Why Does Stochastic Gradient Descent Slow Down in Low-Precision Training?

Vincent-Daniel Yun

Keywords: Optimization, SGD, Low-precision, Quantization, Gradient Descent

TL;DR: Low-precision training systematically shrink gradients, reducing the learning rate and slowing SGD convergence compared to full-precision training.

Abstract: Low-precision training has become crucial for reducing the computational and memory costs of large-scale deep learning. However, quantizing gradients introduces magnitude shrinkage, which can change how stochastic gradient descent (SGD) converges. In this study, we explore SGD convergence under a gradient shrinkage model, where each stochastic gradient is scaled by a factor \( q_k \in (0,1] \). We show that this shrinkage affect the usual stepsize \( \mu_k \) with an effective stepsize \( \mu_k q_k \), slowing convergence when \( q_{\min} < 1 \). With typical smoothness and bounded-variance assumptions, we prove that low-precision SGD still converges, but at a slower pace set by \( q_{\min} \), and with a higher steady error level due to quantization effects. We analyze theoretically how lower numerical precision slows training by treating it as gradient shrinkage within the standard SGD convergence setup.

Submission Number: 5