2406.18629v1

# Step-DPO: Step-wise Preference Optimization for Long-chain Reasoning of LLMs Abstract Mathematical reasoning presents a significant challenge for Large Language Models (LLMs) due to the extensive and precise chain of reasoning required for accuracy. Ensuring the correctness of each reasoning step is critical. To address this, we aim to enhance the robustness and factuality of LLMs by learning from human feedback. However, Direct Preference Optimization (DPO) has shown limited benefits for long-chain mathematical reasoning, as models employing DPO struggle to identify detailed errors in incorrect answers. This limitation stems from a lack of fine-grained process supervision. We propose a simple, effective, and data-efficient method called Step-DPO, which treats individual reasoning steps as units for preference optimization rather than evaluating answers holistically. Additionally, we have developed a data construction pipeline for Step-DPO, enabling the creation of a high-quality dataset containing 10K step-wise preference pairs. We also observe that in DPO, self-generated data is more effective than data generated by humans or GPT-4, due to the latter’s out-of-distribution nature. Our findings demonstrate that as few as 10K preference data pairs and fewer than 500 Step-DPO training steps can yield a nearly 3% gain in accuracy on MATH for models with over 70B parameters. Notably, Step-DPO, when applied to Qwen2-72B-Instruct, achieves scores of 70.8% and 94.0% on the test sets of MATH and GSM8K, respectively, surpassing a series of closed-source models, including GPT-4-1106, Claude-3-Opus, and Gemini-1.5-Pro. Our code, data, and models are available at https://github.com/dvlab-research/Step-DPO. 1 Introduction <details> <summary>x1.png Details</summary>  ### Visual Description ## Scatter Plot: Accuracy on the MATH test set ### Overview The image is a scatter plot visualizing the accuracy of various AI models on a MATH test set. The x-axis represents model size (in billions of parameters), and the y-axis represents accuracy (in percentage). Data points are color-coded and shaped to distinguish between models and their configurations (e.g., "Step-DPO" vs. "Instruct"). The plot includes labeled data points, a legend, and axis markers. --- ### Components/Axes - **Title**: "Accuracy on the MATH test set" - **X-axis**: "Model size" (ranging from 7B to 72B, with markers at 7B, 32B, 47B, 57B, 70B, 72B) - **Y-axis**: "Accuracy (%)" (ranging from 42.5% to 72.5%, with markers at 42.5%, 45%, 47.5%, 50%, 52.5%, 55%, 57.5%, 60%, 62.5%, 65%, 67.5%, 70%) - **Legend**: - **Colors/Symbols**: - **Red triangle**: Qwen2-7B-Step-DPO, Qwen2-32B-Step-DPO, Qwen2-57B-A14B-Step-DPO, Qwen2-72B-Step-DPO - **Green circle**: DeepSeekMath-RL, Qwen2-7B-Instruct, Qwen2-72B-Instruct - **Blue circle**: Qwen1.5-32B-SFT, MAmmoTH2-Mixtral-8x7B - **Yellow circle**: Llama3-70B-Instruct - **Purple circle**: MathGenieLM-Mistral - **Orange circle**: Llama3-70B-Step-DPO - **Brown circle**: Qwen2-72B-Instruct† - **Red triangle with black outline**: Qwen2-72B-Instruct-Step-DPO - **Red triangle with white outline**: Qwen2-72B-Step-DPO - **Red triangle with black outline and white outline**: Qwen2-72B-Instruct-Step-DPO (duplicate label?) --- ### Detailed Analysis #### Data Points and Trends 1. **Model Size vs. Accuracy**: - **Largest models** (70B–72B) generally show higher accuracy: - **Qwen2-72B-Instruct-Step-DPO** (72B, 70%) is the highest. - **Qwen2-72B-Instruct** (72B, 69%) follows closely. - **Llama3-70B-Step-DPO** (70B, 60%) and **Llama3-70B-Instruct** (70B, 50%) are lower but still among the top. - **Mid-sized models** (32B–57B): - **Qwen2-32B-Step-DPO** (32B, 58%) and **Qwen2-57B-A14B-Step-DPO** (57B, 57%) show moderate accuracy. - **Qwen1.5-32B-SFT** (32B, 54%) and **MAmmoTH2-Mixtral-8x7B** (47B, 47%) are lower. - **Smallest models** (7B): - **Qwen2-7B-Step-DPO** (7B, 55%) and **DeepSeekMath-Instruct** (7B, 45%) are the lowest. 2. **Step-DPO vs. Instruct**: - **Step-DPO** (red triangles) generally outperforms **Instruct** (green circles) across similar model sizes: - Qwen2-72B-Step-DPO (65%) > Qwen2-72B-Instruct (69%). - Qwen2-57B-A14B-Step-DPO (57%) > Qwen2-57B-A14B-Instruct (49%). - Exceptions: - **Gemini-1.5-Pro** (68%) and **GPT-4-1106** (65%) are high-performing but not labeled with Step-DPO. 3. **Notable Outliers**: - **MathGenieLM-Mistral** (45%) is the lowest accuracy. - **Qwen2-72B-Instruct†** (69%) is the second-highest accuracy but lacks a Step-DPO label. --- ### Key Observations 1. **Model Size Correlation**: - Larger models (70B–72B) dominate the top accuracy range (65%–70%). - Smaller models (7B–32B) cluster below 60%, with Step-DPO configurations showing marginal improvements. 2. **Step-DPO Effectiveness**: - Step-DPO (red triangles) consistently outperforms Instruct (green circles) for the same model size (e.g., Qwen2-72B-Step-DPO vs. Qwen2-72B-Instruct). 3. **Exceptions**: - **Gemini-1.5-Pro** (68%) and **GPT-4-1106** (65%) achieve high accuracy without Step-DPO, suggesting alternative architectures or training methods. 4. **Llama3 Variants**: - Llama3-70B-Step-DPO (60%) outperforms Llama3-70B-Instruct (50%), but both lag behind Qwen2 models. --- ### Interpretation The data demonstrates a **positive correlation between model size and accuracy**, with larger models (70B–72B) achieving the highest performance. The **Step-DPO method** (red triangles) consistently improves accuracy compared to standard Instruct configurations (green circles) for the same model size. However, exceptions like **Gemini-1.5-Pro** and **GPT-4-1106** suggest that architectural innovations or specialized training data can also drive high performance. The **Qwen2-72B-Instruct-Step-DPO** (70%) stands out as the most accurate model, while **MathGenieLM-Mistral** (45%) highlights the limitations of smaller, less optimized models. The plot underscores the importance of both model scale and training methodology in achieving high accuracy on mathematical tasks. </details> Figure 1: Accuracy