Accelerating Large Language Models with GEMM Optimization in PyTorch

1. Introduction: The Need for Speed in LLMs

Large Language Models (LLMs) have revolutionized natural language processing, powering applications from sophisticated chatbots to advanced content generation. However, their immense size and computational complexity pose significant challenges, particularly regarding inference and training speed. Achieving acceptable performance often requires specialized hardware like GPUs and sophisticated optimization techniques.

At the heart of LLM computation lies a fundamental operation: General Matrix-Matrix Multiplication (GEMM). Understanding and optimizing GEMM is paramount to accelerating LLMs. This report serves as a comprehensive tutorial for developers seeking to enhance LLM performance within the PyTorch framework. Starting with the basics of GEMM and its relevance to LLMs, we will progressively explore optimization strategies, from simple data type adjustments to advanced techniques involving specialized libraries like bitsandbytes, fused operations via scaled_dot_product_attention, and custom kernel development with Triton, including pre-built solutions like Liger. The goal is to equip developers, regardless of their initial expertise level, with the knowledge and practical code examples needed to significantly speed up their LLM workloads.

2. What is GEMM? The Computational Core of LLMs

GEMM stands for General Matrix-Matrix Multiplication. It's a fundamental operation in linear algebra and forms the computational backbone of deep learning, especially for models like LLMs.

The standard GEMM operation is defined by the following equation, often expressed using Basic Linear Algebra Subprograms (BLAS) notation:

$C \leftarrow α \cdot A \cdot B + β \cdot C$

Where:

$A$ is an $m \times k$ matrix.
$B$ is a $k \times n$ matrix.
$C$ is an $m \times n$ matrix (the output matrix, which can also be an input for accumulation).
$α$ and $β$ are scalar values.

In the context of deep learning and LLMs, $α$ is typically 1, and $β$ is often 0 (for a simple multiplication $C = A \cdot B$ ) or 1 (when adding a bias or residual connection, though this is often handled as a separate operation or fused). The core computation involves multiplying matrix $A$ by matrix $B$ to produce matrix $C$ .

Nearly every computationally intensive part of a standard Transformer-based LLM relies heavily on GEMM operations:

Linear Layers (Fully Connected Layers): These layers, defined in PyTorch as torch.nn.Linear, perform the operation $Y = X \cdot W^{T} + b$ , where $X$ is the input, $W$ is the weight matrix, and $b$ is the bias. The core $X \cdot W^{T}$ is a GEMM.
Attention Mechanism Projections: In multi-head self-attention, the input activations ( $X$ ) are projected into Query ( $Q$ ), Key ( $K$ ), and Value ( $V$ ) matrices using separate weight matrices ( $W_{q}, W_{k}, W_{v}$ ). Each projection ( $Q = X \cdot W_{q}$ , $K = X \cdot W_{k}$ , $V = X \cdot W_{v}$ ) is a GEMM [User Material 1]. The output projection, which combines the attention outputs, is also a GEMM [User Material 1].
Feed-Forward Networks (MLP Blocks): Transformer blocks typically contain a position-wise feed-forward network, usually consisting of two linear layers with an activation function in between (e.g., MLP-up, MLP-gate, MLP-down projections). Each of these linear transformations is a GEMM [User Material 1].

Because these operations are repeated across numerous layers and attention heads within an LLM, the model's overall performance becomes critically dependent on the efficiency of the underlying GEMM computations.

3. Why GPUs Excel at GEMM

Graphics Processing Units (GPUs) are the hardware of choice for training and deploying LLMs primarily because their architecture is exceptionally well-suited for the massive parallelism inherent in GEMM operations. Several factors contribute to this efficiency [User Material 2]:

High Arithmetic Intensity: Arithmetic intensity is the ratio of floating-point operations (FLOPs) to bytes of data moved from memory. GEMM exhibits very high arithmetic intensity. Consider the calculation $C_{ij} = \sum_{l = 1}^{k} A_{i l} B_{l j}$ . To compute a single element $C_{ij}$ , we perform $k$ multiplications and $k - 1$ additions (~ $2 k$ FLOPs). However, elements $A_{i l}$ (from row $i$ of $A$ ) and $B_{l j}$ (from column $j$ of $B$ ) can be reused across computations for other elements in the same output row or column. By loading blocks (tiles) of $A$ and $B$ into fast on-chip memory (registers and shared memory), GPUs can perform many computations ( $k$ multiply-accumulates per element pair) before needing to fetch new data from slower main memory (HBM). This reuse minimizes the memory bandwidth bottleneck.
Regular Memory Access Patterns: GEMM involves accessing matrix elements in predictable patterns (rows of $A$ , columns of $B$ ). GPUs can efficiently prefetch and stream these blocks of data ("block-wise" access) into their processing units, ensuring that the compute cores are constantly supplied with data. This contrasts with irregular memory access patterns found in other computations (like sparse operations or graph traversals) which can stall the GPU.
Amenability to Tiling and Parallelism: The GEMM operation can be naturally decomposed into smaller, independent matrix multiplications on sub-blocks (tiles) of the original matrices. This "tiling" strategy allows the overall computation to be split across the thousands of lightweight threads available on a modern GPU. Each thread block can compute a tile of the output matrix $C$ independently, leading to massive parallelism.
Vectorization (SIMT Architecture): GPUs employ a Single Instruction, Multiple Threads (SIMT) execution model. A single instruction can operate on multiple data elements simultaneously (vector processing). Modern GPUs have specialized units (like NVIDIA's Tensor Cores) that can perform small matrix multiplications (e.g., $4 \times 4 \to 4$ ) on lower-precision data types (FP16, BF16, INT8) in a single clock cycle [User Material 2]. This vector-friendly nature allows one instruction to perform the work of many, significantly boosting throughput.

Recognizing the importance of GEMM, hardware vendors provide highly optimized libraries and hardware units specifically for this task:

NVIDIA: Offers cuBLAS/cuBLASLt (highly tuned GEMM library), Tensor Cores (hardware units for accelerated mixed-precision GEMM), and open-source templates like CUTLASS and Triton for building custom GEMM kernels.²
AMD: Provides rocBLAS (similar library for AMD GPUs) [User Material 2].
Intel: Offers oneDNN (formerly MKL-DNN) for optimized primitives on Intel hardware [User Material 2].

These specialized tools ensure that GEMM operations run close to the theoretical peak performance of the hardware.

4. LLMs as "GEMM Farms": The Scale of the Challenge

The sheer number of GEMM operations required during a single forward pass of an LLM highlights why optimizing them is crucial. Let's consider a moderately sized model like Llama 2 7B as an example [User Material 3]. A typical Transformer block in such a model involves several GEMM-based sub-modules:

Self-Attention Projections (Q, K, V): 3 GEMMs to project the input $X$ (shape $m \times k$ ) into $Q, K, V$ (shape $m \times d$ , where $d$ is the head dimension and $m$ often incorporates batch size, number of heads, and sequence length). Matrix shapes: $(m, k) \times (k, d)$ .
Self-Attention Output Projection: 1 GEMM to project the combined attention outputs back to the model dimension. Matrix shape: $(m, d) \times (d, k)$ .
MLP Feed-Forward Network: Typically involves 2 or 3 GEMMs. A common structure uses an up-projection and a gate projection (often to $4 k$ dimensions), followed by an element-wise operation, and then a down-projection back to $k$ dimensions.
- MLP-up: $(m, k) \times (k, 4 k)$
- MLP-gate: $(m, k) \times (k, 4 k)$ (Parallel to MLP-up)
- MLP-down: $(m, 4 k) \times (4 k, k)$

This totals approximately 7 GEMM operations per Transformer block. For a model like Llama 2 7B with 32 blocks, a single forward pass requires roughly $7 \times 32 = 224$ major GEMM calls.

This massive number means that even small percentage improvements in the performance of individual GEMM operations accumulate significantly, leading to noticeable reductions in overall wall-clock time for both training and inference [User Material 3]. LLMs effectively function as "GEMM farms," constantly executing these matrix multiplications. Therefore, ensuring these operations map efficiently onto the underlying hardware is the primary path to LLM acceleration.

5. Level 1 Optimizations: The Low-Hanging Fruit in PyTorch

Fortunately, significant performance gains can often be achieved with relatively simple changes within PyTorch, leveraging the highly optimized libraries and hardware features discussed earlier. These techniques form the foundation of LLM optimization.

Technique 1: Leverage Lower Precision (FP16, BF16, TF32)

Modern GPUs achieve much higher throughput when using lower-precision data types compared to standard 32-bit floating-point (FP32).

FP16 (Half Precision): Uses 16 bits (1 sign, 5 exponent, 10 mantissa). Offers significant speedups (2-8x on Tensor Cores) and memory savings (50% reduction) compared to FP32 [User Material 4]. However, its limited range can sometimes lead to numerical instability (overflow/underflow) during training, often requiring techniques like gradient scaling.
BF16 (Brain Floating Point): Also uses 16 bits (1 sign, 8 exponent, 7 mantissa). It maintains the same dynamic range as FP32 (due to the 8-bit exponent) but has lower precision than FP16. It is generally more stable for training than FP16, especially on hardware that supports it natively (e.g., NVIDIA Ampere and newer, Google TPUs). Offers similar speed and memory benefits as FP16 [User Material 4].
TF32 (TensorFloat-32): An NVIDIA-specific format available on Ampere and later GPUs.³ It uses 19 bits internally for computation (1 sign, 8 exponent, 10 mantissa) while still storing data in 32 bits. It provides a speedup over FP32 by utilizing Tensor Cores for matrix multiplication, while retaining nearly the same numerical range and precision as FP32 [User Material 4]. It's often enabled by default for convolutions but needs explicit enabling for matrix multiplications in PyTorch.

