# Additive Powers-of-Two Quantization: An Efficient Non-uniform Discretization for Neural Networks

arXiv: https://arxiv.org/abs/1909.13144 · HTML: https://mlsanity.com/paper/1909.13144/

## Key facts

- Cited by 148 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2019.

## Built on by (most cited first, 30 of 148)

- [A Survey of Quantization Methods for Efficient Neural Network Inference](https://mlsanity.com/paper/2103.13630/) (2021)
- [BRECQ: Pushing the Limit of Post-Training Quantization by Block Reconstruction](https://mlsanity.com/paper/2102.05426/) (2021)
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- [Nonuniform-to-Uniform Quantization: Towards Accurate Quantization via Generalized Straight-Through Estimation](https://mlsanity.com/paper/2111.14826/) (2021)
- [A Comprehensive Survey on Model Quantization for Deep Neural Networks in Image Classification](https://mlsanity.com/paper/2205.07877/) (2022)
- [Outlier Suppression+: Accurate quantization of large language models by equivalent and optimal shifting and scaling](https://mlsanity.com/paper/2304.09145/) (2023)
- [Oscillation-free Quantization for Low-bit Vision Transformers](https://mlsanity.com/paper/2302.02210/) (2023)
- [Patch Similarity Aware Data-Free Quantization for Vision Transformers](https://mlsanity.com/paper/2203.02250/) (2022)
- [BiT: Robustly Binarized Multi-distilled Transformer](https://mlsanity.com/paper/2205.13016/) (2022)
- [ANT: Exploiting Adaptive Numerical Data Type for Low-bit Deep Neural Network Quantization](https://mlsanity.com/paper/2208.14286/) (2022)
- [Fast-SNN: Fast Spiking Neural Network by Converting Quantized ANN](https://mlsanity.com/paper/2305.19868/) (2023)
- [IntraQ: Learning Synthetic Images with Intra-Class Heterogeneity for Zero-Shot Network Quantization](https://mlsanity.com/paper/2111.09136/) (2021)
- [Learnable Companding Quantization for Accurate Low-bit Neural Networks](https://mlsanity.com/paper/2103.07156/) (2021)
- [PSAQ-ViT V2: Towards Accurate and General Data-Free Quantization for Vision Transformers](https://mlsanity.com/paper/2209.05687/) (2022)
- [MQBench: Towards Reproducible and Deployable Model Quantization Benchmark](https://mlsanity.com/paper/2111.03759/) (2021)
- [Edge-Cloud Polarization and Collaboration: A Comprehensive Survey for AI](https://mlsanity.com/paper/2111.06061/) (2021)
- [Ternary Spike: Learning Ternary Spikes for Spiking Neural Networks](https://mlsanity.com/paper/2312.06372/) (2023)
- [DKM: Differentiable K-Means Clustering Layer for Neural Network Compression](https://mlsanity.com/paper/2108.12659/) (2021)
- [ShiftAddLLM: Accelerating Pretrained LLMs via Post-Training Multiplication-Less Reparameterization](https://mlsanity.com/paper/2406.05981/) (2024)
- [Auto-ViT-Acc: An FPGA-Aware Automatic Acceleration Framework for Vision Transformer with Mixed-Scheme Quantization](https://mlsanity.com/paper/2208.05163/) (2022)
- [Efficient Post-training Quantization with FP8 Formats](https://mlsanity.com/paper/2309.14592/) (2023)
- [AdaLog: Post-Training Quantization for Vision Transformers with Adaptive Logarithm Quantizer](https://mlsanity.com/paper/2407.12951/) (2024)
- [SDQ: Stochastic Differentiable Quantization with Mixed Precision](https://mlsanity.com/paper/2206.04459/) (2022)
- [Accurate Neural Training with 4-bit Matrix Multiplications at Standard Formats](https://mlsanity.com/paper/2112.10769/) (2021)
- [Sharpness-aware Quantization for Deep Neural Networks](https://mlsanity.com/paper/2111.12273/) (2021)
- [Fire Together Wire Together: A Dynamic Pruning Approach with Self-Supervised Mask Prediction](https://mlsanity.com/paper/2110.08232/) (2021)
- [Towards Accurate Post-Training Quantization of Vision Transformers via Error Reduction](https://mlsanity.com/paper/2407.06794/) (2024)
- [Bit-Mixer: Mixed-precision networks with runtime bit-width selection](https://mlsanity.com/paper/2103.17267/) (2021)
- [BiBench: Benchmarking and Analyzing Network Binarization](https://mlsanity.com/paper/2301.11233/) (2023)

## Abstract (arXiv)

We propose Additive Powers-of-Two~(APoT) quantization, an efficient non-uniform quantization scheme for the bell-shaped and long-tailed distribution of weights and activations in neural networks. By constraining all quantization levels as the sum of Powers-of-Two terms, APoT quantization enjoys high computational efficiency and a good match with the distribution of weights. A simple reparameterization of the clipping function is applied to generate a better-defined gradient for learning the clipping threshold. Moreover, weight normalization is presented to refine the distribution of weights to make the training more stable and consistent. Experimental results show that our proposed method outperforms state-of-the-art methods, and is even competitive with the full-precision models, demonstrating the effectiveness of our proposed APoT quantization. For example, our 4-bit quantized ResNet-50 on ImageNet achieves 76.6% top-1 accuracy without bells and whistles; meanwhile, our model reduces 22% computational cost compared with the uniformly quantized counterpart. The code is available at https://github.com/yhhhli/APoT_Quantization.