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

## TL;DR

This paper introduces a statistical framework to determine the minimal accumulation bit-width needed for training deep neural networks at ultra-low precision, enabling more efficient hardware design without sacrificing convergence.

## Contribution

It provides a novel analytical method to accurately estimate the necessary accumulation precision for deep learning training, improving over conservative existing approaches.

## Key findings

- Networks trained with precision set by the proposed equations match baseline accuracy.
- Reducing accumulation bits below the estimated threshold degrades training quality.
- The analysis applies successfully to multiple benchmark networks.

## Abstract

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.