Inside the GPU: How Tensor Cores Revolutionize Matrix Math#

Basic Operation#

The classic definition of matrix multiplication goes like this: if we have two matrices, A (of size M×N) and B (of size N×P), their product C = A × B is an M×P matrix. Each entry c_ij in the matrix C is computed as the dot product of the ith row of A and the jth column of B:

c_ij = ∑ (A_ik × B_kj), for k = 1 to N

Conceptually, for each element in the resulting matrix, you multiply corresponding elements from a row in A and a column in B, then sum those products. The number of total multiplications and additions required for an M×N by N×P multiplication is O(M × N × P). This computation grows quickly and is especially large for deep learning tasks where matrix dimensions may run into the thousands.

Example#

Here is a simple example of a 2×2 by 2×2 multiplication:

Let
A = [ [2, 3],
[1, 4] ]

B = [ [0, 1],
[5, 2] ]

Then C = A × B = [ [2×0 + 3×5, 2×1 + 3×2],
[1×0 + 4×5, 1×1 + 4×2] ]

C = [ [15, 8],
[20, 9] ]

While trivial for small dimensions, real-world neural networks and scientific simulations involve much larger matrices, resulting in billions or trillions of operations. This is why GPUs have become a go-to for acceleration.

Streaming Multiprocessors#

In NVIDIA GPUs, the fundamental building block is the Streaming Multiprocessor (SM). Each SM contains:

• A set of CUDA cores (traditional GPU cores)
• Registers and shared memory
• Additional specialized units, such as arithmetic logic units and floating-point units
• (In newer architectures) Tensor Cores for matrix operations

When a GPU kernel—an instruction set designed to run on the GPU—is launched, it’s distributed across many SMs. Each SM processes a set of threads in parallel. This is why GPUs can handle large matrix multiplication tasks so effectively: each multiplication-addition can be assigned to a thread, which then runs in parallel with thousands of other threads.

The Need for Specialized Hardware#

While conventional GPU cores (often called scalar or CUDA cores) are fast at parallel floating-point operations, increasing demands in machine learning and high-performance computing led hardware manufacturers to introduce specialized units. These specialized units target the most computationally heavy operations—like matrix-matrix multiplication—more directly. On NVIDIA GPUs, these specialized units are the Tensor Cores.

How They Work#

Tensor Cores operate on small matrix blocks, typically 4×4 or 8×8 sub-blocks (depending on the generation). They simultaneously perform multiple multiply-accumulate (MAC) operations. For instance, in some architectures, a single Tensor Core can perform 4×4 FP16 matrix operations in one clock cycle. This is an enormous throughput boost compared to using regular GPU cores to accomplish the same math.

At a low level, a Tensor Core generates a large number of fused multiply-add operations. If you think about the block multiply-accumulate for matrices A, B, and C:

C_mapped = A × B + C_original

This is effectively a single “instruction” for the Tensor Core, but internally it performs multiple multiplications in parallel, accumulating results with minimal stalls in the pipeline. The hardware scheduling is carefully managed by the GPU’s SM, ensuring that each Tensor Core is fed data at the right moment.

Purpose-Built for Mixed-Precision#

One of the defining characteristics of Tensor Cores is their focus on mixed-precision arithmetic. Early versions targeted FP16 (aka half-precision) for multiplication with an FP32 accumulate step. Newer versions support additional data types:

• FP16 (16-bit floating point)
• Bfloat16 (Brain Float, 16-bit floating point with a larger exponent range)
• TensorFloat-32 (TF32) on newer architectures
• INT8 / INT4 for inference acceleration

Mixed-precision works by performing multiplications in a lower precision (e.g., FP16) and accumulating results in a higher precision (FP32). This approach offers a balanced tradeoff between speed and numerical accuracy. The GPU can do more operations at once in FP16 compared to FP32, accelerating tasks like training neural networks without significantly degrading model accuracy (especially when combined with proper loss scaling techniques).

Example of FP16 + FP32 Accumulation#

Imagine a block multiply operation:

A (FP16) × B (FP16) → C (accumulation in FP32)

Pseudo-code:

1. Convert A₁₆, B₁₆ to 16-bit floats in registers.
2. Multiply 16-bit floating-point elements.
3. Accumulate in 32-bit floating point to avoid round-off during summation.
4. Store final results in a 16-bit or 32-bit buffer depending on the user’s needs.

The benefits can be enormous: smaller data size, reduced memory footprint, and faster throughput. Tensor Cores handle these intricacies at the hardware level, shielding the programmer from low-level details.

PyTorch Example#

Modern deep learning frameworks such as PyTorch and TensorFlow automatically use Tensor Cores for supported operations if the hardware is available and the data types are suitable. Below is an example of how you might enable mixed-precision training in PyTorch:

1
import torch
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import torch.nn as nn
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import torch.optim as optim
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model = nn.Sequential(
6
    nn.Linear(512, 512),
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    nn.ReLU(),
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    nn.Linear(512, 10)
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)
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# Move model to GPU
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model = model.cuda()
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# Use an optimizer
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optimizer = optim.Adam(model.parameters(), lr=0.001)
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# Amp (Automatic Mixed Precision)
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scaler = torch.cuda.amp.GradScaler()
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loss_fn = nn.CrossEntropyLoss()
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for epoch in range(10):
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    for images, labels in data_loader:
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        images = images.cuda()
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        labels = labels.cuda()
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        optimizer.zero_grad()
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        # Forward pass in mixed precision
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        with torch.cuda.amp.autocast():
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            outputs = model(images)
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            loss = loss_fn(outputs, labels)
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        # Scale losses
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        scaler.scale(loss).backward()
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        scaler.step(optimizer)
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        scaler.update()

In the snippet above:

1. We import PyTorch libraries and construct a simple feed-forward model.
2. The model is moved to the GPU using model.cuda().
3. Automatic Mixed Precision (AMP) is used to reduce the precision of the forward pass wherever applicable, thus leveraging Tensor Cores for matrix operations. The library will automatically choose FP16 for certain operations, while preserving FP32 for others (e.g., batch normalization or final accumulation).
4. The GradScaler ensures stable gradients by scaling them up before backpropagation and scaling them down afterwards to avoid underflow in FP16.

