# Block Format Error Bounds and Optimal Block Size Selection

arXiv: https://arxiv.org/abs/2210.05470 · HTML: https://mlsanity.com/paper/2210.05470/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 25 references that are in the map.
- First posted on arXiv 11 Oct 2022.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Mixed Precision Training](https://mlsanity.com/paper/1710.03740/) (2017)
- [A Survey of Transformers](https://mlsanity.com/paper/2106.04554/) (2021)
- [Bottleneck Transformers for Visual Recognition](https://mlsanity.com/paper/2101.11605/) (2021)
- [Q-BERT: Hessian Based Ultra Low Precision Quantization of BERT](https://mlsanity.com/paper/1909.05840/) (2019)
- [Q8BERT: Quantized 8Bit BERT](https://mlsanity.com/paper/1910.06188/) (2019)
- [TernaryBERT: Distillation-aware Ultra-low Bit BERT](https://mlsanity.com/paper/2009.12812/) (2020)
- [AMMUS : A Survey of Transformer-based Pretrained Models in Natural Language Processing](https://mlsanity.com/paper/2108.05542/) (2021)
- [GOBO: Quantizing Attention-Based NLP Models for Low Latency and Energy Efficient Inference](https://mlsanity.com/paper/2005.03842/) (2020)
- [VS-Quant: Per-vector Scaled Quantization for Accurate Low-Precision Neural Network Inference](https://mlsanity.com/paper/2102.04503/) (2021)
- [Benchmarking TPU, GPU, and CPU Platforms for Deep Learning](https://mlsanity.com/paper/1907.10701/) (2019)
- [TRADI: Tracking deep neural network weight distributions for uncertainty estimation](https://mlsanity.com/paper/1912.11316/) (2019)

## Built on by (most cited first, 2 of 2)

- [Accurate Block Quantization in LLMs with Outliers](https://mlsanity.com/paper/2403.20137/) (2024)
- [SLaNC: Static LayerNorm Calibration](https://mlsanity.com/paper/2410.10553/) (2024)

## Abstract (arXiv)

The amounts of data that need to be transmitted, processed, and stored by the modern deep neural networks have reached truly enormous volumes in the last few years calling for the invention of new paradigms both in hardware and software development. One of the most promising and rapidly advancing frontiers here is the creation of new numerical formats. In this work we focus on the family of block floating point numerical formats due to their combination of wide dynamic range, numerical accuracy, and efficient hardware implementation of inner products using simple integer arithmetic. These formats are characterized by a block of mantissas with a shared scale factor. The basic Block Floating Point (BFP) format quantizes the block scales into the nearest powers of two on the right. Its simple modification - Scaled BFP (SBFP) - stores the same scales in full precision and thus allows higher accuracy. In this paper, we study the statistical behavior of both these formats rigorously. We develop asymptotic bounds on the inner product error in SBFP- and BFP-quantized normally distributed vectors. Next, we refine those asymptotic results to finite dimensional settings and derive high-dimensional tight bounds for the same errors. Based on the obtained results we introduce a performance measure assessing accuracy of any block format. This measure allows us to determine the optimal parameters, such as the block size, yielding highest accuracy. In particular, we show that if the precision of the BFP format is fixed at 4 bits, the optimal block size becomes 64. All theoretical derivations are supported by numerical experiments and studies on the weights of publicly available pretrained neural networks.