on the MATH test set across models fine-tuned by Step-DPO and other state-of-the-art models. †: reproduced result using our prompt. <details> <summary>x2.png Details</summary>  ### Visual Description ## Line Chart: Accuracy of judging preferred and undesirable outputs ### Overview The chart compares the accuracy of three model configurations (Qwen2-72B-Step-DPO, Qwen2-72B-DPO, Qwen2-7B-DPO) across training steps, showing how accuracy evolves during training. Three distinct lines with different markers and colors represent each configuration. ### Components/Axes - **X-axis**: Training steps (50-250, increments of 50) - **Y-axis**: Accuracy (%) (64-84, increments of 2) - **Legend**: - Purple triangles: Qwen2-72B-Step-DPO - Orange squares: Qwen2-72B-DPO - Orange diamonds: Qwen2-7B-DPO - **Title**: "Accuracy of judging preferred and undesirable outputs" ### Detailed Analysis 1. **Qwen2-72B-Step-DPO (Purple triangles)**: - Starts at 76% accuracy at 50 steps - Peaks at 82% at 100 steps - Dips to 80% at 150 steps - Rises to 82% at 200 steps - Final accuracy: 82% at 250 steps 2. **Qwen2-72B-DPO (Orange squares)**: - Starts at 67% at 50 steps - Rises to 72% at 100 steps - Stabilizes between 72-73% from 100-250 steps 3. **Qwen2-7B-DPO (Orange diamonds)**: - Starts at 66% at 50 steps - Increases to 71% at 100 steps - Fluctuates between 69-71% through 250 steps ### Key Observations - Step-DPO models consistently outperform DPO-only models - 72B-Step-DPO achieves highest accuracy (82%) - 7B-DPO shows gradual improvement but plateaus below 72% - All models show diminishing returns after 150 steps - 72B-Step-DPO maintains >76% accuracy throughout training ### Interpretation The data demonstrates that the Step-DPO training methodology significantly improves output judgment accuracy compared to standard DPO. The 72B-Step-DPO configuration achieves the highest performance, suggesting that both model size and training methodology are critical factors. The 7B-DPO's lower performance highlights the importance of model capacity in complex judgment tasks. The plateauing trends across all models after 150 steps indicate potential optimization limits in the current training framework. The Step-DPO's incremental approach appears to enable more nuanced learning, particularly evident in the 72B model's sustained high performance. </details> <details> <summary>x3.png Details</summary>  ### Visual Description ## Line Chart: Reward margin between preferred and undesirable outputs ### Overview The chart illustrates the reward margin between preferred and undesirable outputs across three model configurations (Qwen2-7B-Step-DPO, Qwen2-72B-Step-DPO, Qwen2-7B-DPO) as a function of training steps. The y-axis represents the margin (0.1–2.1), while the x-axis shows training steps (0–250). Three distinct data series are plotted with different markers and colors. ### Components/Axes - **X-axis (Training steps)**: Labeled "Training steps" with markers at 0, 50, 100, 150, 200, 250. - **Y-axis (Margin)**: Labeled "Margin" with values 0.1, 0.5, 0.9, 1.3, 1.7, 2.1. - **Legend**: Located in the top-right corner, with three entries: - **Orange triangles**: Qwen2-7B-Step-DPO - **Purple triangles**: Qwen2-72B-Step-DPO - **Orange squares**: Qwen2-7B-DPO ### Detailed Analysis 1. **Qwen2-7B-Step-DPO (Orange triangles)**: - Starts at ~0.1 (training step 0). - Increases steadily to ~2.1 by 250 steps. - Key data points: 0.1 (0 steps), 0.9 (50 steps), 1.3 (100 steps), 1.7 (150 steps), 2.1 (200 steps), 2.1 (250 steps). 2. **Qwen2-72B-Step-DPO (Purple triangles)**: - Starts at ~0.4 (training step 0). - Rises to ~1.7 by 150 steps, then plateaus. - Key data points: 0.4 (0 steps), 1.3 (100 steps), 1.7 (150 steps), 1.7 (200 steps), 1.7 (250 steps). 3. **Qwen2-7B-DPO (Orange squares)**: - Starts at ~0.1 (training step 0). - Rises to ~0.6–0.7 and remains flat. - Key data points: 0.1 (0 steps), 0.6 (50 steps), 0.7 (100 steps), 0.6 (150 steps), 0.7 (200 steps), 0.7 (250 steps). ### Key Observations - **Qwen2-7B-Step-DPO** achieves the highest margin, surpassing all other configurations. - **Qwen2-72B-Step-DPO** outperforms the baseline Qwen2-72B-DPO but lags behind the 7B-Step-DPO. - **Qwen2-7B-DPO** shows minimal improvement over training steps, remaining near 0.6–0.7. ### Interpretation The data suggests that the **Step-DPO** training method significantly improves the reward margin compared to standard DPO. The 7B-Step-DPO configuration demonstrates the most substantial gains, indicating that smaller models may benefit more from this approach. The 72B-Step-DPO, while better than its baseline, does not match the 7B-Step-DPO's performance, possibly due to model complexity or other architectural factors. The flat trend of Qwen2-7B-DPO highlights the importance of the Step-DPO technique for optimizing reward margins. </details> Figure 2: Left: Accuracy of judging preferred or undesirable outputs on the validation set during training. Right: Reward margins between preferred and undesirable outputs on the validation set during training. More details about these experiments are given in the appendix. Mathematical reasoning is recognized as a critical long-chain reasoning ability in Large Language Models (LLMs). This task is particularly challenging due to the often extensive chain of thought required, which can include numerous reasoning steps. Any error in these steps can lead to an incorrect final answer. Numerous studies (Yu et al., 2023; Luo et al., 2023; Yue et al., 2023; Liu & Yao, 2024; Lu et al., 2024; Li et al., 2024; Shao et al., 2024; Xin et al., 2024; Yue et al., 2024; Tang et al., 2024) have proposed various data augmentation techniques during the supervised fine-tuning (SFT) stage to enhance alignment. However, models in the SFT process are prone to hallucinations, resulting in saturated performance. A potential reason for this, as highlighted in Hong et al. (2024), is that as the probability of preferred outputs increases, so does the probability of undesirable ones. This phenomenon makes the model more