Under the hood, if the GPU supports Tensor Cores, PyTorch will schedule the recommended sub-block operations in the most efficient manner.

Mixed Precision Training Mechanisms#

When we say “mixed precision,” we’re talking about using multiple floating-point data types within the same model or compute session. This is usually done to balance the performance benefits of lower precision with the accuracy advantages of higher precision.

1. FP16/FP32 Mix: One of the most common forms is to perform multiplications in FP16 but accumulate in FP32.
2. Loss Scaling: Because reducing precision can lead to numerical underflow, frameworks employ dynamic or static scaling of losses during backprop.
3. Batch Normalization: Layers that are particularly sensitive to floating-point errors may remain in FP32 calculations.

TensorFloat-32 (TF32)#

Introduced in NVIDIA Ampere architecture, TF32 is a format that uses FP32 range but with 10-bit mantissa (comparable to FP16’s 10-bit mantissa) for certain matrix operations. Essentially, TF32 tries to strike a balance by maintaining the wide dynamic range of full 32-bit floats while cutting down precision. This allows for accelerated training without a major overhaul of code or hyperparameter settings.

Error Analysis#

A primary concern with reduced-precision arithmetic is numerical stability. Some algorithms are more sensitive to small rounding errors. Key strategies for mitigating issues include:

• Using accumulations in a higher precision.
• Applying Kahan summation or other compensated summation techniques.
• Properly tuning hyperparameters such as learning rate, batch size, and momentum.

In many neural network workloads, small rounding errors may not be catastrophic because the training algorithms are somewhat resilient to noise—sometimes even benefiting from it in a process analogous to regularization. However, in scientific computing, you may need to analyze error propagation carefully, especially in iterative methods or when solutions are sensitive.

1. Speeding Up Transformer Models#

Transformer architectures dominate modern natural language processing tasks, including machine translation, question answering, and large language modeling. They are compute-heavy, especially in the attention mechanism layers, which rely on large batch matrix multiplications (e.g., Q×K^T for the attention scores).

With Tensor Cores:

• Mixed precision training can reduce the memory footprint, allowing larger batch sizes or deeper models.
• FP16 matrix multiplies are up to 8× faster than their FP32 counterparts, depending on the GPU architecture.
• Automatic scaling ensures stable backpropagation.

As a result, training times shrink from weeks to days or even hours, enabling faster experimentation cycles.

2. Large-Scale Simulation in HPC#

Consider fluid dynamics or seismic simulation that discretizes a 3D domain into a grid. Each iteration of the simulation demands matrix-vector or matrix-matrix operations for solving partial differential equations. Historically, HPC codes were done in double precision (FP64) for accuracy. Recently, some HPC workloads have found that partial steps in single or half precision can be used, with final calculations or critical steps computed in double precision to preserve accuracy.

As HPC libraries evolve, they integrate Tensor Core functionality:

1. Iterative Solvers: Many iterative methods (CG, GMRES) can use reduced precision for certain steps, accelerating them on Tensor Cores.
2. Multi-Grid Methods: In multi-grid or domain decomposition methods, partial calculations can be done in lower precision modules, deferring to higher precision only for the final smoothing.
3. Data-Driven HPC: Machine learning surrogates or neural PDE solvers can also benefit from Tensor Cores, bridging HPC with AI.

1. Using Tensor Cores with Sparse Matrices#

A burgeoning research area involves combining sparsity (zero values in matrices) with Tensor Cores. Sparse Tensor Cores can skip computation on zero elements in specialized data representations. This further boosts performance when the data is known to be sparse (common in large neural networks with skip connections, or in certain HPC problems).

2. Multi-GPU and Distributed Training#

Professional deep learning setups often use multiple GPUs in parallel or even entire GPU clusters. Libraries like NVIDIA Collective Communications Library (NCCL) handle data transfer and synchronization across GPUs. When scaling tensor operations to multi-GPU environments, each GPU’s Tensor Cores do heavy matrix multiplication, and partial results are synchronized across the cluster. This distributed approach is central to training models like GPT or large-scale autoencoders.

3. AMP for Production Inference#

Deploying deep learning models in production often emphasizes low latency and minimal resource usage. Tensor Cores accelerate inference by operating on reduced-precision representations (FP16, INT8, or even INT4). This reduces memory bandwidth, which is often a bottleneck in inference, and can drastically cut inference times. Professional frameworks (TensorRT, Triton Inference Server) accommodate these lower-precision formats automatically when exporting or optimizing models.

4. Hybrid HPC-AI Workloads#

Many HPC workloads now integrate elements of machine learning—like data-driven modeling, anomaly detection, or fast approximate solutions via neural networks. Tensor Cores serve both the HPC portion (accelerating large matrix operations in partial-precision steps) and the ML portion (accelerating neural network training). This synergy paves the way for scientific computing frontiers, where HPC and AI merge, forming “AI for Science” pipelines or supercomputers capable of extremely large computations.