在无穷小非组合草图中学习

中文摘要 AI 辅助

本文开发了一个范畴框架——无穷小非组合草图学习（LINCS），用于修复非组合性问题，即图表无法通过提升到切范畴设置的商草图进行分解。机器学习问题被指定为草图，通过通用分解问题的失败来定义非组合性。给定学习草图和模型，定义了基础缺陷和切提升，LINCS被定义为切提升后的分解障碍。本文还介绍了切学习草图，定义了INC自函子，将机器学习表述为寻找相继切展开稳定的余代数不动点。利用Aczel - Mendler定理证明了在特定条件下最终INC余代数的存在性。目前正在多个具体机器学习设置中对LINCS进行详细实验评估。

英文摘要

This paper develops a categorical framework -- Learning in Infinitesimal Non-Compositional Sketches (LINCS) -- as the repair of non-compositionality: failures of diagrams to factor through quotient sketches lifted to the tangent category setting. Machine learning problems are specified as sketches: graphs with commutativity conditions $\mathcal D$, limit cones $\mathcal L$, and colimit cocones $\mathcal K$, generalizing the usual scalarization of loss functions or vector space assumptions. Non-compositionality is defined purely as failure of a universal factorization problem, not as arithmetic error between the desired and actual predictions. Given a learning sketch $\mathbb S=(S,\mathcal D,\mathcal L,\mathcal K)$, whose underlying graph is $S$, and a model $D:J \rightarrow C$, the base defect is the obstruction to factorization $\mbox{Obs}(\mbox{Fact}_{\mathbb S}(D))$. The tangent lift applies the tangent functor $T$ to obtain $TD:J \rightarrow C$, and LINCS is defined as the obstruction $\mbox{Obs}(\mbox{Fact}_{\mathbb S}(TD))$ -- asking whether infinitesimal perturbations preserve the compositionality constraints.The paper also introduces Tangent Learning Sketches, which are sketches equipped with Cockett-Cruttwell tangent structure. The paper defines the INC endofunctor, which iterates the tangent lift, producing a tower $D,TD,T^2D, \cdots$ of factorization problems. ML is thereby formulated as the search for a coalgebraic fixed point where successive tangent unfoldings stabilize ($νT_{\mbox{INC}}$). Using the Aczel--Mendler theorem, we prove existence of a final INC coalgebra whenever $T_{\mbox{INC}}$ admits a set-based class realization that creates its final carrier. A detailed experimental evaluation of LINCS is underway in a number of concrete ML settings, including deep learning, large language models, and reinforcement learning, and is described in companion papers.

发表机构

• Adobe Research（Adobe研究院）
• University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）

机构由 AI 辅助整理，请以论文原文为准。