insight - Matrix computation - # Ozaki scheme for high-precision matrix multiplication on low-precision computing units

Core Concepts

The Ozaki scheme can leverage integer matrix multiplication units (IMMUs) to compute high-precision matrix multiplication faster and more efficiently than using floating-point matrix multiplication units (FMMUs).

Abstract

The paper explores the use of integer matrix multiplication units (IMMUs) for the Ozaki scheme, which computes high-precision matrix multiplication using lower-precision computing units. The key findings are:
Theoretical advantages of using IMMUs over FMMUs:
IMMUs can store more valid bits per byte in a slice, allowing for fewer splits to maintain the same accuracy.
IMMUs require less working memory by reducing duplicated exponent representation and the number of splits.
IMMUs can reduce the number of matrix multiplications squared to the number of splits.
IMMUs typically have higher throughput than FMMUs.
INT8-INT32 IMMUs are the most suitable for the Ozaki scheme.
Experimental evaluation:
Accuracy experiments show that INT8x11 and INT8x13 maintain accuracy even with wide exponent distribution inputs, unlike INT8x9.
Throughput experiments demonstrate that on NVIDIA consumer GPUs, the Ozaki scheme on integer Tensor Cores can outperform cuBLAS DGEMM and an existing implementation on FP16 Tensor Cores by up to 6x.
The Ozaki scheme on integer Tensor Cores is applied to quantum circuit simulation, achieving up to 4.33x throughput improvement over cuBLAS ZGEMM while maintaining FP64 accuracy.

Stats

The number of matrix multiplication operations in the Ozaki scheme is (1 + s) * s/2, where s is the number of splits.
The memory size for storing the matrix slices is proportional to "the number of splits" * "storage size of the slice".

Quotes

"We show the theoretical advantages of using the integer matrix multiplication unit instead of floating point one concerning the accuracy, memory consumption, and the number of operations. It reduces the 50% ∼75% of working memory, a major concern in the Ozaki scheme in practical use, in the middle ∼large size of matrix multiplication."
"Our implementation outperforms cuBLAS DGEMM and the existing implementation up to about 6× on NVIDIA consumer GPUs."
"We have achieved up to 4.33× throughput improvement compared to cuBLAS ZGEMM computation on NVIDIA RTX6000 Ada GPU while maintaining the FP64 accuracy."

Key Insights Distilled From

by Hiroyuki Oot... at **arxiv.org** 04-02-2024

Deeper Inquiries

To further optimize the Ozaki scheme on integer Tensor Cores for higher throughput, several strategies can be implemented:
Memory Optimization: Implementing efficient memory management techniques to reduce the memory footprint of storing matrix slices. This can involve optimizing the data structures used for storing the slices and minimizing redundant data storage.
Parallelization: Utilizing parallel processing techniques to distribute the workload across multiple cores or threads effectively. This can help in maximizing the utilization of the computational resources available on the Tensor Cores.
Algorithmic Improvements: Refining the splitting algorithm used in the Ozaki scheme to minimize the number of arithmetic operations required for matrix multiplication. This can involve optimizing the slicing process and the accumulation of results to reduce computational overhead.
Hardware-Specific Optimization: Leveraging the unique features and capabilities of the integer Tensor Cores to tailor the implementation for optimal performance. This can include utilizing specific instructions or hardware accelerators available on the Tensor Cores.
Precision Adjustment: Fine-tuning the precision settings of the integer arithmetic operations to strike a balance between accuracy and throughput. Adjusting the precision levels based on the specific requirements of the application can help in achieving higher performance.

While the Ozaki scheme on integer Tensor Cores shows promise for accelerating quantum circuit simulations, there are potential limitations and challenges in applying it to other HPC applications:
Data Dependency: Some HPC applications may have complex data dependencies that could impact the efficiency of the Ozaki scheme. Ensuring that the data dependencies are managed effectively to leverage the parallel processing capabilities of the Tensor Cores is crucial.
Algorithm Suitability: The Ozaki scheme may not be suitable for all types of HPC algorithms. Certain algorithms may require different computational approaches that may not align well with the matrix multiplication optimization provided by the Ozaki scheme.
Resource Constraints: The Ozaki scheme relies on the availability of integer Tensor Cores, which may not be present in all HPC hardware configurations. Ensuring compatibility and optimizing the implementation for different hardware architectures can be a challenge.
Accuracy Requirements: Some HPC applications demand high precision and accuracy, which may be challenging to achieve with the Ozaki scheme on integer Tensor Cores. Balancing the trade-off between accuracy and performance is essential for successful implementation.

Combining the Ozaki scheme on integer Tensor Cores with mixed-precision computing techniques can offer enhanced performance-accuracy trade-offs:
Hybrid Precision: Utilizing a combination of integer arithmetic for the matrix multiplication core operations and mixed-precision computing for specific computations can optimize the overall performance. This approach can leverage the strengths of both techniques to achieve better efficiency.
Dynamic Precision Adjustment: Implementing algorithms that dynamically adjust the precision levels based on the computational requirements can optimize the performance-accuracy trade-off. Adapting the precision settings during runtime based on the complexity of the computations can enhance efficiency.
Error Correction Techniques: Integrating error correction mechanisms within the Ozaki scheme on integer Tensor Cores can improve the accuracy of the results. By combining error detection and correction methods with the integer arithmetic operations, a more robust and accurate computation can be achieved.
Performance Profiling: Conducting thorough performance profiling and analysis to identify the optimal precision settings and combinations for different parts of the computation. This can help in fine-tuning the implementation to achieve the best performance-accuracy balance.

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