Group Relative Policy Optimization (GRPO) eliminates the learned critic in PPO by using the mean reward of grouped rollouts as a baseline. We provide a rigorous derivation of GRPO from first principles of the policy gradient theorem, revealing a fundamental credit assignment failure: under output-only reward, every token in a rollout receives identical advantage, collapsing token-level credit to a single scalar. We prove this induces gradient sparsity that intensifies over training, and demonstrate empirically via SVD analysis of GRPO gradients on Nemotron-4B/GSM8K that the gradient matrix has effective rank ≈ 2 regardless of group size R∈{2,4,8}. We formalize this as an intrinsic rank-2 structure arising from the zero-sum constraint on advantages and derive conditions under which GRPO's baseline is optimal. Our results characterize when GRPO's simplicity is theoretically justified and identify the credit assignment bottleneck as the key limitation for multi-step reasoning.
Group Relative Policy Optimization (GRPO) is a standard algorithm for reinforcement learning from verifiable rewards, but its group-mean-centered advantage can fail under binary rewards. The failure mode is gradient starvation: when every response in a group is correct or every response is wrong, the centered advantage is exactly zero and the policy receives no learning signal. We prove that the true degeneracy rate always exceeds the i.i.d. Bernoulli prediction by Jensen's inequality, and observe a 0.69 degeneracy rate at group size four in logged Qwen3.5-9B GSM8K training. We then show that the fixed-reference Sign advantage, A=2r−1, performs pass@G failure descent by increasing the probability that at least one sample in the group succeeds. On the full GSM8K test set across seven seeds, Sign reaches 73.8% accuracy versus 28.4% for standard normalized group-mean DrGRPO at group size four, a 45.4 point gain with p<0.0001. The effect is directionally consistent on Llama-3.1-8B and positive but underpowered on a MATH-500 transfer check. Pass@k analysis indicates that the main benefit is search compression rather than large capacity expansion, aligning the empirical gains with recent RLVR ceiling observations.
Group Relative Policy Optimization (GRPO) is the dominant reinforcement learning algorithm for training reasoning capabilities in large language models, notably adopted by DeepSeek-R1. The recent improvement Dr. GRPO (COLM 2025) identifies the response-level length bias caused by per-trajectory length normalization in GRPO and proposes removing this normalization, claiming the resulting optimizer is "unbiased." We show that this claim is incomplete. Specifically, we establish an impossibility theorem: under the standard outcome reward + GRPO setting, no length-based weighting scheme can simultaneously achieve the following two properties. (P1) Gradient unbiasedness: the gradient estimator is an unbiased estimate of the true policy gradient. (P2) Length invariance: each trajectory's effective contribution to the gradient is independent of its token length. GRPO approximately satisfies P2 but violates P1; Dr. GRPO satisfies P1 but violates P2. We characterize the complete tradeoff spectrum via the parametric family f_alpha(L) = L^{alpha - 1}, where alpha = 0 recovers GRPO, alpha = 1 recovers Dr. GRPO, and provide quantitative analysis showing that Dr. GRPO's length bias can cause longer trajectories to dominate gradient updates by a factor proportional to the length ratio. Our results reveal that neither algorithm is universally "done right"; they occupy opposite ends of a fundamental and unavoidable tradeoff.
Reinforcement learning with verifiable rewards (RLVR), particularly Group Relative Policy Optimization (GRPO), has advanced LLM reasoning. However, GRPO suffers from three credit assignment failures: uniform token-level granularity that ignores heterogeneous informational value, uniform polarity that penalizes correct steps and rewards incorrect ones, and zero-variance collapse that erases outcome-driven gradients. We systematically quantify these failures, revealing highly non-uniform token informativeness, widespread step-level polarity misalignment, and substantial training waste. To address these limitations, we propose Entropy-Progress Aligned GRPO (EP-GRPO), a framework that mines the model's intrinsic information flow for dense, self-supervised guidance. EP-GRPO integrates entropy-gated modulation to prioritize high entropy decision pivots, implicit process signals from policy divergence anchored to outcome advantages for directional token-level feedback without external reward models, and cumulative entropy mapping that enables progress-aligned advantage normalization, naturally maintaining gradient flow under zero reward variance. Extensive experiments on mathematical reasoning benchmarks demonstrate that EP-GRPO achieves superior accuracy and efficiency compared to GRPO and its variants. The code will be available.