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

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.