Fast batched matrix multiplication for small sizes using half-precision arithmetic on GPUS

Matrix multiplication (GEMM) is the most important operation in dense linear algebra. Because it is a compute-bound operation that is rich in data reuse, many applications from different scientific domains cast their most performance-critical stages to use GEMM. With the rise of batch linear algebra, batched GEMM operations have become increasingly popular in domains other than dense linear solvers, such as tensor contractions, sparse direct solvers, and machine learning. In particular for the latter, batched GEMM in reduced precision (i.e., FP16) has been the core operation of many deep learning frameworks. This paper introduces an optimized batched GEMM for FP16 arithmetic (HGEMM) using graphics processing units (GPUs). We provide a detailed design strategy that takes advantage of the Tensor Core technology that was recently introduced in CUDA-enabled GPUs. The developed solution uses low-level APIs provided by the vendor in an optimized design that overcomes the limitations imposed by the hardware (in the form of discrete configurations). The outcome is a highly flexible GPU kernel that provides a lot of controls to the developer, despite the aforementioned restrictions. The paper also pays particular attention to multiplications of very small matrices that cannot fully occupy the Tensor Core units. Our results show that the proposed design can outperform the highly optimized vendor routine for sizes up to 100 by factors between 1.2× and 10× using a Tesla V100 GPU. For extremely small matrices, the observed speedups range between 1.8× and 26×.