  Block Floating Point (BFP) can efficiently support quantization for Deep
Neural Network (DNN) training by providing a wide dynamic range via a shared
exponent across a group of values. In this paper, we propose a Fast First,
Accurate Second Training (FAST) system for DNNs, where the weights,
activations, and gradients are represented in BFP. FAST supports matrix
multiplication with variable precision BFP input operands, enabling incremental
increases in DNN precision throughout training. By increasing the BFP precision
across both training iterations and DNN layers, FAST can greatly shorten the
training time while reducing overall hardware resource usage. Our FAST
Multipler-Accumulator (fMAC) supports dot product computations under multiple
BFP precisions. We validate our FAST system on multiple DNNs with different
datasets, demonstrating a 2-6$\times$ speedup in training on a single-chip
platform over prior work based on \textbf{mixed-precision or block} floating
point number systems while achieving similar performance in validation
accuracy.
