GEOALIGN: Geometric Rollout Curation for Robust LLM Reinforcement Learning

Post-training has split large language model (LLM) alignment into two largely disconnected tracks. Online reinforcement learning (RL) with verifiable rewards drives emergent reasoning on math and code but depends on a programmatic verifier that cannot reach open-ended tasks, while preference optimization handles open-ended generation yet forgoes the continuous exploration that powers online RL. Closing this gap requires a verifier for open-ended quality, but a scalar reward model is the wrong shape for the job. Quality is multi-dimensional, and any scalar score is an incomplete proxy that lets online RL collapse onto whichever axis the score is most sensitive to. We turn instead to the General Preference Model (GPM), which embeds responses into $k$ skew-symmetric subspaces and represents preference as a structured, intransitivity-aware comparison. Building on this, we propose General Preference Reinforcement Learning (GPRL), which carries the $k$-way structure through to the policy update. GPRL computes per-dimension group-relative advantages, normalizes each on its own scale so no axis can dominate, and aggregates them with context-dependent eigenvalues. The same structure powers a closed-loop drift monitor that detects single-axis exploitation and corrects it on the fly by reweighting dimensions and tightening the trust region. Starting from $\texttt{Llama-3-8B-Instruct}$, GPRL reaches a length-controlled win rate of $56.51\%$ on AlpacaEval~2.0 while also outperforming SimPO and SPPO on Arena-Hard, MT-Bench, and WildBench by resisting reward hacking across extended training runs.

Large language model (LLM) alignment via reinforcement learning from human preferences (RLHF) suffers from unstable policy updates, ambiguous gradient directions, poor interpretability, and high gradient variance in mainstream pairwise preference learning paradigms. To systematically address these limitations, we establish a unified theoretical framework for preference-based RL optimization centered on the Pair-GRPO family, comprising two tightly coupled variants: Soft-Pair-GRPO and Hard-Pair-GRPO. Soft-Pair-GRPO is a minimal modification of Group Relative Policy Optimization (GRPO) that replaces group-normalized scalar rewards with binary pairwise preference rewards, retaining GRPO's clipped surrogate and KL-regularized structure. We prove a critical gradient equivalence theorem: under first-order Taylor expansion around the current policy, Soft-Pair-GRPO's gradient is a positive scalar multiple of standard GRPO's gradient, explaining its empirical stability despite discarding continuous reward magnitudes. Building on this foundation, we propose Hard-Pair-GRPO, an advanced variant introducing explicit local probability constraints and constrained KL-fitting optimization to further suppress gradient noise and global policy drift. We provide comprehensive theoretical guarantees for both variants--including monotonic policy improvement, deterministic gradient direction, gradient-variance reduction, and dynamic step-size convergence. Extensive experiments on standard LLM alignment benchmarks (HH-RLHF,UltraFeedback) and the MuJoCo continuous control task HalfCheetah-v4 demonstrate that our Pair-GRPO family consistently outperforms state-of-the-art baselines in alignment quality, human preference win rate, training stability, and generalization to general reinforcement learning. Ablation studies validate the critical contributions of each core component.

On-policy rollout methods such as GRPO are central to post-training of large language models, yet they frequently suffer from training instabilities, cross-task capability degradation, and response-length inflation. Although prior work has characterized the subspace geometry of aggregate updates, the stepwise variation of this geometry and its relationship to model performance remain unclear. We introduce Principal-Subspace Overlap, a dimension-corrected measure of individual rollout updates relative to the dominant singular subspaces of pretrained weights. Despite low average overlap, transient spikes often precede performance degradation. To address this, we propose GCPO (Geometrically Constrained Policy Optimization), which applies hard bilateral orthogonal projections to constrain updates to the complementary subspaces, preventing such excursions by construction. Across mathematical reasoning, code generation, and tool-use tasks on Qwen3-8B and GLM4-9B, GCPO consistently outperforms GRPO and recent variants, including DAPO and GSPO, improving over the base models and the strongest baseline by up to 27.69 and 2.37 points, respectively. Furthermore, GCPO preserves general capabilities, eliminates response-length inflation, and stabilizes policy entropy. Our findings provide a new diagnostic lens and a principled design perspective for stable reinforcement learning post-training.