# Beyond Two-Stage Training: Cooperative SFT and RL for LLM Reasoning

arXiv: https://arxiv.org/abs/2509.06948 · HTML: https://mlsanity.com/paper/2509.06948/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 41 of its 53 references that are in the map.
- First posted on arXiv 8 Sep 2025.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 30 of 41)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [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)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [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)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [Learning to Reason under Off-Policy Guidance](https://mlsanity.com/paper/2504.14945/) (2025)
- [LIMR: Less is More for RL Scaling](https://mlsanity.com/paper/2502.11886/) (2025)
- [A Near-Optimal Algorithm for Stochastic Bilevel Optimization via Double-Momentum](https://mlsanity.com/paper/2102.07367/) (2021)
- [Data-efficient Fine-tuning for LLM-based Recommendation](https://mlsanity.com/paper/2401.17197/) (2024)
- [Learning What Reinforcement Learning Can't: Interleaved Online Fine-Tuning for Hardest Questions](https://mlsanity.com/paper/2506.07527/) (2025)
- [On-Policy RL Meets Off-Policy Experts: Harmonizing Supervised Fine-Tuning and Reinforcement Learning via Dynamic Weighting](https://mlsanity.com/paper/2508.11408/) (2025)
- [On Penalty-based Bilevel Gradient Descent Method](https://mlsanity.com/paper/2302.05185/) (2023)
- [SRFT: A Single-Stage Method with Supervised and Reinforcement Fine-Tuning for Reasoning](https://mlsanity.com/paper/2506.19767/) (2025)
- [Nemotron-Research-Tool-N1: Exploring Tool-Using Language Models with Reinforced Reasoning](https://mlsanity.com/paper/2505.00024/) (2025)
- [SEAL: Safety-enhanced Aligned LLM Fine-tuning via Bilevel Data Selection](https://mlsanity.com/paper/2410.07471/) (2024)
- [On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation](https://mlsanity.com/paper/2309.01753/) (2023)
- [Beyond Factuality: A Comprehensive Evaluation of Large Language Models as Knowledge Generators](https://mlsanity.com/paper/2310.07289/) (2023)
- [Principled Penalty-based Methods for Bilevel Reinforcement Learning and RLHF](https://mlsanity.com/paper/2402.06886/) (2024)
- [Can Language Models Perform Robust Reasoning in Chain-of-thought Prompting with Noisy Rationales?](https://mlsanity.com/paper/2410.23856/) (2024)