Practical Implementation in PyTorch:

You can enable lower precision in several ways:

Set Default Dtype: Change the global default floating-point type. New tensors will be created with this type.
Explicit Casting: Cast specific models or tensors using .half() (for FP16) or .to(torch.bfloat16).
Enable TF32: Use torch.backends.cuda.matmul.allow_tf32 = True to allow PyTorch's cuBLAS integration to use TF32 hardware paths for FP32 matrix multiplications.

Python
import torch

# Check CUDA availability
if not torch.cuda.is_available():
    print("CUDA not available. Skipping dtype examples.")
else:
    print(f"CUDA Device: {torch.cuda.get_device_name(0)}")
    # --- Option 1: Set Default Dtype (Affects new tensors) ---
    # Use BF16 if available (Ampere+ GPU), otherwise FP16
    if torch.cuda.is_bf16_supported():
        print("\nSetting default dtype to BF16")
        original_dtype = torch.get_default_dtype()
        torch.set_default_dtype(torch.bfloat16)
        current_default = torch.get_default_dtype()
    else:
        print("\nSetting default dtype to FP16")
        original_dtype = torch.get_default_dtype()
        torch.set_default_dtype(torch.float16)
        current_default = torch.get_default_dtype()

    print(f"Current default dtype: {current_default}")
    # Example model/tensors will now default to BF16/FP16
    # class MyModel(torch.nn.Module):
    #     def __init__(self):
    #         super().__init__()
    #         self.linear = torch.nn.Linear(10, 10) # Created with default dtype
    # model = MyModel().cuda()
    # input_tensor = torch.randn(5, 10, device='cuda') # Created with default dtype
    # print(f"Model parameter dtype: {next(model.parameters()).dtype}")
    # print(f"Input tensor dtype: {input_tensor.dtype}")

    # Reset default dtype
    torch.set_default_dtype(original_dtype)
    print(f"Reset default dtype to: {torch.get_default_dtype()}")

    # --- Option 2: Explicitly Cast Tensors/Models ---
    print("\nExplicit Casting Example:")
    model_fp32 = torch.nn.Linear(128, 128).cuda() # Starts as FP32
    input_fp32 = torch.randn(16, 128, device='cuda', dtype=torch.float32)
    print(f"Original model dtype: {next(model_fp32.parameters()).dtype}")
    print(f"Original input dtype: {input_fp32.dtype}")

    # Cast model and input to FP16
    model_fp16 = model_fp32.half() #.half() casts to FP16
    input_fp16 = input_fp32.half() # Cast input too
    print(f"Casted model dtype (FP16): {next(model_fp16.parameters()).dtype}")
    print(f"Casted input dtype (FP16): {input_fp16.dtype}")
    # output_fp16 = model_fp16(input_fp16) # Now runs in FP16

    # Cast model and input to BF16 (if supported)
    if torch.cuda.is_bf16_supported():
        model_bf16 = model_fp32.to(torch.bfloat16)
        input_bf16 = input_fp32.to(torch.bfloat16)
        print(f"Casted model dtype (BF16): {next(model_bf16.parameters()).dtype}")
        print(f"Casted input dtype (BF16): {input_bf16.dtype}")
        # output_bf16 = model_bf16(input_bf16) # Now runs in BF16
    else:
        print("BF16 not supported on this GPU.")

    # --- Option 3: Enable TF32 for Matmuls (if using FP32 inputs) ---
    print("\nEnabling TF32 for FP32 Matmuls:")
    # Check if TF32 is available (Compute Capability >= 8.0) and enable it
    if torch.cuda.get_device_capability() >= 8:
        print("GPU supports TF32. Enabling TF32 for matmul.")
        torch.backends.cuda.matmul.allow_tf32 = True
        # Note: TF32 for convolutions is usually enabled by default via cudnn
        # torch.backends.cudnn.allow_tf32 = True
        print(f"torch.backends.cuda.matmul.allow_tf32 = {torch.backends.cuda.matmul.allow_tf32}")
    else:
        print("GPU does not support TF32 (requires Ampere architecture or newer).")
        torch.backends.cuda.matmul.allow_tf32 = False # Ensure it's off

    # Example FP32 matmul potentially using TF32 now
    if torch.backends.cuda.matmul.allow_tf32:
        a_fp32 = torch.randn(512, 512, device='cuda', dtype=torch.float32)
        b_fp32 = torch.randn(512, 512, device='cuda', dtype=torch.float32)
        # This matmul might use TF32 hardware if enabled and available
        # The actual kernel dispatch depends on cuBLAS heuristics
        c_maybe_tf32 = a_fp32 @ b_fp32
        print("Performed FP32 matmul (potentially using TF32). Output dtype:", c_maybe_tf32.dtype)

    # Disable TF32 if needed for specific comparisons or operations
    torch.backends.cuda.matmul.allow_tf32 = False
    print(f"Disabled TF32 for matmul. Current setting: {torch.backends.cuda.matmul.allow_tf32}")

 
 

Technique 2: Aligning with Hardware (Dimension Multiples)

The performance of GEMM operations, especially when using Tensor Cores, is sensitive to the dimensions of the input matrices ( $m, n, k$ ). Tensor Cores and the underlying libraries often process data in fixed-size blocks or tiles (e.g., $8 \times 8$ , $16 \times 16$ , $16 \times 8$ ) [User Material 4]. If the matrix dimensions are not multiples of these tile sizes (commonly 8 or 16 for modern NVIDIA hardware), the hardware execution units (lanes) might be partially idle during computation, leading to wasted cycles and reduced efficiency. Padding is often performed internally by the library, but this adds overhead.

Practical Advice:

During model architecture design or selection, favor dimensions that align well with the hardware. Specifically, choose hidden sizes (d_model), MLP intermediate dimensions, attention head dimensions, and vocabulary sizes that are multiples of 8, or ideally 16. Even better alignment can sometimes be achieved with multiples of 64 or 128, which can also align better with memory transaction sizes [User Material 4].

Python
# Conceptual Example: Choosing Model Dimensions

# Less Optimal (Not multiple of 8/16)
hidden_size_bad = 1000
mlp_intermediate_bad = 3000
vocab_size_unpadded = 32007

print(f"Less Optimal hidden size: {hidden_size_bad}")
print(f"Less Optimal MLP intermediate size: {mlp_intermediate_bad}")
print(f"Less Optimal vocab size: {vocab_size_unpadded}")

# More Optimal (Multiples of 8, 16, 64, or 128)
hidden_size_good = 1024 # Multiple of 128
mlp_intermediate_good = 4096 # Multiple of 128

# Pad vocabulary size to a multiple of 64 or 128 for efficiency
padding_multiple = 64
padded_vocab_size = ((vocab_size_unpadded + padding_multiple - 1) // padding_multiple) * padding_multiple

print(f"\nMore Optimal hidden size: {hidden_size_good} (Multiple of 128)")
print(f"More Optimal MLP intermediate size: {mlp_intermediate_good} (Multiple of 128)")
print(f"Padded vocab size (Multiple of {padding_multiple}): {padded_vocab_size}")

# Rationale: Choosing dimensions like 1024 or 4096 ensures that when these dimensions
# are involved in GEMMs (as M, N, or K), they divide evenly by common tile sizes
# like 8 or 16, maximizing Tensor Core utilization. Padding the vocabulary size
# helps ensure the embedding lookup table and the final output projection layer
# (which often involves a GEMM with the vocabulary size) are well-aligned.

[User Material 4]

Technique 3: Automatic Acceleration with `torch.compile`

PyTorch 2.0 introduced torch.compile, a powerful feature that acts as a just-in-time (JIT) compiler for PyTorch models.⁴ By simply wrapping your model with torch.compile(), you can often achieve significant speedups with minimal code changes.

How it Works: Under the hood, torch.compile uses several components ⁵:

TorchDynamo: Captures the PyTorch computational graph from your Python code safely and with low overhead. It identifies parts of the graph that can be compiled.
AOTAutograd: Traces the autograd engine to capture the backward pass ahead-of-time, allowing the compiler to optimize both forward and backward computations.
PrimTorch: Canonicalizes the ~2000+ PyTorch operators into a smaller set of ~250 primitive operators, providing a stable target for backends.
TorchInductor: The default compiler backend. It takes the graph of primitive operators and generates optimized low-level code. For GPUs, TorchInductor heavily relies on OpenAI Triton ⁵ to generate fast custom kernels, often fusing operations together automatically.⁵

This means torch.compile can automatically perform many optimizations, including generating efficient, potentially fused GEMM kernels tailored to your model and hardware, without requiring manual kernel writing.

Usage:

The basic usage is straightforward. torch.compile offers different modes that trade off compilation time for runtime performance 5:

mode="default": Good balance, optimized for large models, low compile overhead.
mode="reduce-overhead": Faster compilation, uses more memory, good for smaller models where framework overhead dominates.
mode="max-autotune": Longest compilation time, potentially fastest runtime by extensively searching for optimal kernel configurations.

