# Network Quantization with Element-wise Gradient Scaling

arXiv: https://arxiv.org/abs/2104.00903 · HTML: https://mlsanity.com/paper/2104.00903/

## Key facts

- Cited by 87 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 48 references that are in the map.
- First posted on arXiv 2 Apr 2021.

## Builds on (most cited first, 30 of 38)

- [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)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [SSD: Single Shot MultiBox Detector](https://mlsanity.com/paper/1512.02325/) (2015)
- [Rich feature hierarchies for accurate object detection and semantic segmentation](https://mlsanity.com/paper/1311.2524/) (2013)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks](https://mlsanity.com/paper/1603.05279/) (2016)
- [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)
- [PACT: Parameterized Clipping Activation for Quantized Neural Networks](https://mlsanity.com/paper/1805.06085/) (2018)
- [HAQ: Hardware-Aware Automated Quantization with Mixed Precision](https://mlsanity.com/paper/1811.08886/) (2018)
- [HAWQ: Hessian AWare Quantization of Neural Networks with Mixed-Precision](https://mlsanity.com/paper/1905.03696/) (2019)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [LQ-Nets: Learned Quantization for Highly Accurate and Compact Deep Neural Networks](https://mlsanity.com/paper/1807.10029/) (2018)
- [HAWQ-V2: Hessian Aware trace-Weighted Quantization of Neural Networks](https://mlsanity.com/paper/1911.03852/) (2019)
- [Differentiable Soft Quantization: Bridging Full-Precision and Low-Bit Neural Networks](https://mlsanity.com/paper/1908.05033/) (2019)
- [Understanding Straight-Through Estimator in Training Activation Quantized Neural Nets](https://mlsanity.com/paper/1903.05662/) (2019)
- [Incremental Network Quantization: Towards Lossless CNNs with Low-Precision Weights](https://mlsanity.com/paper/1702.03044/) (2017)
- [Trained Ternary Quantization](https://mlsanity.com/paper/1612.01064/) (2016)
- [Ternary Weight Networks](https://mlsanity.com/paper/1605.04711/) (2016)
- [LSQ+: Improving low-bit quantization through learnable offsets and better initialization](https://mlsanity.com/paper/2004.09576/) (2020)
- [Forward and Backward Information Retention for Accurate Binary Neural Networks](https://mlsanity.com/paper/1909.10788/) (2019)
- [Learning to Quantize Deep Networks by Optimizing Quantization Intervals with Task Loss](https://mlsanity.com/paper/1808.05779/) (2018)

