Day 76: Mixed Precision & Tensor Cores (If Supported)

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Modern NVIDIA GPUs (Volta and later) include Tensor Cores that accelerate matrix operations at mixed precision, typically FP16 or BF16 arithmetic. Harnessing these can dramatically improve throughput in HPC and deep learning workloads, but reduced precision can introduce accuracy trade-offs if not carefully managed. In Day 76, we explore mixed precision usage, focusing on FP16 matrix multiplication on Tensor Cores, and discuss best practices for balancing speed and accuracy. Multiple conceptual diagrams illustrate how data flows through Tensor Cores and how partial sums at lower precision might affect final results.

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Table of Contents

1. Overview
2. Why Mixed Precision and Tensor Cores?
3. Precision Trade-Offs
4. Implementation Steps for FP16 Matrix Multiply
   • a) Data Conversion (FP32 ↔ FP16)
   • b) Using Tensor Cores via WMMA API
   • c) Launch Configuration & Mixed Precision Accumulation
5. Code Example: FP16 GEMM with Tensor Cores (WMMA)
   • Explanation & Comments
6. Common Pitfalls & Accuracy Issues
7. Multiple Conceptual Diagrams
   • Diagram 1: Mixed Precision Data Flow
   • Diagram 2: Tensor Core WMMA Tile Mapping
   • Diagram 3: Accumulation at Higher Precision
8. References & Further Reading
9. Conclusion
10. Next Steps

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1. Overview

NVIDIA Tensor Cores introduced in Volta (and improved in later architectures) accelerate matrix-matrix multiplications by operating on FP16 or BF16 input tiles, often accumulating partial sums in higher precision (FP32). This mixed precision approach enables significantly higher throughput for dense matrix ops, widely used in deep learning or HPC. However, reduced precision can cause errors if the data range is large or if the algorithm is sensitive to rounding.

Key Focus: Implementing a FP16 matrix multiply on Tensor Cores, discussing how to feed half-precision data, use the WMMA (Warp Matrix Multiply-Accumulate) intrinsics, and accumulate partial sums in FP32 to preserve final result accuracy.

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2. Why Mixed Precision and Tensor Cores?

1. Higher Throughput: Tensor Cores can deliver multiple TFLOPS for matrix ops, far outstripping conventional FP32 or FP64.
2. Lower Memory Footprint: Storing half-precision data halves memory usage for matrix inputs, often doubling effective memory bandwidth.
3. AI & HPC: Many deep learning tasks rely on FP16, while some HPC workflows can tolerate partial or final mixed precision for performance gains.

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3. Precision Trade-Offs

• Range & Rounding: FP16 has fewer exponent bits and mantissa bits, limiting representable range and precision. Large or extremely small values can degrade accuracy.
• Accumulate in FP32: Commonly, partial sums are done in FP32 to mitigate catastrophic rounding.
• Validation: For HPC tasks requiring tight error bounds, carefully check if the final results meet domain-specific precision requirements.

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4. Implementation Steps for FP16 Matrix Multiply

a) Data Conversion (FP32 ↔ FP16)

• Loading: If your data is in FP32, convert to FP16 on the host or device. If the data size is large, consider pinned memory for faster transfers.
• Storing: After the operation, you may convert the result back to FP32 for consumption by other parts of the pipeline that require higher precision.

b) Using Tensor Cores via WMMA API

• WMMA (Warp Matrix Multiply-Accumulate) intrinsics: Provide warp-level ops for 16×16 or 8×8 tile multiplications.
• Tile Mismatch: You must tile your matrices in multiples of the warp-level matrix shapes (e.g., 16×16). If partial blocks remain, handle them separately.

c) Launch Configuration & Mixed Precision Accumulation

• Block Layout: Usually each warp processes a tile. Possibly multiple warps per block.
• Accumulation: Many HPC codes keep partial sums in FP32 to reduce rounding errors, while input & intermediate are FP16.

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5. Code Example: FP16 GEMM with Tensor Cores (WMMA)

