SoD: Algorithmic Differentiation of Functional Programs

CCF-0438806 Jeffrey Mark SiskindPurdue大学函数式程序的微分自动微分（AD）是一个成熟的企业，它试图通过符号操作而不是有限差分来获取指定为命令式程序的函数的导数。 将研究一个将AD技术应用于现代函数式编程语言而不是命令式程序的框架。 这个框架将把AD操作符视为将函数对象映射到函数对象的高阶函数。 这些操作符将是合成的，并且将应用于比传统AD变换受限的真实的向量之间的映射更一般的函数。 该框架提供了几个优点：-被呼叫者可以执行必要的AD，而不是坚持要求呼叫者 提供适当的转换功能，使其切实可行， 组合构建基于导数的高效函数库-高阶导数可以自然构造，无需特殊的 机制-梯度可以通过本身涉及基于AD的 优化或近似迭代到定点运算符，如PDE solversThe拟议的工作将在工程设计，科学计算和机器学习等不同领域产生广泛的影响，研究人员经常需要对复杂的程序进行衍生。 它具有智力价值，因为它提供了一种新的综合微分与lambda演算。

CCF-0438806Jeffrey Mark SiskindPurdue UniversityAlgorithmic Differentiation of Functional ProgramsAutomatic Differentiation (AD) is an established enterprise that seeks to take the derivatives of functions specified as imperative programs through symbolic manipulation rather than finite differencing. A framework for applying AD techniques to modern functional-programming languages, instead of imperativeprograms, will be investigated. This framework will treat AD operators as higher-order functions that map function objects to function objects. These operators will be compositional and will apply to more general functions than the maps between real vectors to which traditional AD transformations are limited. This framework offers several advantages:- a callee can perform the necessary AD rather than insisting that callers provide appropriately transformed functions, making it practical to compositionally build an efficient library of functions based on derivatives - higher-order derivatives can be constructed naturally, without special mechanisms- gradients can be taken through processes that themselves involve AD-based optimization or approximate iterate-to-fixedpoint operators such as PDE solversThe proposed work will have broad impact in such diverse fields as engineering design, scientific computing, and machine learning, where researchers regularly need to take derivatives of complex programs. It has intellectual merit as it offers a novel synthesis of the differential calculus with the lambda calculus.