Python
import torch
import time
import copy
import torch.nn as nn

# Define a simple model (e.g., a couple of linear layers mimicking an MLP block)
class SimpleMLP(nn.Module):
    def __init__(self, dim=1024):
        super().__init__()
        # Ensure dimensions are multiples of 8/16 for good baseline performance
        assert dim % 16 == 0, "Dimension should be multiple of 16"
        intermediate_dim = 4 * dim
        assert intermediate_dim % 16 == 0, "Intermediate dimension should be multiple of 16"

        self.linear1 = nn.Linear(dim, intermediate_dim)
        self.relu = nn.ReLU()
        self.linear2 = nn.Linear(intermediate_dim, dim)

    def forward(self, x):
        x = self.linear1(x)
        x = self.relu(x)
        x = self.linear2(x)
        return x

# Ensure we are on a GPU
if not torch.cuda.is_available():
    print("CUDA not available, skipping torch.compile example.")
elif torch.__version__ < "2.0":
    print("torch.compile requires PyTorch 2.0 or later. Skipping example.")
else:
    device = 'cuda'
    dim_size = 2048 # Use a multiple of 16
    batch_size = 64
    # For demonstration, treat sequence as part of batch dim for a simple 2D input
    # In a real LLM, input might be (batch_size * seq_len, dim_size)
    num_tokens = batch_size * 128

    # Create model instance and move to GPU
    model = SimpleMLP(dim=dim_size).to(device)
    # Use appropriate dtype (e.g., float16 or bfloat16 if supported)
    dtype = torch.bfloat16 if torch.cuda.is_bf16_supported() else torch.float16
    model = model.to(dtype)
    input_tensor = torch.randn(num_tokens, dim_size, device=device, dtype=dtype)

    print(f"PyTorch Version: {torch.__version__}")
    print(f"Model and tensor created on GPU with dtype: {dtype}.")
    print(f"Input shape: {input_tensor.shape}")

    # --- Baseline Eager Execution ---
    print("\nRunning Eager Mode...")
    # Warmup runs
    for _ in range(5):
        _ = model(input_tensor)
    torch.cuda.synchronize() # Wait for GPU work to finish
    start_time = time.time()
    num_runs = 20
    for _ in range(num_runs):
        _ = model(input_tensor)
    torch.cuda.synchronize()
    eager_time = time.time() - start_time
    print(f"Eager execution time ({num_runs} runs): {eager_time:.4f} seconds")

    # --- Using torch.compile ---
    print("\nCompiling model with torch.compile (mode='default')...")
    # Make a copy to compile, leave original intact if needed
    # Note: Compilation happens on the first call(s) to the compiled model
    compiled_model = torch.compile(copy.deepcopy(model)) # Default mode
    # Other modes:
    # compiled_model = torch.compile(model, mode="reduce-overhead")
    # compiled_model = torch.compile(model, mode="max-autotune")

    print("Running Compiled Mode...")
    # Warmup runs for compiled model (includes compilation time on first run)
    for _ in range(5):
        _ = compiled_model(input_tensor)
    torch.cuda.synchronize()
    start_time = time.time()
    for _ in range(num_runs):
        _ = compiled_model(input_tensor)
    torch.cuda.synchronize()
    compiled_time = time.time() - start_time
    print(f"Compiled execution time ({num_runs} runs): {compiled_time:.4f} seconds")

    # Compare results (optional, should be numerically close)
    # Use no_grad for fair comparison if model includes dropout/batchnorm
    with torch.no_grad():
        model.eval() # Set model to evaluation mode
        compiled_model.eval()
        eager_output = model(input_tensor)
        compiled_output = compiled_model(input_tensor)
        # Use appropriate tolerance for the chosen dtype
        atol = 1e-2 if dtype == torch.float16 else 1e-1
        rtol = 1e-2 if dtype == torch.float16 else 1e-1
        if torch.allclose(eager_output, compiled_output, atol=atol, rtol=rtol):
            print("\n✅ Outputs match between eager and compiled model (within tolerance).")
        else:
            print("\n❌ Outputs differ significantly between eager and compiled model.")
            print(f"   Max absolute difference: {torch.abs(eager_output - compiled_output).max().item()}")
            print(f"   Mean absolute difference: {torch.abs(eager_output - compiled_output).mean().item()}")


    if compiled_time > 0:
        print(f"\nSpeedup factor (Eager / Compiled): {eager_time / compiled_time:.2f}x")
    else:
        print("\nCompiled time was zero or negative, cannot calculate speedup.")

⁴

Using lower precision, ensuring dimension alignment, and leveraging torch.compile represent the first line of defense in optimizing LLM performance. They are relatively easy to implement and often yield substantial gains by better utilizing the underlying hardware and compiler technologies.

6. Level 2 Optimizations: Getting More Advanced

Beyond the basic techniques, further performance can be unlocked by employing strategies that require more specific library usage or a deeper understanding of the computational patterns within LLMs. These include batching small operations, quantizing models, and fusing sequences of operations.

Technique 4: Batching Small Matmuls (`torch.bmm`)

LLMs often involve operations that result in many small, independent matrix multiplications. For example, in multi-head attention, computations might be performed independently for each head before results are combined. Executing these small GEMMs one by one using standard matrix multiplication (@ or torch.matmul) within a Python loop is highly inefficient. Each call incurs kernel launch latency – the overhead associated with telling the GPU to start a new computation. This latency can dominate the actual computation time for small matrices [User Material 4].

GPUs thrive on parallelism and perform best when given large, contiguous chunks of work. PyTorch provides torch.bmm (Batch Matrix Multiply) specifically for this scenario. It takes batches of matrices as input (shapes (B, m, k) and (B, k, n), where B is the batch dimension) and performs B independent matrix multiplications ( $m \times k$ times $k \times n$ ) using a single, optimized kernel launch [User Material 4, User Material 5.2]. This significantly reduces the kernel launch overhead and improves GPU utilization.

Reproducible Code:

Python
import torch
import time

# Setup: Batch of small matrices
batch_size = 256 # Larger batch size makes the overhead difference more apparent
m, k, n = 32, 64, 48 # Small dimensions, but keep them multiples of 8

# Ensure CUDA is available
if not torch.cuda.is_available():
    print("CUDA not available. Skipping torch.bmm example.")
else:
    device = 'cuda'
    dtype = torch.float16 # Use half precision for speed

    # Create batches of matrices on GPU
    A_batch = torch.randn(batch_size, m, k, device=device, dtype=dtype)
    B_batch = torch.randn(batch_size, k, n, device=device, dtype=dtype)
    # Pre-allocate output tensors
    C_loop = torch.empty(batch_size, m, n, device=device, dtype=dtype)
    # C_bmm will be created by the bmm operation

    print(f"Matrices: Batch={batch_size}, M={m}, K={k}, N={n}, dtype={dtype}")
    print(f"Input A shape: {A_batch.shape}")
    print(f"Input B shape: {B_batch.shape}")

    # --- Method 1: Python Loop (Inefficient) ---
    print("\nRunning with Python loop...")
    # Warmup - run a few iterations first
    for i in range(min(batch_size, 10)):
         _ = A_batch[i] @ B_batch[i]
    torch.cuda.synchronize() # Wait for warmup to finish

    start_time = time.time()
    num_runs = 10 # Run the loop multiple times for averaging if needed
    for _ in range(num_runs):
        for i in range(batch_size):
            C_loop[i] = A_batch[i] @ B_batch[i] # Standard matmul inside loop
    torch.cuda.synchronize() # Wait for all computations to finish
    loop_time = (time.time() - start_time) / num_runs
    print(f"Python loop average time: {loop_time:.6f} seconds")

    # --- Method 2: Batched Matrix Multiply (torch.bmm) ---
    print("\nRunning with torch.bmm...")
    # Warmup
    _ = torch.bmm(A_batch, B_batch)
    torch.cuda.synchronize()

    start_time = time.time()
    for _ in range(num_runs):
        C_bmm = torch.bmm(A_batch, B_batch) # Single kernel launch for the whole batch
    torch.cuda.synchronize()
    bmm_time = (time.time() - start_time) / num_runs
    print(f"torch.bmm average time: {bmm_time:.6f} seconds")

    # Check results for correctness
    if torch.allclose(C_loop, C_bmm, atol=1e-3): # Adjust tolerance for FP16
         print("\n✅ Outputs match between loop and bmm.")
    else:
         print("\n❌ Outputs differ between loop and bmm.")
         print(f"   Max difference: {torch.abs(C_loop - C_bmm).max().item()}")

    if bmm_time > 0:
        print(f"\nSpeedup factor (loop time / bmm time): {loop_time / bmm_time:.2f}x")
    else:
        print("\nCannot calculate speedup factor.")

[User Material 4, User Material 5.2]

Whenever you encounter a situation requiring identical matrix multiplications across a batch or group dimension, consider using torch.bmm or related functions like torch.einsum (which can express batch matrix multiplication and more complex tensor contractions) to consolidate the work into fewer, larger kernel launches.

