学习者的语言

David I. Spivak
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引用次数: 9

摘要

在“Backprop as函子”中,作者表明深度学习的基本元素——梯度下降和反向传播——可以被概念化为一个强单函数子Para(Euc) $\to$从参数化欧几里德空间的范畴学习到学习者的范畴,一个明确开发的范畴捕捉参数更新和反向传播。很快就意识到有一个同构的Learn $\cong$ Para(Slens),其中Slens是函数式编程中使用的简单透镜的对称单面类别。在这个笔记中,我们观察到Slens是Poly的一个完整的子范畴,Poly是一个变量多项式函子的范畴,通过函子$A\mapsto Ay^A$。利用(Poly, $\otimes$)是单轴封闭的事实,我们证明了Para(Slens)中的映射$A\to B$在动力系统(更准确地说,是广义摩尔机)方面具有自然的解释,其接口是内homtype $[Ay^A,By^B]$。最后,我们回顾了在任意$p \in$ Poly上的动力系统的范畴p-Coalg形成一个拓扑的事实,并考虑了可以用其内部语言表述的逻辑命题。最后以梯度下降法为例,讨论了今后的研究方向。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Learners' languages
In"Backprop as functor", the authors show that the fundamental elements of deep learning -- gradient descent and backpropagation -- can be conceptualized as a strong monoidal functor Para(Euc)$\to$Learn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism Learn$\cong$Para(Slens), where Slens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that Slens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor $A\mapsto Ay^A$. Using the fact that (Poly,$\otimes$) is monoidal closed, we show that a map $A\to B$ in Para(Slens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type $[Ay^A,By^B]$. Finally, we review the fact that the category p-Coalg of dynamical systems on any $p \in$ Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.
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