# 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
\n
## Scatter Plot: Dimensionality Reduction of Benchmarks
### Overview
This image presents a scatter plot visualizing the distribution of three different benchmarks – GSM8k, MATH, and OlympiadBench – across two dimensions (Dimension x and Dimension y). The plot appears to be the result of a dimensionality reduction technique (like PCA or t-SNE) applied to some higher-dimensional data representing these benchmarks. The points are colored to distinguish between the benchmarks, and semi-transparent shading indicates density.
### Components/Axes
* **X-axis:** Labeled "Dimension x", ranging from approximately -40 to 50.
* **Y-axis:** Labeled "Dimension y", ranging from approximately -20 to 25.
* **Legend:** Located in the top-center of the plot.
* GSM8k: Represented by teal circles.
* MATH: Represented by orange circles.
* OlympiadBench: Represented by blue circles.
* **Data Points:** Scatter plot points representing individual data instances from each benchmark.
* **Shading:** Semi-transparent shading around each cluster of points, indicating density.
### Detailed Analysis
The plot shows three distinct clusters of points, corresponding to the three benchmarks.
* **GSM8k (Teal):** This cluster is located in the right portion of the plot, centered around Dimension x = 30 and Dimension y = 5. The points are spread out, with a range of Dimension x values from approximately 15 to 45 and Dimension y values from approximately -5 to 15. The density appears highest around (30, 5) and decreases as you move away from this point.
* **MATH (Orange):** This cluster is located in the left-center portion of the plot, centered around Dimension x = -20 and Dimension y = -10. The points are relatively tightly clustered, with a range of Dimension x values from approximately -35 to -10 and Dimension y values from approximately -15 to -5. The density is highest around (-20, -10).
* **OlympiadBench (Blue):** This cluster is located in the left portion of the plot, centered around Dimension x = -10 and Dimension y = 5. The points are spread out, with a range of Dimension x values from approximately -25 to 10 and Dimension y values from approximately 0 to 15. The density appears highest around (-10, 5) and decreases as you move away from this point.
There are a few outliers for each benchmark, points that are distant from the main cluster. For example, there are a few blue points (OlympiadBench) with Dimension x values greater than 20.
### Key Observations
* The three benchmarks are clearly separable in this two-dimensional space, suggesting that the dimensionality reduction has successfully captured some underlying differences between them.
* MATH appears to be the most tightly clustered benchmark, indicating that its data instances are more similar to each other than those of GSM8k or OlympiadBench.
* GSM8k and OlympiadBench have more spread-out distributions, suggesting greater diversity within those benchmarks.
* The shading indicates that GSM8k and OlympiadBench have higher densities in certain regions, while MATH has a more uniform density within its cluster.
### Interpretation
This plot likely represents a visualization of embeddings generated from a language model or a similar system applied to the three benchmarks. The fact that the benchmarks are separable suggests that the model learns different representations for each benchmark. The tightness of the MATH cluster could indicate that the problems in the MATH benchmark are more homogeneous in terms of the skills or knowledge required to solve them. The spread of GSM8k and OlympiadBench could reflect the greater variety of problem types and difficulty levels within those benchmarks.
The dimensionality reduction technique used (likely PCA or t-SNE) has reduced the complexity of the original data while preserving the relative distances between data points. This allows for a visual assessment of the relationships between the benchmarks. The outliers could represent unusual or challenging instances within each benchmark.
The plot provides a high-level overview of the characteristics of each benchmark and how they differ from each other. It could be used to inform further analysis or to guide the development of more effective models for solving problems in these domains.
</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
\n
## Diagram: Problem Solving Process Comparison
### Overview
The image presents a comparative diagram illustrating the problem-solving processes of three different Large Language Models (LLMs): ChatGPT-4o, Qwen2.5-Math-72B-instruct, and Qwen2.5-Math-1.5B-instruct. The problem posed is a mathematical word problem: "Five different numbers are added together in pairs, and the results are 101, 102, 103, 104, 105, 106, 107, 108, 109, 111. Which is the largest of the five numbers?". The diagram visually compares the steps, chain length, solution style, and answer provided by each model.
### Components/Axes
The diagram is structured into three main columns, one for each LLM. Each column is further divided into sections detailing:
* **Process:** A textual description of the model's approach to solving the problem.