Technique 5: Quantization (INT8/INT4) with `bitsandbytes`

Quantization is a powerful technique, especially for optimizing LLM inference. It involves converting model parameters (weights) and sometimes activations from higher-precision floating-point formats (like FP32 or FP16) to lower-precision integer formats, typically 8-bit integers (INT8) or even 4-bit integers (INT4).¹

Benefits:

Reduced Memory Footprint: INT8 weights require 4x less memory than FP32 and 2x less than FP16. INT4 offers even greater savings (8x vs FP32, 4x vs FP16). This allows larger models to fit into limited GPU memory.⁸
Reduced Memory Bandwidth: Moving less data from HBM to the compute units alleviates memory bandwidth bottlenecks, which is often the limiting factor in LLM inference [User Material 4].
Faster Computation: Many GPUs have specialized hardware instructions for performing integer arithmetic faster than floating-point operations. INT8 computations can leverage Tensor Cores for significant speedups [User Material 4].

Trade-offs:

Potential Accuracy Loss: Representing values with fewer bits inevitably introduces approximation errors (quantization noise). While techniques are designed to minimize this, there can be a drop in model accuracy, which needs careful evaluation.⁸
Complexity: Implementing quantization correctly often involves calibration (determining the optimal mapping from float to int ranges) or Quantization-Aware Training (QAT), although libraries like bitsandbytes simplify Post-Training Quantization (PTQ) significantly.⁸

The bitsandbytes Library:

bitsandbytes is a popular open-source library that makes advanced quantization techniques accessible within the PyTorch ecosystem.8 It's particularly well-known for:

LLM.int8(): An 8-bit quantization scheme primarily for inference. It typically involves quantizing nn.Linear layer weights to INT8 while keeping activations in FP16. It uses mixed-precision decomposition during the GEMM computation to maintain accuracy, especially handling outlier activation values that are crucial for LLM performance.¹¹
4-bit Quantization (NF4, FP4): bitsandbytes also supports 4-bit quantization, notably the NF4 (NormalFloat 4-bit) data type, which was introduced with the QLoRA method for efficient fine-tuning.¹
Ease of Use: It integrates smoothly with Hugging Face's transformers library, often enabling 8-bit or 4-bit loading with simple flags like load_in_8bit=True or load_in_4bit=True passed via a BitsAndBytesConfig.⁸ It also provides standalone quantized layer implementations like bnb.nn.Linear8bitLt and bnb.nn.Linear4bit.¹

Reproducible Code (Using Linear8bitLt directly):

This example shows how to replace a standard nn.Linear layer with the bitsandbytes 8-bit version for inference.

Python
import torch
import torch.nn as nn
import os
import importlib

# Check if bitsandbytes is installed and seems compatible
try:
    import bitsandbytes as bnb
    from bitsandbytes.nn import Linear8bitLt
    # Perform a basic check that might fail if CUDA setup is incorrect
    if torch.cuda.is_available():
        p = torch.nn.Parameter(torch.rand(10, 10).cuda())
        adam = bnb.optim.Adam8bit([p]) # Try creating an 8-bit optimizer
        print("bitsandbytes imported and basic check passed.")
        bnb_available = True
    else:
        print("bitsandbytes imported, but CUDA not available.")
        bnb_available = False
except ImportError:
    print("bitsandbytes not installed.")
    bnb_available = False
except Exception as e:
    print(f"bitsandbytes might not be compatible or installed correctly: {e}")
    bnb_available = False

if bnb_available and torch.cuda.is_available():
    print("\nRunning bitsandbytes Linear8bitLt Example...")
    # Define model dimensions (use multiples of 8/16)
    in_features = 256
    out_features = 512
    batch_size = 32
    num_tokens = batch_size * 16 # Example token count

    # 1. Create or load a standard FP16 model/layer
    # In a real scenario, you'd load pretrained weights
    fp16_linear_layer = nn.Linear(in_features, out_features, bias=False).cuda().half()
    fp16_input = torch.randn(num_tokens, in_features, device='cuda', dtype=torch.float16)
    print(f"Input shape: {fp16_input.shape}")
    print(f"Original FP16 Layer: {fp16_linear_layer}")

    # 2. Create the equivalent layer using bitsandbytes Linear8bitLt for INT8 inference
    # For the common LLM.int8() inference pattern (INT8 weights, FP16 activations):
    # - Set has_fp16_weights=False: This tells the layer to store weights internally as INT8.
    # - threshold=6.0: This parameter is specific to the LLM.int8() mixed-precision
    #   decomposition technique to handle activation outliers. It determines the threshold
    #   above which activation values are treated separately in FP16.
    int8_linear_layer = Linear8bitLt(
        in_features,
        out_features,
        bias=False, # Match the original layer's bias setting
        has_fp16_weights=False, # Critical for INT8 weight storage
        threshold=6.0 # Default threshold for LLM.int8()
    ).cuda() # Move layer to GPU *before* loading state_dict or using it

    print(f"Bitsandbytes INT8 Layer: {int8_linear_layer}")

    # 3. Load the state dict from the FP16 layer into the INT8 layer
    # bitsandbytes automatically handles the quantization of the weights
    # when the state dict is loaded into the Linear8bitLt layer moved to CUDA.
    int8_linear_layer.load_state_dict(fp16_linear_layer.state_dict())
    print("Loaded FP16 weights into INT8 layer (quantization applied).")

    # Verify weight storage (internal attribute, may change)
    if hasattr(int8_linear_layer, 'weight') and int8_linear_layer.weight.dtype == torch.int8:
         print(f"Verified internal weight dtype is: {int8_linear_layer.weight.dtype}")
    else:
         print("Could not verify internal weight dtype (might be stored differently).")


    # 4. Perform inference using the INT8 layer
    # Input activations should typically remain in FP16 for LLM.int8()
    start_time = time.time()
    int8_output = int8_linear_layer(fp16_input)
    torch.cuda.synchronize()
    int8_time = time.time() - start_time

    print(f"\nOutput shape (INT8 Layer): {int8_output.shape}")
    # The output is typically returned in FP16 after the mixed-precision computation
    print(f"Output dtype (INT8 Layer): {int8_output.dtype}")
    print(f"Inference time (INT8 Layer): {int8_time:.6f} seconds")

    # Optional: Compare output and speed with the original FP16 layer
    start_time = time.time()
    fp16_output = fp16_linear_layer(fp16_input)
    torch.cuda.synchronize()
    fp16_time = time.time() - start_time
    print(f"Inference time (FP16 Layer): {fp16_time:.6f} seconds")

    diff = torch.abs(fp16_output - int8_output).mean()
    print(f"Mean absolute difference between FP16 and INT8 outputs: {diff.item():.4f}")
    if fp16_time > 0 :
        print(f"Speedup factor (INT8 vs FP16): {fp16_time / int8_time:.2f}x")


    # --- Using Hugging Face Integration (Conceptual Code) ---
    print("\n--- Conceptual Hugging Face Integration ---")
    print("# This requires transformers, accelerate, and bitsandbytes installed.")
    print("# Example:")
    print("# from transformers import AutoModelForCausalLM, BitsAndBytesConfig")
    print("# ")
    print("# model_id = 'meta-llama/Llama-2-7b-hf' # Or any compatible model")
    print("# ")
    print("# # Configure 8-bit quantization")
    print("# quantization_config = BitsAndBytesConfig(")
    print("#     load_in_8bit=True,")
    print("#     # Optional: Specify compute dtype if needed")
    print("#     # bnb_4bit_compute_dtype=torch.bfloat16 ")
    print("# )")
    print("# ")
    print("# # Load the model with quantization enabled")
    print("# model = AutoModelForCausalLM.from_pretrained(")
    print("#     model_id,")
    print("#     quantization_config=quantization_config,")
    print("#     device_map='auto' # Handles placing layers on devices")
    print("# )")
    print("# ")
    print("# print(f'Model {model_id} loaded with 8-bit quantization via Hugging Face!')")
    print("# print(model)")

else:
    print("Skipping bitsandbytes Linear8bitLt example due to unavailability.")

Quantization, particularly using libraries like bitsandbytes, offers a compelling way to reduce the memory demands and potentially accelerate the inference of large models, making them more deployable in resource-constrained environments.

Technique 6: Fused Operations (FlashAttention & xFormers via SDPA)

Kernel fusion is a powerful optimization technique where multiple distinct computational steps are combined into a single GPU kernel [User Material 4].

Why Fuse?

Modern GPUs have vastly more computational power (FLOPs) than memory bandwidth. Many operations in neural networks are "memory-bound," meaning the time taken is limited by how fast data can be read from and written to the GPU's main memory (HBM), rather than by the speed of the calculations themselves. Fusion combats this by 13:

Reducing Kernel Launch Overhead: Launching one fused kernel is faster than launching multiple smaller ones sequentially.
Minimizing HBM Traffic: By keeping intermediate results within the GPU's much faster on-chip memory (SRAM or registers) between fused operations, fusion avoids costly round trips to HBM. Data is loaded once, processed through multiple steps, and then written back.

Attention Fusion: The Killer App

The self-attention mechanism in Transformers is a prime candidate for fusion. Standard attention calculation involves several steps:

Calculate $Q \cdot K^{T}$ (GEMM).
Scale the result.
Apply attention mask (element-wise).
Apply softmax (element-wise).
Apply dropout (element-wise, during training).
Multiply by $V$ (GEMM).

Crucially, the intermediate attention score matrix ( $Q \cdot K^{T}$ after scaling and masking) can be very large, especially for long sequences (size $S e q L e n \times S e q L e n$ ). In a naive implementation, this large matrix must be written to and read back from HBM, consuming significant memory ( $O (SeqLen^{2})$ ) and bandwidth [User Material 4].

