Optimizing Sparse Matrix-Vector Multiplications on An ARMv8-based Many-Core Architecture

Sparse matrix–vector multiplications (SpMV) are common in scientific and HPC applications but are hard to be optimized. While the ARMv8-based processor IP is emerging as an alternative to the traditional x64 HPC processor design, there is little study on SpMV performance on such new many-cores. To design efficient HPC software and hardware, we need to understand how well SpMV performs. This work develops a quantitative approach to characterize SpMV performance on a recent ARMv8-based many-core architecture, Phytium FT-2000 Plus (FTP). We perform extensive experiments involved over 9500 distinct profiling runs on 956 sparse datasets and five mainstream sparse matrix storage formats, and compare FTP against the Intel Knights Landing many-core. We experimentally show that picking the optimal sparse matrix storage format and parameters is non-trivial as the correct decision requires expert knowledge of the input matrix and the hardware. We address the problem by proposing a machine learning based model that predicts the best storage format and parameters using input matrix features. The model automatically specializes to the many-core architectures we considered. The experimental results show that our approach achieves on average 93% of the best-available performance without incurring runtime profiling overhead

Characterizing Scalability of Sparse Matrix–Vector Multiplications on Phytium FT-2000+

Understanding the scalability of parallel programs is crucial for software optimization and hardware architecture design. As HPC hardware is moving towards many-core design, it becomes increasingly difficult for a parallel program to make effective use of all available processor cores. This makes scalability analysis increasingly important. This paper presents a quantitative study for characterizing the scalability of sparse matrix–vector multiplications (SpMV) on Phytium FT-2000+, an ARM-based HPC many-core architecture. We choose SpMV as it is a common operation in scientific and HPC applications. Due to the newness of ARM-based many-core architectures, there is little work on understanding the SpMV scalability on such hardware design. To close the gap, we carry out a large-scale empirical evaluation involved over 1000 representative SpMV datasets. We show that, while many computation-intensive SpMV applications contain extensive parallelism, achieving a linear speedup is non-trivial on Phytium FT-2000+. To better understand what software and hardware parameters are most important for determining the scalability of a given SpMV kernel, we develop a performance analytical model based on the regression tree. We show that our model is highly effective in characterizing SpMV scalability, offering useful insights to help application developers for better optimizing SpMV on an emerging HPC architecture

The HPCG benchmark: analysis, shared memory preliminary improvements and evaluation on an Arm-based platform

The High-Performance Conjugate Gradient (HPCG) benchmark complements the LINPACK benchmark in the performance evaluation coverage of large High-Performance Computing (HPC) systems. Due to its lower arithmetic intensity and higher memory pressure, HPCG is recognized as a more representative benchmark for data-center and irregular memory access pattern workloads, therefore its popularity and acceptance is raising within the HPC community. As only a small fraction of the reference version of the HPCG benchmark is parallelized with shared memory techniques (OpenMP), we introduce in this report two OpenMP parallelization methods. Due to the increasing importance of Arm architecture in the HPC scenario, we evaluate our HPCG code at scale on a state-of-the-art HPC system based on Cavium ThunderX2 SoC. We consider our work as a contribution to the Arm ecosystem: along with this technical report, we plan in fact to release our code for boosting the tuning of the HPCG benchmark within the Arm community.Postprint (author's final draft

Reformulating the direct convolution for high-performance deep learning inference on ARM processors

We present two high-performance implementations of the convolution operator via the direct algorithm that outperform the so-called lowering approach based on the im2col transform plus the gemm kernel on an ARMv8-based processor. One of our methods presents the additional advantage of zero-memory overhead while the other employs an additional yet rather moderate workspace, substantially smaller than that required by the im2col+gemm solution. In contrast with a previous implementation of a similar zero-memory overhead direct convolution, this work exhibits the key advantage of preserving the conventional NHWC data layout for the input/output activations of the convolution layers.Funding for open access charge: CRUE-Universitat Jaume

