# Accumulation Bit-Width Scaling For Ultra-Low Precision Training Of Deep Networks

arXiv: https://arxiv.org/abs/1901.06588 · HTML: https://mlsanity.com/paper/1901.06588/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jan 2019.

## Built on by (most cited first, 11 of 11)

- [Optimal Clipping and Magnitude-aware Differentiation for Improved Quantization-aware Training](https://mlsanity.com/paper/2206.06501/) (2022)
- [A2Q: Accumulator-Aware Quantization with Guaranteed Overflow Avoidance](https://mlsanity.com/paper/2308.13504/) (2023)
- [Low-Precision Training of Large Language Models: Methods, Challenges, and Opportunities](https://mlsanity.com/paper/2505.01043/) (2025)
- [Compacting Deep Neural Networks for Internet of Things: Methods and Applications](https://mlsanity.com/paper/2103.11083/) (2021)
- [Accumulator-Aware Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2409.17092/) (2024)
- [Mixed precision accumulation for neural network inference guided by componentwise forward error analysis](https://mlsanity.com/paper/2503.15568/) (2025)
- [Multiplication-Free Transformer Training via Piecewise Affine Operations](https://mlsanity.com/paper/2305.17190/) (2023)
- [Towards Cheaper Inference in Deep Networks with Lower Bit-Width Accumulators](https://mlsanity.com/paper/2401.14110/) (2024)
- [Quantized Neural Networks for Low-Precision Accumulation with Guaranteed Overflow Avoidance](https://mlsanity.com/paper/2301.13376/) (2023)
- [Numerical Stability of DeepGOPlus Inference](https://mlsanity.com/paper/2212.06361/) (2022)
- [One QK Channel, Many Sources: Tracing Low-Precision Attention Collapse](https://mlsanity.com/paper/2608.02091/) (2026)

## Abstract (arXiv)

Efforts to reduce the numerical precision of computations in deep learning training have yielded systems that aggressively quantize weights and activations, yet employ wide high-precision accumulators for partial sums in inner-product operations to preserve the quality of convergence. The absence of any framework to analyze the precision requirements of partial sum accumulations results in conservative design choices. This imposes an upper-bound on the reduction of complexity of multiply-accumulate units. We present a statistical approach to analyze the impact of reduced accumulation precision on deep learning training. Observing that a bad choice for accumulation precision results in loss of information that manifests itself as a reduction in variance in an ensemble of partial sums, we derive a set of equations that relate this variance to the length of accumulation and the minimum number of bits needed for accumulation. We apply our analysis to three benchmark networks: CIFAR-10 ResNet 32, ImageNet ResNet 18 and ImageNet AlexNet. In each case, with accumulation precision set in accordance with our proposed equations, the networks successfully converge to the single precision floating-point baseline. We also show that reducing accumulation precision further degrades the quality of the trained network, proving that our equations produce tight bounds. Overall this analysis enables precise tailoring of computation hardware to the application, yielding area- and power-optimal systems.