## Built on by (most cited first, 30 of 87)

- [Overcoming Oscillations in Quantization-Aware Training](https://mlsanity.com/paper/2203.11086/) (2022)
- [AffineQuant: Affine Transformation Quantization for Large Language Models](https://mlsanity.com/paper/2403.12544/) (2024)
- [Towards Accurate Post-training Quantization for Diffusion Models](https://mlsanity.com/paper/2305.18723/) (2023)
- [Q-VLM: Post-training Quantization for Large Vision-Language Models](https://mlsanity.com/paper/2410.08119/) (2024)
- [IntraQ: Learning Synthetic Images with Intra-Class Heterogeneity for Zero-Shot Network Quantization](https://mlsanity.com/paper/2111.09136/) (2021)
- [BitsFusion: 1.99 bits Weight Quantization of Diffusion Model](https://mlsanity.com/paper/2406.04333/) (2024)
- [DKM: Differentiable K-Means Clustering Layer for Neural Network Compression](https://mlsanity.com/paper/2108.12659/) (2021)
- [NOLA: Compressing LoRA using Linear Combination of Random Basis](https://mlsanity.com/paper/2310.02556/) (2023)
- [AZ-NAS: Assembling Zero-Cost Proxies for Network Architecture Search](https://mlsanity.com/paper/2403.19232/) (2024)
- [CADyQ: Content-Aware Dynamic Quantization for Image Super-Resolution](https://mlsanity.com/paper/2207.10345/) (2022)
- [Sharpness-aware Quantization for Deep Neural Networks](https://mlsanity.com/paper/2111.12273/) (2021)
- [Instance-Aware Group Quantization for Vision Transformers](https://mlsanity.com/paper/2404.00928/) (2024)
- [Quantization without Tears](https://mlsanity.com/paper/2411.13918/) (2024)
- [Advances in Multimodal Adaptation and Generalization: From Traditional Approaches to Foundation Models](https://mlsanity.com/paper/2501.18592/) (2025)
- [CABM: Content-Aware Bit Mapping for Single Image Super-Resolution Network with Large Input](https://mlsanity.com/paper/2304.06454/) (2023)
- [Hard Sample Matters a Lot in Zero-Shot Quantization](https://mlsanity.com/paper/2303.13826/) (2023)
- [Quantized Feature Distillation for Network Quantization](https://mlsanity.com/paper/2307.10638/) (2023)
- [Retraining-free Model Quantization via One-Shot Weight-Coupling Learning](https://mlsanity.com/paper/2401.01543/) (2024)
- [Quantization Robust Federated Learning for Efficient Inference on Heterogeneous Devices](https://mlsanity.com/paper/2206.10844/) (2022)
- [PDP: Parameter-free Differentiable Pruning is All You Need](https://mlsanity.com/paper/2305.11203/) (2023)
- [Poster: Self-Supervised Quantization-Aware Knowledge Distillation](https://mlsanity.com/paper/2309.13220/) (2023)
- [Low-bit Model Quantization for Deep Neural Networks: A Survey](https://mlsanity.com/paper/2505.05530/) (2025)
- [Q-LIC: Quantizing Learned Image Compression with Channel Splitting](https://mlsanity.com/paper/2205.14510/) (2022)
- [Class-based Quantization for Neural Networks](https://mlsanity.com/paper/2211.14928/) (2022)
- [Self-Adapting Large Visual-Language Models to Edge Devices across Visual Modalities](https://mlsanity.com/paper/2403.04908/) (2024)
- [QCore: Data-Efficient, On-Device Continual Calibration for Quantized Models -- Extended Version](https://mlsanity.com/paper/2404.13990/) (2024)
- [Towards Accurate Binary Neural Networks via Modeling Contextual Dependencies](https://mlsanity.com/paper/2209.01404/) (2022)
- [BARVINN: Arbitrary Precision DNN Accelerator Controlled by a RISC-V CPU](https://mlsanity.com/paper/2301.00290/) (2022)
- [Reducing the Side-Effects of Oscillations in Training of Quantized YOLO Networks](https://mlsanity.com/paper/2311.05109/) (2023)
- [Custom Gradient Estimators are Straight-Through Estimators in Disguise](https://mlsanity.com/paper/2405.05171/) (2024)

## Abstract (arXiv)

Network quantization aims at reducing bit-widths of weights and/or activations, particularly important for implementing deep neural networks with limited hardware resources. Most methods use the straight-through estimator (STE) to train quantized networks, which avoids a zero-gradient problem by replacing a derivative of a discretizer (i.e., a round function) with that of an identity function. Although quantized networks exploiting the STE have shown decent performance, the STE is sub-optimal in that it simply propagates the same gradient without considering discretization errors between inputs and outputs of the discretizer. In this paper, we propose an element-wise gradient scaling (EWGS), a simple yet effective alternative to the STE, training a quantized network better than the STE in terms of stability and accuracy. Given a gradient of the discretizer output, EWGS adaptively scales up or down each gradient element, and uses the scaled gradient as the one for the discretizer input to train quantized networks via backpropagation. The scaling is performed depending on both the sign of each gradient element and an error between the continuous input and discrete output of the discretizer. We adjust a scaling factor adaptively using Hessian information of a network. We show extensive experimental results on the image classification datasets, including CIFAR-10 and ImageNet, with diverse network architectures under a wide range of bit-width settings, demonstrating the effectiveness of our method.