# Parallel Complexity of Forward and Backward Propagation

arXiv: https://arxiv.org/abs/1712.06577 · HTML: https://mlsanity.com/paper/1712.06577/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2017.

## Built on by (most cited first, 3 of 3)

- [DeepPCR: Parallelizing Sequential Operations in Neural Networks](https://mlsanity.com/paper/2309.16318/) (2023)
- [A Unifying Framework for Parallelizing Sequential Models with Linear Dynamical Systems](https://mlsanity.com/paper/2509.21716/) (2025)
- [Accelerate TarFlow Sampling with GS-Jacobi Iteration](https://mlsanity.com/paper/2505.12849/) (2025)

## Abstract (arXiv)

We show that the forward and backward propagation can be formulated as a solution of lower and upper triangular systems of equations. For standard feedforward (FNNs) and recurrent neural networks (RNNs) the triangular systems are always block bi-diagonal, while for a general computation graph (directed acyclic graph) they can have a more complex triangular sparsity pattern. We discuss direct and iterative parallel algorithms that can be used for their solution and interpreted as different ways of performing model parallelism. Also, we show that for FNNs and RNNs with k layers and τ time steps the backward propagation can be performed in parallel in O(log k) and O(log k log τ) steps, respectively. Finally, we outline the generalization of this technique using Jacobians that potentially allows us to handle arbitrary layers.