# A First-Principles Derivation of LLM Policy Optimization: From Expected Reward to GRPO and Its Structural Extensions

arXiv: https://arxiv.org/abs/2606.16733 · HTML: https://mlsanity.com/paper/2606.16733/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 69 of its 130 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Placed in

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

## Builds on (most cited first, 30 of 69)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [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)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [GLM-4.5V and GLM-4.1V-Thinking: Towards Versatile Multimodal Reasoning with Scalable Reinforcement Learning](https://mlsanity.com/paper/2507.01006/) (2025)
- [Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs](https://mlsanity.com/paper/2506.14245/) (2025)
- [VAPO: Efficient and Reliable Reinforcement Learning for Advanced Reasoning Tasks](https://mlsanity.com/paper/2504.05118/) (2025)
- [REINFORCE++: Stabilizing Critic-Free Policy Optimization with Global Advantage Normalization](https://mlsanity.com/paper/2501.03262/) (2025)
- [ZeroSearch: Incentivize the Search Capability of LLMs without Searching](https://mlsanity.com/paper/2505.04588/) (2025)
- [ThinkPrune: Pruning Long Chain-of-Thought of LLMs via Reinforcement Learning](https://mlsanity.com/paper/2504.01296/) (2025)
- [A Survey of On-Policy Distillation for Large Language Models](https://mlsanity.com/paper/2604.00626/) (2026)
- [Self-Distilled RLVR](https://mlsanity.com/paper/2604.03128/) (2026)
- [SWEET-RL: Training Multi-Turn LLM Agents on Collaborative Reasoning Tasks](https://mlsanity.com/paper/2503.15478/) (2025)
- [Agentic Reasoning and Tool Integration for LLMs via Reinforcement Learning](https://mlsanity.com/paper/2505.01441/) (2025)
- [Reinforcement Learning for Long-Horizon Interactive LLM Agents](https://mlsanity.com/paper/2502.01600/) (2025)
- [Soft Adaptive Policy Optimization](https://mlsanity.com/paper/2511.20347/) (2025)
- [Reinforcement Learning for LLM Post-Training: A Survey](https://mlsanity.com/paper/2407.16216/) (2024)
- [Scaling Reasoning Efficiently via Relaxed On-Policy Distillation](https://mlsanity.com/paper/2603.11137/) (2026)
- [ARPO:End-to-End Policy Optimization for GUI Agents with Experience Replay](https://mlsanity.com/paper/2505.16282/) (2025)
- [Reinforcement Learning Enhanced LLMs: A Survey](https://mlsanity.com/paper/2412.10400/) (2024)
- [Agent World Model: Infinity Synthetic Environments for Agentic Reinforcement Learning](https://mlsanity.com/paper/2602.10090/) (2026)
- [AgentRL: Scaling Agentic Reinforcement Learning with a Multi-Turn, Multi-Task Framework](https://mlsanity.com/paper/2510.04206/) (2025)
- [Blending Supervised and Reinforcement Fine-Tuning with Prefix Sampling](https://mlsanity.com/paper/2507.01679/) (2025)
- [Simulating Environments with Reasoning Models for Agent Training](https://mlsanity.com/paper/2511.01824/) (2025)
- [Self-Distilled Agentic Reinforcement Learning](https://mlsanity.com/paper/2605.15155/) (2026)

## Built on by (most cited first, 1 of 1)

- [GRPO, Dr. GRPO, and DAPO Are Three Operations on One Number: The Group-Standard-Deviation Identity](https://mlsanity.com/paper/2607.00152/) (2026)

## Abstract (arXiv)

Policy gradient algorithms for language models optimize the same objective J(θ) = mathbbE*τ ~ p*θ(τ)[R(τ)], which has exactly two factors: the trajectory probability pθ(τ) and the reward R(τ). Every method from REINFORCE to PPO to GRPO and their descendants modifies one or both factors to address a specific failure in the preceding formulation. Existing surveys organize these methods by domain or chronology, which obscures the rationale behind each design choice and the precise location of its intervention within the gradient estimator. This survey revisits the landscape of LLM policy optimization from J(θ) on first principles and uses the trajectory side, induced by pθ(τ), and the reward side, induced by R(τ), as the two axes along which methods are located. It covers the path from REINFORCE and PPO to GRPO, as well as post-GRPO variants, Agentic RL, and GRPO-OPD. The resulting framework is unified, diagnostic, and extensible: it analyzes methods from a shared objective, identifies which side each method modifies and why, and applies the same trajectory and reward axes across these settings. Across these settings, the framework also exposes compound failures that no single-side fix resolves and that therefore require joint design of the trajectory side and the reward side. The boundary cases and coupled failures identified by this map mark where existing solutions run out and provide a principled starting point for designing the next generation of LLM policy optimization algorithms.