition**:
- Role: Math teacher
- Task: Judge solution steps for correctness
- **Input Section**:
- Question: "How many seconds are in 5.5 minutes?"
- Process Steps:
1. 5.5 minutes = 5 minutes + 0.5 minutes
2. 5 minutes = 300 seconds (5 × 60)
3. 0.5 minutes = 30 seconds (0.5 × 60)
- **Output**: "+" (indicating all steps are correct)
### Detailed Analysis
1. **Step 1**: Correct. 5.5 minutes is accurately decomposed into 5 minutes and 0.5 minutes.
2. **Step 2**: Correct. 5 minutes × 60 seconds/minute = 300 seconds.
3. **Step 3**: Correct. 0.5 minutes × 60 seconds/minute = 30 seconds.
4. **Final Calculation**: Implicitly correct (300 + 30 = 330 seconds total).
### Key Observations
- All steps follow logical unit conversion principles.
- The output "+" confirms no errors in the provided solution.
- The system enforces strict binary feedback (+/-), limiting evaluator nuance.
### Interpretation
This system demonstrates a clear, step-by-step approach to unit conversion problems. The solution correctly applies the conversion factor (60 seconds/minute) to both whole and fractional minute values. The "+" output validates the methodology, suggesting this could serve as an effective template for automated or human evaluation of mathematical reasoning processes. The strict feedback mechanism ensures focus on fundamental correctness rather than alternative solution paths.
</details>
Figure 5: The illustration of PRM input template.
<details>
<summary>x6.png Details</summary>
### Visual Description
## Screenshot: Math Problem Evaluation System
### Overview
The image depicts a structured interface for evaluating mathematical problem-solving steps. It includes a system prompt, reference questions with solutions, a target question, and a step-by-step analysis framework. The system emphasizes correctness judgment using `+` (correct) and `-` (incorrect) labels.
### Components/Axes
- **System Prompt**: Instructions for the AI to act as a math teacher, evaluating solution steps for correctness.
- **Reference Questions**:
- **Question 1**: "What is the equivalent number of seconds in 7.8 minutes?"
- **Process**: Correct calculation (`7.8 * 60 = 46` seconds).
- **Label**: `(+)`
- **Question 2**: Incomplete (truncated with `...`).
- **Target Question**: "How many seconds are in 5.5 minutes?"
- **Process**:
- **Step 1**: "5.5 minutes is the same as 5 minutes and 0.5 minutes."
- **Step 2**: "Since there are 60 seconds in a minute, then there are 300 seconds in 5 minutes."
- **Reference Steps**:
- **Step 1**: "0.3 hours equal to 0.3 * 60 = 18 minutes. This reference step is correct."
- **Step 2**: Truncated (`...`).
- **Target Step 3**: "And since there are 60 seconds in a minute, there are 30 seconds in 0.5 minutes."
- **Label**: `(+)`
### Detailed Analysis
- **System Instructions**:
- The AI must judge each step as `+` (correct) or `-` (incorrect).
- Irrelevant or unhelpful steps are labeled `-`.
- **Reference Question 1**:
- Correctly calculates seconds in 7.8 minutes using multiplication by 60.
- **Target Question**:
- Breaks 5.5 minutes into 5 + 0.5 minutes.
- Step 2 calculates 5 minutes as 300 seconds (correct).
- Step 3 calculates 0.5 minutes as 30 seconds (correct).
- **Output**: Final judgment for Target Step 3 is `+`.
### Key Observations
- All provided steps in Reference Question 1 and Target Steps 1–3 are correct.
- The system uses explicit labels (`+`, `-`) to denote correctness.
- The structure emphasizes incremental validation of problem-solving logic.
### Interpretation
This system is designed to automate the evaluation of mathematical reasoning, ensuring each step adheres to logical and arithmetic principles. By breaking problems into smaller steps (e.g., decomposing 5.5 minutes into 5 + 0.5), it mirrors pedagogical techniques to verify understanding. The use of `+` and `-` labels provides immediate feedback, critical for educational tools or automated grading systems. The truncation of Reference Question 2 and Step 2 suggests incomplete data, but the visible steps align with standard unit conversion methods.
</details>
Figure 6: The illustration of RetrievalPRM input template.
- Skywork-PRM o1 Team (2024) is a Qwen2.5-Math-based PRM published by KunLun.
- Qwen2.5-PRM Zheng et al. (2024) is trained by fine-tuning the Qwen2.5-Math-7B-Instruct model on the PRM800K dataset.
- Math-Shepherd Wang et al. (2024b) generates process labels for each step by estimating the empirical probability that a given step leads to the correct final answer and trains a PRM based on their published dataset.
- RLHFlow-PRM Xiong et al. (2024) is an 8-billion-parameter reward model trained with process supervision.
A.2 Language Models as Critic
- Llama Dubey et al. (2024) is an open-source model developed by Meta (formerly Facebook), designed for natural language understanding and generation tasks.
- Qwen2 Yang et al. (2024b) is a large language model developed by Alibaba Cloud, offering multilingual support and strong capabilities in language understanding and generation.
