[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86569-en":3,"doc-seo-86569-105":30,"detail-sidebar-cat-0-en-105":83},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86569,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Agentic Skill Optimization over Lie Algebroids","Agentic systems self-improve by editing skills—prompts, rubrics, plans, tool contracts, examples, validators, and traces—but skill edits behave as local repairs to structured artifacts rather than independent vector coordinates. Different edits may share the same immediate visible effect while differing in routing context, template state, guardrail scope, or future composability, and edit order can change outcomes. The paper introduces LASKO, modeling typed anchored Markdown skills as a Lie algebroid and using its anchor/kernel and bracket to analyze visible versus latent structure.","arXiv :2607 . 1 1493v 1 [ cs .LG] 13 Jul 2026  \nAGENTIC SKILL OPTIMIZATION OVER LIE ALGEBROIDS ∗  \nA PREPRINT  \nSridhar Mahadevan  \nAdobe Research and University of Massachusetts, Amherst  \n[smahadev@adobe.com](smahadev@adobe.com) , [mahadeva@umass.edu](mahadeva@umass.edu)  \nJuly 14, 2026  \nABSTRACT  \nAgentic systems increasingly improve themselves by editing skills: prompts, rubrics, plans, tool contracts, examples, validators, and traces. Skill edits are not independent coordinates in a vector space: they are local repairs to structured artifacts whose effects are observed only after rollout, validation, and critique. Distinct edits can have the same immediate visible effect while differing in routing context, template state, guardrail scope, or future composability. The order of edits can matter as well: repairing a schema before a normalization rule need not be equivalent to applying the same edits in the reverse order. This paper introduces a new framework for skill optimization called LASKO, for Lie Algebroid SKill Optimization. LASKO models typed, anchored Markdown skills as the base category and available edit policies as sections of a controlled Lie algebroid A → Md with anchor ρ . The anchor maps an edit policy to its visible Markdown effect; the kernel ker(ρ) represents latent template, routing, or implementation structure; and the algebroid bracket measures noncommuting edit composition.  \nAs shown in the paper, LASKO achieves order-of-magnitude speedups in skill optimization in our preliminary benchmark results, primarily because it substitutes inexpensive Lie-bracket screening tests that run in microseconds, before investing in expensive validations that require running large language models. On a causal extraction from natural language task, LASKO achieved a speedup of almost 15 × compared to a brute-force approach that validated all edits by running them through a DeepSeek V3.1 4-bit model with 671B parameters.  \nKeywords Skill optimization · Large Language Models · Lie Algebroids · Tangent Categories · Infinitesimal causality  \n1 Introduction  \nAgentic skill optimization, exemplified by the SKILLOPT framework of Yang et al. (2026), is currently one of the most important frontiers of agentic AI. SKILLOPT makes a decisive step away from one-shot prompt writing and loosely controlled self-revision: it treats a skill as external textual state of a frozen agent, asks an optimizer model to propose bounded add/delete/replace edits, and accepts an edit only when it improves held-out validation performance. This discipline has a practical virtue that is easy to understate. It turns agent improvement into an experimental loop: propose a skill repair, roll out the agent, gate the edit on validation performance, and repeat.  \nThe difficulty is that the search space is not flat. A skill is a structured artifact: a prompt, rubric, schema, plan, tool contract, validator, example set, or trace summary. A failed rollout may be caused by any one of these anchors, or by an interaction among several of them. Two edits can look locally reasonable and still fail as a pair; conversely, two weak edits may become productive only in the right order. Repairing a schema before a normalization rule, adding an abstention policy before a formatting contract, or installing a validator before a final-answer template are order-sensitive operations. A short-horizon SKILLOPT loop that validates only single edits can therefore miss the actual repair, while exhaustive enumeration of ordered edit programs quickly becomes the wrong place to spend expensive rollout calls.  \n∗Draft under revision.  \nA PREPRINT-JULY 14, 2026  \nThis paper proposes that this is the central geometric problem of agentic skill optimization. We call the proposed generic framework LASKO: Lie Algebroid SKill Optimization. The aim is not merely to formalize an already successful method after the fact. The aim is to expose the geometry that can improve systems such as SKILLOPT: ","cbCaioQWLp3yNTz3","https://ap.wps.com/l/cbCaioQWLp3yNTz3","pdf",353578,4,1,20,"English","en",105,"# Introduction\n## Agentic skill optimization as an experimental loop\n## Non-flat search space and order sensitivity\n## LASKO framework overview\n## Infinitesimal causality via Lie algebroids","[{\"question\":\"Why does LASKO speed up skill optimization?\",\"answer\":\"LASKO enables Lie-bracket screening tests that run in microseconds before performing expensive validations requiring large language model executions. In a causal extraction benchmark, it reports nearly a 15× speedup versus validating all edits using a DeepSeek V3.1 4-bit model.\"}]",1784212693,50,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":28},"agentic-skill-optimization-over-lie-algebroids","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/agentic-skill-optimization-over-lie-algebroids/86569/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"Why does LASKO speed up skill optimization?","Question",{"text":75,"@type":76},"LASKO enables Lie-bracket screening tests that run in microseconds before performing expensive validations requiring large language model executions. 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