# MEC-Quant: Maximum Entropy Coding for Extremely Low Bit Quantization-Aware Training

arXiv: https://arxiv.org/abs/2509.15514 · HTML: https://mlsanity.com/paper/2509.15514/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in MoE large language models, one of 12 directions in Mixture-of-experts.
- Builds on 28 of its 33 references that are in the map.
- First posted on arXiv 19 Sep 2025.

## Placed in

- Mixture-of-experts / [MoE large language models](https://mlsanity.com/moe/moe-large-language-models/)

## Builds on (most cited first, 28 of 28)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications](https://mlsanity.com/paper/1704.04861/) (2017)
- [Deep Compression: Compressing Deep Neural Networks with Pruning, Trained Quantization and Huffman Coding](https://mlsanity.com/paper/1510.00149/) (2015)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference](https://mlsanity.com/paper/1712.05877/) (2017)
- [Neural Architecture Search with Reinforcement Learning](https://mlsanity.com/paper/1611.01578/) (2016)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Learned Step Size Quantization](https://mlsanity.com/paper/1902.08153/) (2019)
- [Up or Down? Adaptive Rounding for Post-Training Quantization](https://mlsanity.com/paper/2004.10568/) (2020)
- [BRECQ: Pushing the Limit of Post-Training Quantization by Block Reconstruction](https://mlsanity.com/paper/2102.05426/) (2021)
- [PACT: Parameterized Clipping Activation for Quantized Neural Networks](https://mlsanity.com/paper/1805.06085/) (2018)
- [Model compression via distillation and quantization](https://mlsanity.com/paper/1802.05668/) (2018)
- [LSQ+: Improving low-bit quantization through learnable offsets and better initialization](https://mlsanity.com/paper/2004.09576/) (2020)
- [QDrop: Randomly Dropping Quantization for Extremely Low-bit Post-Training Quantization](https://mlsanity.com/paper/2203.05740/) (2022)
- [Overcoming Oscillations in Quantization-Aware Training](https://mlsanity.com/paper/2203.11086/) (2022)
- [Q-ViT: Accurate and Fully Quantized Low-bit Vision Transformer](https://mlsanity.com/paper/2210.06707/) (2022)
- [Network Quantization with Element-wise Gradient Scaling](https://mlsanity.com/paper/2104.00903/) (2021)
- [Nonuniform-to-Uniform Quantization: Towards Accurate Quantization via Generalized Straight-Through Estimation](https://mlsanity.com/paper/2111.14826/) (2021)
- [Differentiable Model Compression via Pseudo Quantization Noise](https://mlsanity.com/paper/2104.09987/) (2021)
- [MQBench: Towards Reproducible and Deployable Model Quantization Benchmark](https://mlsanity.com/paper/2111.03759/) (2021)
- [Self-Supervised Learning via Maximum Entropy Coding](https://mlsanity.com/paper/2210.11464/) (2022)
- [Q-DETR: An Efficient Low-Bit Quantized Detection Transformer](https://mlsanity.com/paper/2304.00253/) (2023)
- [Quantization Mimic: Towards Very Tiny CNN for Object Detection](https://mlsanity.com/paper/1805.02152/) (2018)
- [Resilient Binary Neural Network](https://mlsanity.com/paper/2302.00956/) (2023)
- [Synergistic Self-supervised and Quantization Learning](https://mlsanity.com/paper/2207.05432/) (2022)

## Abstract (arXiv)

Quantization-Aware Training (QAT) has driven much attention to produce efficient neural networks. Current QAT still obtains inferior performances compared with the Full Precision (FP) counterpart. In this work, we argue that quantization inevitably introduce biases into the learned representation, especially under the extremely low-bit setting. To cope with this issue, we propose Maximum Entropy Coding Quantization (MEC-Quant), a more principled objective that explicitly optimizes on the structure of the representation, so that the learned representation is less biased and thus generalizes better to unseen in-distribution samples. To make the objective end-to-end trainable, we propose to leverage the minimal coding length in lossy data coding as a computationally tractable surrogate for the entropy, and further derive a scalable reformulation of the objective based on Mixture Of Experts (MOE) that not only allows fast computation but also handles the long-tailed distribution for weights or activation values. Extensive experiments on various tasks on computer vision tasks prove its superiority. With MEC-Qaunt, the limit of QAT is pushed to the x-bit activation for the first time and the accuracy of MEC-Quant is comparable to or even surpass the FP counterpart. Without bells and whistles, MEC-Qaunt establishes a new state of the art for QAT.