- Qwen2.5 Team (2024) is an advanced iteration of the Qwen series, pretrained on 18 trillion tokens, enhancing knowledge retention, programming, and mathematical reasoning.
- Qwen2.5-MATH Yang et al. (2024a) is a specialized model for mathematical problem-solving, trained on extensive math-focused data and incorporating Chain-of-Thought (CoT) and Tool-Integrated Reasoning (TIR).
- Qwen2.5-Coder Hui et al. (2024) is a programming-oriented model trained on 5.5 trillion code-related tokens, excelling in code generation, debugging, and multilingual programming tasks.
- GPT-4o OpenAI et al. (2024) is a multimodal AI model developed by OpenAI that processes and generates text, audio, and images in real-time, with enhanced speed and natural interaction capabilities.
Appendix B Implementation Details
B.1 Basemodel and Training hyperparameters
We selected Qwen-2.5-Math-7b-instruct Team (2024) as the foundational large language model (LLM) for our experiments. All computations were performed using H100 GPUs. To enhance training resource efficiency, we employed Parameter-Efficient Fine-tuning techniques LoRA. The LoRA configuration was set with a rank of 32, an alpha value of 64, and dropout set to 0.1. LoRA update matrices were specifically applied to the query and value projection matrices within the attention blocks.
We use PRM800K as our training data and both PRM800K and Math-Shepherd as our retrieval pool. The training process was carried out with batch sizes chosen from $\{64,128,256,512\}$ and initial learning rates selected from $\{1× 10^{-3},1× 10^{-4},3× 10^{-4},1× 10^{-5},3× 10^{%
-5}\}$ using a linear scheduler.
B.2 Prompts
In this section, we show our training prompts for PRM in details as is shown in Figure 5 and Figure 6.
Appendix C Datasets
GSM8K Cobbe et al. (2021): Grade School Math is a dataset for basic to intermediate math problems, covering arithmetic, algebra, geometry and other fields. Its difficulty is suitable for math problems in elementary to middle school.
MATH Hendrycks et al. (2021): The MATH dataset contains a variety of math problems from basic to university level, covering multiple mathematical fields such as algebra, geometry, calculus, number theory, etc.
OlympiadBench He et al. (2024): The Olympiadbench dataset contains questions from the Mathematical Olympiad. The questions are of high difficulty and involve complex combinatorial mathematics, number theory, geometry and other advanced mathematical fields.
Omni-MATH Gao et al. (2024b): Omni-MATH is a general Olympiad-level mathematics benchmark dataset for large language models, covering multi-domain and high-difficulty mathematics problems, and is designed to evaluate the reasoning ability of models in various mathematical fields.
Except for GSM8K, which focuses on grade school math problems, the other three datasets feature problems of competition or Olympiad-level difficulty.
Appendix D Supplementary Evaluation Results
In this section, we show the breakdown of our main results in Table 3 and ablation results in Table 4
Table 3: Breakdown of evaluation results of different models on ProcessBench. The best result is given in bold, and the second-best value is underlined.
| Model | GSM8k | MATH | OlympiadBench | OmniMATH | | | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| error | correct | F1 | error | correct | F1 | error | correct | F1 | error | correct | F1 | | |
| Open-source PRM | RetrievalPRM-7B(Ours) | 64.7 | 88.1 | 74.6 | 67.2 | 75.6 | 71.1 | 56.0 | 65.2 | 60.2 | 52.8 | 62.65 | 57.33 |
