# 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 with Clusters: GSM8k, MATH, and OlympiadBench Datasets
### Overview
The image is a 2D scatter plot visualizing the distribution of data points from three distinct datasets: GSM8k, MATH, and OlympiadBench. The plot reveals clear clustering patterns, with each dataset occupying a distinct region in the 2D space defined by "Dimension x" and "Dimension y". The data points are semi-transparent, allowing density to be inferred from color saturation. Each dataset cluster is also overlaid with a shaded, semi-transparent polygon (a convex hull) that outlines the approximate boundary of its main data cloud.
### Components/Axes
* **Chart Type:** Scatter plot with cluster boundaries (convex hulls).
* **X-Axis:** Labeled "Dimension x". Scale ranges from -40 to 50, with major tick marks every 10 units.
* **Y-Axis:** Labeled "Dimension y". Scale ranges from -20 to 25, with major tick marks every 5 units.
* **Legend:** Positioned at the top center of the chart area. It contains three entries:
* A green circle labeled "GSM8k"
* An orange circle labeled "MATH"
* A blue circle labeled "OlympiadBench"
* **Data Series & Visual Encoding:**
* **GSM8k:** Represented by green dots. Its cluster is bounded by a light green shaded polygon.
* **MATH:** Represented by orange dots. Its cluster is bounded by a light orange shaded polygon.
* **OlympiadBench:** Represented by blue dots. Its cluster is bounded by a light blue shaded polygon.
### Detailed Analysis
**Spatial Distribution and Cluster Characteristics:**
1. **GSM8k (Green):**
* **Trend/Placement:** Forms a dense, vertically elongated cluster on the right side of the plot.
* **Approximate Center:** (x ≈ 30, y ≈ 0).
* **Range:** Primarily spans Dimension x from ~15 to ~40, and Dimension y from ~-15 to ~15.
* **Density:** High density in the core region (x=25-35, y=-5 to 5). Points become sparser towards the edges of its convex hull.
2. **MATH (Orange):**
* **Trend/Placement:** Forms a dense cluster in the bottom-left quadrant.
* **Approximate Center:** (x ≈ -20, y ≈ -8).
* **Range:** Primarily spans Dimension x from ~-35 to ~-5, and Dimension y from ~-18 to ~5.
* **Density:** Very high density in its core (x=-25 to -15, y=-12 to -4). It shows significant overlap with the lower portion of the OlympiadBench cluster.
3. **OlympiadBench (Blue):**
* **Trend/Placement:** Forms a dense cluster in the top-left quadrant.
* **Approximate Center:** (x ≈ -15, y ≈ 5).
* **Range:** Primarily spans Dimension x from ~-30 to ~5, and Dimension y from ~-5 to ~18.
* **Density:** High density in its core (x=-20 to -10, y=0 to 10). It overlaps substantially with the upper portion of the MATH cluster.
**Inter-Cluster Relationships:**
* There is a clear and wide separation along the x-axis between the **GSM8k** cluster (positive x) and the clusters for **MATH** and **OlympiadBench** (negative x).
* The **MATH** and **OlympiadBench** clusters are adjacent and overlap significantly in the region around (x ≈ -15, y ≈ -2). Their convex hulls merge in this area.
* A sparse scattering of outlier points from all three datasets exists in the central region of the plot (x ≈ -5 to 15, y ≈ -5 to 5), between the main clusters.
### Key Observations
1. **Distinct Separation of GSM8k:** The most prominent feature is the isolation of the GSM8k dataset along the positive x-axis, suggesting its data points have fundamentally different characteristics in this 2D projection compared to the other two datasets.
2. **Overlap of MATH and OlympiadBench:** These two datasets share a significant portion of the feature space, particularly in the negative x, mid-y region. This indicates potential similarity in the underlying properties being measured for a subset of their data.
3. **Cluster Shape:** The GSM8k cluster is vertically oriented, while the MATH and OlympiadBench clusters are more horizontally oriented and blob-like.
4. **Outliers:** All datasets have points that lie outside their main convex hull, scattered primarily in the central "gap" between the three primary clusters.