likely to make errors in long-chain reasoning. Therefore, it is essential to develop methods to suppress the likelihood of undesirable outputs. Recently, Direct Preference Optimization (DPO) (Rafailov et al., 2024) has been proposed for alignment using pair-wise preference data and is popular due to its simplicity. Despite its effectiveness in chat benchmarks (Tunstall et al., 2023; Zheng et al., 2024), DPO offers minimal benefits for long-chain mathematical reasoning. As shown in Fig. 2 (left), models using vanilla DPO perform poorly in distinguishing between preferred and undesirable outputs, failing to identify errors in rejected answers. Additionally, Fig. 2 (right) shows that the reward margin (i.e., the gap between the rewards of preferred and undesirable outputs) is limited for models using vanilla DPO and plateaus with further training. These findings indicate that models fine-tuned with vanilla DPO cannot pinpoint detailed errors in incorrect answers, hindering the improvement of reasoning abilities. In this work, we introduce Step-DPO, where each intermediate reasoning step is treated as the basic unit for preference optimization. As illustrated in Fig. 3, unlike vanilla DPO, which only considers preference optimization between complete answers (i.e., $p(y_{win}|x)$ and $p(y_{lose}|x)$ ), Step-DPO examines the step-by-step answer (i.e., $y=s_{1},...,s_{n}$ ) and specifically targets the first erroneous reasoning step. Step-DPO aims to select a correct reasoning step and reject an incorrect one, given a math problem and several initial correct reasoning steps (i.e., maximize $p(s_{win}|x;s_{1},s_{2},...,s_{k-1})$ and minimize $p(s_{lose}|x;s_{1},s_{2},...,s_{k-1})$ ). This transition allows the model to easily locate erroneous tokens for effective optimization, significantly enhancing long-chain reasoning. Moreover, we present an effective and economical pipeline to collect pair-wise preference data, resulting in a high-quality dataset for Step-DPO. This dataset contains approximately 10K samples, each consisting of: 1) a mathematical problem, 2) prior reasoning steps, 3) the chosen step, and 4) the rejected step. Our three-step pipeline for dataset construction includes: 1) Error collection, 2) Step localization, and 3) Rectification. Notably, the chosen reasoning step is generated by the model itself, as we find that in-distribution data (i.e., self-generated data) is more effective than out-of-distribution data (e.g., data written by humans or GPT-4) for Step-DPO, as shown in Table 4. With this curated dataset, mathematical reasoning performance can be significantly boosted with only hundreds of training steps, as demonstrated in Fig. 6. For instance, fine-tuning Qwen-72B-Instruct with Step-DPO results in a model achieving 70.8% accuracy on MATH and 94.0% on GSM8K, surpassing a series of closed-source models, including GPT-4-1106, Claude-3-Opus, and Gemini-1.5-Pro. <details> <summary>x4.png Details</summary>  ### Visual Description ## Diagram: DPO vs Step-DPO Preference Optimization ### Overview The image compares two optimization frameworks for language models: **DPO** (Direct Preference Optimization) and **Step-DPO** (Step-wise Preference Optimization). Both use preference data to train language models but differ in their approach to processing preferences. --- ### Components/Axes #### DPO Section (Left) - **Input**: - Two speech bubbles labeled `y_w` (with a crown icon) and `y_t`, connected by `>` (indicating preference: `y_w > y_t`). - Text: "preference data" below the speech bubbles. - **Output**: - Arrow labeled "maximum likelihood" pointing to a language model diagram. - Language model represented by interconnected nodes: - **Colors**: Yellow (positive), Red (negative), Blue (neutral). - **Placement**: Right-aligned, with arrows indicating influence between nodes. #### Step-DPO Section (Right) - **Input**: - Sequential states labeled `S₁` → `S₂` (top path) and `S₁` → `S_{k-1}` → `S_{lose}` (bottom path). - Green checkmark (`✓`) on `S_win` (success state), red cross (`✗`) on `S_lose` (failure state). - Text: "step-wise preference data" below the states. - **Output**: - Arrow labeled "maximum likelihood" pointing to the same language model diagram as in DPO. - Language model nodes retain the same color scheme (yellow, red, blue). --- ### Detailed Analysis #### DPO Workflow 1. **Preference Data**: A single preference pair (`y_w > y_t`) is used as input. 2. **Optimization**: The preference data is optimized for "maximum likelihood," directly influencing the language model. 3. **Language Model**: - Nodes are colored yellow (likely high-confidence/positive), red (low-confidence/negative), and blue (neutral/unknown). - Arrows suggest iterative refinement of node states based on preferences. #### Step-DPO Workflow 1. **Step-wise Preferences**: - States `S₁` and `S₂` represent sequential steps in a process. - `S_win` (green) and `S_lose` (red) indicate terminal outcomes of the sequence. 2. **Optimization**: - The stepwise preferences are optimized for "maximum likelihood," similar to DPO. - The language model integrates these stepwise outcomes, refining node states accordingly. --- ### Key Observations 1. **Shared Language Model**: Both frameworks use identical node colors (yellow, red, blue) and connectivity, suggesting a common underlying model architecture. 2. **Preference Handling**: - DPO uses a single preference pair (`y_w > y_t`). - Step-DPO uses a sequence of states (`S₁ → S₂` or `S₁ → S_{k-1} → S_{lose}`) to model preferences. 3. **Outcome Indicators**: - `S_win` (green checkmark) and `S_lose` (red cross) explicitly mark success/failure states in Step-DPO. 4. **Flow Direction**: - DPO follows a linear flow from preference data to the language model. - Step-DPO introduces branching paths and intermediate states before reaching the language model. --- ### Interpretation 1. **DPO vs. Step-DPO**: - **DPO** simplifies preference optimization by directly using pairwise comparisons (`y_w > y_t`), likely for efficiency. - **Step-DPO** models preferences as sequential processes (e.g., `S₁ → S₂`), which may better capture complex, multi-step decision-making but requires more computational steps. 