| Qwen2.5-Math-7B-PRM800K | 53.1 | 95.3 | 68.2 | 48.0 | 90.1 | 62.6 | 35.7 | 87.3 | 50.7 | 29.8 | 86.3 | 44.3 | |
| Skywork-PRM-7B | 61.8 | 82.9 | 70.8 | 43.8 | 69.2 | 53.6 | 17.9 | 31.9 | 22.9 | 14.0 | 41.9 | 21.0 | |
| RLHFlow-PRM-Mistral-8B | 33.8 | 99.0 | 50.4 | 21.7 | 72.2 | 33.4 | 8.2 | 43.1 | 13.8 | 9.6 | 45.2 | 15.8 | |
| RLHFlow-PRM-Deepseek-8B | 24.2 | 98.4 | 38.8 | 21.4 | 80.0 | 33.8 | 10.1 | 51.0 | 16.9 | 10.1 | 51.9 | 16.9 | |
| Skywork-PRM-1.5B | 50.2 | 71.5 | 59.0 | 37.9 | 65.3 | 48.0 | 15.4 | 26.0 | 19.3 | 13.6 | 32.8 | 19.2 | |
| Math-Shepherd-PRM-7B | 32.4 | 91.7 | 47.9 | 18.0 | 82.0 | 29.5 | 15.0 | 71.1 | 24.8 | 14.2 | 73.0 | 23.8 | |
| Language Models | QwQ-32B-Preview | 81.6 | 95.3 | 88.0 | 78.1 | 79.3 | 78.7 | 61.4 | 54.6 | 57.8 | 55.7 | 68.0 | 61.3 |
| GPT-4o | 70.0 | 91.2 | 79.2 | 54.4 | 76.6 | 63.6 | 45.8 | 58.4 | 51.4 | 45.2 | 53.5 | 61.9 | |
| Qwen2.5-72B-Instruct | 62.8 | 96.9 | 76.2 | 46.3 | 93.1 | 61.8 | 38.7 | 92.6 | 54.6 | 36.6 | 90.9 | 52.2 | |
| Llama-3.3-70B-Instruct | 72.5 | 96.9 | 82.9 | 43.3 | 94.6 | 59.4 | 31.0 | 94.1 | 46.7 | 28.2 | 90.5 | 43.0 | |
| Qwen2.5-32B-Instruct | 49.3 | 97.9 | 65.6 | 36.7 | 95.8 | 53.1 | 25.3 | 95.9 | 40.0 | 24.1 | 92.5 | 38.3 | |
| Qwen2.5-14B-Instruct | 54.6 | 94.8 | 69.3 | 38.4 | 87.4 | 53.3 | 31.5 | 78.8 | 45.0 | 28.3 | 76.3 | 41.3 | |
| Qwen2.5-Coder-32B-Instruct | 54.1 | 94.8 | 68.9 | 44.9 | 90.6 | 60.1 | 33.4 | 91.2 | 48.9 | 31.5 | 87.6 | 46.3 | |
| Qwen2.5-Coder-14B-Instruct | 33.8 | 96.4 | 50.1 | 25.4 | 92.4 | 39.9 | 20.7 | 94.1 | 34.0 | 15.9 | 94.2 | 27.3 | |
| Qwen2.5-Coder-7B-Instruct | 7.7 | 100.0 | 14.3 | 3.4 | 98.3 | 6.5 | 2.1 | 99.1 | 4.1 | 0.9 | 98.3 | 1.8 | |
| Qwen2.5-Math-72B-Instruct | 49.8 | 96.9 | 65.8 | 36.0 | 94.3 | 52.1 | 19.5 | 97.3 | 32.5 | 19.0 | 96.3 | 31.7 | |
| Qwen2.5-Math-7B-Instruct | 15.5 | 100.0 | 26.8 | 14.8 | 96.8 | 25.7 | 7.7 | 91.7 | 14.2 | 6.9 | 88.0 | 12.7 | |
| Llama-3.1-70B-Instruct | 64.3 | 89.6 | 74.9 | 35.4 | 75.6 | 48.2 | 35.1 | 69.9 | 46.7 | 30.7 | 61.8 | 41.0 | |
| Meta-Llama-3-70B-Instruct | 35.7 | 96.9 | 52.2 | 13.0 | 93.3 | 22.8 | 12.0 | 92.0 | 21.2 | 11.2 | 91.7 | 20.0 | |
| Qwen2-72B-Instruct | 57.0 | 82.9 | 67.6 | 37.7 | 70.9 | 49.2 | 34.0 | 55.2 | 42.1 | 32.3 | 53.1 | 40.2 | |
| Qwen2.5-7B-Instruct | 40.6 | 33.2 | 36.5 | 30.8 | 45.1 | 36.6 | 26.5 | 33.9 | 29.7 | 26.2 | 28.6 | 27.4 | |
| Qwen2-7B-Instruct | 40.6 | 4.7 | 8.4 | 30.5 | 13.8 | 19.0 | 22.4 | 10.9 | 14.7 | 20.0 | 8.7 | 12.1 | |
| Llama-3.1-8B-Instruct | 44.4 | 6.2 | 10.9 | 41.9 | 2.7 | 5.1 | 32.4 | 1.5 | 2.8 | 32.0 | 0.8 | 1.6 | |
| Meta-Llama-3-8B-Instruct | 42.5 | 7.8 | 13.1 | 28.6 | 9.1 | 13.8 | 27.1 | 2.7 | 4.8 | 26.1 | 8.3 | 12.6 | |
Table 4: Breakdown of evaluation results of different variants of RetrievalPRM on ProcessBench. We remove different components of RetrievalPRM to evaluate the contribution of each part to the model. The best result is given in bold, and the second-best value is underlined.
| Retrieval Components | GSM8k | MATH | OlympiadBench | OmniMATH | Avg.F1 | | | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Question-level | Step-level | error | correct | F1 | error | correct | F1 | error | correct | F1 | error | correct | F1 | |
| ✓ | ✓ | 64.7 | 88.1 | 74.6 | 67.2 | 75.6 | 71.1 | 56.0 | 65.2 | 60.2 | 52.8 | 62.65 | 57.33 | 65.8 |
| ✓ | $×$ | 61.8 | 94.8 | 74.9 | 62.1 | 83.3 | 71.2 | 48.7 | 77.3 | 59.8 | 43.2 | 73.4 | 54.4 | 65.0 |
| $×$ | ✓ | 51.7 | 97.4 | 67.5 | 57.2 | 87.4 | 69.2 | 46.0 | 82.0 | 58.9 | 43.9 | 78.4 | 56.3 | 63.0 |
| $×$ | $×$ | 50.7 | 92.7 | 65.6 | 57.9 | 81.0 | 67.5 | 46.9 | 68.7 | 55.8 | 39.7 | 71.0 | 50.9 | 59.9 |