### Interpretation
This scatter plot likely visualizes the output of a dimensionality reduction technique (like t-SNE or PCA) applied to features from three mathematical reasoning benchmarks. The spatial arrangement suggests:
* **GSM8k (Grade School Math 8k)** problems form a distinct category in the feature space. Their separation implies the model's internal representations or the problem characteristics for grade-school level math are consistently different from more advanced math.
* **MATH** and **OlympiadBench** datasets, which both target more advanced mathematical problem-solving, are not fully distinct from each other. Their overlap indicates that a significant portion of problems from these benchmarks are processed similarly by the model or share underlying features. The OlympiadBench cluster extends slightly higher on the y-axis, which might hint at a subset of problems with even greater complexity or a different sub-type.
* The **central region of outliers** could represent ambiguous problems, problems that bridge the characteristics of the main clusters, or noise in the embedding process.
The visualization effectively demonstrates that while advanced math problems (MATH, OlympiadBench) share a common feature subspace, grade-school math (GSM8k) occupies a separate region, providing a clear visual distinction between problem difficulty levels in the analyzed embedding space.
</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: Comparative Problem-Solving Processes of Three AI Models
### Overview
The image is a technical diagram comparing the step-by-step reasoning processes of three different AI models (ChatGPT-4o, Qwen2.5-Math-72B-instruct, and Qwen2.5-Math-1.5B-instruct) as they solve the same mathematical word problem. The diagram visually contrasts their methodologies, chain lengths, solution styles, and final answers.
### Components/Axes
The diagram is structured into four main regions:
1. **Header (Top):** Contains the problem statement.
2. **Main Comparison Area (Center):** Three vertical panels, each dedicated to one model's process.
3. **Connecting Annotations (Between Panels):** Text and arrows indicating the axes of comparison.
4. **Footer (Bottom of each panel):** A summary box for each model.
**Labels and Text Elements:**
* **Problem Statement (Top):** "Question: 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?"
* **Model Panel Headers (Left to Right):**
* "ChatGPT-4o Process:" (in blue text)
* "Qwen2.5-Math-72B-instruct Process:" (in blue text)
* "Qwen2.5-Math-1.5B-instruct Process:" (in blue text)
* **Connecting Annotations:**
* Between the first and second panel: A double-headed arrow labeled "Model Type Difference".
* Between the second and third panel: A double-headed arrow labeled "Model Size Difference".
* **Footer Summary Boxes (Yellow background, bottom of each panel):**
* **Left (ChatGPT-4o):** "Chain Length: 6 steps", "Solution Style: Analyze, Calculate", "Answer: 56"
* **Center (Qwen2.5-Math-72B-instruct):** "Chain Length: 10 steps", "Solution Style: Solve Equations", "Answer: 57"
* **Right (Qwen2.5-Math-1.5B-instruct):** "Chain Length: 3 steps", "Solution Style: Enumerate", "Answer: 56"
* **Logos:** Each panel contains a small logo in its bottom-right corner (OpenAI logo for ChatGPT-4o, Qwen logo for the other two).
### Detailed Analysis
**1. ChatGPT-4o Process (Left Panel):**
* **Methodology:** Begins by denoting the five numbers as `a,b,c,d,e` with `a<b<c<d<e`.
* **Step 1:** Calculates the sum of all pairwise sums. It states the total is `4(a+b+c+d+e)` and defines `S = a+b+c+d+e`, concluding the total sum is `4S`.
* **Step 2:** Calculates the sum of the given pairwise sums (101 through 111, excluding 110). It states the total is 1066, leading to the equation `4S=1066`, so `S=266.5`.
* **Process End:** The text ends with "..." indicating omitted steps before the final answer.
* **Summary:** Chain Length: 6 steps. Solution Style: Analyze, Calculate. Final Answer: **56**.
**2. Qwen2.5-Math-72B-instruct Process (Center Panel):**
* **Methodology:** Also starts by denoting the numbers as `a,b,c,d,e` with `a<b<c<d<e`.
* **Step 1:** Identifies specific pairwise sums: `a+b=101` (smallest), `d+e=111` (largest), `a+c=102`, `c+e=109`, `a+d=103`, `b+e=108`.
