The I/O Complexity of Attention, or How Optimal is Flash Attention?

Self-attention is at the heart of the popular Transformer architecture, yet suffers from quadratic time and memory complexity. The breakthrough FlashAttention algorithm revealed I/O complexity as the true bottleneck in scaling Transformers. Given two levels of memory hierarchy, a fast cache (e.g. GPU on-chip SRAM) and a slow memory (e.g. GPU high-bandwidth memory), the I/O complexity measures the number of accesses to memory. FlashAttention computes attention using $\frac{N^2d^2}{M}$ I/O operations where $N$ is the dimension of the attention matrix, $d$ the head-dimension and $M$ the cache size. However, is this I/O complexity optimal? The known lower bound only rules out an I/O complexity of $o(Nd)$ when $M=\Theta(Nd)$, since the output that needs to be written to slow memory is $\Omega(Nd)$. This leads to the main question of our work: Is FlashAttention I/O optimal for all values of $M$? We resolve the above question in its full generality by showing an I/O complexity lower bound that matches the upper bound provided by FlashAttention for any values of $M \geq d^2$ within any constant factors. Further, we give a better algorithm with lower I/O complexity for $M < d^2$, and show that it is optimal as well. Moreover, our lower bounds do not rely on using combinatorial matrix multiplication for computing the attention matrix. We show even if one uses fast matrix multiplication, the above I/O complexity bounds cannot be improved. We do so by introducing a new communication complexity protocol for matrix compression, and connecting communication complexity to I/O complexity. To the best of our knowledge, this is the first work to establish a connection between communication complexity and I/O complexity, and we believe this connection could be of independent interest and will find many more applications in proving I/O complexity lower bounds in the future.

Predicting the Stereoselectivity of Chemical Transformations by Machine Learning

Stereoselective reactions (both chemical and enzymatic reactions) have been essential for origin of life, evolution, human biology and medicine. Since late 1960s, there have been numerous successes in the exciting new frontier of asymmetric catalysis. However, most industrial and academic asymmetric catalysis nowadays do follow the trial-and-error model, since the energetic difference for success or failure in asymmetric catalysis is incredibly small. Our current understanding about stereoselective reactions is mostly qualitative that stereoselectivity arises from differences in steric effects and electronic effects in multiple competing mechanistic pathways. Quantitatively understanding and modulating the stereoselectivity of for a given chemical reaction still remains extremely difficult. As a proof of principle, we herein present a novel machine learning technique, which combines a LASSO model and two Random Forest model via two Gaussian Mixture models, for quantitatively predicting stereoselectivity of chemical reactions. Compared to the recent ground-breaking approach [1], our approach is able to capture interactions between features and exploit complex data distributions, which are important for predicting stereoselectivity. Experimental results on a recently published dataset demonstrate that our approach significantly outperform [1]. The insight obtained from our results provide a solid foundation for further exploration of other synthetically valuable yet mechanistically intriguing stereoselective reactions.