一句话：将反向传播从一阶计算图扩展到带线性否定的简单类型 λ-演算，并保持一阶反向传播的效率。
可微编程希望用带控制流和高阶组合子（如 map、fold）的编程语言更模块化地表达计算图。论文以带线性否定的简单类型 λ-演算为范式语言，定义了一种组合式程序变换：把源程序变换为同一演算中带线性否定的程序，并证明该变换计算源程序的梯度，且复杂度与一阶反向传播相同。变换完全无副作用，从而给出反向传播动态的纯逻辑解释。
Backpropagation is a classic automatic differentiation algorithm computing the gradient of functions specified by a certain class of simple, first-order programs, called computational graphs. It is a fundamental tool in several fields, most notably machine learning, where it is the key for efficiently training (deep) neural networks. Recent years have witnessed the quick growth of a research field called differentiable programming, the aim of which is to express computational graphs more synthetically and modularly by resorting to actual programming languages endowed with control flow operators and higher-order combinators, such as map and fold. In this paper, we extend the backpropagation algorithm to a paradigmatic example of such a programming language: we define a compositional program transformation from the simply-typed lambda-calculus to itself augmented with a notion of linear negation, and prove that this computes the gradient of the source program with the same efficiency as first-order backpropagation. The transformation is completely effect-free and thus provides a purely logical understanding of the dynamics of backpropagation.
@article{paperbot575,
title = {Backpropagation in the simply typed lambda-calculus with linear negation},
author = {Aloïs Brunel and Damiano Mazza and Michele Pagani},
journal = {Proceedings of the ACM on Programming Languages},
volume = {4},
number = {POPL},
year = {2019},
doi = {10.1145/3371132}
}