* **Step 2:** Proposes solving these equations step-by-step. It shows the first subtraction: `(a+c)-(a+b)=102-101` leading to `c-b=1` or `c=b+1`.
* **Process End:** The text ends with "..." indicating omitted steps.
* **Summary:** Chain Length: 10 steps. Solution Style: Solve Equations. Final Answer: **57**.
**3. Qwen2.5-Math-1.5B-instruct Process (Right Panel):**
* **Methodology:** Starts by listing the given sums: "101,102,103,104,105,106,107,108,109,111".
* **Step 1:** States the goal is to identify the pair producing the highest sum.
* **Step 2:** Begins to "list all possible pairs" starting with "(101,101), (101,102), (101,103) (101,104), (101,105), (101,106)..." This approach appears to be an enumeration of combinations from the given sums, which is a different strategy.
* **Process End:** The text ends with "..." indicating omitted steps.
* **Summary:** Chain Length: 3 steps. Solution Style: Enumerate. Final Answer: **56**.
### Key Observations
1. **Divergent Answers:** The most significant observation is the discrepancy in the final answer. The 72B-parameter model concludes the largest number is **57**, while both ChatGPT-4o and the smaller 1.5B model conclude it is **56**.
2. **Contrasting Methodologies:**
* **ChatGPT-4o** uses an **analytical/calculative** approach, focusing on the total sum of all pairs.
* **Qwen2.5-Math-72B** uses an **algebraic equation-solving** approach, setting up and manipulating specific equations derived from the smallest and largest sums.
* **Qwen2.5-Math-1.5B** uses an **enumerative** approach, attempting to list pairs directly from the given sums.
3. **Chain Length vs. Model Size:** The largest model (72B) has the longest described chain (10 steps), while the smallest model (1.5B) has the shortest (3 steps). ChatGPT-4o is intermediate (6 steps). This suggests a correlation between model size/capability and the complexity of the reasoning chain it generates for this problem.
4. **Spatial Layout:** The diagram uses a left-to-right flow to compare models. The "Model Type Difference" arrow contrasts the proprietary model (ChatGPT-4o) with the open-weight Qwen models. The "Model Size Difference" arrow contrasts the two Qwen models of different scales (72B vs. 1.5B parameters).
### Interpretation
This diagram serves as a case study in **AI model reasoning divergence**. It demonstrates that different AI architectures, training paradigms, and model sizes can lead to fundamentally different problem-solving strategies for the same logical puzzle, even resulting in different final answers.
* **What the Data Suggests:** The core mathematical problem has a single correct answer. The fact that two models agree on 56 and one (the largest) disagrees with 57 creates an investigative scenario. One must verify the correct solution to determine which model's reasoning is flawed. The 72B model's algebraic setup appears more rigorous at first glance, but its answer differs. This highlights the challenge of evaluating AI reasoning without a ground truth.
* **Relationship Between Elements:** The connecting arrows frame the comparison. The "Type Difference" suggests that the fundamental design of ChatGPT versus Qwen influences their approach. The "Size Difference" within the Qwen family isolates the effect of scale, showing that a larger model (72B) adopts a more complex, equation-based method compared to the smaller model's (1.5B) simplistic enumeration.
* **Notable Anomalies:** The primary anomaly is the answer split. A secondary observation is that the enumerative approach of the 1.5B model, while short, is likely incorrect or incomplete for this problem type, as it seems to be enumerating pairs *from the sum list* rather than deducing the original numbers. The diagram effectively exposes how model "confidence" (implied by a detailed chain) does not guarantee correctness, and how different "solution styles" are employed as heuristics.
</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 $D=\{(q_i,s_i,y_i)\}_i=1^N$ , where $N$ is the number of data instances. The input $q_i$ is the $i^th$ Math question. $s_i=\{s^1_i,s^2_i,…,s^n_i_i\}$ are the solution steps, where $n_i$ is the step number of solution $s_i$ . $y_i=\{y^1_i,y^2_i,…,y^n_i_i\}$ and the label $y^j_i$ indicates the correctness from the $1^st$ step to the $j^th$ step.
