# Toward INT4 Fixed-Point Training via Exploring Quantization Error for Gradients

arXiv: https://arxiv.org/abs/2407.12637 · HTML: https://mlsanity.com/paper/2407.12637/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 45 references that are in the map.
- First posted on arXiv 17 Jul 2024.

## Builds on (most cited first, 30 of 31)

- [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)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks](https://mlsanity.com/paper/1506.01497/) (2015)
- [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)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Aggregated Residual Transformations for Deep Neural Networks](https://mlsanity.com/paper/1611.05431/) (2016)
- [Enhanced Deep Residual Networks for Single Image Super-Resolution](https://mlsanity.com/paper/1707.02921/) (2017)
- [Mixed Precision Training](https://mlsanity.com/paper/1710.03740/) (2017)
- [Learned Step Size Quantization](https://mlsanity.com/paper/1902.08153/) (2019)
- [DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients](https://mlsanity.com/paper/1606.06160/) (2016)
- [Deep Gradient Compression: Reducing the Communication Bandwidth for Distributed Training](https://mlsanity.com/paper/1712.01887/) (2017)
- [PACT: Parameterized Clipping Activation for Quantized Neural Networks](https://mlsanity.com/paper/1805.06085/) (2018)
- [Deep Learning with Limited Numerical Precision](https://mlsanity.com/paper/1502.02551/) (2015)
- [Differentiable Soft Quantization: Bridging Full-Precision and Low-Bit Neural Networks](https://mlsanity.com/paper/1908.05033/) (2019)
- [Training Deep Neural Networks with 8-bit Floating Point Numbers](https://mlsanity.com/paper/1812.08011/) (2018)
- [Learning to Quantize Deep Networks by Optimizing Quantization Intervals with Task Loss](https://mlsanity.com/paper/1808.05779/) (2018)
- [Quantization Networks](https://mlsanity.com/paper/1911.09464/) (2019)
- [Network Quantization with Element-wise Gradient Scaling](https://mlsanity.com/paper/2104.00903/) (2021)
- [Towards Unified INT8 Training for Convolutional Neural Network](https://mlsanity.com/paper/1912.12607/) (2019)
- [A Statistical Framework for Low-bitwidth Training of Deep Neural Networks](https://mlsanity.com/paper/2010.14298/) (2020)
- [Mixed Precision Training of Convolutional Neural Networks using Integer Operations](https://mlsanity.com/paper/1802.00930/) (2018)
- [Flexpoint: An Adaptive Numerical Format for Efficient Training of Deep Neural Networks](https://mlsanity.com/paper/1711.02213/) (2017)
- [Training High-Performance and Large-Scale Deep Neural Networks with Full 8-bit Integers](https://mlsanity.com/paper/1909.02384/) (2019)
- [Distribution Adaptive INT8 Quantization for Training CNNs](https://mlsanity.com/paper/2102.04782/) (2021)
- [Shifted and Squeezed 8-bit Floating Point format for Low-Precision Training of Deep Neural Networks](https://mlsanity.com/paper/2001.05674/) (2020)
- [Distance-aware Quantization](https://mlsanity.com/paper/2108.06983/) (2021)
- [Exploring the Potential of Low-bit Training of Convolutional Neural Networks](https://mlsanity.com/paper/2006.02804/) (2020)

## Built on by (most cited first, 1 of 1)

- [Distribution-Aware Hadamard Quantization for Hardware-Efficient Implicit Neural Representations](https://mlsanity.com/paper/2508.13478/) (2025)

## Abstract (arXiv)

Network quantization generally converts full-precision weights and/or activations into low-bit fixed-point values in order to accelerate an inference process. Recent approaches to network quantization further discretize the gradients into low-bit fixed-point values, enabling an efficient training. They typically set a quantization interval using a min-max range of the gradients or adjust the interval such that the quantization error for entire gradients is minimized. In this paper, we analyze the quantization error of gradients for the low-bit fixed-point training, and show that lowering the error for large-magnitude gradients boosts the quantization performance significantly. Based on this, we derive an upper bound of quantization error for the large gradients in terms of the quantization interval, and obtain an optimal condition for the interval minimizing the quantization error for large gradients. We also introduce an interval update algorithm that adjusts the quantization interval adaptively to maintain a small quantization error for large gradients. Experimental results demonstrate the effectiveness of our quantization method for various combinations of network architectures and bit-widths on various tasks, including image classification, object detection, and super-resolution.