Enhancing the intelligibility and interpretability of machine learning is a crucial task in responding to the demand for Explicability as an AI principle, and in promoting the better social implementation of AI. The aim of our research is to contribute to this improvement by reformulating machine learning models through the lens of category theory, thereby developing a semantic framework for structuring and understanding AI systems. Our categorical modeling in this paper clarifies and formalizes the structural interplay between residuals and parameters in supervised learning. The present paper focuses on the multiple linear regression model, which represents the most basic form of supervised learning. By defining two Lawvere-enriched categories corresponding to parameters and data, along with an adjoint pair of functors between them, we introduce our categorical formulation of supervised learning. We show that the essential structure of this framework is captured by what we call the Gauss-Markov Adjunction. Within this setting, the dual flow of information can be explicitly described as a correspondence between variations in parameters and residuals. The ordinary least squares estimator for the parameters and the minimum residual are related via the preservation of limits by the right adjoint functor. Furthermore, we position this formulation as an instance of extended denotational semantics for supervised learning, and propose applying a semantic perspective developed in theoretical computer science as a formal foundation for Explicability in AI.

翻译：提升机器学习的可理解性与可解释性，是响应可解释性作为人工智能原则的需求、并促进人工智能更好社会实施的关键任务。我们研究的目标是通过范畴论视角重构机器学习模型，从而为构建和理解人工智能系统发展一种语义框架，以此推动这一改进。本文中的范畴建模澄清并形式化了监督学习中残差与参数之间的结构相互作用。本文聚焦于多元线性回归模型，它代表了监督学习最基本的形式。通过定义分别对应于参数和数据的两个Lawvere-丰富范畴，以及它们之间的一个伴随函子对，我们引入了监督学习的范畴化表述。我们证明该框架的本质结构可由我们称为高斯-马尔可夫伴随的结构所刻画。在此设定下，信息的对偶流动可被明确描述为参数变化与残差之间的对应关系。参数的最小二乘估计量与最小残差通过右伴随函子对极限的保持相联系。此外，我们将此表述定位为监督学习的扩展指称语义的一个实例，并提议将理论计算机科学中发展的语义视角作为人工智能可解释性的形式化基础。