FTTN: Feature-Targeted Testing for Numerical Properties of NVIDIA & AMD Matrix Accelerators

Core Concepts

The author presents a rigorous methodology to create discriminatory tests for NVIDIA and AMD GPUs, focusing on feature differences in matrix accelerators to ensure reliable code porting.

Abstract

The content discusses the importance of understanding hardware-specific computational feature differences in GPUs, highlighting the risks of porting code across different platforms without considering these variations. The author provides detailed insights into testing methodologies targeting key attributes like subnormal support, rounding modes, extra bits, accumulation order control, and FMA unit width. By analyzing results from various GPUs, the content emphasizes the need for comprehensive testing to avoid significant discrepancies in computed results.

Stats

Examples of Dij values computed using a simple GEMM implementation on different GPUs: 0 on NVIDIA A100, V100, AMD MI250X and CPU; 255.875 on AMD MI100; and 191.875 on NVIDIA H100. All GPU models use RTN-TE rounding mode for outputting FP16 and BF16. For products involving FP32/FP64 inputs, all GPUs utilize the RTN-TE mode.

Quotes

"There is no prior work that shows that a simple test like this can produce three different answers on five different platforms—raising concern that one carefully understand Matrix Accelerator features before porting code across them." "We demonstrate that the lack of information on how these features differ across two matrix accelerators can make it impossible to reliably port codes across GPUs containing these differing accelerators." "Our primary contribution in this paper is a rigorous methodology that has enabled us to create such discriminatory tests for NVIDIA and AMD GPUs."

Key Insights Distilled From

FTTN

by Xinyi Li,Ang... at arxiv.org 03-04-2024

https://arxiv.org/pdf/2403.00232.pdf

Deeper Inquiries

How can manufacturers balance cost considerations with supporting more demanding rounding modes?

Manufacturers can balance cost considerations with supporting more demanding rounding modes by carefully evaluating the trade-offs between hardware complexity and precision requirements. Implementing more demanding rounding modes, such as round-to-nearest ties to even (RTN-TE), often requires additional resources in terms of extra precision bits and intricate circuitry to handle complex arithmetic operations. To manage costs while still meeting the demands for precise floating-point arithmetic, manufacturers may opt for a tiered approach where different GPUs within their product lineup offer varying levels of support for advanced rounding modes. For example, they might reserve full support for RTN-TE only on high-end models that cater to applications requiring utmost accuracy, while lower-tier GPUs could utilize simpler rounding schemes like truncation or round towards zero. By strategically allocating resources based on market needs and performance targets, manufacturers can strike a balance between providing robust numerical capabilities and controlling production costs effectively.

What implications do the findings have for developers working with mixed-precision algorithms like GMRES?

The findings from the study have significant implications for developers working with mixed-precision algorithms like GMRES (Generalized Minimal Residual Method). Understanding the subtle differences in numerical behavior across various GPU architectures is crucial when porting code or designing algorithms that rely on precise floating-point computations. Developers need to be aware of potential variations in results due to differences in features such as subnormal handling, extra precision bit support, accumulation order control, and FMA unit width among GPUs. Failure to account for these discrepancies could lead to unexpected errors or inaccuracies in algorithm outputs when transitioning between different hardware platforms. For GMRES specifically, which involves iterative refinement techniques using matrix multiplications and updates similar to those demonstrated in the study's matrix multiplication example, developers must consider how GPU-specific characteristics impact computation results. Adjustments may be necessary to ensure consistent performance across diverse GPU architectures while maintaining numerical stability and accuracy.

How might formal methods be applied to generalize and enhance the discriminatory tests developed in this study?

Formal methods can play a crucial role in generalizing and enhancing the discriminatory tests developed in this study by providing rigorous mathematical frameworks for verifying correctness properties of floating-point computations across different GPUs. Here are some ways formal methods could be applied: Property Specification: Formal specifications can define desired properties related to IEEE floating-point standards compliance, ensuring that tests cover essential aspects like subnormal number handling, rounding modes consistency, accumulation order control verification systematically. Model Checking: By creating formal models representing GPU architectures' numerical behaviors under specific conditions (e.g., input ranges), developers can use model checking techniques to exhaustively analyze all possible states and transitions within these models against defined properties. Theorem Proving: Utilizing theorem proving tools allows developers to formally prove mathematical assertions about floating-point operations' correctness under varying scenarios encountered during testing procedures. Abstraction Techniques: Abstract interpretation techniques enable simplification of complex numerical behaviors into more manageable representations without losing critical information needed for discrimination testing purposes. Automated Testing: Formal methods facilitate automated test generation based on specified properties derived from formal models or proofs, streamlining the process of identifying feature differences impacting computational results efficiently. By integrating formal methods into test development workflows, researchers can establish a solid foundation for comprehensive analysis of GPU-specific numeric features while ensuring robustness and reliability across diverse hardware environments.