# Step-Controlled DPO: Leveraging Stepwise Error for Enhanced Mathematical Reasoning

arXiv: https://arxiv.org/abs/2407.00782 · HTML: https://mlsanity.com/paper/2407.00782/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 38 of its 56 references that are in the map.
- First posted on arXiv 30 Jun 2024.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)
- LLM alignment / RLHF / preference optimisation / [Process reward models for reasoning](https://mlsanity.com/alignment/process-reward-models-for-reasoning/)

## Builds on (most cited first, 30 of 38)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Constitutional AI: Harmlessness from AI Feedback](https://mlsanity.com/paper/2212.08073/) (2022)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [MathVista: Evaluating Mathematical Reasoning of Foundation Models in Visual Contexts](https://mlsanity.com/paper/2310.02255/) (2023)
- [Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Tasks](https://mlsanity.com/paper/2211.12588/) (2022)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [Self-Rewarding Language Models](https://mlsanity.com/paper/2401.10020/) (2024)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [Measuring Multimodal Mathematical Reasoning with MATH-Vision Dataset](https://mlsanity.com/paper/2402.14804/) (2024)
- [Zephyr: Direct Distillation of LM Alignment](https://mlsanity.com/paper/2310.16944/) (2023)
- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://mlsanity.com/paper/2309.05653/) (2023)
- [Scaling Relationship on Learning Mathematical Reasoning with Large Language Models](https://mlsanity.com/paper/2308.01825/) (2023)
- [SLiC-HF: Sequence Likelihood Calibration with Human Feedback](https://mlsanity.com/paper/2305.10425/) (2023)
- [InternLM2 Technical Report](https://mlsanity.com/paper/2403.17297/) (2024)
- [Llemma: An Open Language Model For Mathematics](https://mlsanity.com/paper/2310.10631/) (2023)
- [Statistical Rejection Sampling Improves Preference Optimization](https://mlsanity.com/paper/2309.06657/) (2023)
- [ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving](https://mlsanity.com/paper/2309.17452/) (2023)
- [KTO: Model Alignment as Prospect Theoretic Optimization](https://mlsanity.com/paper/2402.01306/) (2024)
- [Is Reinforcement Learning (Not) for Natural Language Processing: Benchmarks, Baselines, and Building Blocks for Natural Language Policy Optimization](https://mlsanity.com/paper/2210.01241/) (2022)
- [From r to Q^*: Your Language Model is Secretly a Q-Function](https://mlsanity.com/paper/2404.12358/) (2024)
- [Smaug: Fixing Failure Modes of Preference Optimisation with DPO-Positive](https://mlsanity.com/paper/2402.13228/) (2024)
- [Solving Challenging Math Word Problems Using GPT-4 Code Interpreter with Code-based Self-Verification](https://mlsanity.com/paper/2308.07921/) (2023)
- [MathCoder: Seamless Code Integration in LLMs for Enhanced Mathematical Reasoning](https://mlsanity.com/paper/2310.03731/) (2023)

