Transformers are slow and memory-hungry on long sequences, since the time and memory complexity of self-attention are quadratic in sequence length. Approximate attention methods have attempted to address this problem by trading off model quality to reduce the compute complexity, but often do not achieve wall-clock speedup. We argue that a missing principle is making attention algorithms IO-aware -- accounting for reads and writes between levels of GPU memory. We propose FlashAttention, an IO-aware exact attention algorithm that uses tiling to reduce the number of memory reads/writes between GPU high bandwidth memory (HBM) and GPU on-chip SRAM. We analyze the IO complexity of FlashAttention, showing that it requires fewer HBM accesses than standard attention, and is optimal for a range of SRAM sizes. We also extend FlashAttention to block-sparse attention, yielding an approximate attention algorithm that is faster than any existing approximate attention method. FlashAttention trains Transformers faster than existing baselines: 15% end-to-end wall-clock speedup on BERT-large (seq. length 512) compared to the MLPerf 1.1 training speed record, 3$\times$ speedup on GPT-2 (seq. length 1K), and 2.4$\times$ speedup on long-range arena (seq. length 1K-4K). FlashAttention and block-sparse FlashAttention enable longer context in Transformers, yielding higher quality models (0.7 better perplexity on GPT-2 and 6.4 points of lift on long-document classification) and entirely new capabilities: the first Transformers to achieve better-than-chance performance on the Path-X challenge (seq. length 16K, 61.4% accuracy) and Path-256 (seq. length 64K, 63.1% accuracy).

FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness

Semiseparable matrices and many other rank-structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We represent HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS forms. Fast HSS algorithms including new HSS structure generation and HSS form Cholesky factorization are developed. Moreover, we provide a new linear complexity explicit ULV factorization algorithm for symmetric positive definite HSS matrices with a low-rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the algorithms. All these algorithms have nice data locality. They are useful in developing fast-structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc. Some applications are shown. Copyright q 2009 John Wiley & Sons, Ltd.

https://www.math.purdue.edu/~xiaj/work/fasthss.pdf

Fast algorithms for hierarchically semiseparable matrices

We consider an algebraic representation that is useful for matrices with off-diagonal blocks of low numerical rank. A fast and stable solver for linear systems of equations in which the coefficient matrix has this representation is presented. We also present a fast algorithm to construct the hierarchically semiseparable representation in the general case.

A Fast $ULV$ Decomposition Solver for Hierarchically Semiseparable Representations

Methods for the solution of boundary integral equations have changed significantly during the last two decades. This is due, in part, to improvements in computer hardware, but more importantly, to the development of fast algorithms which scale linearly or nearly linearly with the number of degrees of freedom required. These methods are typically iterative, based on coupling fast matrix-vector multiplication routines with conjugate-gradient-type schemes. Here, we discuss methods that are currently under development for the fast, direct solution of boundary integral equations in three dimensions. After reviewing the mathematical foundations of such schemes, we illustrate their performance with some numerical examples, and discuss the potential impact of the overall approach in a variety of settings.

/pdf/fast-direct-solvers-for-integral-equations-in-complex-three-3wwdj2pbew.pdf

Fast direct solvers for integral equations in complex three-dimensional domains

We consider the modeling of (polarized) seismic wave propagation on a rectangular domain via the discretization and solution of the inhomogeneous Helmholtz equation in 3D, by exploiting a parallel multifrontal sparse direct solver equipped with Hierarchically Semi-Separable (HSS) structure to reduce the computational complexity and storage. In particular, we are concerned with solving this equation on a large domain, for a large number of different forcing terms in the context of seismic problems in general, and modeling in particular. We resort to a parsimonious mixed grid finite differences scheme for discretizing the Helmholtz operator and Perfect Matched Layer boundaries, resulting in a non-Hermitian matrix. We make use of a nested dissection based domain decomposition, and introduce an approximate direct solver by developing a parallel HSS matrix compression, factorization, and solution approach. We cast our massive parallelization in the framework of the multifrontal method. The assembly tree is partitioned into local trees and a global tree. The local trees are eliminated independently in each processor, while the global tree is eliminated through massive communication. The solver for the inhomogeneous equation is a parallel hybrid between multifrontal and HSS structure. The computational complexity associated with the factorization is almost linear in the size, n say, of the matrix, viz. between O(nlogn )a ndO(n 4/3 logn), while the storage is almost linear as well, between O(n )a ndO(nlogn). We exploit the use of a regular (Cartesian) mesh common in many seismic applications.

https://www.math.purdue.edu/~xiaj/work/Helmholtz3D.pdf

On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver

In this paper we continue the study of a class of structured matrices which admit a linear complexity inversion algorithm. This class contains in particular diagonal plus semiseparable matrices of order one and tridiagonal matrices. For this class explicit inversion formulas are obtained and linear complexity inversion algorithms are developed. The implementation of algorithms is illustrated by numerical experiments. The new class which is studied here appears naturally as the class of matrices of input output operators for descriptor linear systems.

Linear complexity inversion algorithms for a class of structured matrices

Here are considered matrices represented as a sum of diagonal and semiseparable ones. These matrices belong to the class of structured matrices which arises in numerous applications. FastO(N) algorithms for their inversion were developed before under additional restrictions which are a source of instability. Our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms. In this paper we obtain such algorithms with the only requirement that the considered matrix is invertible and its determinant is not close to zero. The case of semiseparable matrices of order one was considered in detail in an earlier paper of the authors.

https://link.springer.com/content/pdf/10.1007/BF01191530.pdf

Fast inversion algorithms for diagonal plus semiseparable matrices

LetH be a Hilbert space andR∶H→H be a bounded linear operator represented by an operator matrix which is a sum of a diagonal and of a semiseparable type of order one operator matrices. We consider three methods for solution of the operator equationRx=y. The obtained results yields new algorithms for solution of integral equations and for inversion of matrices.

Algorithms for inversion of diagonal plus semiseparable operator matrices

In this paper we continue the study of structured matrices which admit a linear complexity inversion algorithm. The new class which is studied here appears naturally as the class of matrices of input output operators for discrete time dependent descriptor linear systems. The algebra of such operators is analyzed. Multiplication and inversion algorithms of linear complexity are presented and their implementation is illustrated.

On a new class of structured matrices

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Papers

Linear complexity inversion algorithms for a class of structured matrices

Fast inversion algorithms for diagonal plus semiseparable matrices

Algorithms for inversion of diagonal plus semiseparable operator matrices

On a new class of structured matrices