2. **Language Model Role**: - The shared language model acts as a unifying component, integrating preferences (direct or stepwise) to refine its internal states (nodes). - Color-coded nodes (yellow/red/blue) likely represent confidence levels or activation states, adjusted during training. 3. **Practical Implications**: - DPO may be preferable for scenarios with limited preference data. - Step-DPO could outperform in tasks requiring nuanced, multi-step reasoning (e.g., dialogue systems, code generation). --- ### Notes - No numerical values or quantitative data are present; the diagram focuses on architectural and procedural differences. - The crown icon on `y_w` may symbolize a "gold standard" or high-priority preference in DPO. </details> Figure 3: Comparison between DPO and Step-DPO. 2 Related Works 2.1 Mathematical Reasoning Large Language Models (LLMs) have exhibited substantial reasoning capabilities, primarily due to their auto-regressive nature, which allows them to predict the next token based on contextual information. However, these models still struggle with long-chain reasoning tasks, particularly in mathematical contexts. Several prior studies (Yao et al., 2024; Chen et al., 2024; Yoran et al., 2023; Li et al., 2023; Tong et al., 2024; Fu et al., 2022; Zhou et al., 2022) have attempted to enhance the Chain-of-Thought (CoT) inference framework (Wei et al., 2022) to address this issue. While these efforts have led to significant improvements in certain tasks, they have not fully mitigated common hallucinations and have limited generalizability across all reasoning tasks. Another research direction (Yu et al., 2023; Luo et al., 2023; Yue et al., 2023; Liu & Yao, 2024; Lu et al., 2024; Xu et al., 2024; Li et al., 2024; Shao et al., 2024; Xin et al., 2024; Zhou et al., 2024; Liu et al., 2023; Ying et al., 2024; Yue et al., 2024; Tang et al., 2024; Mitra et al., 2024; Yuan et al., 2023) focuses on various data augmentation techniques, such as rephrasing, extension, and evolution, for supervised fine-tuning (SFT). These methods have significantly enhanced the reasoning abilities of LLMs, but their performance plateaus once the data reaches a certain volume. Additionally, methods like those proposed by Wang et al. (2023a); Liao et al. (2024); Toshniwal et al. (2024); Gou et al. (2023) employ external tools, such as Python, to substantially reduce calculation errors. Other approaches (Azerbayev et al., 2023; Shao et al., 2024; Lin et al., 2024; Ying et al., 2024; Wang et al., 2023c) involve continued pre-training on extensive, high-quality math-related datasets, which markedly improve mathematical reasoning capabilities. Recent studies (Xu et al., 2024; Ying et al., 2024) have explored reinforcement learning to mitigate hallucinations in mathematical reasoning. Works like Lightman et al. (2023); Shao et al. (2024); Wang et al. (2023b) emphasize the importance of step-by-step verification in reinforcement learning for mathematical problems. However, these methods still rely on the quality of the reward model and require the complex training pipelines of RLHF. Building on this line of research, we propose Step-DPO, a simpler, more effective, and more efficient method. 2.2 Reinforcement Learning from Human Feedback Supervised fine-tuning (SFT) can align models with human preferences. However, as the probability of preferred outputs increases, so does the likelihood of undesirable ones, leading to hallucinations. To generate more reliable outputs, Reinforcement Learning from Human Feedback (RLHF) (Christiano et al., 2017; Ouyang et al., 2022) has been introduced for LLM alignment. This approach involves training a reward model with comparison data and then using this reward model to optimize the policy model. The final performance heavily depends on the quality of the reward model, and the training pipeline is quite complex. To simplify this process, Direct Preference Optimization (DPO) (Rafailov et al., 2024) was proposed, which directly uses pair-wise preference data for model optimization. This transition significantly streamlines the training pipeline. While DPO has proven effective in chat benchmarks, it offers only marginal benefits for mathematical reasoning. Inheriting the principles of DPO, Step-DPO is specifically designed for long-chain reasoning and has shown significant performance improvements in solving math word problems. 3 Step-DPO In this section, we elaborate on the proposed Step-DPO. First, we present step-wise formulation in Sec. 3.1, a novel approach designed to enhance long-chain reasoning abilities by building on DPO. Next, in Sec. 3.2, we illustrate a pipeline for constructing the step-wise preference dataset for Step-DPO. Both components are essential for achieving the desired performance improvements. 3.1 Step-wise Formulation Preliminary. Reinforcement Learning from Human Feedback (RLHF) (Christiano et al., 2017) is an effective approach for enhancing the robustness, factuality, and safety of LLMs (Ouyang et al., 2022). RLHF consists of two training phases: 1) reward model training, and 2) policy model training. However, the final performance of RLHF is highly sensitive to various hyperparameters in both phases, necessitating meticulous tuning. To avoid this complex training pipeline, Rafailov et al. (2024) proposed Direct Preference Optimization (DPO), which directly uses pair-wise preference data to optimize the policy model with an equivalent optimization objective. Specifically, given an input prompt $x$ , and a preference data pair $(y_{win},y_{lose})$ , DPO aims to maximize the probability of the preferred output $y_{win}$ and minimize that of the undesirable output $y_{lose}$ . The optimization objective is formulated as: $$ \displaystyle\begin{aligned} \mathcal{L}_{DPO}(\theta)=-\mathbb{E}_{(x,y_{win}% ,y_{lose})\sim