* **Model Type:** Categorization of the model.
* **Chain Length:** The number of steps in the model's solution process.
* **Solution Style:** The method used to arrive at the solution.
* **Answer:** The numerical answer provided by the model.
A "Difference" label is present between each model's column.
### Detailed Analysis or Content Details
**ChatGPT-4o Process:**
* Step 1: Sum of all pairwise sums. The total of all ten pairwise sums is…
* Is equivalent to: 4(a+b+c+d+e)
* Let S=a+b+c+d+e. Then, the total sum of all pairwise sums is 4S.
* Step 2: Find the sum of the pairwise sums.
* Now, calculate the total sum of the pairwise sums given in the problem… Thus, we have: 4S=1066 so S=1066/4=266.5
* Chain Length: 6 steps
* Solution Style: Analyze, Calculate
* Answer: 56
**Qwen2.5-Math-72B-instruct Process:**
* To solve this problem, Let’s denote the five numbers as a,b,c,d and e such that a<b<c<d<e.
* Step 1: First, we note that the smallest sum a+b=101 and the largest sum d+e=111. The next smallest sum is a+c=102 and the next largest sum is c+d=109. The third smallest sum is a+d=103 and the third largest sum is b+e=108.
* Step 2: We can solve these equations step by step. First, subtract the first equation from the second: (a+c)-(a+b)=102-101 => c-b=1 => c=b+1
* Chain Length: 10 steps
* Solution Style: Solve Equations
* Answer: 57
**Qwen2.5-Math-1.5B-instruct Process:**
* Step 1: To determine which number is the largest among the five given sums: 101,102,103,104,105,106,107,108, 109,111,
* we need to identify the pair that produces the highest sum when each number is added to itself.
* Step 2: First, let’s list all possible pairs: (101,101), (101,102), (101,103) (101,104), (101,105), (101,106)…
* Chain Length: 3 steps
* Solution Style: Enumerate
* Answer: 56
### Key Observations
* The models differ significantly in their approach to solving the problem. ChatGPT-4o uses a summation-based approach, Qwen2.5-Math-72B-instruct employs an equation-solving method, and Qwen2.5-Math-1.5B-instruct attempts an enumeration strategy.
* The chain lengths vary considerably, with Qwen2.5-Math-72B-instruct having the longest chain (10 steps) and Qwen2.5-Math-1.5B-instruct the shortest (3 steps).
* Two models (ChatGPT-4o and Qwen2.5-Math-1.5B-instruct) arrive at the same answer (56), while Qwen2.5-Math-72B-instruct provides a different answer (57).
* The "Model Type" is labeled as "Difference" between each model, suggesting a comparison of their underlying architectures or capabilities.
### Interpretation
The diagram highlights the diversity in problem-solving strategies employed by different LLMs. The varying chain lengths and solution styles suggest different levels of complexity and efficiency in their reasoning processes. The discrepancy in answers indicates that not all models are equally reliable in solving mathematical problems. The diagram serves as a visual comparison of the strengths and weaknesses of each model, providing insights into their respective capabilities and limitations. The use of different solution styles (analytical calculation, equation solving, and enumeration) demonstrates the flexibility of LLMs in approaching the same problem from multiple angles. The fact that two models agree on the answer suggests a degree of convergence in their reasoning, while the outlier answer from Qwen2.5-Math-72B-instruct warrants further investigation. This comparison is valuable for understanding the current state of LLM technology and identifying areas for improvement.
</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. $\mathbf{s}_{i}=\{s^{1}_{i},s^{2}_{i},\ldots,s^{n_{i}}_{i}\}$ are the solution steps, where $n_{i}$ is the step number of solution $\mathbf{s}_{i}$ . $\mathbf{y}_{i}=\{y^{1}_{i},y^{2}_{i},\ldots,y^{n_{i}}_{i}\}$ 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 $\hat{y}_{i}^{j}\in[0,1]$ 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.
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<summary>x3.png Details</summary>
### Visual Description
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## Diagram: Retrieval Framework for Mathematical Problem Solving
### Overview
This diagram illustrates three different frameworks for solving mathematical problems: Traditional Prompting (PRM), a Two-stage Retrieval-enhanced Mechanism, and a Retrieval Framework. It focuses on how each framework handles a target question ("How many seconds are in 5.5 minutes?") and assesses the correctness of solution steps. The diagram uses flowcharts to depict the process within each framework, and includes example questions, steps, and confidence scores.