$$
y^j_i=\begin{cases}1,~{}(s^1_i,…,s^j_i)~{}is correct
for~{}q_i;\\
0,~{}otherwise.\end{cases} \tag{1}
$$
### 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
$$
∀{i},\hat{y}^n_i_i=ORM(q_i,s^1_i,…,s^n_i_i). \tag{2}
$$
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.
$$
∀{i,j},\hat{y}^j_i=PRM(q_i,s^1_i,…,s^j_i). \tag{3}
$$
### 2.3 Large Language Model for PRM scoring
When directly adopting LLMs as the PRM for scoring, we need to convert the data $(q_i,s_i,y_i)$ 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 $s_i$ , 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:
$$
\hat{y}_i^j=\frac{\exp(l_i,+)}{\exp(l_i,+)+\exp(l_i,
-)}∈(0,1). \tag{4}
$$
where $l_i,+$ and $l_i,-$ 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 Three Process Reward Model (PRM) Frameworks for Mathematical Reasoning
### Overview
The image is a technical diagram comparing three different architectural approaches for evaluating the correctness of mathematical solution steps. It is divided into three distinct panels, each illustrating a different framework: "Traditional PRM" (left), "Two-stage Retrieval-enhanced Mechanism" (center), and "Retrieval PRM Framework" (right). The diagram uses flowcharts, text boxes, arrows, and icons to explain the workflow and components of each system.
### Components/Axes
The diagram is organized into three vertical panels, each with a title bar at the top.
**Panel 1: Traditional PRM (Left)**
* **Title:** "Traditional PRM"
* **Components (Top to Bottom):**
1. **System Prompt (Green Box):** "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."
2. **Target Question (Oval):** "How many seconds are in 5.5 minutes?"
3. **Solution Steps (Three Rectangles in Sequence):**
* "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 50 seconds in 0.5 minutes."
4. **Target Step (Oval):** "Is that step correct?" (Pointing to Step 3)
5. **Judgment Output:** A sad face emoji (😞) next to two boxes: "Yes" with a value of "0.9" and "No" with a value of "0.1".
**Panel 2: Two-stage Retrieval-enhanced Mechanism (Center)**
* **Title:** "Two-stage Retrieval-enhanced Mechanism"
* **Components:**
1. **Input:** "Target Question" labeled with a red "Q" pointing to a "Question Pool" database icon.
2. **Question Retrieval:** The Question Pool connects to multiple document icons, representing retrieved similar questions.
3. **Reference Question Box (Yellow):** Contains an example.
* **Header:** "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 = 46 seconds. The answer is: 46 (-)"
* **Note:** "Reference Question 2: Process: ..."
4. **Step Retrieval:** "Target Step" labeled with a red "S" points to a "New Step Pool" database icon.
5. **Reference Step Cloud (Light Blue):** Contains an example.
* "Reference Step 1: 0.3 hours equal to 0.3 * 60 = 18 minutes. This reference step is correct."
* "Reference Step 2: ..."
**Panel 3: Retrieval PRM Framework (Right)**
* **Title:** "Retrieval PRM Framework"
* **Components:**
1. **System Prompt (Green Box):** "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.**"
2. **Reference Flow:**
* "Reference Question 1" (Yellow Box) points to "Reference Question 2" (Yellow Box).
* "Reference Question 1" also points down to "Step 1 : 5.5 minutes is the same as 5 minutes and 0.5 minutes." (White Box).
* "Reference Question 2" points down to the "Target Question" oval: "How many seconds are in 5.5 minutes?"
3. **Step Evaluation Flow:**
* The "Step 1" box points to "Step 2 : Since there are 60 seconds in a minute, then there are 300 seconds in 5 minutes." (White Box).
* "Step 2" points to "Step 3 : And since there are 60 seconds in a minute, there are 50 seconds in 0.5 minutes." (White Box).
* A red text note states: "I will give you some steps for reference".
* Two light blue clouds labeled "Reference Step2" and "Reference Step1" point to a decision box.
4. **Judgment:** The decision box "Is the target step correct?" (referring to Step 3) leads to:
* "Yes" with a value of "0.2"
* "No" with a value of "0.8"
* A happy face emoji (😊) is shown next to the "No" outcome.