Model-Architecture Co-design of Deep Neural Networks for Embedded Systems

In deep learning, a convolutional neural network (ConvNet or CNN) is a powerful tool for building interesting embedded applications that use data to make predictions. An application running on an embedded system typically has limited access to memory resources, processing power, and storage. Implementing deep convolutional neural network-based inference on resource-constrained devices can be very challenging, as these environments cannot usually make use of the massive computing power and storage that are present in cloud server environments. Furthermore, the constantly evolving nature of modern deep network architecture aggravates the problem by making it necessary to balance flexibility against specialisation to avoid the inability to adapt. However, much of the baseline architecture of a deep convolutional neural network stayed the same. With careful optimisation of the most common and widely occurring layer architectures, it is typically possible to accelerate these emerging workloads for resource-constrained embedded systems. This thesis makes four contributions. I first developed a lossy three-stage low-rank approximation scheme that can reduce the computational complexity of a pre-trained model by 3-5x and up to 8-9x for individual convolutional layers. This scheme requires restructuring of the convolutional layers and generally suits the scenario where both the training data and trained model are available. In many scenarios, the training data is not available for fine-tuning any loss in prediction accuracy if structural changes are made to a model as a post-processing step. Besides the lack of availability of training data, there are other situations where the architecture of a model cannot be changed after training. My second contribution handles this scenario by using a low-level optimisation scheme that requires no changes to the model architecture, unlike the low-rank approximation scheme. This novel scheme uses a modified version of the Cook-Toom algorithm to reduce the computational intensity of commonly occurring dense and spatial convolutional layers and speedup inference time by 2-4x. My third contribution is an efficient implementation of the Cook-Toom class of algorithms on ubiquitous Arm's low-power Cortex processor. Unlike the direct convolution, computing convolutions using the modified Cook-Toom algorithm requires a different data processing pipeline as it involves pre- and post-transformations of the intermediate activations. I introduced a multi-channel multi-region (MCMR) scheme to enable an efficient implementation of the fast Cook-Toom algorithm. I demonstrate that by effectively using SIMD instructions and the MCMR scheme an average 2-3x and a peak 4x per layer speedup is easily achievable. My final contribution is the Cook-Toom accelerator, a custom hardware architecture for modern convolutional neural networks. This accelerator architecture is designed from the ground up to address some of the limitations of a resource-constrained SIMD processor. I also illustrate how new emerging layer types can be mapped efficiently to the same flexible architecture without any modification

Adaptive Hybrid Storage Format for Sparse Matrix–Vector Multiplication on Multi-Core SIMD CPUs

Optimizing sparse matrix–vector multiplication (SpMV) is challenging due to the non-uniform distribution of the non-zero elements of the sparse matrix. The best-performing SpMV format changes depending on the input matrix and the underlying architecture, and there is no “one-size-fit-for-all” format. A hybrid scheme combining multiple SpMV storage formats allows one to choose an appropriate format to use for the target matrix and hardware. However, existing hybrid approaches are inadequate for utilizing the SIMD cores of modern multi-core CPUs with SIMDs, and it remains unclear how to best mix different SpMV formats for a given matrix. This paper presents a new hybrid storage format for sparse matrices, specifically targeting multi-core CPUs with SIMDs. Our approach partitions the target sparse matrix into two segmentations based on the regularities of the memory access pattern, where each segmentation is stored in a format suitable for its memory access patterns. Unlike prior hybrid storage schemes that rely on the user to determine the data partition among storage formats, we employ machine learning to build a predictive model to automatically determine the partition threshold on a per matrix basis. Our predictive model is first trained off line, and the trained model can be applied to any new, unseen sparse matrix. We apply our approach to 956 matrices and evaluate its performance on three distinct multi-core CPU platforms: a 72-core Intel Knights Landing (KNL) CPU, a 128-core AMD EPYC CPU, and a 64-core Phytium ARMv8 CPU. Experimental results show that our hybrid scheme, combined with the predictive model, outperforms the best-performing alternative by 2.9%, 17.5% and 16% on average on KNL, AMD, and Phytium, respectively

Cache Optimization and Performance Modeling of Batched, Small, and Rectangular Matrix Multiplication on Intel, AMD, and Fujitsu Processors

Factorization and multiplication of dense matrices and tensors are critical, yet extremely expensive pieces of the scientific toolbox. Careful use of low rank approximation can drastically reduce the computation and memory requirements of these operations. In addition to a lower arithmetic complexity, such methods can, by their structure, be designed to efficiently exploit modern hardware architectures. The majority of existing work relies on batched BLAS libraries to handle the computation of many small dense matrices. We show that through careful analysis of the cache utilization, register accumulation using SIMD registers and a redesign of the implementation, one can achieve significantly higher throughput for these types of batched low-rank matrices across a large range of block and batch sizes. We test our algorithm on 3 CPUs using diverse ISAs -- the Fujitsu A64FX using ARM SVE, the Intel Xeon 6148 using AVX-512 and AMD EPYC 7502 using AVX-2, and show that our new batching methodology is able to obtain more than twice the throughput of vendor optimized libraries for all CPU architectures and problem sizes