### Components/Axes
The diagram is divided into three main columns, each representing a different framework. Each column is further divided into sections for "System Prompt", "Target Question", "Solution Steps", and "Target Step". There are also visual elements like question pools, step pools, and decision points (Yes/No). Confidence scores are represented by bars with numerical values.
### Detailed Analysis or Content Details
**1. Traditional PRM (Left Column)**
* **System Prompt:** "I want you to act as a math teacher. I will provide a mathematical question and several solution steps, and it will be your job to judge whether these steps are correct or not."
* **Target Question:** "How many seconds are in 5.5 minutes?"
* **Solution Steps:**
* 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."
* Step 3: "And since there are 60 seconds in a minute, there are 30 seconds in 0.5 minutes."
* **Target Step:** "Is that step correct?"
* **Confidence Score:** A bar graph shows a confidence score of approximately 0.9 (90%) for "Yes" and 0.1 (10%) for "No".
**2. Two-stage Retrieval-enhanced Mechanism (Middle Column)**
* **System Prompt:** Not explicitly shown, but implied to be related to retrieval.
* **Target Question:** Represented by a "Q" icon, leading to a "Question Pool".
* **Reference Question 1:** "What is the equivalent number of seconds in 7.8 minutes?"
* **Process:** "Since there are 60 seconds in a minute, we can find the number of seconds by multiplying the number of minutes by 60. (+/-) So, 7.8 minutes is equal to 7.8 * 60 = 468 seconds. The answer is: 468 (+/-)"
* **Reference Question 2:** "Process:" (text is incomplete, but implies a similar calculation).
* **Target Step:** Represented by an "S" icon, leading to a "New Step Pool".
* **Reference Step 1:** "0.3 hours equal to 0.3 * 60 = 18 minutes." - Correct.
* **Reference Step 2:** (text is incomplete).
**3. Retrieval Framework (Right Column)**
* **System Prompt:** "I want you to act as a math teacher. I will... judge whether these steps are correct or not. First I will give you some similar questions and their steps for reference. For each step, if the step is correct, the step is labeled as +. If the step is wrong, the step is labeled as -. If there is no relevant or helpful information in the provided questions and steps, try to answer yourself."
* **Target Question:** "How many seconds are in 5.5 minutes?"
* **Reference Question 1:**
* Step 1: "5.5 minutes is the same as 5 minutes and 0.5 minutes."
* **Reference Question 2:**
* Step 1: "How many seconds are in 5.5 minutes?"
* **Reference Step 1:** "I will give you some steps for reference"
* **Decision Point:** "Is the target step correct?"
* **Confidence Score:** A bar graph shows a confidence score of approximately 0.2 (20%) for "Yes" and 0.8 (80%) for "No".
### Key Observations
* The Traditional PRM framework shows a high confidence in the correctness of the steps (90% Yes).
* The Retrieval Framework shows a low confidence in the correctness of the steps (20% Yes, 80% No).
* The Two-stage Retrieval-enhanced Mechanism includes reference questions and steps, suggesting a process of comparison and validation.
* The diagram highlights the importance of providing reference material for evaluating the correctness of solution steps.
* The confidence scores vary significantly between the frameworks, indicating different levels of certainty in the solution.
### Interpretation
The diagram demonstrates a progression in problem-solving frameworks, from a simple prompting approach to more sophisticated retrieval-enhanced methods. The Traditional PRM relies solely on the model's internal knowledge, while the Retrieval Framework leverages external information (reference questions and steps) to assess correctness. The Two-stage Retrieval-enhanced Mechanism appears to be an intermediate step, utilizing retrieval to provide context for the model.
The differing confidence scores suggest that the retrieval-enhanced frameworks may be more critical in their assessment of solution steps, potentially identifying errors that the Traditional PRM might overlook. The lower confidence in the Retrieval Framework could indicate that the reference material is not sufficiently relevant or helpful, or that the model struggles to effectively integrate the retrieved information.
