Parallel Complexity of Forward and Backward Propagation

2017/12/18 by Maxim Naumov, Naumov, Maxim · 2 citations
Computer Science · Physics and Astronomy · #15B99 (Secondary) #68T05 (Primary) 65F99 #Artificial Intelligence (cs.AI) #FOS: Computer and information sciences #FOS: Mathematics #I.2.6 #I.5.0 #Image and Signal Denoising Methods #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Neural Networks and Applications #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1712.06577

openalex publication_date 2017/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Distributed Deep Learning Using Synchronous Stochastic Gradient Descent
• Optimization Methods for Large-Scale Machine Learning
• Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour
• Feedforward and Recurrent Neural Networks Backward Propagation and Hessian in Matrix Form
• Large Batch Training of Convolutional Networks
• AdaBatch: Adaptive Batch Sizes for Training Deep Neural Networks

Cited by

• A Unifying Framework for Parallelizing Sequential Models with Linear Dynamical Systems
• Accelerate TarFlow Sampling with GS-Jacobi Iteration

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