D}[\log\sigma(\beta\log\frac{\pi_{\theta}(y_{win}|x)}{\pi_{ref}% (y_{win}|x)}-\beta\log\frac{\pi_{\theta}(y_{lose}|x)}{\pi_{ref}(y_{lose}|x)})]% ,\end{aligned} \tag{1} $$ where $D$ is the pair-wise preference dataset, $\sigma$ is the sigmoid function, $\pi_{\theta}(·|x)$ is the policy model to be optimized, $\pi_{ref}(·|x)$ is the reference model kept unchanged during training, and the hyperparameter $\beta$ controls the distance from the reference model. Our Solution. While DPO has proven effective in chat benchmarks, it brings only marginal improvements for long-chain reasoning tasks such as mathematical problems, as shown in Fig. 2 and Table 3. This limitation arises because most undesirable answers in these tasks do not contain errors initially; the first error often appears midway through the reasoning process. Rejecting an entire undesirable answer in DPO may also discard preceding correct reasoning steps, introducing significant noise and negatively impacting training. Analogous to how teachers correct students by pinpointing specific errors rather than dismissing entire answers, our proposed Step-DPO provides more detailed supervision by identifying the specific erroneous reasoning step. This granular focus allows the model to swiftly locate, rectify, and avoid erroneous steps. Specifically, the answer $y$ can be decomposed into a sequence of reasoning steps $y=s_{1},...,s_{n}$ , where $s_{i}$ is the $i$ -th reasoning step. As illustrated in Fig. 3, given a prompt $x$ and a series of initial correct reasoning steps $s_{1\sim k-1}=s_{1},...,s_{k-1}$ , Step-DPO aims to maximize the probability of the correct next reasoning step $s_{win}$ and minimize the probability of the incorrect one $s_{lose}$ . This objective can be formulated as: $$ \displaystyle\begin{aligned} \mathcal{L}(\theta)=-\mathbb{E}_{(x,s_{1\sim k-1}% ,s_{win},s_{lose})\sim D}[\log\sigma(\beta\log\frac{\pi_{\theta}(s_{win}|x;s_{% 1\sim k-1})}{\pi_{ref}(s_{win}|x;s_{1\sim k-1})}-\beta\log\frac{\pi_{\theta}(s% _{lose}|x;s_{1\sim k-1})}{\pi_{ref}(s_{lose}|x;s_{1\sim k-1})})].\end{aligned} \tag{2} $$ 3.2 In-distribution Data Construction <details> <summary>x5.png Details</summary>  ### Visual Description ## Flowchart: Error Detection and Correction Process in Model Responses ### Overview The image depicts a three-stage flowchart illustrating the process of identifying, localizing, and rectifying errors in model-generated responses to math problems. It combines textual instructions, neural network diagrams, and symbolic verification elements to demonstrate error handling in computational systems. ### Components/Axes 1. **Error Collection Section** (Top) - **Math Problem Preparation**: Text box with instruction to find the least positive integer x satisfying (x + 4609) ≡ 2104 (mod 12) - **Model Completion**: Neural network diagram (y ~ π_ref(y|x)) - **Wrong Answer Collection**: Icon of magnifying glass over X symbol 2. **Step Localization Section** (Left) - **Step-by-Step Verification**: Checklist with mixed green checkmarks and red X - **Critical Error**: Step 4 marked with red X (incorrect calculation: -2505 + 12·210 = 30) 3. **Rectification Section** (Right) - **Correct Answer Collection**: Text box showing corrected response (answer = 3) - **Model Completion**: Neural network diagram (y_cont ~ π_ref(y|x; s₁₋ₖ₋₁)) - **Flow Arrows**: Orange bidirectional arrows connecting error detection to correction ### Detailed Analysis 1. **Error Collection** - Initial problem: Find x where (x + 4609) ≡ 2104 (mod 12) - Incorrect model response: x = 6 (Step 6) - Error visualization: Red X over incorrect answer 2. **Step Localization** - Step 1-3: Correct calculations (congruence transformations) - Step 4: Critical error in arithmetic (incorrect remainder handling) - Step 5-6: Correct conclusion despite earlier error 3. **Rectification** - Corrected response: x = 3 - Revised model completion with additional context (s₁₋ₖ₋₁) - Orange arrows show correction pathway from error to solution ### Key Observations - **Error Propagation**: Single arithmetic error in Step 4 invalidates subsequent steps despite correct congruence logic - **Verification Gaps**: Step 5-6 shows model's ability to recover from intermediate errors - **Model Adaptation**: Rectification phase introduces contextual parameters (s₁₋ₖ₋₁) for improved completion - **Visual Hierarchy**: Color coding (green/red) and iconography (checkmarks/X) create immediate error detection cues ### Interpretation This flowchart demonstrates a systematic approach to error handling in computational problem-solving: 1. **Error Collection** establishes baseline problem parameters and initial (incorrect) model output 2. **Step Localization** uses structured verification to pinpoint error origins (Step 4) 3. **Rectification** shows model adaptation through contextual parameter inclusion (s₁₋ₖ₋₁) The process highlights the importance of: - Step-by-step verification in complex calculations - Model architecture modifications for error recovery - Visual feedback mechanisms (color coding, icons) for error detection - Contextual parameterization to improve model robustness The correction from x=6 to x=3 demonstrates how intermediate errors can be resolved through systematic verification and model adaptation, emphasizing the value of multi-stage error handling in computational systems. </details> Figure 4: Data construction pipeline for Step-DPO. <details> <summary>x6.png Details</summary>  ### Visual Description ## Code Snippet: Modular Arithmetic Problem Solving ### Overview The image displays a code snippet containing a mathematical problem-solving process for finding the least positive integer `x` that satisfies the congruence `x + 4609 ≡ 2104 (mod 12)`. The text includes reasoning steps, calculations, and a comparison between correct and rejected approaches. ### Components/Axes - **Sections**: - `prompt`: Problem statement - `initial_reasoning_steps`: Initial incorrect approach - `chosen_step`: Correct adjustment method - `rejected_step`: Incorrect alternative approach ### Detailed Analysis #### Prompt - Problem: "Determine the least positive integer `x` for which the sum of `x` and 4609 gives a remainder of 2104 when divided by 12." #### Initial Reasoning Steps 1. **Step 1**: `x + 4609 ≡ 2104 (mod 12)` 2. **Step 2**: Rewritten as `x ≡ 2104 - 4609 (mod 12)` 3. **Step 3**: Calculation: `2104 - 4609 = -2505` 4. **Step 4**: Not explicitly shown but implied as the result of the subtraction. #### Chosen Step - **Goal**: Find the equivalent positive integer for `-2505 (mod 12)`. - **Process**: - Division: `-2505 ÷ 12 = -208` remainder `-9` - Adjustment: Add 12 to `-9` to get a positive remainder: `-9 + 12 = 3` - Final congruence: `-2505 ≡ 3 (mod 12)` #### Rejected Step - **Approach**: Adding 12 repeatedly to `-2505` until positive: - Calculation: `-2505 + 12 × 210 = 30` - **Error**: This results in `30`, which is not the minimal positive solution. ### Key Observations 1. The correct approach uses modular arithmetic properties to find the smallest positive remainder. 2. The rejected step incorrectly assumes adding 12 multiple times yields the minimal solution, but `30` is not the least positive integer satisfying the original congruence. ### Interpretation The problem demonstrates modular arithmetic adjustments for negative remainders. The correct method isolates the remainder and adjusts it to the positive range by adding the modulus (12). The rejected step fails because it does not recognize that the minimal solution requires only a single adjustment (`-9 + 12 = 3`), not repeated additions. The final answer should be `x = 3`, as `-2505 ≡ 3 (mod 12)` is the smallest positive equivalent. </details> Figure 5: An example of preference data sample for Step-DPO. According to the optimization target of Step-DPO, we need to create a corresponding high-quality pair-wise preference dataset. Each data sample should comprise four entries: 1) prompt $x$ ; 2) initial reasoning steps $s_{1\sim k-1}=s_{1},...,s_{k-1}$ ; 3) preferred reasoning step $s_{win}$ ; 4) undesirable reasoning step $s_{lose}$ , as shown in Fig. 5. To obtain a high-quality dataset, we propose a data construction pipeline illustrated in Fig. 4, which includes the following three steps. Error collection. First, we collect a set $D_{0}=\{(x,\hat{y})\}$ of mathematical problems $x$ with ground-truth answers $\hat{y}$ . Each mathematical problem $x$ is then used as a prompt to infer answers using the initial model $\pi_{ref}$ . Before inference, we add the step-wise Chain-of-Thought (CoT) prefix for prompting, i.e., "Let’s think step by step. Step 1:". This ensures that the model’s inference results are structured into multiple reasoning steps, with each step explicitly starting with "Step i:". Upon completion of inference, we obtain the model answers $y$ for each mathematical problem $x$ . We then select instances where the final answer $y$ differs from the ground truth $\hat{y}$ , resulting in a dataset of erroneous inference results, denoted as $D_{1}=\{(x,\hat{y},y)|x∈ D_{0}\}$ . Step localization. Given that each erroneous inference result is explicitly presented as a sequence of reasoning steps $y=s_{1},s_{2},...,s_{n}$ , we proceed to verify the correctness of each reasoning step until we find the first error and record its step number $k$ . This process can be done manually or using GPT-4. We select $s_{k}$ as the erroneous reasoning step $s_{lose}$ , resulting in a dataset that contains the erroneous steps, denoted as $D_{2}=\{(x,\hat{y},s_{1\sim k-1},s_{lose})|x∈ D_{1}\}$ . Rectification. To obtain the corresponding correct reasoning step for each sample in $D_{2}$ , we need to sample multiple outputs $y_{cont}$ by inferring the model $\pi_{ref}$ with the prompt $x$ and the preceding correct reasoning steps $s_{1\sim k-1}$ . This process is formulated as: $$ \displaystyle\begin{aligned} y_{cont}\sim\pi_{ref}(y|x;s_{1\sim k-1}).\end{aligned} \tag{3} $$ We retain those outputs where the final answer matches the ground truth. Among the remaining outputs, we select the first reasoning step in $y_{cont}$ as $s_{win}$ , resulting in the final dataset $D=\{(x,s_{1\sim k-1},s_{lose},s_{win})|x∈ D_{2}\}$ . An example of a resulting data sample is shown in Fig. 5. Notably, some cases may have correct final answers but erroneous intermediate reasoning steps. Therefore, we may need to further filter out samples where $s_{win}$ is incorrect, which can be done manually or by GPT-4. We omit this process in the notations for simplicity, and more details are provided in the appendix. It is important to note that the data pipeline is user-friendly. In this data pipeline, humans or GPT-4 are only required to locate errors and rank answers, and they do not need to write answers or rectifications by themselves. We also note that the use of in-distribution data is crucial. When selecting $s_{win}$ , we use outputs generated by the model $\pi_{ref}$ rather than answers rectified by humans or GPT-4. Since human or GPT-4 rectified answers $s_{win}^{ood}$ are out-of-distribution (OOD) regarding the model $\pi_{ref}$ , the log-probability of outputting $s_{win}^{ood}$ (i.e., $\log\pi_{ref}(s_{win}^{ood}|x)$ ) is significantly lower than that of an in-distribution (ID) output $\log\pi_{ref}(s_{win}^{id}|x)$ . Moreover, it is challenging for the policy model $\pi_{\theta}$ to learn to increase the probability of $s_{win}^{ood}$ due to gradient decay issues (detailed in the appendix). Consequently, adopting self-generated in-distribution data as the preferred answer proves to be a more effective way of aligning with human preferences. 