We present the design and optimization of a solver for efficient and high-throughput computation of the marginalized graph kernel on General Purpose GPUs. The graph kernel is computed using the conjugate gradient method to solve a generalized Laplacian of the tensor product between a pair of graphs. To cope with the large gap between the instruction throughput and the memory bandwidth of the GPUs, our solver forms the graph tensor product on-the-fly without storing it in memory . This is achieved by using threads in a warp cooperatively to 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. We propose a new partition-based reordering algorithm for aggregating nonzero elements of the graphs into fewer but denser tiles to further exploit sparsity . 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.
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.
R
solver for
linear solver on General Purpose GPUs for the
R
computation
evaluation
R
on General Purpose GPUs. The graph kernel is computed using the conjugate gradient method to solve
between pairs of labeled graphs. The solver implements a preconditioned conjugate gradient (PCG) method to compute the solution to
R
of
equation associated with
R
between a pair of
of two
D
large
R
the
current generation
R
graph tensor product
tensor product linear system
R
. This is achieved
when performing matrix-vector dot product operations in PCG. Such on-the-fly computation is accomplished
R
cooperatively to
to cooperatively
R
We
Besides, we
R
further exploit sparsity
improve the efficiency of the sparse format