A High-Throughput Solver for Marginalized Graph Kernels on GPU

We present the design and optimization of a linear solver on General Purpose GPUs for the efficient and high-throughput evaluation of the marginalized graph kernel between pairs of labeled graphs. The solver implements a preconditioned conjugate gradient (PCG) method to compute the solution to a generalized Laplacian equation associated with the tensor product of two graphs. To cope with the gap between the instruction throughput and the memory bandwidth of current generation GPUs, our solver forms the tensor product linear system on-the-fly without storing it in memory when performing matrix-vector dot product operations in PCG. Such on-the-fly computation is accomplished by using threads in a warp to cooperatively stream the adjacency and edge label matrices of individual graphs by small square matrix blocks called tiles, which are then staged in registers and the shared memory for later reuse. Warps across a thread block can further share tiles via the shared memory to increase data reuse. We exploit the sparsity of the graphs hierarchically by storing only non-empty tiles using a coordinate format and nonzero elements within each tile using bitmaps. Besides, we propose a new partition-based reordering algorithm for aggregating nonzero elements of the graphs into fewer but denser tiles to improve the efficiency of the sparse format.We carry out extensive theoretical analyses on the graph tensor product primitives for tiles of various density and evaluate their performance on synthetic and real-world datasets. Our solver delivers three to four orders of magnitude speedup over existing CPU-based solvers such as GraKeL and GraphKernels. The capability of the solver enables kernel-based learning tasks at unprecedented scales

Topology-aware optimization of big sparse matrices and matrix multiplications on main-memory systems

Since data sizes of analytical applications are continuously growing, many data scientists are switching from customized micro-solutions to scalable alternatives, such as statistical and scientific databases. However, many algorithms in data mining and science are expressed in terms of linear algebra, which is barely supported by major database vendors and big data solutions. On the other side, conventional linear algebra algorithms and legacy matrix representations are often not suitable for very large matrices. We propose a strategy for large matrix processing on modern multicore systems that is based on a novel, adaptive tile matrix representation (AT MATRIX). Our solution utilizes multiple techniques inspired from database technology, such as multidimensional data partitioning, cardinality estimation, indexing, dynamic rewrites, and many more in order to optimize the execution time. Based thereon we present a matrix multiplication operator ATMULT, which outperforms alternative approaches. The aim of our solution is to overcome the burden for data scientists of selecting appropriate algorithms and matrix storage representations. We evaluated AT MATRIX together with ATMULT on several real-world and synthetic random matrices

Polyhedral+Dataflow Graphs

This research presents an intermediate compiler representation that is designed for optimization, and emphasizes the temporary storage requirements and execution schedule of a given computation to guide optimization decisions. The representation is expressed as a dataflow graph that describes computational statements and data mappings within the polyhedral compilation model. The targeted applications include both the regular and irregular scientific domains. The intermediate representation can be integrated into existing compiler infrastructures. A specification language implemented as a domain specific language in C++ describes the graph components and the transformations that can be applied. The visual representation allows users to reason about optimizations. Graph variants can be translated into source code or other representation. The language, intermediate representation, and associated transformations have been applied to improve the performance of differential equation solvers, or sparse matrix operations, tensor decomposition, and structured multigrid methods

Input-sensitive dense-sparse primitive compositions for GNN acceleration

Graph neural networks (GNN) have become an important class of neural network models that have gained popularity in domains such as social and financial network analysis. Different phases of GNN computations can be modeled using both dense and sparse matrix operations. There have been many frameworks and optimization techniques proposed in the literature to accelerate GNNs. However, getting consistently high performance across many input graphs with different sparsity patterns and GNN embedding sizes has remained difficult. In this paper, we propose different algebraic reassociations of GNN computations that lead to novel dense and sparse matrix primitive selections and compositions. We show that the profitability of these compositions depends on the input graph, embedding size, and the target hardware. We developed SENSEi, a system that uses a data-driven adaptive strategy to select the best composition given the input graph and GNN embedding sizes. Our evaluations on a wide range of graphs and embedding sizes show that SENSEi achieves geomean speedups of $1.105\times$ (up to $2.959\times$) and $1.187\times$ (up to $1.99\times$) on graph convolutional networks and geomean speedups of $2.307\times$ (up to $35.866\times$) and $1.44\times$ (up to $5.69\times$) on graph attention networks on CPUs and GPUs respectively over the widely used Deep Graph Library. Further, we show that the compositions yield notable synergistic performance benefits on top of other established sparse optimizations such as sparse matrix tiling by evaluating against a well-tuned baseline