The diagram highlights the potential benefits of retrieval-augmented generation (RAG) in improving the accuracy and reliability of mathematical problem-solving systems. It also suggests that the quality and relevance of the retrieved information are crucial factors in the success of these systems. The use of "+" and "-" labels for reference steps indicates a nuanced evaluation process, going beyond simple correctness judgments. The incomplete text in some sections suggests that the diagram is a work in progress or a simplified representation of a more complex system.
</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 $e_{q_{i}}\in\mathbb{R}^{D}$ 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 $e^{\prime}_{q_{i}}\in\mathbb{R}^{d}$ 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 |
| ✓ | $\times$ | 77.8 | 74.9 | 70.7 | 71.2 | 58.4 | 59.8 | 50.5 | 54.4 | 65.0 |
| $\times$ | ✓ | 73.8 | 67.5 | 69.5 | 69.2 | 58.2 | 58.9 | 52.2 | 56.3 | 63.0 |
| $\times$ | $\times$ | 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
\n
## Bar Chart: F1 Score vs. Number of Retrieval Questions
### Overview
This bar chart compares the F1 scores of four datasets (GSM8k, MATH, OlympiadBench, and OmniMATH) across different numbers of retrieval questions (Top-0, Top-1, Top-2, and Top-3). A black dashed line represents the average F1 score across all datasets. The chart visually demonstrates how performance changes as the number of retrieved questions increases.
### Components/Axes
* **X-axis:** Number of Retrieval Questions (Top-0, Top-1, Top-2, Top-3)
* **Y-axis:** F1 Score (ranging from 50 to 80)
* **Legend (Top-right):**
* GSM8k (Light Gray)
* MATH (Gray)
* OlympiadBench (Teal)
* OmniMATH (Orange)
* Average F1 (Black, dashed line with diamond markers)
### Detailed Analysis
The chart consists of grouped bar plots for each dataset at each retrieval question level, along with a line plot representing the average F1 score.
* **Top-0:**
* GSM8k: Approximately 65.60
* MATH: Approximately 55.80
* OlympiadBench: Approximately 50.90
* OmniMATH: Approximately 67.50
* Average F1: Approximately 58.20
* **Top-1:**
* GSM8k: Approximately 72.60
* MATH: Approximately 60.80
* OlympiadBench: Approximately 56.90
* OmniMATH: Approximately 70.50
* Average F1: Approximately 65.20
* **Top-2:**
* GSM8k: Approximately 74.90
* MATH: Approximately 59.80
* OlympiadBench: Approximately 54.40
* OmniMATH: Approximately 71.20
* Average F1: Approximately 65.10
* **Top-3:**
* GSM8k: Approximately 72.30
* MATH: Approximately 57.30
* OlympiadBench: Approximately 56.70
* OmniMATH: Approximately 71.60
* Average F1: Approximately 64.50
**Trends:**
* The Average F1 score initially increases from Top-0 to Top-1, then plateaus between Top-1 and Top-3. The line slopes upward from approximately 58.20 to 65.20, then remains relatively flat.
* GSM8k consistently exhibits the highest F1 scores across all retrieval question levels.
* MATH generally has lower F1 scores compared to GSM8k and OmniMATH.
* OlympiadBench consistently has the lowest F1 scores.
* OmniMATH shows a significant increase in F1 score from Top-0 to Top-1, and then a slight decrease from Top-1 to Top-3.
### Key Observations
* Retrieving even a single question (Top-1) significantly improves the average F1 score.
* The benefit of adding more retrieved questions diminishes after the first question (Top-1).
* GSM8k is the most robust dataset, showing consistent high performance.
* OlympiadBench is the most challenging dataset, with consistently low performance.
### Interpretation
The data suggests that retrieving a small number of relevant questions can substantially improve the performance of a model on these math datasets. However, there's a diminishing return to scale; adding more questions beyond the first few doesn't lead to significant further improvements. This could indicate that the initial retrieval step is the most critical, and subsequent questions provide only marginal benefit. The differences in performance between the datasets highlight the varying difficulty levels of the problems within each dataset. GSM8k's consistently high scores suggest it contains relatively simpler problems, while OlympiadBench's low scores indicate it presents more challenging problems. The plateauing of the average F1 score suggests that the model's ability to utilize additional retrieved information is limited, potentially due to factors like the quality of the retrieved questions or the model's capacity to process them effectively.
</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.
