OrderGrad: Optimizing Beyond the Mean with Order-Statistic Policy Gradient Estimation

Reinforcement learning with verifiable rewards is widely used for post-training reasoning models, but sparse outcome rewards make exploration difficult. A complementary approach is to optimize inference-time objectives such as pass@K and max@K directly, yet existing policy-gradient estimators for these objectives use different signals, baselines, and normalizations, making their relationships unclear. We study this issue through baseline design and advantage centering. Starting from the advantage estimator of a leading method in the field, we show that it is policy-gradient unbiased but yields a non-centered advantage. We then introduce a Leave-Two-Out baseline that preserves policy-gradient unbiasedness while making realized batch advantages exactly centered. The resulting method, MaxPO, has an efficient quadratic-time implementation and integrates naturally into group-based RL for LLM post-training. We further derive the canonical finite-batch advantage for max@K, providing a unified view of existing advantage estimators. Empirically, we verify that the L2O baseline reduces gradient variance and outperforms non-centered alternatives.

The alignment of Large Language Models (LLMs) utilizes Reinforcement Learning from AI Feedback (RLAIF) for non-verifiable domains such as long-form question answering and open-ended instruction following. These domains often rely on LLM based auto-raters to provide granular, multi-tier discrete rewards (e.g., 1-10 rubrics) that are inherently stochastic due to prompt sensitivity and sampling randomness. We empirically verify the stochasticity of auto-raters that can propagate and corrupt standard advantage estimators like GRPO and MaxRL, as a noisy reward samples can skew normalization statistics and degrade the global learning signal. Empirically, sampling more rewards and taking majority voting may reduce the noise and improve performance, but this approach is computationally expensive. To address this bottleneck, we introduce $\textbf{O}$rdinal $\textbf{D}$ecomposition for $\textbf{R}$obust $\textbf{P}$olicy $\textbf{O}$ptimization ($\textbf{ODRPO}$), a framework that structurally isolates evaluation noise by decomposing discrete rewards into a sequence of ordinal binary indicators. By independently computing and accumulating advantages across these progressively challenging success thresholds, ODRPO prevents outlier evaluations from corrupting the global update while establishing an implicit, variance-aware learning curriculum. Empirically, ODRPO achieves robust performance on Qwen2.5-7B and Qwen3-4B models, outperforming baselines with relative improvements of upto 14.8% on FACTS-grounding-v2 and 7.5% on Alpaca-Evals. Critically, these gains are achieved with negligible training-time overhead, as ODRPO requires no additional compute per step compared to standard estimators. Supported by theoretical analysis confirming its optimization stability, ODRPO provides a scalable and robust framework for aligning models within the noisy, discrete evaluation landscape of modern RLAIF.

In policy gradient reinforcement learning, access to a differentiable model enables 1st-order gradient estimation that accelerates learning compared to relying solely on derivative-free 0th-order estimators. However, discontinuous dynamics cause bias and undermine the effectiveness of 1st-order estimators. Prior work addressed this bias by constructing a confidence interval around the REINFORCE 0th-order gradient estimator and using these bounds to detect discontinuities. However, the REINFORCE estimator is notoriously noisy, and we find that this method requires task-specific hyperparameter tuning and has low sample efficiency. This paper asks whether such bias is the primary obstacle and what minimal fixes suffice. First, we re-examine standard discontinuous settings from prior work and introduce DDCG, a lightweight test that switches estimators in nonsmooth regions; with a single hyperparameter, DDCG achieves robust performance and remains reliable with small samples. Second, on differentiable robotics control tasks, we present IVW-H, a per-step inverse-variance implementation that stabilizes variance without explicit discontinuity detection and yields strong results. Together, these findings indicate that while estimator switching improves robustness in controlled studies, careful variance control often dominates in practical deployments.