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

- [A Survey of Reinforcement Learning for Large Reasoning Models](https://mlsanity.com/paper/2509.08827/) (2025)
- [Asymmetric On-Policy Distillation: Bridging Exploitation and Imitation at the Token Level](https://mlsanity.com/paper/2605.06387/) (2026)
- [RL Squeezes, SFT Expands: A Comparative Study of Reasoning LLMs](https://mlsanity.com/paper/2509.21128/) (2025)
- [Lightning OPD: Efficient Post-Training for Large Reasoning Models with Offline On-Policy Distillation](https://mlsanity.com/paper/2604.13010/) (2026)
- [Dynamic-TreeRPO: Breaking the Independent Trajectory Bottleneck with Structured Sampling](https://mlsanity.com/paper/2509.23352/) (2025)
- [Advancing Multi-agent Traffic Simulation via R1-Style Reinforcement Fine-Tuning](https://mlsanity.com/paper/2509.23993/) (2025)
- [EEPO: Exploration-Enhanced Policy Optimization via Sample-Then-Forget](https://mlsanity.com/paper/2510.05837/) (2025)
- [Data-regularized Reinforcement Learning for Diffusion Models at Scale](https://mlsanity.com/paper/2512.04332/) (2025)
- [Adaptive Defense against Harmful Fine-Tuning for Large Language Models via Bayesian Data Scheduler](https://mlsanity.com/paper/2510.27172/) (2025)
- [On the Non-decoupling of Supervised Fine-tuning and Reinforcement Learning in Post-training](https://mlsanity.com/paper/2601.07389/) (2026)
- [Towards a Mechanistic Understanding of Large Reasoning Models: A Survey of Training, Inference, and Failures](https://mlsanity.com/paper/2601.19928/) (2026)
- [Lightning OPD 2.0: Mitigating Style Bias in Cross-Teacher On-Policy Distillation for Large Reasoning Models](https://mlsanity.com/paper/2607.28449/) (2026)
- [A Survey on Unlearning in Large Language Models](https://mlsanity.com/paper/2510.25117/) (2025)
- [Learning Agentic Policy from Action Guidance](https://mlsanity.com/paper/2605.12004/) (2026)
- [Toward Honest Language Models for Deductive Reasoning](https://mlsanity.com/paper/2511.09222/) (2025)
- [MapTab: A Diagnostic Benchmark for Long-Horizon Multi-Criteria Multimodal Reasoning on Heterogeneous Topological Graphs](https://mlsanity.com/paper/2602.18600/) (2026)
- [HART: High-Resolution Annotation-Free Reasoning Technique through a Closed-loop Framework](https://mlsanity.com/paper/2602.23615/) (2026)
- [Model-based Offline RL via Robust Value-Aware Model Learning with Implicitly Differentiable Adaptive Weighting](https://mlsanity.com/paper/2603.08118/) (2026)
- [WavAlign: Enhancing Intelligence and Expressiveness in Spoken Dialogue Models via Adaptive Hybrid Post-Training](https://mlsanity.com/paper/2604.14932/) (2026)
- [PACT: Privileged Trace Co-Training for Multi-Turn Tool-Use Agents](https://mlsanity.com/paper/2606.16215/) (2026)
- [RLoop: An Self-Improving Framework for Reinforcement Learning with Iterative Policy Initialization](https://mlsanity.com/paper/2511.04285/) (2025)
- [OraPO: Oracle-educated Reinforcement Learning for Data-efficient and Factual Radiology Report Generation](https://mlsanity.com/paper/2509.18600/) (2025)
- [Bridging SFT and RL: Dynamic Policy Optimization for Robust Reasoning](https://mlsanity.com/paper/2604.08926/) (2026)
- [Enhancing LLM Metacognition via Cognitive Pairwise Training](https://mlsanity.com/paper/2606.00869/) (2026)
- [One Tool Is Enough: Reinforcement Learning for Repository-Level LLM Agents](https://mlsanity.com/paper/2512.20957/) (2025)
- [Enhancing Reinforcement Learning Fine-Tuning with an Online Refiner](https://mlsanity.com/paper/2603.18088/) (2026)
- [AIPO: Learning to Reason from Active Interaction](https://mlsanity.com/paper/2605.08401/) (2026)
- [LoRA Scaffolded Policy Optimization (LSPO): A Sampling-Time Low-Rank Scaffold for Recovering Reinforcement-Learning Gradient on Zero-Reward Cliff Prompts](https://mlsanity.com/paper/2607.27787/) (2026)
- [Knowing When to Quit: Diagnosing and Training LLMs to Abort Futile Reasoning](https://mlsanity.com/paper/2607.29211/) (2026)
- [HuatuoGPT-3: RL-Only Domain Adaptation from Base Models](https://mlsanity.com/paper/2610.05966/) (2026)

## Abstract (arXiv)

Supervised fine-tuning (SFT) and reinforcement learning with verifiable rewards (RLVR) are two widely used post-training paradigms for improving the reasoning ability of large language models (LLMs). Recent methods attempt to integrate SFT and RLVR in a single stage by reweighting or scheduling their objectives. However, such coupling can be counterproductive because supervised updates are not uniformly beneficial for reward optimization. To address this, we propose BRIDGE, a scalable framework in which SFT learns to supervise RL by selectively transferring knowledge that improves reward optimization. Specifically, BRIDGE alternates two updates at each meta-training step: a base-model update that fuses the SFT and RL gradients, and an update to a lightweight low-rank adapter (LoRA) that coordinates the two objectives by maximizing a cooperative-gain signal, defined as the reward of joint SFT-RL training over an RL-only baseline. Across five mathematical reasoning benchmarks, BRIDGE consistently outperforms two-stage cold start, naive mixing, and representative single-stage integration baselines, yielding over three points average absolute improvement and more stable training dynamics. We further show that BRIDGE extends to logical reasoning and generalizes out-of-distribution to code and science without additional training, while staying robust under noisy rewards.