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Agentic Skill Optimization over Lie Algebroids

🕒 Published (v1): 2026-07-13 12:48 UTC · Source: Arxiv · link

Why this paper was selected

Mahadevan (UMass); formal Lie algebroid framework for structured skill editing in agentic systems

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TL;DR

LASKO (Lie Algebroid SKill Optimization) introduces a geometric framework that models agentic skill edits (prompts, schemas, validators, plans) as sections of a controlled Lie algebroid \(A \xrightarrow{\rho} T\mathcal{M}_d\), where the anchor \(\rho\) maps a controlled edit policy to its visible Markdown effect and the bracket \([\cdot,\cdot]_A\) measures edit order-sensitivity. By using inexpensive Lie-bracket screening (microseconds) to filter which edit pairs warrant expensive LLM rollout validation, LASKO achieves ~15Ă— speedup over brute-force validation on a preliminary causal extraction benchmark.

Problem

Existing skill optimizers like SkillOpt treat each proposed edit as an independent probe, ignoring two structural facts: (1) two edits can have the same visible document effect while carrying different latent template, routing, or implementation state that alters future edit composition; and (2) edit order matters—repairing a schema before a normalization rule need not equal the reverse sequence. Exhaustive enumeration of ordered edit programs quickly becomes intractable when each validation requires an expensive LLM rollout.

Method

LASKO lifts skill optimization from flat vector spaces to controlled Lie algebroids over a typed Markdown base category \(\mathcal{M}_d\).

  • Base category \(\mathcal{M}_d\): objects are tuples \(M = (A, P, \tau, \sigma, \chi)\) — anchor poset, Markdown parse trees indexed by anchors, block-type assignment, semantic metadata, and local contracts (schema validity, tool preconditions, safety constraints). Morphisms are anchor-respecting rewrite programs.
  • Algebroid \(A \to \mathcal{M}_d\): fiber \(A_x\) contains controlled edit modes at skill state \(x\) (e.g., strengthen guardrail, repair schema, add example). The anchor \(\rho: A \to T\mathcal{M}_d\) maps each controlled edit to its visible first-order Markdown effect. The kernel \(\ker(\rho)_x = \{a \in A_x : \rho(a) = 0\}\) captures latent template variables, unfilled slots, and routing choices with no immediate visible effect.
  • Screening criterion: the algebroid bracket \([s_i, s_j]_A\) is computed cheaply (microseconds). A high-bracket pair indicates order-sensitive composition and is prioritized for validation; low-bracket pairs are skipped. This substitutes an algebraic test for expensive rollout.
  • Visible Frobenius residual: \(R_{\text{vis}}(i,j) = (I - P_{\hat{V}})[\rho(s_i), \rho(s_j)]\) measures whether the observed edit library closes under the ambient tangent structure. Non-closure signals a missing repair direction or hidden workflow condition.
  • Anchor defect residual: \(R_{\text{anchor}}(i,j) = \hat{\rho}([s_i, s_j]_{\hat{A}}) - [\hat{\rho}(s_i), \hat{\rho}(s_j)]\) measures whether the estimated controlled intervention algebra explains the observed visible bracket.
  • The tangent functor \(T\) on \(\mathcal{M}_d\) is defined conditionally: \(TM = (M, E(M))\) with pushforward \(Tf(M, \delta) = (N, Df_M(\delta))\), requiring edit fibers to form an additive commutative monoid and rewrites to preserve edit composition—raw Markdown lacks sufficient structure.

Key Contributions

  • Formal embedding of agentic skill optimization into the theory of Lie algebroids and tangent categories, giving the first geometric account of edit order-dependence and latent workflow state.
  • Identification of \(\ker(\rho)\) as the algebraically precise notion of "latent implementation structure" in skill editing—distinct from unobserved statistical variables.
  • Lie-bracket screening as an order-of-magnitude computational shortcut: bracket tests run in microseconds and gate which edit pairs receive expensive LLM rollouts.
  • Definition of typed Markdown as a tangent category (\(\mathcal{M}_d\)) and the conditional tangent structure proposition (Proposition 1).
  • Decomposition of the optimization problem into regular, equivalence/Atiyah-style, and singular cases, with singular algebroids handling context-conditional edit policies (safety guardrails, tool contracts) that activate only in certain workflow regimes.

Results

  • On a causal extraction from natural language task with a 10-anchor benchmark, LASKO achieved a ~15Ă— speedup over brute-force validation that ran all edits through DeepSeek V3.1 4-bit (671B parameters).
  • LASKO-prioritized policy recovered the exhaustive ordered-pair solution with a small validation budget; greedy single-edit search missed the interaction.
  • Preliminary benchmarks show order-of-magnitude speedups in skill optimization overall.
  • (No additional numeric baselines or ablations are reported in the provided text.)

Limitations

  • Paper is explicitly marked as a "Draft under revision"; results are described as "preliminary."
  • Only one benchmark task (causal extraction) is reported with quantitative speedup numbers; breadth of evaluation is thin.
  • The tangent category structure on \(\mathcal{M}_d\) requires anchors, types, admissible edit fibers, and a merge/conflict policy—applicability to unstructured or legacy prompts is unclear.
  • Singular rank-change behavior (variable-rank anchor across contexts) is identified but not fully characterized or experimentally studied.
  • Algebroidic extensions for related systems (BRIDGE/SKFM for causal discovery, ALLORA/LICKET for neural adapter composition) are explicitly deferred to future work.
  • The bracket screening criterion assumes the bracket can be computed tractably; no analysis of bracket estimation cost or approximation error is provided.

Relevance to Agentic AI / LLM Agents

LASKO directly formalizes a core loop in modern agentic self-improvement—skill editing via structured artifact modification—and provides a principled way to prune the exponentially growing space of ordered edit sequences by using algebraic screening instead of repeated LLM rollouts. This connects to the growing literature on automated prompt optimization (SkillOpt, DSPy, TextGrad) by exposing why greedy, order-blind edit search fails: the composition structure of edits is fundamentally non-abelian, and the 15× reported speedup suggests the bracket test is a practically useful approximation. For researchers building self-improving or tool-using agents, the framework also formalizes "latent workflow state" (\(\ker(\rho)\))—the hidden routing, template, and precondition structure that exists above the visible prompt and can cause mysterious edit interaction failures—a concept that has been observed empirically but lacked a clean theoretical handle.