Below is a simplified snippet demonstrating how to do a 16×16 tile multiplication using half precision on Tensor Cores. This example uses the cuda::wmma namespace introduced in newer CUDA versions. (Truncated for brevity.)

```cpp
// Mixed Precision (FP16) Matrix Multiply on Tensor Cores
// File: tensorcore_wmma_example.cu

#include <cuda_fp16.h>
#include <mma.h>
using namespace nvcuda::wmma;

__global__ void wmmaGemmKernel(half *A, half *B, float *C,
                               int M, int N, int K) {
    // Each warp will compute a tile of C using wmma
    // NOTE: M, N, K must be multiples of 16 for simplicity

    // 1) Create fragment objects
    fragment<matrix_a, 16, 16, 16, half, col_major> aFrag;
    fragment<matrix_b, 16, 16, 16, half, col_major> bFrag;
    fragment<accumulator, 16, 16, 16, float> cFrag;

    fill_fragment(cFrag, 0.0f); // init to zero

    // 2) Map warp to 16x16 tile in row/col space
    int warpM = (blockIdx.x * blockDim.x + threadIdx.x) / 32; // example
    // This is a simplistic approach ignoring partial warps, so adjust as needed.

    // 3) Load A, B sub-matrices
    load_matrix_sync(aFrag, A + warpM*16, K); // if col_major
    load_matrix_sync(bFrag, B + warpM*16, K);

    // 4) Perform the MMA operation
    mma_sync(cFrag, aFrag, bFrag, cFrag);

    // 5) Store results (accumulator is float)
    store_matrix_sync(C + warpM*16, cFrag, N, mem_col_major);
}

int main() {
    // host data, device memory, etc.
    // A, B in half, C in float
    // call wmmaGemmKernel<<<...>>>(A, B, C, M, N, K);

    return 0;
}

Explanation & Comments

1. FP16: The __half or half type is used for A and B. The final accumulation uses float.
2. wmma: We create fragment objects for each operand, then mma_sync() runs the fused multiply-accumulate on Tensor Cores.
3. Tiling: This snippet is incomplete—real code must carefully map each warp to a sub-tile, handle partial edges, and do a gather/scatter for the full matrix.

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6. Common Pitfalls & Accuracy Issues

• FP16 Range: If input data spans large magnitudes, some values may underflow or overflow in half precision, losing important bits.
• Alignment & Tiling: Tensor Cores require alignment to 16×16 tile boundaries. Non-multiple matrix dimensions need separate handling.
• Intermediate Storage: Partial sums typically in FP32 to reduce rounding, but final results might remain in FP16 for memory savings.
• API Evolution: The wmma API may differ slightly across CUDA versions. Always consult the latest documentation.

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7. Multiple Conceptual Diagrams

Diagram 1: Mixed Precision Data Flow

flowchart LR
    A[FP32 Host Data] --> B[Convert/Store as FP16 on device]
    B --> C[Tensor Cores operate in half]
    C --> D[Accumulate partial sums in FP32 internally]
    D --> E[Result can be FP32 or FP16]

Explanation: The data is loaded in FP16 for multiply, partial sums stored in FP32, final result can be kept in either precision as needed.

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Diagram 2: Tensor Core WMMA Tile Mapping

flowchart TD
    subgraph Warp
    A16x16[A sub-tile: 16x16 half]
    B16x16[B sub-tile: 16x16 half]
    C16x16[Output tile: 16x16 float]
    end

    A16x16 -- MMA --> C16x16
    B16x16 -- MMA --> C16x16

Explanation: Each warp handles a 16×16 sub-tile for A and B, performing the matrix multiply-accumulate to produce a 16×16 sub-tile in the output matrix.

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Diagram 3: Accumulation at Higher Precision

flowchart LR
    A[FP16: a, b]
    B[(a * b) => partial in FP16?]
    C[accumulate in FP32 => sum]
    D[final result => either float or half]

    A --> B --> C --> D

Explanation: Even though inputs are half, the actual multiply-accumulate path often promotes partial sums to FP32 to reduce catastrophic rounding.

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8. References & Further Reading

• CUDA C Programming Guide – “Tensor Core Programming”
• NVIDIA Developer Blog – Mixed Precision & HPC
• PyTorch / cuBLAS HPC Guides for FP16 GEMM
• Nsight Systems for Analyzing Tensor Core usage

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9. Conclusion

Day 76 underscores how mixed precision with Tensor Cores can drastically accelerate matrix multiply performance, albeit with potential precision trade-offs. By operating in FP16 for inputs while accumulating in FP32, you capture the benefits of lower precision and faster throughput from Tensor Cores, typically sustaining near-peak performance on modern NVIDIA GPUs. The key is verifying that numerical accuracy remains acceptable. Tools like WMMA intrinsics or libraries (like cuBLAS) can streamline usage, while being mindful of alignment constraints and data layout.

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10. Next Steps

1. Convert a Real HPC Kernel to half precision, measure performance speedup, and track any changes in result accuracy.
2. Profile: Use Nsight Compute to see if the kernel saturates Tensor Cores or if memory bandwidth remains the bottleneck.
3. Double-Check: If domain demands high accuracy, partial or final re-accumulation in FP32 or FP64 may be required.
4. Tiling: Ensure matrix dimensions align with the 16×16 block structure needed for wmma to avoid overhead in partial tiles.
5. Expand: Evaluate BF16 or other low-precision formats if your GPU or library supports them.

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