### Detailed Analysis
* **Traditional PRM Flow:** A linear process. A system prompt defines the task. A target question is presented with its solution steps. The model must judge a specific target step (Step 3) in isolation. The output is a probability distribution (Yes: 0.9, No: 0.1), with a sad emoji suggesting an incorrect or low-confidence judgment for the given example.
* **Two-stage Retrieval Flow:** A parallel, retrieval-based process. It first retrieves similar questions from a pool based on the target question (Q). It then retrieves similar steps from a separate pool based on the target step (S). These retrieved items ("Reference Question" and "Reference Step") are provided as context, containing their own processes and correctness labels (+/-).
* **Retrieval PRM Framework Flow:** An integrated process that combines retrieval and judgment. The system prompt is modified to instruct the model to use provided references. It shows a chain where reference questions and their steps are provided alongside the target question and its steps. The model is explicitly given "Reference Steps" to aid in judging the target step. The output probability (Yes: 0.2, No: 0.8) with a happy emoji suggests a more confident and correct judgment ("No") for the same target step (Step 3) compared to the Traditional PRM.
### Key Observations
1. **Evolution of Context:** The core difference is the amount of contextual information provided to the judge. Traditional PRM uses none, the Two-stage Mechanism retrieves it separately, and the Retrieval PRM Framework integrates it directly into the prompt.
2. **Judgment Confidence:** For the identical target step ("Step 3: ... there are 50 seconds in 0.5 minutes"), the Traditional PRM assigns a high probability (0.9) to "Yes" (incorrect), while the Retrieval PRM Framework assigns a high probability (0.8) to "No" (correct). This visually demonstrates the claimed improvement of the retrieval-enhanced approach.
3. **Prompt Engineering:** The system prompt in the Retrieval PRM Framework is significantly more detailed, explicitly instructing the model on how to use the provided reference examples and their labels (+/-).
4. **Visual Cues:** The use of emojis (😞 vs. 😊) provides an immediate, non-numerical indicator of the perceived quality or correctness of the model's output in each framework.
### Interpretation
This diagram argues for the superiority of retrieval-augmented methods in training Process Reward Models for mathematical reasoning. The **Traditional PRM** is depicted as limited, making judgments in a vacuum, which can lead to confident errors (as shown by the high "Yes" probability for a wrong step). The **Two-stage Retrieval-enhanced Mechanism** introduces the concept of gathering relevant external knowledge (similar questions and steps) but presents it as a separate, preparatory stage.
The **Retrieval PRM Framework** is presented as the most advanced solution. It seamlessly integrates the retrieved references into the model's context window, transforming the task from pure judgment to **judgment-by-analogy**. The model is no longer just a math teacher but a teacher with a textbook of solved examples open in front of them. The dramatic shift in the probability distribution for the same step (from 0.9/0.1 to 0.2/0.8) is the central piece of evidence, suggesting that providing relevant, labeled reference cases significantly improves the model's ability to discern correct reasoning steps. The diagram implies that this approach leads to more reliable and accurate reward signals for training reasoning models.
</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 $D_q=\{q_i\}_i=1^N$ . During retrieval process, we treat:
- Query: the target question $q_t$ .
- Key: all $q_i$ in the retrieval pool.
- Value: all the $(q_i,s_i)$ 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.
$$
\{e_q_{i}\}_i=1^N=SentenceBERT(\{q_i\}_i=1^N) \tag{5}
$$
where $e_q_{i}∈ℝ^D$ 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.
$$
\{e^\prime_q_{i}\}_i=1^N=PCA(\{e_q_{i}\}_i=1^N) \tag{6}
$$
where $e^\prime_q_{i}∈ℝ^d$ 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.
$$
\displaystyle⟨ q_i,q_t⟩ \displaystyle=\frac{e^\prime_q_{t}· e^\prime_q_{i}}{|e^\prime_q
_{t}|·|e^\prime_q_{i}|}. \tag{7}
$$
Now we sort the vector $\{⟨ q_i,q_t⟩\}_i=1^N$ of similarity and choose top- k $(q_i,s_i)$ pairs as reference questions $q_r$ and put them in RetrievalPRM’s input together with the target question. Furthermore, we store all the solutions $\{s_i\}_i=1^m$ 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 $D_s$ . 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 $q_i^r$ and reference steps as $s_i^r$ . The whole input $x_i^j$ of predicting the $j_th$ step of $q_i$ in RetrievalPRM can be formulated as:
$$
\displaystylex^j_i=(RetSP,q^r_i, \displaystyle q_i,s^1_i,…,s^j-1_i,s^r_i,s^j_i,
y^j_i), \displaystyle\hat{y}^j_i= \displaystylePRM(x^j_i) \tag{8}
$$
where $s^j_i$ is the $j_th$ step of solution $s_i$ .