## Built on by (most cited first, 30 of 35)

- [Agent Q: Advanced Reasoning and Learning for Autonomous AI Agents](https://mlsanity.com/paper/2408.07199/) (2024)
- [Improve Vision Language Model Chain-of-thought Reasoning](https://mlsanity.com/paper/2410.16198/) (2024)
- [Can We Generate Images with CoT? Let's Verify and Reinforce Image Generation Step by Step](https://mlsanity.com/paper/2501.13926/) (2025)
- [Process Reward Model with Q-Value Rankings](https://mlsanity.com/paper/2410.11287/) (2024)
- [Self-rewarding correction for mathematical reasoning](https://mlsanity.com/paper/2502.19613/) (2025)
- [Building Math Agents with Multi-Turn Iterative Preference Learning](https://mlsanity.com/paper/2409.02392/) (2024)
- [Evaluating Step-by-step Reasoning Traces: A Survey](https://mlsanity.com/paper/2502.12289/) (2025)
- [MathCoder2: Better Math Reasoning from Continued Pretraining on Model-translated Mathematical Code](https://mlsanity.com/paper/2410.08196/) (2024)
- [A Survey of Direct Preference Optimization](https://mlsanity.com/paper/2503.11701/) (2025)
- [MathFusion: Enhancing Mathematical Problem-solving of LLM through Instruction Fusion](https://mlsanity.com/paper/2503.16212/) (2025)
- [A Comprehensive Survey of Direct Preference Optimization: Datasets, Theories, Variants, and Applications](https://mlsanity.com/paper/2410.15595/) (2024)
- [LEMMA: Learning from Errors for MatheMatical Advancement in LLMs](https://mlsanity.com/paper/2503.17439/) (2025)
- [Iterative Tool Usage Exploration for Multimodal Agents via Step-wise Preference Tuning](https://mlsanity.com/paper/2504.21561/) (2025)
- [Improving Multi-Step Reasoning Abilities of Large Language Models with Direct Advantage Policy Optimization](https://mlsanity.com/paper/2412.18279/) (2024)
- [Full-Step-DPO: Self-Supervised Preference Optimization with Step-wise Rewards for Mathematical Reasoning](https://mlsanity.com/paper/2502.14356/) (2025)
- [Autoregressive Multi-trait Essay Scoring via Reinforcement Learning with Scoring-aware Multiple Rewards](https://mlsanity.com/paper/2409.17472/) (2024)
- [Debate, Reflect, and Distill: Multi-Agent Feedback with Tree-Structured Preference Optimization for Efficient Language Model Enhancement](https://mlsanity.com/paper/2506.03541/) (2025)
- [Two Heads Are Better Than One: Dual-Model Verbal Reflection at Inference-Time](https://mlsanity.com/paper/2502.19230/) (2025)
- [Improving Rationality in the Reasoning Process of Language Models through Self-playing Game](https://mlsanity.com/paper/2506.22920/) (2025)
- [PrivAct: Internalizing Contextual Privacy Preservation via Multi-Agent Preference Training](https://mlsanity.com/paper/2602.13840/) (2026)
- [Self-Evolved Preference Optimization for Enhancing Mathematical Reasoning in Small Language Models](https://mlsanity.com/paper/2503.04813/) (2025)
- [Alignment with Fill-In-the-Middle for Enhancing Code Generation](https://mlsanity.com/paper/2508.19532/) (2025)
- [Putting on the Thinking Hats: A Survey on Chain of Thought Fine-tuning from the Perspective of Human Reasoning Mechanism](https://mlsanity.com/paper/2510.13170/) (2025)
- [Mars-PO: Multi-Agent Reasoning System Preference Optimization](https://mlsanity.com/paper/2411.19039/) (2024)
- [Multi-Step Alignment as Markov Games: An Optimistic Online Gradient Descent Approach with Convergence Guarantees](https://mlsanity.com/paper/2502.12678/) (2025)
- [Eeyore: Realistic Depression Simulation via Supervised and Preference Optimization](https://mlsanity.com/paper/2503.00018/) (2025)
- [Beyond Rejection Sampling: Trajectory Fusion for Scaling Mathematical Reasoning](https://mlsanity.com/paper/2602.04391/) (2026)
- [Local Look-Ahead Guidance via Verifier-in-the-Loop for Automated Theorem Proving](https://mlsanity.com/paper/2503.09730/) (2025)
- [MARGE: Improving Math Reasoning for LLMs with Guided Exploration](https://mlsanity.com/paper/2505.12500/) (2025)
- [PIPA: Preference Alignment as Prior-Informed Statistical Estimation](https://mlsanity.com/paper/2502.05773/) (2025)

## Abstract (arXiv)

Direct Preference Optimization (DPO) has proven effective at improving the performance of large language models (LLMs) on downstream tasks such as reasoning and alignment. In this work, we propose Step-Controlled DPO (SCDPO), a method for automatically providing stepwise error supervision by creating negative samples of mathematical reasoning rationales that start making errors at a specified step. By applying these samples in DPO training, SCDPO can better align the model to understand reasoning errors and output accurate reasoning steps. We apply SCDPO to both code-integrated and chain-of-thought solutions, empirically showing that it consistently improves the performance compared to naive DPO on three different SFT models, including one existing SFT model and two models we finetuned. Qualitative analysis of the credit assignment of SCDPO and DPO demonstrates the effectiveness of SCDPO at identifying errors in mathematical solutions. We then apply SCDPO to an InternLM2-20B model, resulting in a 20B model that achieves high scores of 88.5% on GSM8K and 58.1% on MATH, rivaling all other open-source LLMs, showing the great potential of our method.