# Automatic Differentiation in PCF

arXiv: https://arxiv.org/abs/2011.03335 · HTML: https://mlsanity.com/paper/2011.03335/

## Key facts

- Cited by 4 papers in the map (data through 13 Oct 2026).
- First posted on arXiv 6 Nov 2020.

## Built on by (most cited first, 4 of 4)

- [On the Correctness of Automatic Differentiation for Neural Networks with Machine-Representable Parameters](https://mlsanity.com/paper/2301.13370/) (2023)
- [ωPAP Spaces: Reasoning Denotationally About Higher-Order, Recursive Probabilistic and Differentiable Programs](https://mlsanity.com/paper/2302.10636/) (2023)
- [Combinatory Adjoints and Differentiation](https://mlsanity.com/paper/2207.00847/) (2022)
- [Differentiable Quantum Programming with Unbounded Loops](https://mlsanity.com/paper/2211.04507/) (2022)

## Abstract (arXiv)

We study the correctness of automatic differentiation (AD) in the context of a higher-order, Turing-complete language (PCF with real numbers), both in forward and reverse mode. Our main result is that, under mild hypotheses on the primitive functions included in the language, AD is almost everywhere correct, that is, it computes the derivative or gradient of the program under consideration except for a set of Lebesgue measure zero. Stated otherwise, there are inputs on which AD is incorrect, but the probability of randomly choosing one such input is zero. Our result is in fact more precise, in that the set of failure points admits a more explicit description: for example, in case the primitive functions are just constants, addition and multiplication, the set of points where AD fails is contained in a countable union of zero sets of non-identically-zero polynomials.