A Geometric Analysis of Skill Parameterization

Agent skills enable LLMs to execute complex, domain-specific, multi-step procedures without modifying model weights. Typically, agents inject skills as tokens into the prompt at inference, incurring token costs proportional to their length. Parametric retrieval-augmented generation (PRAG) [2] eliminates the token overhead by encoding each document into a LoRA adapter offline, trained via document augmentation. ParametricSkills [4] scales skill parameterization by training a hypernetwork to generate adapter weights from skill documents in a single forward pass using skill exploitation trajectories. From another angle, SkillOpt [3] refines skill documents in text space through scored trajectory feedback before parameterization. Yet whether structurally similar skills cluster together in adapter space, whether this structure explains parameterization quality, and whether it can be improved through skill refinement, remains unexplored. We investigate this through three questions. RQ1: How do different skill parameterization techniques shape the geometry of the LoRA adapter space? RQ2: How are the geometric properties of the adapter space related to skill parameterization quality? RQ3: How does skill refinement alter the geometric properties of adapters? We study these questions through a geometric analysis of the LoRA adapter space across ten parameterization conditions on SRA-Bench [1], a benchmark of human-written skills covering six skill types. Adapter weight matrices are represented via QR decomposition and visualised using UMAP to reveal the geometry of the resulting adapter space. We vary the training objective (next token prediction, NTP) on raw skill documents and question-answer (QA) document augmentation, LoRA rank ($k=2, k=8$), and granularity (per-skill, per-benchmark), training 483 adapters per skill condition and 4 per benchmark condition (3,872 total). For RQ1, training objective is the dominant factor shaping adapter geometry at $k=8$: NTP and QA adapters occupy distinct regions of the adapter space (cosine distance $= 0.28$; Figure (in pdf)), consistent with an $8.2$pp gap in parametric accuracy. Skill types are separable across all training objectives, but separability increases with rank: silhouette scores rise from 0.21--0.38 at $k=2$ to 0.36--0.41 at $k=8$ for skill-granularity adapters, and 0.33--0.86 for benchmark-granularity adapters. For RQ2, we correlate geometric properties, effective rank and Frobenius norm with parametric accuracy across 483 skills. For benchmark-granularity adapters, effective rank correlates with accuracy at low rank but this relationship vanishes at higher rank, while Frobenius norm correlates consistently across both ranks, with smaller weight updates predicting higher accuracy. No meaningful relationships hold for skill-granularity adapters ($|r| < 0.13$ across all conditions). Ongoing work: For RQ3, we apply SkillOpt [3] to refine gold skills and re-parameterize them using the same training objectives, to test whether refinement shifts the adapters toward geometric properties associated with higher accuracy. Since training objective dominates adapter geometry (RQ1), we train and analyze a hypernetwork conditioned on skill content [4]. We further observe that NTP adapters appear more dispersed than QA adapters in Figure (in pdf), and plan to quantify this dispersion to test whether it explains NTP's higher accuracy. [1] W. Su, J. Long, Q. Ai, Q. He, Y. Tang, C. Wang, Y. Tu, Y. Wang, and Y. Liu. Skill retrieval augmentation for agentic ai. arXiv preprint, 2026. [2] W. Su, Y. Tang, Q. Ai, J. Yan, C. Wang, H. Wang, Z. Ye, Y. Zhou, and Y. Liu. Parametric retrieval augmented generation. In Proc. 48th ACM SIGIR, page 1240–1250, 2025. [3] Y. Yang, Z. Gong, W. Huang, Q. Yang, Z. Zhou, Z. Huang, Y. Li, X. Gao, Q. Dai, B. Liu, K. Qiu, Y. Yang, D. Chen, X. Yang, and C. Luo. Skillopt: Executive strategy for self-evolving agent skills. arXiv preprint, 2026. [4] X. Zhao, H. He, Q. Yang, M. Li, J. Ye, Z. Tan, B. Wan, and P. Ye. Parametric skills. arXiv preprint, 2026.