# Parallel Dual-Numbers Reverse AD

arXiv: https://arxiv.org/abs/2207.03418 · HTML: https://mlsanity.com/paper/2207.03418/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jul 2022.

## Built on by (most cited first, 2 of 2)

- [On the Correctness of Automatic Differentiation for Neural Networks with Machine-Representable Parameters](https://mlsanity.com/paper/2301.13370/) (2023)
- [nablaSD: Differentiable Programming for Sparse Tensors](https://mlsanity.com/paper/2303.07030/) (2023)

## Abstract (arXiv)

Where dual-numbers forward-mode automatic differentiation (AD) pairs each scalar value with its tangent value, dual-numbers reverse-mode AD attempts to achieve reverse AD using a similarly simple idea: by pairing each scalar value with a backpropagator function. Its correctness and efficiency on higher-order input languages have been analysed by Brunel, Mazza and Pagani, but this analysis used a custom operational semantics for which it is unclear whether it can be implemented efficiently. We take inspiration from their use of linear factoring to optimise dual-numbers reverse-mode AD to an algorithm that has the correct complexity and enjoys an efficient implementation in a standard functional language with support for mutable arrays, such as Haskell. Aside from the linear factoring ingredient, our optimisation steps consist of well-known ideas from the functional programming community. We demonstrate the use of our technique by providing a practical implementation that differentiates most of Haskell98. Where previous work on dual numbers reverse AD has required sequentialisation to construct the reverse pass, we demonstrate that we can apply our technique to task-parallel source programs and generate a task-parallel derivative computation.