# A Theoretical View of Linear Backpropagation and Its Convergence

arXiv: https://arxiv.org/abs/2112.11018 · HTML: https://mlsanity.com/paper/2112.11018/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 36 references that are in the map.
- First posted on arXiv 21 Dec 2021.

## Builds on (most cited first, 29 of 29)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [DeepFool: a simple and accurate method to fool deep neural networks](https://mlsanity.com/paper/1511.04599/) (2015)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Adversarial Machine Learning at Scale](https://mlsanity.com/paper/1611.01236/) (2016)
- [Practical Black-Box Attacks against Machine Learning](https://mlsanity.com/paper/1602.02697/) (2016)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [RobustBench: a standardized adversarial robustness benchmark](https://mlsanity.com/paper/2010.09670/) (2020)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Geometry-aware Instance-reweighted Adversarial Training](https://mlsanity.com/paper/2010.01736/) (2020)
- [An Exponential Learning Rate Schedule for Deep Learning](https://mlsanity.com/paper/1910.07454/) (2019)
- [Backpropagating Linearly Improves Transferability of Adversarial Examples](https://mlsanity.com/paper/2012.03528/) (2020)
- [An Analytical Formula of Population Gradient for two-layered ReLU network and its Applications in Convergence and Critical Point Analysis](https://mlsanity.com/paper/1703.00560/) (2017)

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

- [Improved Generation of Adversarial Examples Against Safety-aligned LLMs](https://mlsanity.com/paper/2405.20778/) (2024)
- [Improving Transferability of Adversarial Examples via Bayesian Attacks](https://mlsanity.com/paper/2307.11334/) (2023)

## Abstract (arXiv)

Backpropagation (BP) is widely used for calculating gradients in deep neural networks (DNNs). Applied often along with stochastic gradient descent (SGD) or its variants, BP is considered as a de-facto choice in a variety of machine learning tasks including DNN training and adversarial attack/defense. Recently, a linear variant of BP named LinBP was introduced for generating more transferable adversarial examples for performing black-box attacks, by Guo et al. Although it has been shown empirically effective in black-box attacks, theoretical studies and convergence analyses of such a method is lacking. This paper serves as a complement and somewhat an extension to Guo et al.'s paper, by providing theoretical analyses on LinBP in neural-network-involved learning tasks, including adversarial attack and model training. We demonstrate that, somewhat surprisingly, LinBP can lead to faster convergence in these tasks in the same hyper-parameter settings, compared to BP. We confirm our theoretical results with extensive experiments.