FlashAttention and xFormers:

Libraries and algorithms have been developed to optimize attention by fusing these steps and avoiding the materialization of the full attention matrix:

FlashAttention: An algorithm that computes the exact attention output without ever storing the full $S e q L e n \times S e q L e n$ attention matrix in HBM.¹³ It works by loading blocks (tiles) of $Q$ , $K$ , and $V$ into the fast on-chip SRAM, computing the attention output for that block (including the softmax), and writing the result for that block back to HBM. It uses clever numerical techniques (softmax rescaling) to ensure the final result is correct even though it's computed block by block.¹³ This reduces the memory complexity from $O (SeqLen^{2})$ to $O (SeqLen)$ and significantly speeds up computation by dramatically reducing HBM access.¹³ FlashAttention-2 and FlashAttention-3 offer further performance improvements on newer hardware.¹³
xFormers: A library from Meta AI containing various optimized building blocks for Transformers, including highly efficient memory-optimized attention implementations.³ These implementations also aim to reduce memory usage and improve speed compared to naive attention.

PyTorch Integration: scaled_dot_product_attention (SDPA)

Recognizing the importance of fused attention, PyTorch 2.0 introduced torch.nn.functional.scaled_dot_product_attention (SDPA). This function provides a unified interface for calculating scaled dot product attention.19 When running on CUDA with appropriate hardware and inputs, SDPA automatically dispatches to highly optimized fused kernels, including 4:

FlashAttention (v1, v2, or v3 depending on PyTorch version and hardware).¹³
Memory-Efficient Attention kernels (inspired by or directly from xFormers).
A fallback PyTorch C++ implementation if fused kernels are unavailable.

Using F.scaled_dot_product_attention is now the standard and recommended way to implement attention in PyTorch to automatically benefit from these powerful fusion optimizations.

Reproducible Code (Using SDPA):

Python
import torch
import torch.nn.functional as F
import time
import math

# Attempt to import SDPA context manager, handle older PyTorch versions
sdpa_kernel, SDPBackend = None, None
try:
    from torch.backends.cuda import sdp_kernel, SDPBackend
except ImportError:
    print("Warning: Could not import sdp_kernel. Explicit backend control disabled.")
    print("         (Requires PyTorch 2.0+ with CUDA support)")

# Setup Parameters
batch_size = 8
num_heads = 12
seq_len = 2048 # Longer sequence length benefits more from fusion
head_dim = 64
dtype = torch.float16 # Fused kernels often optimized for FP16/BF16
device = 'cuda'

# Check prerequisites
if not torch.cuda.is_available():
    print("CUDA not available, skipping SDPA example.")
elif torch.__version__ < "2.0":
    print("Fused attention via SDPA requires PyTorch 2.0 or later. Skipping.")
else:
    print(f"PyTorch Version: {torch.__version__}")
    print(f"Using device: {device}, dtype: {dtype}")
    print(f"Batch={batch_size}, Heads={num_heads}, SeqLen={seq_len}, HeadDim={head_dim}")

    # Create realistic input tensors
    query = torch.randn(batch_size, num_heads, seq_len, head_dim, device=device, dtype=dtype)
    key = torch.randn(batch_size, num_heads, seq_len, head_dim, device=device, dtype=dtype)
    value = torch.randn(batch_size, num_heads, seq_len, head_dim, device=device, dtype=dtype)
    # Example: Causal mask for decoder self-attention
    is_causal_flag = True

    # --- Using torch.nn.functional.scaled_dot_product_attention ---
    print("\nRunning with torch.nn.functional.scaled_dot_product_attention...")
    num_runs = 20
    warmup_runs = 5

    results = {}

    # --- Backend Option 1: Math Kernel (Reference, Non-Fused) ---
    if sdp_kernel: # Only run if context manager is available
        print("\nRunning with Math backend (Reference)...")
        try:
            # Force use of the basic math implementation
            with sdp_kernel(enable_flash=False, enable_math=True, enable_mem_efficient=False):
                # Warmup
                for _ in range(warmup_runs):
                    _ = F.scaled_dot_product_attention(query, key, value, is_causal=is_causal_flag)
                torch.cuda.synchronize()
                start_time = time.time()
                for _ in range(num_runs):
                    output_math = F.scaled_dot_product_attention(query, key, value, is_causal=is_causal_flag)
                torch.cuda.synchronize()
            math_time = (time.time() - start_time) / num_runs
            results['math'] = math_time
            print(f"Math backend average time: {math_time:.6f} seconds")
        except RuntimeError as e:
            print(f"Math backend failed or not applicable: {e}")
            results['math'] = float('inf')
    else:
        print("\nSkipping Math backend (sdp_kernel not available).")
        results['math'] = float('inf')


    # --- Backend Option 2: FlashAttention Kernel ---
    if sdp_kernel:
        print("\nRunning with FlashAttention backend (if available)...")
        try:
            # Force use of FlashAttention
            with sdp_kernel(enable_flash=True, enable_math=False, enable_mem_efficient=False):
                 # Warmup
                for _ in range(warmup_runs):
                    _ = F.scaled_dot_product_attention(query, key, value, is_causal=is_causal_flag)
                torch.cuda.synchronize()
                start_time = time.time()
                for _ in range(num_runs):
                    output_flash = F.scaled_dot_product_attention(query, key, value, is_causal=is_causal_flag)
                torch.cuda.synchronize()
            flash_time = (time.time() - start_time) / num_runs
            results['flash'] = flash_time
            print(f"FlashAttention backend average time: {flash_time:.6f} seconds")
            if not math.isinf(results['math']) and flash_time > 0:
                print(f"  Speedup vs Math: {results['math'] / flash_time:.2f}x")
        except RuntimeError as e:
            print(f"FlashAttention backend not supported or failed: {e}")
            results['flash'] = float('inf')
    else:
        print("\nSkipping FlashAttention backend (sdp_kernel not available).")
        results['flash'] = float('inf')


    # --- Backend Option 3: Memory Efficient Attention Kernel ---
    if sdp_kernel:
        print("\nRunning with Memory Efficient backend (if available)...")
        try:
            # Force use of Memory Efficient Attention
            with sdp_kernel(enable_flash=False, enable_math=False, enable_mem_efficient=True):
                 # Warmup
                for _ in range(warmup_runs):
                    _ = F.scaled_dot_product_attention(query, key, value, is_causal=is_causal_flag)
                torch.cuda.synchronize()
                start_time = time.time()
                for _ in range(num_runs):
                    output_mem = F.scaled_dot_product_attention(query, key, value, is_causal=is_causal_flag)
                torch.cuda.synchronize()
            mem_efficient_time = (time.time() - start_time) / num_runs
            results['mem_efficient'] = mem_efficient_time
            print(f"Memory Efficient backend average time: {mem_efficient_time:.6f} seconds")
            if not math.isinf(results['math']) and mem_efficient_time > 0:
                 print(f"  Speedup vs Math: {results['math'] / mem_efficient_time:.2f}x")
        except RuntimeError as e:
            print(f"Memory Efficient backend not supported or failed: {e}")
            results['mem_efficient'] = float('inf')
    else:
        print("\nSkipping Memory Efficient backend (sdp_kernel not available).")
        results['mem_efficient'] = float('inf')

    # --- Automatic Backend Selection (Recommended Usage) ---
    print("\nRunning SDPA with automatic backend selection...")
    # No context manager needed, PyTorch chooses the best available fused kernel
    try:
        # Warmup
        for _ in range(warmup_runs):
            _ = F.scaled_dot_product_attention(query, key, value, is_causal=is_causal_flag)
        torch.cuda.synchronize()
        start_time = time.time()
        for _ in range(num_runs):
            output_auto = F.scaled_dot_product_attention(query, key, value, is_causal=is_causal_flag)
        torch.cuda.synchronize()
        auto_time = (time.time() - start_time) / num_runs
        results['auto'] = auto_time
        print(f"Automatic backend average time: {auto_time:.6f} seconds")
        # Compare automatic time to the fastest explicit fused kernel time
        best_fused_time = min(results.get('flash', float('inf')), results.get('mem_efficient', float('inf')))
        if not math.isinf(best_fused_time) and best_fused_time > 0:
             print(f"  Ratio vs best explicit fused backend: {auto_time / best_fused_time:.2f}")
        if not math.isinf(results['math']) and auto_time > 0:
             print(f"  Speedup vs Math: {results['math'] / auto_time:.2f}x")

    except Exception as e:
        print(f"Automatic SDPA failed: {e}")
        results['auto'] = float('inf')

    # (Optional) Check if outputs match across successful backends
    # Note: Numerical differences might exist due to computation order/precision used in fused kernels
    print("\nChecking output consistency (if multiple backends ran successfully)...")
    outputs_to_compare =
    if not math.isinf(results['math']): outputs_to_compare.append(output_math)
    if not math.isinf(results['flash']): outputs_to_compare.append(output_flash)
    if not math.isinf(results['mem_efficient']): outputs_to_compare.append(output_mem)
    if not math.isinf(results['auto']): outputs_to_compare.append(output_auto)

    if len(outputs_to_compare) > 1:
        base_output = outputs_to_compare
        match = True
        for i in range(1, len(outputs_to_compare)):
            if not torch.allclose(base_output, outputs_to_compare[i], atol=1e-2, rtol=1e-2): # Tolerance for FP16
                print(f"❌ Mismatch detected between backend outputs!")
                match = False
                break
        if match:
            print("✅ Outputs appear consistent across successful backends (within tolerance).")
    else:
        print("Not enough successful backend runs to compare outputs.")

