  While low-precision optimization has been widely used to accelerate deep
learning, low-precision sampling remains largely unexplored. As a consequence,
sampling is simply infeasible in many large-scale scenarios, despite providing
remarkable benefits to generalization and uncertainty estimation for neural
networks. In this paper, we provide the first study of low-precision Stochastic
Gradient Langevin Dynamics (SGLD), showing that its costs can be significantly
reduced without sacrificing performance, due to its intrinsic ability to handle
system noise. We prove that the convergence of low-precision SGLD with
full-precision gradient accumulators is less affected by the quantization error
than its SGD counterpart in the strongly convex setting. To further enable
low-precision gradient accumulators, we develop a new quantization function for
SGLD that preserves the variance in each update step. We demonstrate that
low-precision SGLD achieves comparable performance to full-precision SGLD with
only 8 bits on a variety of deep learning tasks.
