Composing Automatic Differentiation with Custom Derivatives of Higher-Order Functions

Recent theoretical work on automatic differentiation (autodiff) has focused on characteristics such as correctness and efficiency while assuming that all derivatives are automatically generated by autodiff using program transformation, with the exception of a fixed set of derivatives for primitive operations. However, in practice this assumption is insufficient: the programmer often needs to provide custom derivatives for composite functions to achieve efficiency and numerical stability. In this work, we start from the untyped lambda calculus with a reverse-mode autodiff operator, extend it with an operator to attach manual derivatives, and demonstrate its utility via several examples.