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 $1∼ n-1$ .
## 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
## Grouped Bar Chart with Line Overlay: F1 Score by Number of Retrieval Questions
### Overview
The image displays a grouped bar chart with an overlaid line graph. It compares the F1 Score performance of four different datasets across four categories representing the "Number of Retrieval Questions" (Top-0, Top-1, Top-2, Top-3). An "Average F1" trend line is superimposed on the chart.
### Components/Axes
* **Chart Type:** Grouped bar chart with a line graph overlay.
* **X-Axis:** Labeled "Number of Retrieval Questions". It has four categorical groups: `Top-0`, `Top-1`, `Top-2`, and `Top-3`.
* **Y-Axis:** Labeled "F1 Score". The scale runs from 50 to 80, with major gridlines at intervals of 5.
* **Legend:** Located in the top-left corner of the chart area. It defines the following series:
* **GSM8k:** Light gray bar.
* **MATH:** Blue-gray bar.
* **OlympiadBench:** Teal/green-gray bar.
* **OmniMATH:** Tan/light brown bar.
* **Average F1:** Black dashed line with circular markers.
* **Data Labels:** Numerical F1 Score values are printed directly above each bar.
### Detailed Analysis
**Data Series and Values (by X-Axis Category):**
1. **Top-0:**
* GSM8k (Light Gray): 65.60
* MATH (Blue-Gray): 67.50
* OlympiadBench (Teal): 55.80
* OmniMATH (Tan): 50.90
* *Average F1 (Black Line Marker):* Positioned at 60.00.
2. **Top-1:**
* GSM8k (Light Gray): 70.50
* MATH (Blue-Gray): 72.60
* OlympiadBench (Teal): 60.80
* OmniMATH (Tan): 56.90
* *Average F1 (Black Line Marker):* Positioned at 65.00.
3. **Top-2:**
* GSM8k (Light Gray): 74.90
* MATH (Blue-Gray): 71.20
* OlympiadBench (Teal): 59.80
* OmniMATH (Tan): 54.40
* *Average F1 (Black Line Marker):* Positioned at 65.00.
4. **Top-3:**
* GSM8k (Light Gray): 72.30
* MATH (Blue-Gray): 71.60
* OlympiadBench (Teal): 57.30
* OmniMATH (Tan): 56.70
* *Average F1 (Black Line Marker):* Positioned at approximately 64.50.
**Trend Verification (Visual Description):**
* **GSM8k (Light Gray):** The bar height increases from Top-0 to Top-2, reaching a peak, then shows a slight decrease at Top-3.
* **MATH (Blue-Gray):** The bar height increases from Top-0 to Top-1, then shows a gradual decrease through Top-2 and Top-3.
* **OlympiadBench (Teal):** The bar height increases from Top-0 to Top-1, then decreases at Top-2 and further at Top-3.
* **OmniMATH (Tan):** The bar height increases from Top-0 to Top-1, decreases at Top-2, and then increases again at Top-3.
* **Average F1 (Black Dashed Line):** The line shows a clear upward slope from Top-0 to Top-1, then plateaus horizontally to Top-2, followed by a very slight downward slope to Top-3.
### Key Observations
1. **Performance Hierarchy:** Across all categories, the `GSM8k` and `MATH` datasets consistently achieve higher F1 Scores (generally above 65) compared to `OlympiadBench` and `OmniMATH` (generally below 61).
2. **Peak Performance:** The highest individual F1 Score recorded is 74.90 for `GSM8k` at `Top-2`. The highest score for `MATH` is 72.60 at `Top-1`.