Fusion, especially in memory-intensive operations like attention, is a cornerstone of modern LLM optimization. By minimizing data movement, fused kernels like those automatically invoked by PyTorch's SDPA significantly boost performance and enable models to handle much longer input sequences.

7. Level 3: Unleashing Custom Power with Triton

While libraries like cuBLAS, bitsandbytes, and the fused kernels within PyTorch provide significant acceleration, there are scenarios where even more performance is desired, or where novel operations need to be implemented efficiently. This is where tools like OpenAI Triton come into play, allowing developers to write custom, high-performance GPU kernels directly within Python.

What is Triton?

Triton is an open-source programming language and compiler designed to make writing efficient GPU code easier and more productive than traditional CUDA C++.6 It provides a Pythonic syntax, allowing developers familiar with NumPy and PyTorch to write parallel GPU kernels.6 Triton's compiler takes this high-level Python code, performs numerous optimizations (like instruction scheduling, memory access coalescing, and automatic management of shared memory), and compiles it down to low-level machine code (PTX for NVIDIA GPUs or other backends) using frameworks like LLVM/MLIR.22

Why Use Triton?

Custom Kernel Fusion: Triton excels at creating custom fused kernels. You can combine sequences of operations specific to your model (e.g., GEMM + bias + custom activation + residual add) into a single kernel, minimizing memory traffic beyond what standard libraries or even torch.compile might achieve automatically.²²
Rapid Prototyping: The Python-based syntax allows for much faster iteration and experimentation with different kernel implementations compared to CUDA C++.²²
Performance: Triton aims to generate kernels with performance comparable to those written by CUDA experts, often achieving near hardware limits, especially for operations like GEMM and attention.²² It includes an auto-tuner to find optimal configurations (like block sizes) for specific hardware.
Accessibility: It significantly lowers the barrier to entry for GPU programming, enabling more researchers and engineers to write custom kernels without deep CUDA expertise.⁶
Integration with PyTorch: Triton kernels can be seamlessly called from PyTorch. Furthermore, TorchInductor (the default backend for torch.compile in PyTorch 2.0+) heavily utilizes Triton to generate its optimized GPU kernels.⁵ Understanding Triton can provide insights into how torch.compile works.

Core Triton Concepts and Syntax:

Triton kernels operate on blocks of data (tiles) processed in parallel by GPU thread blocks. Key elements include:

@triton.jit: The JIT compiler decorator for kernel functions.⁶
tl.program_id(axis): Returns the unique ID of the current thread block executing the kernel instance.⁶ Used for partitioning work.
tl.arange(start, end): Creates a 1D tensor representing a range of indices, often used for calculating memory offsets within a block.⁶
tl.load(pointer, mask=..., other=...): Loads a tile of data from global GPU memory (HBM) into faster on-chip memory (SRAM/registers). The mask handles boundary conditions safely.⁶
tl.store(pointer, value, mask=...): Stores a tile of data from on-chip memory back to HBM, again using mask for safety.⁶
tl.dot(a, b): Performs block-level matrix multiplication, typically leveraging Tensor Cores for acceleration. Assumes a and b are tiles loaded into SRAM.⁶
tl.* element-wise ops: Functions like tl.exp, tl.maximum, tl.sum, etc., operate element-wise on Triton tensors.⁶
tl.constexpr: Marks variables (like BLOCK_SIZE) as compile-time constants, allowing the compiler to generate more specialized and optimized code.²⁵

Simple Triton Kernel Example (Fused ReLU):

Before tackling GEMM, let's look at a simpler example: applying the ReLU activation function. While PyTorch's built-in ReLU is fast, this illustrates the basic structure of a Triton kernel.

Python
import torch
import triton
import triton.language as tl

@triton.jit
def relu_kernel(x_ptr, # Pointer to input tensor
                y_ptr, # Pointer to output tensor
                n_elements, # Total number of elements in the tensor
                BLOCK_SIZE: tl.constexpr # Size of blocks processed by each kernel instance
                ):
    """
    Triton kernel for element-wise ReLU activation: y = max(0, x).
    """
    # 1. Calculate the range of elements this kernel instance will process
    pid = tl.program_id(axis=0) # Get the unique ID (0, 1, 2,...) for this instance
    # Calculate the starting offset for this instance
    block_start = pid * BLOCK_SIZE
    # Create a range of offsets relative to the block start
    offsets = block_start + tl.arange(0, BLOCK_SIZE) # e.g., [1024, 1025,..., 2047] if pid=1, BLOCK_SIZE=1024

    # 2. Create a mask to handle elements potentially out of bounds
    # This is important if n_elements is not perfectly divisible by BLOCK_SIZE
    mask = offsets < n_elements

    # 3. Load the input data block safely using the mask
    # Elements outside the mask will be loaded with 0.0 (or another specified value)
    x = tl.load(x_ptr + offsets, mask=mask, other=0.0)

    # 4. Apply the ReLU activation element-wise
    # tl.maximum performs element-wise max operation
    relu_output = tl.maximum(x, 0.0)

    # 5. Store the result back to the output tensor, using the mask
    # Only elements within the valid range (mask=True) are written
    tl.store(y_ptr + offsets, relu_output, mask=mask)

def triton_relu(x: torch.Tensor):
    """
    Wrapper function to launch the Triton ReLU kernel.
    """
    # Allocate output tensor
    y = torch.empty_like(x)
    # Ensure tensors are contiguous in memory, as many kernels assume this
    assert x.is_contiguous(), "Input tensor must be contiguous"
    assert y.is_contiguous(), "Output tensor must be contiguous"

    n_elements = x.numel() # Get the total number of elements

    # Define kernel launch parameters
    # BLOCK_SIZE determines how many elements each kernel instance handles.
    # This is a tunable parameter affecting performance.
    BLOCK_SIZE = 1024 # Powers of 2 are common

    # Grid size determines how many kernel instances to launch.
    # We need enough instances to cover all elements.
    # triton.cdiv(a, b) computes ceil(a / b)
    grid = (triton.cdiv(n_elements, BLOCK_SIZE),) # Launch a 1D grid of instances

    # Launch the kernel
    # Pass pointers, dimensions, and constexpr block size
    relu_kernel[grid](x, y, n_elements, BLOCK_SIZE=BLOCK_SIZE)

    return y

# Example Usage
if torch.cuda.is_available():
    # Create a sample tensor on GPU
    input_tensor = torch.randn(2048 * 2, device='cuda') * 10 - 5 # Include negative values

    # Run Triton ReLU
    output_triton = triton_relu(input_tensor)

    # Run standard PyTorch ReLU for comparison
    output_torch = torch.relu(input_tensor)

    print("--- Triton ReLU Example ---")
    print(f"Input shape: {input_tensor.shape}")
    # print(f"Input sample: {input_tensor[:8]}")
    # print(f"Triton output sample: {output_triton[:8]}")
    # print(f"PyTorch output sample: {output_torch[:8]}")

    # Verify correctness
    if torch.allclose(output_triton, output_torch):
        print("✅ Triton and Torch ReLU match")
    else:
        print("❌ Triton and Torch ReLU differ")
        print(f"   Max difference: {torch.abs(output_triton - output_torch).max().item()}")
else:
    print("CUDA not available, skipping Triton ReLU example.")

Triton GEMM Example:

Now, let's look at the structure of a basic GEMM kernel in Triton. This kernel implements C=A×B. It divides the output matrix C into blocks and assigns each block to a kernel instance (program ID). Each instance then iterates through blocks of the K dimension, loading tiles of A and B into fast memory, performing a block-level matrix multiplication using tl.dot, and accumulating the result.

Python
import torch
import triton
import triton.language as tl

@triton.jit
def matmul_kernel(
        # Pointers to matrices
        a_ptr, b_ptr, c_ptr,
        # Matrix dimensions
        M, N, K,
        # Strides for row-major layout
        stride_am, stride_ak, # Stride to move to next row (M) or next column (K) in A
        stride_bk, stride_bn, # Stride to move to next row (K) or next column (N) in B
        stride_cm, stride_cn, # Stride to move to next row (M) or next column (N) in C
        # Tile dimensions (compile-time constants)
        BLOCK_SIZE_M: tl.constexpr, BLOCK_SIZE_N: tl.constexpr, BLOCK_SIZE_K: tl.constexpr
        # Optional: Grouping for L2 cache locality (not shown in this basic version)
        # GROUP_SIZE_M: tl.constexpr
        ):
    """
    Kernel for computing the matrix multiplication C = A x B.
    A has shape (M, K), B has shape (K, N) and C has shape (M, N)
    Each kernel instance computes one BLOCK_SIZE_M x BLOCK_SIZE_N block of C.
    """
    # -----------------------------------------------------------
    # Map program ids `pid` to the block of C it should compute.
    # We use a 1D launch grid, so pid is a scalar.
    pid = tl.program_id(axis=0)
    # Calculate how many blocks are needed grid-wise
    num_pid_m = tl.cdiv(M, BLOCK_SIZE_M) # Number of blocks in M dimension
    num_pid_n = tl.cdiv(N, BLOCK_SIZE_N) # Number of blocks in N dimension
    # Calculate the grid indices for the current block
    pid_m = pid // num_pid_n # Row index of the block in the grid
    pid_n = pid % num_pid_n  # Column index of the block in the grid