4 Experiments In this section, we first introduce the experimental setup in Sec. 4.1. Then, we present the main results in Sec. 4.2, which include an exhaustive performance comparison. Moreover, we conduct an extensive ablation study in Sec. 4.3. Finally, a few demonstrations are shown in Sec. 4.4 to further understand Step-DPO. 4.1 Experimental Setup Network Architecture. Our experiments are based on various base models, including the Qwen2 and Qwen1.5 series (Bai et al., 2023), Meta-Llama-3-70B (Touvron et al., 2023), and deepseek-math-7b-base (Shao et al., 2024). Table 1: Math reasoning performance comparison on MATH and GSM8K across various models. general: general-purpose model. open: open-source. - $\;\;{}^{\dagger}$ Supervised fine-tuned models with our 299K SFT data based on the open-source base model. - $\;\;{}^{\ddagger}$ Reproduced using our prompt Table 2: Math reasoning performance comparison on compitition-level math problems, i.e., AIME 2024 and Odyssey-MATH. Note that the training data for Step-DPO is the same as before. | Model | size | open | AIME | Odyssey-MATH (%) | | --- | --- | --- | --- | --- | | Gemini-1.5-Pro (Reid et al., 2024) | - | ✗ | 2 / 30 | 45.0 | | Claude-3-Opus | - | ✗ | 2 / 30 | 40.6 | | GPT-4-1106 (Achiam et al., 2023) | - | ✗ | 1 / 30 | 49.1 | | GPT-4-Turbo-0409 (Achiam et al., 2023) | - | ✗ | 3 / 30 | 46.8 | | GPT-4o-0513 | - | ✗ | 2 / 30 | 53.2 | | DeepSeek-Coder-V2-Lite-Instruct (Zhu et al., 2024) | 16B | ✓ | 0 / 30 | 44.4 | | Llama-3-70B-Instruct (Touvron et al., 2023) | 70B | ✓ | 1 / 30 | 27.9 | | DeepSeek-Coder-V2-Instruct (Zhu et al., 2024) | 236B | ✓ | 4 / 30 | 53.7 | | Qwen2-72B-SFT † | 72B | ✓ | 1 / 30 | 44.2 | | Qwen2-72B-SFT + Step-DPO | 72B | ✓ | 3 / 30 | 47.0 (+2.8) | | Qwen2-72B-Instruct (Bai et al., 2023) | 72B | ✓ | 5 / 30 | 47.0 | | Qwen2-72B-Instruct + Step-DPO | 72B | ✓ | 4 / 30 | 50.1 (+3.1) | - $\;\;{}^{\dagger}$ Supervised fine-tuned models with our 299K SFT data based on the open-source base model. Datasets. In supervised fine-tuning (SFT), we use augmented mathematical problems from MetaMath (Yu et al., 2023) and MMIQC (Liu & Yao, 2024) to infer step-by-step responses with DeepSeekMath, as the SFT data used in DeepSeekMath (Shao et al., 2024) is not publicly available. After filtering out responses with erroneous final answers, we obtain 374K SFT data. Of these, 299K are used for SFT, and the remainder is used for further Step-DPO training. In the Step-DPO phase, alongside the remaining SFT data, we also incorporate a subset of AQuA (Ling et al., 2017). These datasets are processed as described in Sec. 3.2, resulting in 10K pair-wise preference data for Step-DPO. For evaluation, we use the widely adopted MATH (Hendrycks et al., 2021) and GSM8K (Cobbe et al., 2021) datasets. Accuracy in these datasets serves as the evaluation metric. The MATH test set contains 5000 mathematical problems spanning 5 difficulty levels and 7 subjects, including algebra, counting and probability, geometry, intermediate algebra, number theory, prealgebra, and precalculus. The GSM8K test set includes 1319 mathematical problems, each with a step-by-step solution and a ground-truth answer. The problems in GSM8K are generally easier than those in MATH. Besides, we also use completition-level problems in American Invitational Mathematics Examination (AIME) (MAA, 2024) and Odyssey-MATH (Netmind.AI, 2024) to evaluate the math reasoning capabilities in solving hard problems. Implementation Details. First, we use the 299K SFT data for supervised fine-tuning on the base models, obtaining the SFT models. We train 7B models for 3 epochs and models larger than 30B for 2 epochs. The global batch size is set to 256, and the learning rate is set to 5e-6. We use the AdamW optimizer with a linear decay learning rate scheduler, setting the warmup ratio to 0.03. DeepSpeed ZeRO3 with CPU offload is used to reduce GPU memory usage during training. Next, we perform Step-DPO based on the SFT models. For Step-DPO, we train 7B models for 8 epochs and models larger than 30B for 4 epochs. The global batch size is set to 128, and the learning rate is set to 5e-7. The hyperparameter $\beta$ is set to 0.5 for the 72B model and 0.4 for others. We use the AdamW optimizer and a cosine learning rate scheduler, with the warmup ratio set to 0.1. 4.2 Results Table 3: Performance comparison between DPO and Step-DPO. We use only 5K data for training in this ablation study. | Model MATH (%) Model | Qwen2-7B-SFT 54.8 Qwen2-72B-SFT | Qwen2-7B-SFT + DPO (5K) 55.0 Qwen2-72B-SFT + DPO (5K) | Qwen2-7B-SFT + Step-DPO (5K) 55.8 Qwen2-72B-SFT + Step-DPO (5K) | | --- | --- | --- | --- | | MATH (%) | 61.7 | 62.5 | 64.1 | Table 4: Performance comparison between out-of-distribution and in-distribution data. OOD: out-of-distribution data. ID: in-distribution data. | MATH (%) | 54.8 | 55.1 | 55.8 | | --- | --- | --- | --- | Applying on open-source instruct models. Table 1 presents a comprehensive comparison of various models, encompassing both open-source and closed-source models. Notably, Step-DPO can be directly integrated into open-source instruction models, such as DeepSeekMath-RL and Qwen2-72B-Instruct, leading to significant performance enhancements even after their prior RLHF training phase. This indicates that Step-DPO complements RLHF effectively. Specifically, when applied to Qwen2-72B-Instruct, Step-DPO achieves scores of 70.8% and 94.0% on the MATH and GSM8K test sets, respectively, surpassing a series of closed-source models, including GPT-4-1106, Claude-3-Opus, and Gemini-1.5-Pro. Applying on SFT models. To further substantiate the efficacy of Step-DPO, we applied it to SFT models. Initially, we performed supervised fine-tuning on the 299K SFT dataset mentioned in Sec. 4.1, resulting in models such as DeepSeekMath-Base-SFT, Qwen2-7B-SFT, Qwen1.5-32B-SFT, Llama3-70B-SFT, and Qwen2-72B-SFT. Step-DPO proved highly effective, yielding significant improvements across various model sizes. Particularly, for models exceeding 70B parameters (i.e., Llama-3-70B-SFT and Qwen-2-72B-SFT), Step-DPO achieved approximately a 3% performance boost on the MATH test set. Interestingly, larger models exhibited greater performance gains from Step-DPO. We hypothesize that larger models have untapped potential that Step-DPO can exploit. If the performance ceiling is not reached through supervised fine-tuning (SFT), Step-DPO can help models approach their optimal performance. Results on math competition problems. To further illustrate the superiority of Step-DPO in mathematical reasoning, we evaluated the models on competition-level math problems, specifically AIME 2024 and Odyssey-MATH, as shown in Fig. 2. Despite the increased difficulty of these problems compared to MATH and GSM8K, Step-DPO significantly enhanced performance. On Odyssey-MATH, Step-DPO applied to Qwen2-72B-Instruct achieved 50.1% accuracy, narrowing the performance gap with GPT-4o. Notably, the models used the same Step-DPO training data for these competition-level problems as for problems of normal difficulty, highlighting Step-DPO’s robust generalization capability. 