3. **Average Trend:** The "Average F1" line indicates that, on average, performance improves significantly when moving from no retrieval (`Top-0`) to one retrieval question (`Top-1`). The average performance then stabilizes between `Top-1` and `Top-2` before a marginal decline at `Top-3`.
4. **Dataset-Specific Patterns:** `GSM8k` shows the most pronounced benefit from retrieval, peaking at `Top-2`. `MATH` peaks earlier at `Top-1`. `OmniMATH` shows a non-linear pattern, dipping at `Top-2` before recovering at `Top-3`.
### Interpretation
This chart demonstrates the impact of incorporating retrieved information (quantified by the number of retrieval questions) on model performance across different mathematical reasoning benchmarks. The data suggests a clear, positive effect of retrieval, with the most substantial gain occurring with the introduction of the first retrieval question (`Top-0` to `Top-1`). The plateau in the average score from `Top-1` to `Top-2` implies diminishing returns; adding a second retrieval question does not yield a significant average improvement. The slight average decline at `Top-3` could indicate potential noise or interference from too much retrieved context, though the effect is small.
The variance in optimal points across datasets (`GSM8k` at `Top-2`, `MATH` at `Top-1`) highlights that the ideal amount of retrieval may be task or dataset-dependent. The consistently lower scores for `OlympiadBench` and `OmniMATH` suggest these benchmarks present a greater challenge for the evaluated system, regardless of the retrieval configuration. Overall, the visualization argues for the strategic use of retrieval in mathematical problem-solving, with a recommendation to consider 1 or 2 retrieval questions as a potentially optimal range for balancing performance and complexity.
</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
## Screenshot: Structured AI Interaction Prompt
### Overview
The image is a screenshot of a structured text document, likely representing a prompt or a template for an AI interaction. It is divided into three distinct sections by horizontal lines: "System," "Input," and "Output." The text is in English and presents a scenario where an AI is instructed to act as a math teacher to evaluate solution steps.
### Components/Axes
The document has a clear, box-like layout with three primary sections:
1. **System Section (Top):**
* **Header:** "System:" (in bold)
* **Content:** A single paragraph defining the AI's role and task.
2. **Input Section (Middle):**
* **Header:** "Input:" (in bold)
* **Content:** A structured math problem containing:
* A "Question:" sub-header.
* A "Process:" sub-header.
* Three numbered steps (Step 1, Step 2, Step 3).
* A final instruction line.
3. **Output Section (Bottom):**
* **Header:** "Output:" (in bold)
* **Content:** A single character response.
### Detailed Analysis / Content Details
**Full Text Transcription:**
**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 document uses a consistent format with bold section headers and plain text content.
* The "Input" section contains a complete, self-contained math problem with a clear question, a multi-step process, and specific instructions for the expected output format ("+ or -").
* The "Output" section shows a sample response ("+."), which aligns with the instruction to only provide a "+" or "-" symbol. The period after the "+" may be a formatting artifact or part of the sample.
* The math problem involves unit conversion (minutes to seconds) and is broken down into logical, verifiable steps.
### Interpretation
This image depicts a **prompt engineering template** or a **few-shot example** for configuring an AI model to perform a specific, constrained task: evaluating the correctness of mathematical solution steps.
* **Purpose:** The structure is designed to teach or test an AI's ability to follow precise instructions and perform logical validation. The "System" prompt sets the persona, the "Input" provides the test case, and the "Output" demonstrates the required minimal response format.
* **Data Relationship:** The "Input" is the core data. The three steps logically decompose the problem `5.5 minutes * 60 seconds/minute`. Step 1 correctly splits the decimal. Step 2 correctly calculates `5 * 60 = 300`. Step 3 correctly calculates `0.5 * 60 = 30`. Therefore, the process is mathematically sound, which is correctly reflected by the "+" in the "Output."
* **Notable Pattern:** The instruction "You should ONLY tell me + or -" is a critical constraint, forcing the AI to bypass explanatory text and output a pure, machine-readable judgment. This is common in evaluation pipelines or reinforcement learning from human feedback (RLHF) setups.
* **Anomaly/Clarity:** The final line in the Input, "Is that Step Correct?", is slightly ambiguous. It could be interpreted as asking about the last step (Step 3) only, or the entire process. Given the context and the sample output "+", it is logically interpreted as asking about the entire provided process. The sample output confirms this interpretation.
