Cross-Entropy Games and Frost Training

Policy gradient algorithms for language models optimize the same objective $J(θ) = \mathbb{E}*{τ\sim p*θ(τ)}[R(τ)]$, which has exactly two factors: the trajectory probability $p_θ(τ)$ 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 $p_θ(τ)$, 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.

Post-training large language models with reinforcement learning is bottlenecked by the reward signal. Existing approaches require either ground-truth verifiable rewards, restricting training to domains with automatic correctness checks (e.g., mathematics, code execution), or human preference labels, which are expensive to collect and prone to reward hacking. Recent label-free methods replace ground-truth verifiers with self-referential signals like majority voting or token entropy over a model's own outputs, but risk reinforcing a model's own errors. In this work we propose Cross-Model Entropy (CME), the mean log-likelihood of a generator's response under a separate verifier model, as a label-free reward signal for RL post-training. CME is continuous, training-free, and grounded in the principle that responses a verifier finds unsurprising are likely correct or high quality. Because the verifier is independent of the generator, the signal cannot be gamed through self-consistency. We integrate CME into GRPO with no other changes to the training loop, extending label-free RL to open-ended instruction following -- a regime where self-referential signals are inapplicable or poorly suited. On open-ended instruction following (UltraFeedback prompts, evaluated on AlpacaEval 2.0), CME rewards beat the untrained base in head-to-head LLM-as-Judge comparisons across four model families (Qwen, Llama, Gemma, OLMo) and three training regimes (pretrained, SFT, and instruction-tuned), with tie-adjusted win rates ranging from 52.5% to 71.4%. Code will be released upon publication.

The standard LLM training pipeline applies reinforcement learning (RL) only after pre-training and supervised fine-tuning (SFT). We question this status quo by training a LLM from scratch and applying RL, SFT, and SFT followed by RL directly to intermediate pre-training checkpoints. We find that RL is effective very early, and often matches the full SFT$\to$RL pipeline early as well. Through experiments on harder problems, we find that targeted pre-training data composition is a strong lever for RL effectiveness, even more so than model scale. Beyond reasoning accuracy, applying RL directly to base checkpoints expands the model's distribution; the sharpening effect reported in recent work arises only when RL follows SFT. The general capabilities of the model remain essentially unchanged by RL, while they degrade following SFT. Finally, we merge RL and SFT objectives by parallel averaging, which outperforms across all other training methods discussed, across metrics, while preserving general capabilities. Together, these results suggest that LLM training might benefit from an expanded use of RL.