Reinforcement Learning with Verifiable Rewards (RLVR) is crucial for advancing large-scale reasoning models. However, existing parameter-efficient methods, such as PiSSA and MiLoRA, are designed for Supervised Fine-Tuning (SFT) and do not account for the distinct optimization dynamics and geometric structures of RLVR. Applying these methods directly leads to spectral collapse and optimization instability, which severely limit model performance. Meanwhile, alternative approaches that leverage update sparsity encounter significant efficiency bottlenecks on modern hardware due to unstructured computations. To address these challenges, we propose GeoRA (Geometry-Aware Low-Rank Adaptation), which exploits the anisotropic and compressible nature of RL update subspaces. GeoRA initializes adapters by extracting principal directions via Singular Value Decomposition (SVD) within a geometrically constrained subspace while freezing the residual components. This method preserves the pre-trained geometric structure and enables efficient GPU computation through dense operators. Experiments on Qwen and Llama demonstrate that GeoRA mitigates optimization bottlenecks caused by geometric misalignment. It consistently outperforms established low-rank baselines on key mathematical benchmarks, achieving state-of-the-art (SOTA) results. Moreover, GeoRA shows superior generalization and resilience to catastrophic forgetting in out-of-domain tasks.

翻译：基于可验证奖励的强化学习（RLVR）对于推进大规模推理模型至关重要。然而，现有的参数高效方法（如PiSSA和MiLoRA）是为监督微调（SFT）设计的，并未考虑RLVR独特的优化动态和几何结构。直接应用这些方法会导致谱崩溃和优化不稳定，从而严重限制模型性能。同时，利用更新稀疏性的替代方法由于非结构化计算，在现代硬件上遇到了显著的效率瓶颈。为解决这些挑战，我们提出了GeoRA（几何感知低秩自适应），它利用了RL更新子空间的各向异性和可压缩特性。GeoRA通过在几何约束的子空间内通过奇异值分解（SVD）提取主方向来初始化适配器，同时冻结残差分量。该方法保留了预训练的几何结构，并通过密集算子实现了高效的GPU计算。在Qwen和Llama上的实验表明，GeoRA缓解了由几何失配引起的优化瓶颈。它在关键数学基准测试中持续优于既有的低秩基线，实现了最先进的（SOTA）结果。此外，在领域外任务中，GeoRA展现出卓越的泛化能力和对灾难性遗忘的鲁棒性。