# f-GRPO and Beyond: Divergence-Based Reinforcement Learning Algorithms for General LLM Alignment

arXiv: https://arxiv.org/abs/2602.05946 · HTML: https://mlsanity.com/paper/2602.05946/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 21 of its 34 references that are in the map.
- First posted on arXiv 5 Feb 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-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, 21 of 21)

- [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)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Universal and Transferable Adversarial Attacks on Aligned Language Models](https://mlsanity.com/paper/2307.15043/) (2023)
- [Constitutional AI: Harmlessness from AI Feedback](https://mlsanity.com/paper/2212.08073/) (2022)
- [Qwen2 Technical Report](https://mlsanity.com/paper/2407.10671/) (2024)
- [MMLU-Pro: A More Robust and Challenging Multi-Task Language Understanding Benchmark](https://mlsanity.com/paper/2406.01574/) (2024)
- [Instruction-Following Evaluation for Large Language Models](https://mlsanity.com/paper/2311.07911/) (2023)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [ToxiGen: A Large-Scale Machine-Generated Dataset for Adversarial and Implicit Hate Speech Detection](https://mlsanity.com/paper/2203.09509/) (2022)
- [KTO: Model Alignment as Prospect Theoretic Optimization](https://mlsanity.com/paper/2402.01306/) (2024)
- [SALAD-Bench: A Hierarchical and Comprehensive Safety Benchmark for Large Language Models](https://mlsanity.com/paper/2402.05044/) (2024)
- [MuSR: Testing the Limits of Chain-of-thought with Multistep Soft Reasoning](https://mlsanity.com/paper/2310.16049/) (2023)
- [On Prompt-Driven Safeguarding for Large Language Models](https://mlsanity.com/paper/2401.18018/) (2024)
- [VerlTool: Towards Holistic Agentic Reinforcement Learning with Tool Use](https://mlsanity.com/paper/2509.01055/) (2025)
- [Reinforcement Learning for Reasoning in Small LLMs: What Works and What Doesn't](https://mlsanity.com/paper/2503.16219/) (2025)
- [Towards Understanding Jailbreak Attacks in LLMs: A Representation Space Analysis](https://mlsanity.com/paper/2406.10794/) (2024)
- [What is the Alignment Objective of GRPO?](https://mlsanity.com/paper/2502.18548/) (2025)
- [LLM Safety Alignment is Divergence Estimation in Disguise](https://mlsanity.com/paper/2502.00657/) (2025)

## Abstract (arXiv)

Recent work shows that preference alignment objectives can be interpreted as divergence estimators between aligned (preferred) & unaligned (less-preferred) distributions, yielding a principled recipe for designing alignment losses. However, this view has so far been limited to preference-based supervision. We extend it to general LLM alignment, including reinforcement learning with verifiable rewards (RLVR), where alignment feedback is given only as scalar rewards. We introduce f-Group Relative Policy Optimization (f-GRPO), a class of on-policy RL objectives, and f-Hybrid Alignment Loss (f-HAL), which combines on-policy reward optimization with off-policy preference supervision. We show that these objectives estimate f-divergences between reward-aligned & reward-unaligned distributions induced by above- & below-average reward responses, and prove expected reward improvement after alignment. Empirically, f-GRPO improves over GRPO on math-reasoning RLVR tasks, while hybrid f-HAL mitigates reward hacking in on-policy safety alignment when verifiable rewards are unavailable and learned reward models must be used.