    # ----------------------------------------------------------
    # Create pointers for the first blocks of A and B.
    # We need ranges of indices for the M, N, and K dimensions within a block.
    offs_m = pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M) # Row indices for the C block
    offs_n = pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N) # Col indices for the C block
    offs_k = tl.arange(0, BLOCK_SIZE_K) # Indices for the K dimension

    # Calculate pointers for the initial K-block of A and B
    # Use broadcasting to create pointers for the entire tile
    # a_ptrs shape: (BLOCK_SIZE_M, BLOCK_SIZE_K)
    # b_ptrs shape: (BLOCK_SIZE_K, BLOCK_SIZE_N)
    a_ptrs = a_ptr + (offs_m[:, None] * stride_am + offs_k[None, :] * stride_ak)
    b_ptrs = b_ptr + (offs_k[:, None] * stride_bk + offs_n[None, :] * stride_bn)

    # -----------------------------------------------------------
    # Iterate to compute a block of the C matrix.
    # We accumulate results in high precision (float32)
    accumulator = tl.zeros((BLOCK_SIZE_M, BLOCK_SIZE_N), dtype=tl.float32)

    # Loop over the K dimension in steps of BLOCK_SIZE_K
    for k in range(0, tl.cdiv(K, BLOCK_SIZE_K)):
        # Load the current blocks of A and B
        # Masks handle potential out-of-bounds accesses in the K dimension
        # 'other=0.0' pads with zeros if out of bounds
        a = tl.load(a_ptrs + k * BLOCK_SIZE_K * stride_ak, # Advance pointer in K dim
                    mask=(offs_k[None, :] + k * BLOCK_SIZE_K < K), other=0.0)
        b = tl.load(b_ptrs + k * BLOCK_SIZE_K * stride_bk, # Advance pointer in K dim
                    mask=(offs_k[:, None] + k * BLOCK_SIZE_K < K), other=0.0)

        # Perform block-level matrix multiplication and accumulate
        accumulator += tl.dot(a, b)

        # Pointers `a_ptrs` and `b_ptrs` themselves are NOT advanced here
        # because the offset `k * BLOCK_SIZE_K * stride_ak` is added inside tl.load.
        # Alternatively, one could advance the base pointers like:
        # a = tl.load(a_ptrs, mask=...)
        # b = tl.load(b_ptrs, mask=...)
        # accumulator += tl.dot(a, b)
        # a_ptrs += BLOCK_SIZE_K * stride_ak
        # b_ptrs += BLOCK_SIZE_K * stride_bk

    # Convert accumulator to the desired output type (e.g., float16)
    c = accumulator.to(c_ptr.dtype.element_ty)

    # -----------------------------------------------------------
    # Write back the block of the output matrix C with masks.
    # Calculate pointers for the C block
    c_ptrs = c_ptr + stride_cm * offs_m[:, None] + stride_cn * offs_n[None, :]
    # Create masks to prevent writing out of bounds for the M and N dimensions
    c_mask = (offs_m[:, None] < M) & (offs_n[None, :] < N)
    tl.store(c_ptrs, c, mask=c_mask)


def triton_matmul(a, b):
    """
    Wrapper function to launch the Triton Matmul kernel.
    """
    # Check constraints.
    assert a.shape[1] == b.shape, "Incompatible dimensions"
    assert a.is_contiguous(), "Matrix A must be contiguous"
    assert b.is_contiguous(), "Matrix B must be contiguous"
    M, K = a.shape
    K, N = b.shape
    # Allocates output.
    c = torch.empty((M, N), device=a.device, dtype=a.dtype) # Match input dtype

    # Define block sizes (these are crucial for performance and can be tuned)
    # Common values, but optimal ones depend on GPU architecture and matrix shapes
    BLOCK_SIZE_M = 128
    BLOCK_SIZE_N = 128
    BLOCK_SIZE_K = 32
    # GROUP_SIZE_M = 8 # Example for potential L2 cache optimization

    # Launch grid: 1D grid where each program instance computes one block of C
    grid = (triton.cdiv(M, BLOCK_SIZE_M) * triton.cdiv(N, BLOCK_SIZE_N), )

    # Launch the kernel
    matmul_kernel[grid](
        a, b, c,
        M, N, K,
        a.stride(0), a.stride(1),
        b.stride(0), b.stride(1),
        c.stride(0), c.stride(1),
        BLOCK_SIZE_M=BLOCK_SIZE_M, BLOCK_SIZE_N=BLOCK_SIZE_N, BLOCK_SIZE_K=BLOCK_SIZE_K
        # GROUP_SIZE_M=GROUP_SIZE_M # If using grouping
    )
    return c

# Example Usage
if torch.cuda.is_available():
    torch.manual_seed(0)
    # Use dimensions that are multiples of block sizes if possible
    M, N, K = 512, 1024, 256
    a = torch.randn((M, K), device='cuda', dtype=torch.float16)
    b = torch.randn((K, N), device='cuda', dtype=torch.float16)

    print("\n--- Triton GEMM Example ---")
    print(f"Input shapes: A=({M}, {K}), B=({K}, {N})")
    print(f"Using dtype: {a.dtype}")

    # Time Triton GEMM
    # Warmup
    _ = triton_matmul(a, b)
    torch.cuda.synchronize()
    start_time = time.time()
    triton_output = triton_matmul(a, b)
    torch.cuda.synchronize()
    triton_time = time.time() - start_time
    print(f"Triton GEMM time: {triton_time:.6f} seconds")

    # Time PyTorch GEMM
    # Warmup
    _ = torch.matmul(a, b)
    torch.cuda.synchronize()
    start_time = time.time()
    torch_output = torch.matmul(a, b)
    torch.cuda.synchronize()
    torch_time = time.time() - start_time
    print(f"Torch GEMM time: {torch_time:.6f} seconds")

    # Verify correctness
    if torch.allclose(triton_output, torch_output, atol=1e-2, rtol=0.01): # Adjust tolerance for FP16
        print("✅ Triton and Torch GEMM match")
    else:
        print("❌ Triton and Torch GEMM differ")
        print(f"   Max difference: {torch.abs(triton_output - torch_output).max().item()}")

    if torch_time > 0:
        print(f"Speedup factor (Torch / Triton): {torch_time / triton_time:.2f}x")
        # Note: Torch matmul often calls highly optimized cuBLAS kernels.
        # Beating it requires careful Triton tuning and potentially fusion.
        # This basic kernel might not outperform cuBLAS without further optimization.

else:
    print("CUDA not available, skipping Triton GEMM example.")

⁶

Triton represents a significant step towards making custom GPU kernel development more accessible. While torch.compile leverages it automatically, understanding Triton allows developers to manually craft highly optimized kernels for specific bottlenecks or novel operations, pushing performance beyond standard libraries when necessary.

8. Leveraging Pre-Optimized Kernels: Introduction to Liger

While Triton empowers developers to write custom kernels, another approach is to leverage pre-built, optimized Triton kernels packaged into libraries. Liger-Kernel is a prominent example specifically targeting LLM acceleration.¹⁴

What is Liger?

Liger-Kernel is an open-source library developed by LinkedIn containing a collection of efficient Triton kernels designed to accelerate common operations found in LLM training and inference pipelines.14 It aims to provide drop-in replacements for standard Hugging Face Transformer components, offering performance improvements with minimal code changes.

Key Features & Optimizations:

Liger achieves performance gains primarily through kernel fusion, similar in principle to FlashAttention but applied to different parts of the Transformer block:

Fused Kernels: It provides optimized Triton implementations for operations like:
- RMSNorm: Fuses the normalization calculation with element-wise operations.
- RoPE (Rotary Position Embeddings): Fuses the embedding calculation and application.
- SwiGLU / GeGLU: Fuses the gated linear unit activations (often involving multiple element-wise operations and GEMMs).
- CrossEntropy Loss: Can fuse the final linear projection with the cross-entropy loss calculation, avoiding materialization of the full logit tensor, which is beneficial for large vocabularies.³¹
- Other potential fusions depending on the library version. By fusing these operations, Liger reduces memory I/O and kernel launch overhead.¹⁴
Performance Gains: Liger reports significant throughput increases (e.g., up to ~20-26% in multi-GPU training) and substantial memory reduction (e.g., up to ~60%) compared to baseline Hugging Face implementations. This can enable training with larger batch sizes or longer context lengths on the same hardware.¹⁴
Ease of Use: Liger is designed for easy integration. It offers:
- AutoLigerKernelForCausalLM: An AutoModel class that automatically patches supported Hugging Face models upon loading.³⁰
- Model-Specific Patching APIs (e.g., apply_liger_kernel_to_llama): Functions to apply Liger kernels to specific model architectures before instantiation.³⁰
- Individual Kernel Modules: For advanced users composing custom models.³²
Lightweight Dependency: Requires only PyTorch and Triton, avoiding complex dependencies.¹⁴
Compatibility: Supports multi-GPU training setups (FSDP, DeepSpeed) and both NVIDIA CUDA and AMD ROCm platforms.¹⁴

Reproducible Code (Applying Liger to HF Model):

