# Probability-Consistent Preference Optimization for Enhanced LLM Reasoning

arXiv: https://arxiv.org/abs/2505.23540 · HTML: https://mlsanity.com/paper/2505.23540/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 32 of its 50 references that are in the map.
- First posted on arXiv 29 May 2025.

## 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 32)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [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)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [Constitutional AI: Harmlessness from AI Feedback](https://mlsanity.com/paper/2212.08073/) (2022)
- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [PaLM 2 Technical Report](https://mlsanity.com/paper/2305.10403/) (2023)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Self-Rewarding Language Models](https://mlsanity.com/paper/2401.10020/) (2024)
- [Large Language Models: A Survey](https://mlsanity.com/paper/2402.06196/) (2024)
- [Large Language Models for Mathematical Reasoning: Progresses and Challenges](https://mlsanity.com/paper/2402.00157/) (2024)
- [Iterative Reasoning Preference Optimization](https://mlsanity.com/paper/2404.19733/) (2024)
- [Inverse Scaling: When Bigger Isn't Better](https://mlsanity.com/paper/2306.09479/) (2023)
- [Token-level Direct Preference Optimization](https://mlsanity.com/paper/2404.11999/) (2024)
- [Rho-1: Not All Tokens Are What You Need](https://mlsanity.com/paper/2404.07965/) (2024)
- [Some things are more CRINGE than others: Iterative Preference Optimization with the Pairwise Cringe Loss](https://mlsanity.com/paper/2312.16682/) (2023)
- [Critical Tokens Matter: Token-Level Contrastive Estimation Enhances LLM's Reasoning Capability](https://mlsanity.com/paper/2411.19943/) (2024)
- [Consistency of a Recurrent Language Model With Respect to Incomplete Decoding](https://mlsanity.com/paper/2002.02492/) (2020)
- [Self-Consistency Preference Optimization](https://mlsanity.com/paper/2411.04109/) (2024)
- [RLCD: Reinforcement Learning from Contrastive Distillation for Language Model Alignment](https://mlsanity.com/paper/2307.12950/) (2023)
- [Towards a Theoretical Understanding of Synthetic Data in LLM Post-Training: A Reverse-Bottleneck Perspective](https://mlsanity.com/paper/2410.01720/) (2024)

## Built on by (most cited first, 2 of 2)

- [Confidence-Aware Alignment Makes Reasoning LLMs More Reliable](https://mlsanity.com/paper/2605.07353/) (2026)
- [MultiTalk: Scaling Full-Duplex Speech Models to Long, Multi-Party, Bilingual Conversation](https://mlsanity.com/paper/2609.36903/) (2026)

## Abstract (arXiv)

Recent advances in preference optimization have demonstrated significant potential for improving mathematical reasoning capabilities in large language models (LLMs). While current approaches leverage high-quality pairwise preference data through outcome-based criteria like answer correctness or consistency, they fundamentally neglect the internal logical coherence of responses. To overcome this, we propose Probability-Consistent Preference Optimization (PCPO), a novel framework that establishes dual quantitative metrics for preference selection: (1) surface-level answer correctness and (2) intrinsic token-level probability consistency across responses. Extensive experiments show that our PCPO consistently outperforms existing outcome-only criterion approaches across a diverse range of LLMs and benchmarks. Our code is publicly available at https://github.com/YunqiaoYang/PCPO.