4.3 Ablation Study To validate the effectiveness of Step-DPO and its data construction process, we conducted an extensive ablation study. DPO vs. Step-DPO. As discussed in Sec. 3.1, models utilizing vanilla DPO struggle to accurately identify errors in incorrect answers, providing only marginal benefits to mathematical reasoning performance. To verify this, we compared vanilla DPO and Step-DPO in terms of both accuracy in judging preferred versus undesirable outputs (left side of Fig. 2) and the reward margin between them (right side of Fig. 2). We also reported the final mathematical reasoning performance on the MATH test set in Table 3. The results indicated that the benefits of DPO are limited and significantly less than those of Step-DPO. In this experiment, we used only 5K Step-DPO training data and 3K for evaluation in Fig. 2. Out-of-Distribution vs. In-Distribution Data. The importance of in-distribution data was emphasized in Sec. 3.2. To illustrate its effectiveness, we compared out-of-distribution and in-distribution data in Table 4. Out-of-distribution data was generated by using GPT-4 to correct erroneous reasoning steps in incorrect answers, whereas in-distribution data was generated through the pipeline described in Sec. 3.2. The results in Table 4 underscore the critical role of in-distribution data in enhancing performance. 4.4 Demonstrations <details> <summary>x7.png Details</summary>  ### Visual Description ## Text-Based Problem Solution Analysis ### Overview The image contains two side-by-side solutions to a mathematical problem involving integer values of *t* that satisfy the condition: **2 < √t < 3.5**. Both solutions follow similar steps but differ in their final answers (7 vs. 8 integer values) due to nuanced interpretations of inequality boundaries. --- ### Components/Axes - **Problem Statement**: "The square root of *t* is greater than 2 and less than 3.5. How many integer values of *t* satisfy this condition?" - **Solution Sections**: 1. **Qwen2-72B-Instruct** (Left Column) 2. **Qwen2-72B-Instruct + Step-DPO** (Right Column) - **Key Elements**: - Inequalities (e.g., **2 < √t < 3.5**) - Squared inequalities (e.g., **4 < t < 12.25**) - Integer value ranges (e.g., **5, 6, 7, 8, 9, 10, 11, 12**) - Color-coded annotations (red for exclusion, green for inclusion) --- ### Detailed Analysis #### Qwen2-72B-Instruct (Left Column) 1. **Step 1**: Translate condition to **2 < √t < 3.5**. 2. **Step 2**: Square all parts → **4 < t < 12.25**. 3. **Step 3**: Integer values between 4 and 12.25: **5, 6, 7, 8, 9, 10, 11, 12**. 4. **Step 4**: Exclude **12** (red text) because **t must be strictly less than 12.25**. 5. **Step 5**: Final count: **7 integers** (**5–11**). #### Qwen2-72B-Instruct + Step-DPO (Right Column) 1. **Step 1**: Translate condition to **2 < √t < 3.5**. 2. **Step 2**: Square all parts → **4 < t < 12.25**. 3. **Step 3**: Integer values between 4 and 12.25: **5, 6, 7, 8, 9, 10, 11, 12**. 4. **Step 4**: Include **12** (green text) because **12 < 12.25** satisfies the strict inequality. 5. **Step 5**: Final count: **8 integers** (**5–12**). --- ### Key Observations - **Discrepancy in Final Answers**: - Left solution excludes **12** (red text), resulting in **7 integers**. - Right solution includes **12** (green text), resulting in **8 integers**. - **Critical Difference**: Interpretation of **t < 12.25**. - Left: **12** is excluded because **12 ≮ 12.25** (strict inequality). - Right: **12** is included because **12 < 12.25** (strict inequality holds). --- ### Interpretation The divergence between the two solutions hinges on whether **12** is included in the solution set. - **Mathematical Justification**: - **12² = 144**, and **√144 = 12**, which satisfies **2 < 12 < 3.5**? - No! **12 > 3.5**, so **12** does **not** satisfy the original condition. - **Correction**: The squared inequality **4 < t < 12.25** implies **t** must be **less than 12.25**, but **√t < 3.5** requires **t < 12.25** (since **3.5² = 12.25**). - **Final Valid Range**: **4 < t < 12.25** → **t = 5, 6, 7, 8, 9, 10, 11** (7 integers). - **Error in Right Solution**: The inclusion of **12** is incorrect because **√12 ≈ 3.464**, which **does** satisfy **2 < √12 < 3.5**. - **Correction**: **12** should be included, making the total **8 integers** (**5–12**). --- ### Conclusion The correct answer is **8 integer values** (**5–12**), as **12** satisfies **2 < √12 < 3.5**. The left solution contains an error in excluding **12**, while the right solution correctly includes it. </details> Figure 6: An example of comparison between Qwen2-72B-Instruct and Qwen2-72B-Instruct-Step-DPO. As shown in Fig. 6, we demonstrate an example of comparison between Qwen2-72B-Instruct and Qwen2-72B-Instruct-Step-DPO. It turns out that Step-DPO does well in correcting minor mistakes in previous models. More comparisons are provided in the appendix. 5 Conclusion In this work, we proposed a simple, effective, and data-efficient method called Step-DPO. Unlike DPO, which compares preferences between holistic answers, Step-DPO uses a single reasoning step as the fundamental unit for preference comparison. This transition enables fine-grained process supervision for LLMs, facilitating the quick localization of errors within incorrect answers. Additionally, we introduced a data construction pipeline for Step-DPO, creating a dataset with 10K preference data pairs. 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