FlashFFTConv: Efficient Convolutions for Long Sequences with Tensor
  Cores
Authors: name: Daniel Y. Fu, name: Hermann Kumbong, name: Eric Nguyen, name: Christopher Ré

Convolution models with long filters have demonstrated state-of-the-art
reasoning abilities in many long-sequence tasks but lag behind the most
optimized Transformers in wall-clock time. A major bottleneck is the Fast
Fourier Transform (FFT)--which allows long convolutions to run in O(N logN)
time in sequence length N but has poor hardware utilization. In this paper,
we study how to optimize the FFT convolution. We find two key bottlenecks: the
FFT does not effectively use specialized matrix multiply units, and it incurs
expensive I/O between layers of the memory hierarchy. In response, we propose
FlashFFTConv. FlashFFTConv uses a matrix decomposition that computes the FFT
using matrix multiply units and enables kernel fusion for long sequences,
reducing I/O. We also present two sparse convolution algorithms--1) partial
convolutions and 2) frequency-sparse convolutions--which can be implemented
simply by skipping blocks in the matrix decomposition, enabling further
opportunities for memory and compute savings. FlashFFTConv speeds up exact FFT
convolutions by up to 7.93times over PyTorch and achieves up to 4.4times
speedup end-to-end. Given the same compute budget, FlashFFTConv allows
Hyena-GPT-s to achieve 2.3 points better perplexity on the PILE and
M2-BERT-base to achieve 3.3 points higher GLUE score--matching models with
twice the parameter count. FlashFFTConv also achieves 96.1% accuracy on
Path-512, a high-resolution vision task where no model had previously achieved
better than 50%. Furthermore, partial convolutions enable longer-sequence
models--yielding the first DNA model that can process the longest human genes
(2.3M base pairs)--and frequency-sparse convolutions speed up pretrained models
while maintaining or improving model quality.