# Optimizing Chain-of-Thought Reasoners via Gradient Variance Minimization in Rejection Sampling and RL

arXiv: https://arxiv.org/abs/2505.02391 · HTML: https://mlsanity.com/paper/2505.02391/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Placed in Prompting and chain-of-thought reasoning, one of 17 directions in LLM reasoning.
- Builds on 29 of its 55 references that are in the map.
- First posted on arXiv 5 May 2025.

## Placed in

- LLM reasoning / [Prompting and chain-of-thought reasoning](https://mlsanity.com/lr/prompting-and-chain-of-thought-reasoning/)

## Builds on (most cited first, 29 of 29)

- [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)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [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)
- [A General Theoretical Paradigm to Understand Learning from Human Preferences](https://mlsanity.com/paper/2310.12036/) (2023)
- [Reinforced Self-Training (ReST) for Language Modeling](https://mlsanity.com/paper/2308.08998/) (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)
- [Beyond Human Data: Scaling Self-Training for Problem-Solving with Language Models](https://mlsanity.com/paper/2312.06585/) (2023)
- [RLHF Workflow: From Reward Modeling to Online RLHF](https://mlsanity.com/paper/2405.07863/) (2024)
- [REINFORCE++: Stabilizing Critic-Free Policy Optimization with Global Advantage Normalization](https://mlsanity.com/paper/2501.03262/) (2025)
- [Generalized Preference Optimization: A Unified Approach to Offline Alignment](https://mlsanity.com/paper/2402.05749/) (2024)
- [A Minimalist Approach to LLM Reasoning: from Rejection Sampling to Reinforce](https://mlsanity.com/paper/2504.11343/) (2025)
- [DPO Meets PPO: Reinforced Token Optimization for RLHF](https://mlsanity.com/paper/2404.18922/) (2024)
- [Back to Basics: Revisiting REINFORCE Style Optimization for Learning from Human Feedback in LLMs](https://mlsanity.com/paper/2402.14740/) (2024)
- [DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving](https://mlsanity.com/paper/2407.13690/) (2024)
- [Efficient Reinforcement Finetuning via Adaptive Curriculum Learning](https://mlsanity.com/paper/2504.05520/) (2025)
- [Self-rewarding correction for mathematical reasoning](https://mlsanity.com/paper/2502.19613/) (2025)
- [Unveiling the Statistical Foundations of Chain-of-Thought Prompting Methods](https://mlsanity.com/paper/2408.14511/) (2024)
- [Accelerating Minibatch Stochastic Gradient Descent using Stratified Sampling](https://mlsanity.com/paper/1405.3080/) (2014)
- [BRiTE: Bootstrapping Reinforced Thinking Process to Enhance Language Model Reasoning](https://mlsanity.com/paper/2501.18858/) (2025)

## Built on by (most cited first, 24 of 24)

- [Perception, Reason, Think, and Plan: A Survey on Large Multimodal Reasoning Models](https://mlsanity.com/paper/2505.04921/) (2025)
- [Improving Sampling Efficiency in RLVR through Adaptive Rollout and Response Reuse](https://mlsanity.com/paper/2509.25808/) (2025)
- [Reinforce-Ada: An Adaptive Sampling Framework under Non-linear RL Objectives](https://mlsanity.com/paper/2510.04996/) (2025)
- [Knapsack RL: Unlocking Exploration of LLMs via Optimizing Budget Allocation](https://mlsanity.com/paper/2509.25849/) (2025)
- [Self-Hinting Language Models Enhance Reinforcement Learning](https://mlsanity.com/paper/2602.03143/) (2026)
- [CurES: From Gradient Analysis to Efficient Curriculum Learning for Reasoning LLMs](https://mlsanity.com/paper/2510.01037/) (2025)
- [XRPO: Pushing the limits of GRPO with Targeted Exploration and Exploitation](https://mlsanity.com/paper/2510.06672/) (2025)
- [Multi-Turn On-Policy Distillation with Prefix Replay](https://mlsanity.com/paper/2607.04763/) (2026)
- [How to Allocate, How to Learn? Dynamic Rollout Allocation and Advantage Modulation for Policy Optimization](https://mlsanity.com/paper/2602.19208/) (2026)
- [Future-KL Regularized GRPO: Process-Level Credit Assignment from f-Divergence Regularization](https://mlsanity.com/paper/2601.10201/) (2026)
- [CoBA-RL: Capability-Oriented Budget Allocation for Reinforcement Learning in LLMs](https://mlsanity.com/paper/2602.03048/) (2026)
- [Learning to Hint for Reinforcement Learning](https://mlsanity.com/paper/2604.00698/) (2026)
- [VI-CuRL: Stabilizing Verifier-Independent RL Reasoning via Confidence-Guided Variance Reduction](https://mlsanity.com/paper/2602.12579/) (2026)
- [Longer Context, Deeper Thinking: Uncovering the Role of Long-Context Ability in Reasoning](https://mlsanity.com/paper/2505.17315/) (2025)
- [SIGMA: Refining Large Language Model Reasoning via Sibling-Guided Monte Carlo Augmentation](https://mlsanity.com/paper/2506.06470/) (2025)
- [Supervised Fine-Tuning versus Reinforcement Learning: A Study of Post-Training Methods for Large Language Models](https://mlsanity.com/paper/2603.13985/) (2026)
- [Rollout-Level Advantage-Prioritized Experience Replay for GRPO](https://mlsanity.com/paper/2606.04560/) (2026)
- [LEEPS: Latent-Guided Explore-Exploit Prompt Sampling for Efficient RLVR in Large Language Models](https://mlsanity.com/paper/2607.28077/) (2026)
- [Constraints-of-Thought: A Framework for Constrained Reasoning in Language-Model-Guided Search](https://mlsanity.com/paper/2510.08992/) (2025)
- [Variance-Aware Baselines and Adaptive Learning Rates for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2511.23310/) (2025)
- [Learning to Solve Hard Problems in RL for LLMs by Never Giving Up](https://mlsanity.com/paper/2609.13443/) (2026)
- [Know Thy Reasoner: Not All Language Models Explore Alike](https://mlsanity.com/paper/2604.10827/) (2026)
- [SERA: Scale-Equalized Rollout Allocation for Maximum Likelihood Reinforcement Learning](https://mlsanity.com/paper/2609.36552/) (2026)
- [Variance-Optimal State Resampling for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2610.15888/) (2026)

## Abstract (arXiv)

Chain-of-thought (CoT) reasoning in large language models (LLMs) can be formalized as a latent variable problem, where the model needs to generate intermediate reasoning steps. While prior approaches such as iterative reward-ranked fine-tuning (RAFT) have relied on such formulations, they typically apply uniform inference budgets across prompts, which fails to account for variability in difficulty and convergence behavior. This work identifies the main bottleneck in CoT training as inefficient stochastic gradient estimation due to static sampling strategies. We propose GVM-RAFT, a prompt-specific Dynamic Sample Allocation Strategy designed to minimize stochastic gradient variance under a computational budget constraint. The method dynamically allocates computational resources by monitoring prompt acceptance rates and stochastic gradient norms, ensuring that the resulting gradient variance is minimized. Our theoretical analysis shows that the proposed dynamic sampling strategy leads to accelerated convergence guarantees under suitable conditions. Experiments on mathematical reasoning show that GVM-RAFT achieves a 2-4x speedup and considerable accuracy improvements over vanilla RAFT. The proposed dynamic sampling strategy is general and can be incorporated into other reinforcement learning algorithms, such as GRPO, leading to similar improvements in convergence and test accuracy. Our code is available at https://github.com/RLHFlow/GVM.