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## Appendix A Baselines
### A.1 Open-source PRM
<details>
<summary>x5.png Details</summary>
### Visual Description
\n
## Text Block: Math Problem & Solution Verification
### Overview
The image presents a text-based interaction between a "System" (acting as a prompt provider) and an "Output" (representing a response). The System poses a mathematical question and provides a step-by-step solution process, requesting the Output to verify the correctness of each step.
### Content Details
The text content is as follows:
**System:**
I want you to act as a math teacher. I will provide a mathematical question and several solution steps, and it will be your job to judge whether these steps are correct or not.
**Input:**
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.
Step 3 : And since there are 60 seconds in a minute, there are 30 seconds in 0.5 minutes.
Is that Step Correct? You should ONLY tell me + or -.
**Output:**
+.
### Key Observations
The interaction is structured as a question-and-answer format. The System provides a clear instruction and a specific question. The Input section details the question and the proposed solution steps. The Output provides a binary response (+ or -) indicating the correctness of the step.
### Interpretation
The image demonstrates a simple verification task. The System is testing the ability of an agent (represented by the Output) to assess the logical correctness of mathematical steps. The positive response (+) suggests that the agent correctly identified the steps as valid. The entire interaction is a test case for a system designed to evaluate mathematical reasoning. The simplicity of the problem suggests it's a foundational test, likely part of a larger evaluation suite. The constraint to respond with only "+" or "-" indicates a focus on binary correctness assessment, rather than detailed explanations.
</details>
Figure 5: The illustration of PRM input template.
<details>
<summary>x6.png Details</summary>
### Visual Description
\n
## Textual Document: Math Problem & Solution Steps
### Overview
The image presents a series of math problems and solution steps, framed as a tutoring scenario where a "System" (presumably a teacher) asks for validation of each step. The problems involve unit conversions (minutes to seconds, hours to minutes). The task is to determine if each step is correct (+) or incorrect (-).
### Components/Axes
The document is structured into sections:
* **Reference Question 1:** A solved example problem.
* **Reference Question 2:** An incomplete problem.
* **Target Question:** The problem to be solved.
* **Target Process:** The steps taken to solve the target question.
* **Reference Step 1 & 2:** Steps from other problems used for comparison.
There are no axes or charts in this document. It is purely textual.
### Detailed Analysis or Content Details
**Reference Question 1:**
* **Question:** What is the equivalent number of seconds in 7.8 minutes?
* **Process:** "Since there are 60 seconds in a minute, we can find the number of seconds by multiplying the number of minutes by 60."
* **Calculation:** 7.8 * 60 = 46 seconds.
* **Assessment:** The answer is marked as incorrect (-). The correct answer is 7.8 * 60 = 468 seconds.
**Reference Question 2:**
* **Process:** This section is incomplete.
**Target Question:**
* **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 Step 1:**
* 0.3 hours equal to 0.3 * 60 = 18 minutes. This reference step is correct.
* **Reference Step 2:**
* This section is incomplete.
* **Target Step 3:**
* And since there are 60 seconds in a minute, there are 30 seconds in 0.5 minutes.
* **Assessment Request:** Is the Step Correct?
### Key Observations
* The initial reference question has an arithmetic error. 7.8 * 60 is not 46, but 468.
* The target question is broken down into logical steps.
* The reference steps seem to be included to provide context or similar examples.
* The final step (Target Step 3) is awaiting validation.
### Interpretation
The document is designed to test understanding of unit conversion and arithmetic skills. The inclusion of a deliberately incorrect answer in the first reference question suggests the system is testing the ability to identify errors, not just perform calculations. The breakdown of the target question into steps indicates a focus on the *process* of problem-solving. The request for validation of the final step implies the system is seeking confirmation that the user understands the underlying principles. The reference steps, while seemingly unrelated, might be intended to distract or assess the ability to filter relevant information.
The final step, "And since there are 60 seconds in a minute, there are 30 seconds in 0.5 minutes," is correct. 0.5 * 60 = 30.
</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 $\{1\times 10^{-3},1\times 10^{-4},3\times 10^{-4},1\times 10^{-5},3\times 10^{ -5}\}$ 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 |
| ✓ | $\times$ | 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 |
| $\times$ | ✓ | 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 |
| $\times$ | $\times$ | 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 |