</details>
Figure 5: The illustration of PRM input template.
<details>
<summary>x6.png Details</summary>
### Visual Description
## System Interface Screenshot: Math Teacher AI Evaluation Prompt
### Overview
The image is a screenshot of a structured text interface, likely a prompt or a user interface for an AI system designed to act as a math teacher. The interface is divided into three distinct sections: **System**, **Input**, and **Output**. The text is presented in a monospaced font on a light background, with specific instructions and content highlighted in different colors (red, blue, magenta).
### Components/Axes
The interface is organized into three labeled sections:
1. **System:** Contains the core instructions for the AI's role and behavior.
2. **Input:** Contains the problem data, including reference examples and a target problem to evaluate.
3. **Output:** Contains the AI's final response to the evaluation task.
### Detailed Analysis
#### **System Section**
* **Text (Black):** "System:"
* **Instruction Text (Black):** "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."
* **Instruction Text (Red):** "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."
#### **Input Section**
* **Text (Black):** "Input:"
* **Reference Question 1 (Blue Label):**
* **Question (Black):** "What is the equivalent number of seconds in 7.8 minutes?"
* **Process (Black):** "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 = 46 seconds.The answer is: 46 (-)"
* *Note: The process contains two inline labels: a correct step `(+)` and an incorrect final answer `(-)`.*
* **Reference Question 2 (Blue Label):**
* **Question (Black):** "..."
* **Process (Black):** "..."
* *Note: This section is truncated with ellipses, indicating omitted content.*
* **Target Question (Magenta Label):**
* **Question (Black):** "How many seconds are in 5.5 minutes?"
* **Process (Black):**
* "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 Step1 (Blue Label):**
* **Text (Black):** "0.3 hours equal to 0.3 * 60 = 18 minutes. This reference step is correct."
* **Reference Step2 (Blue Label):**
* **Text (Black):** "..."
* *Note: This section is truncated with ellipses.*
* **Target Step 3 (Magenta Label):**
* **Text (Black):** "And since there are 60 seconds in a minute, there are 30 seconds in 0.5 minutes."
* **Final Instruction (Black):** "Is the Step Correct? You should ONLY tell me + or -."
#### **Output Section**
* **Text (Black):** "Output:"
* **Response (Black):** "+."
### Key Observations
1. **Color-Coded Structure:** The interface uses color to differentiate between system instructions (red), reference material labels (blue), and the target problem labels (magenta).
2. **Evaluation Logic:** The system prompt establishes a clear binary evaluation scheme (`+` for correct, `-` for incorrect) for mathematical solution steps.
3. **Reference-Based Learning:** The AI is instructed to use provided reference questions and steps as a guide for its judgment, falling back to its own knowledge if references are unhelpful.
4. **Truncated Content:** Both "Reference Question 2" and "Reference Step2" are shown as ellipses (`...`), indicating that the full context is not displayed in this screenshot.
5. **Target Step Evaluation:** The final step presented for evaluation ("Target Step 3") is a correct mathematical statement (0.5 minutes * 60 seconds/minute = 30 seconds). The AI's output (`+.`) correctly identifies it as such.
### Interpretation
This image depicts a **few-shot learning or chain-of-thought prompting setup** for an AI model fine-tuned or instructed to perform step-by-step verification of mathematical reasoning. The structure is pedagogical:
* **Goal:** To train or prompt an AI to act as a meticulous math tutor that validates each logical step in a solution, not just the final answer.
* **Method:** It provides a clear rubric (`+`/`-`), concrete examples (Reference Question 1 shows a correct step and an incorrect final answer), and a specific target task.
* **Underlying Task:** The AI must parse natural language math problems, understand unit conversion logic (minutes to seconds), and assess the correctness of intermediate reasoning steps. The output `+` confirms the AI successfully identified the logical validity of converting 0.5 minutes to 30 seconds.
* **Purpose:** This setup is likely used for creating training data, evaluating model performance on reasoning tasks, or as part of a interactive tutoring system where the AI provides feedback on a student's work. The presence of truncated references suggests this is a snippet from a longer, more comprehensive prompt.
</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 |