# Boosting Entropy with Bell Box Quantization

arXiv: https://arxiv.org/abs/2603.01599 · HTML: https://mlsanity.com/paper/2603.01599/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 50 references that are in the map.
- First posted on arXiv 2 Mar 2026.

## Builds on (most cited first, 30 of 34)

- [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)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Learned Step Size Quantization](https://mlsanity.com/paper/1902.08153/) (2019)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [A White Paper on Neural Network Quantization](https://mlsanity.com/paper/2106.08295/) (2021)
- [OmniQuant: Omnidirectionally Calibrated Quantization for Large Language Models](https://mlsanity.com/paper/2308.13137/) (2023)
- [8-bit Optimizers via Block-wise Quantization](https://mlsanity.com/paper/2110.02861/) (2021)
- [LLM-QAT: Data-Free Quantization Aware Training for Large Language Models](https://mlsanity.com/paper/2305.17888/) (2023)
- [The Era of 1-bit LLMs: All Large Language Models are in 1.58 Bits](https://mlsanity.com/paper/2402.17764/) (2024)
- [SqueezeLLM: Dense-and-Sparse Quantization](https://mlsanity.com/paper/2306.07629/) (2023)
- [Learning in the Frequency Domain](https://mlsanity.com/paper/2002.12416/) (2020)
- [DuQuant: Distributing Outliers via Dual Transformation Makes Stronger Quantized LLMs](https://mlsanity.com/paper/2406.01721/) (2024)
- [Understanding Batch Normalization](https://mlsanity.com/paper/1806.02375/) (2018)
- [Scaling Laws for Precision](https://mlsanity.com/paper/2411.04330/) (2024)
- [Nonuniform-to-Uniform Quantization: Towards Accurate Quantization via Generalized Straight-Through Estimation](https://mlsanity.com/paper/2111.14826/) (2021)
- [AffineQuant: Affine Transformation Quantization for Large Language Models](https://mlsanity.com/paper/2403.12544/) (2024)
- [QLLM: Accurate and Efficient Low-Bitwidth Quantization for Large Language Models](https://mlsanity.com/paper/2310.08041/) (2023)
- [PV-Tuning: Beyond Straight-Through Estimation for Extreme LLM Compression](https://mlsanity.com/paper/2405.14852/) (2024)
- [Training Transformers with 4-bit Integers](https://mlsanity.com/paper/2306.11987/) (2023)
- [ParetoQ: Improving Scaling Laws in Extremely Low-bit LLM Quantization](https://mlsanity.com/paper/2502.02631/) (2025)
- [Learnable Companding Quantization for Accurate Low-bit Neural Networks](https://mlsanity.com/paper/2103.07156/) (2021)
- [Quartet: Native FP4 Training Can Be Optimal for Large Language Models](https://mlsanity.com/paper/2505.14669/) (2025)
- [Learning from Students: Applying t-Distributions to Explore Accurate and Efficient Formats for LLMs](https://mlsanity.com/paper/2405.03103/) (2024)

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

- [Quantization-Aware Pre-Training with Constrained Empirical Weight Distribution](https://mlsanity.com/paper/2609.32659/) (2026)

## Abstract (arXiv)

Quantization-Aware Pre-Training (QAPT) is an effective technique to reduce the compute and memory overhead of Deep Neural Networks while improving their energy efficiency on edge devices. Existing QAPT methods produce models stored in compute-efficient data types (e.g. integers) that are not information theoretically optimal (ITO). On the other hand, existing ITO data types (e.g. Quantile/NormalFloat Quantization) are not compute-efficient. We propose BBQ, the first ITO quantization method that is also compute-efficient. BBQ builds on our key insight that since learning is domain-agnostic, the output of a quantizer does not need to reside in the same domain as its input. BBQ performs ITO quantization in its input domain, and returns its output in a compute-efficient domain where ITO data types are mapped to compute-efficient data types. Without sacrificing compute efficiency, BBQ outperforms prior SOTA QAPT methods by a perplexity reduction of up to 2 points for 4-bit models, up to 4 points for 3-bit models, up to 5 points for 2-bit models, and up to 18 points for 1-bit models. Code is available at https://github.com/1733116199/bbq.