Python
# NOTE: Requires Liger-Kernel: pip install liger-kernel
# NOTE: Requires transformers: pip install transformers
# NOTE: Requires accelerate for device_map='auto': pip install accelerate
# NOTE: Replace "meta-llama/Llama-2-7b-hf" with an actual model path or identifier
#       accessible to your environment (requires appropriate access/download).

import torch
import transformers
import os
import importlib

# Check if Liger is installed and CUDA is available
try:
    import liger_kernel
    # Check for specific components to be more certain
    from liger_kernel.transformers import AutoLigerKernelForCausalLM, apply_liger_kernel_to_llama
    liger_available = True
    print(f"Liger-Kernel version {liger_kernel.__version__} found.")
except ImportError:
    liger_available = False
    print("Liger-Kernel not installed. Skipping Liger example.")

if not torch.cuda.is_available():
     print("CUDA not available. Skipping Liger example.")
     liger_available = False # Ensure flag is false if no CUDA

if liger_available:
    # --- Option 1: Using AutoLigerKernelForCausalLM (Simplest) ---
    print("\n--- Method 1: Using AutoLigerKernelForCausalLM ---")
    # Use a known supported model identifier (replace if necessary)
    # Ensure you have access permission and enough disk space/memory.
    model_name_or_path = "meta-llama/Llama-2-7b-hf"
    print(f"Attempting to load '{model_name_or_path}' with AutoLigerKernelForCausalLM...")
    print("Note: This may download the model if not cached locally.")

    try:
        # This AutoModel wrapper automatically patches if the model type (e.g., Llama) is supported by Liger.
        model_auto = AutoLigerKernelForCausalLM.from_pretrained(
            model_name_or_path,
            torch_dtype=torch.bfloat16 if torch.cuda.is_bf16_supported() else torch.float16,
            device_map='auto' # Requires 'accelerate' library
        )
        print("\nModel loaded successfully using AutoLigerKernelForCausalLM!")
        print("Liger kernels should be automatically applied where supported.")
        # Example: Check if a layer known to be patched exists and is of Liger type (conceptual)
        # This requires knowing Liger's internal class names, which might change.
        # Example check for RMSNorm in the first block of a Llama model:
        if hasattr(model_auto, 'model') and hasattr(model_auto.model, 'layers') and len(model_auto.model.layers) > 0:
             first_layer = model_auto.model.layers
             if hasattr(first_layer, 'input_layernorm'):
                 norm_layer = first_layer.input_layernorm
                 if 'liger_kernel' in str(norm_layer.__class__).lower():
                      print(f"Detected Liger RMSNorm in layer 0: {type(norm_layer)}")
                 else:
                      print(f"Layer 0 RMSNorm type: {type(norm_layer)} (Not Liger?)")

        # Clean up memory
        del model_auto
        torch.cuda.empty_cache()

    except Exception as e:
        print(f"\n[Error] Could not load model with AutoLigerKernelForCausalLM: {e}")
        print("  Possible reasons: Model not found, insufficient memory/disk, missing dependencies (transformers, accelerate), or access restrictions.")


    # --- Option 2: Using Model-Specific Patching API (for Llama) ---
    print("\n--- Method 2: Using apply_liger_kernel_to_llama ---")
    # Check if the model name suggests it's a Llama model for this specific patcher
    if "llama" in model_name_or_path.lower():
        print(f"Applying Liger kernel patches for Llama architecture...")
        try:
            # Apply patches *before* loading the model.
            # Defaults usually apply all relevant kernels. Can be specific:
            apply_liger_kernel_to_llama(
                rms_norm=True,
                rope=True,
                swiglu=True,
                # cross_entropy=True, # Only relevant for models with a value head, etc.
                # fused_linear_cross_entropy=False
            )
            print("Liger patches applied to transformers Llama classes.")

            print(f"\nAttempting to load '{model_name_or_path}' with standard AutoModelForCausalLM...")
            # Load the model using the standard Hugging Face class AFTER patching
            model_patched = transformers.AutoModelForCausalLM.from_pretrained(
                model_name_or_path,
                torch_dtype=torch.bfloat16 if torch.cuda.is_bf16_supported() else torch.float16,
                device_map='auto'
            )
            print("\nModel loaded successfully using standard AutoModel after patching!")
            print("The instantiated model should now use Liger kernels where applicable.")

            # Example check (similar to above)
            if hasattr(model_patched, 'model') and hasattr(model_patched.model, 'layers') and len(model_patched.model.layers) > 0:
                 first_layer = model_patched.model.layers
                 if hasattr(first_layer, 'input_layernorm'):
                     norm_layer = first_layer.input_layernorm
                     if 'liger_kernel' in str(norm_layer.__class__).lower():
                          print(f"Detected Liger RMSNorm in layer 0: {type(norm_layer)}")
                     else:
                          print(f"Layer 0 RMSNorm type: {type(norm_layer)} (Not Liger?)")

            # Clean up memory
            del model_patched
            torch.cuda.empty_cache()

        except Exception as e:
            print(f"\n[Error] Could not load model after applying Liger patches: {e}")
            print("  Possible reasons: Model not found, insufficient memory/disk, missing dependencies, or access restrictions.")
    else:
        print(f"Model name '{model_name_or_path}' does not contain 'llama', skipping Llama-specific patching example.")

¹⁴

Libraries like Liger-Kernel represent a valuable layer in the optimization stack. By leveraging Triton internally, they provide the performance benefits of custom kernels without requiring end-users to write or tune Triton code themselves. This modular approach allows the community to build and share optimized components, accelerating progress in LLM efficiency.

9. When GEMM Isn't the Whole Story

While this tutorial has focused heavily on GEMM optimization due to its dominance in standard dense Transformer LLMs, it's important to acknowledge scenarios where other operations might become bottlenecks, or where the computational pattern shifts away from GEMM entirely [User Material 6].

Convolutions: In Convolutional Neural Networks (CNNs) or hybrid architectures incorporating convolutional layers, the convolution operation itself is key. Libraries like NVIDIA's cuDNN (used by torch.nn.Conv2d) employ various algorithms for convolutions, including:
- Implicit GEMM: Techniques like im2col or im2row transform the convolution into a large GEMM operation, which can then be accelerated by cuBLAS or Tensor Cores.
- Direct Methods: Algorithms like Winograd or FFT-based convolutions can be faster than implicit GEMM for certain kernel sizes and hardware. cuDNN selects the best algorithm based on heuristics.
Sparse Models: Techniques like Mixture-of-Experts (MoE) or model pruning introduce sparsity into the computations. Instead of dense GEMM, these models rely heavily on Sparse Matrix Multiplication (SpMM) kernels. Optimizing SpMM involves different strategies focused on handling irregular memory access and load balancing for the non-zero elements [User Material 6].
Alternative Architectures & Long Sequences: For handling extremely long sequences where the quadratic complexity of standard attention becomes prohibitive even with FlashAttention, alternative architectures are being explored. State-Space Models (SSMs) like Mamba, or certain recurrent architectures, may rely less on traditional GEMM and more on operations like parallel scans or structured matrix-vector multiplications [User Material 6]. Even within attention, techniques like FlashAttention are critical adaptations for long sequences, fundamentally changing the memory access pattern compared to naive GEMM-based attention [User Material 4].

While GEMM optimization is crucial for today's prevalent dense LLMs, developers should be aware that the critical bottlenecks can shift depending on the model architecture and specific application. Profiling the model remains essential to identify where computational time is actually spent.

10. Conclusion: Your GEMM Optimization Toolkit

Accelerating Large Language Models hinges significantly on optimizing the underlying General Matrix-Matrix Multiplication (GEMM) operations that dominate their computational cost. This tutorial has journeyed from the fundamentals of GEMM and its importance in LLMs to a multi-level approach for optimization within the PyTorch ecosystem.

We began by understanding why GPUs are so effective at GEMM, leveraging their massive parallelism, high arithmetic intensity, and specialized hardware like Tensor Cores. We then explored practical optimization techniques, starting with the foundational Level 1 strategies:

Using lower-precision data types (FP16, BF16, TF32) to boost throughput and reduce memory usage.
Aligning matrix dimensions with hardware tile sizes (multiples of 8/16) to maximize utilization.
Leveraging torch.compile for automatic kernel fusion and optimization, often utilizing Triton under the hood.

Moving to Level 2, we examined more advanced techniques requiring specific libraries or structural awareness:

Employing torch.bmm to batch many small matrix multiplications into fewer, larger kernel calls.
Utilizing quantization (INT8/INT4) via libraries like bitsandbytes to drastically cut memory footprint and potentially speed up inference.
Leveraging fused attention through PyTorch's scaled_dot_product_attention, which automatically uses backends like FlashAttention to overcome memory bandwidth limitations and enable longer sequence processing.

Finally, at Level 3, we introduced Triton as a powerful tool for writing custom, high-performance GPU kernels directly in Python, enabling bespoke fusions and fine-grained control. We also looked at Liger-Kernel as an example of a library providing pre-built, optimized Triton kernels for common LLM operations beyond attention, simplifying access to advanced fusion techniques.

The core principles underlying these optimizations are consistent: reduce data movement between slow and fast memory (HBM vs SRAM/registers), maximize the utilization of specialized hardware units (like Tensor Cores), and exploit the massive parallelism offered by GPUs.

Choosing the right technique depends on the specific context, performance requirements, and development resources available. Profiling your model is crucial to identify the true bottlenecks before applying optimizations. Start with the Level 1 techniques, as they often provide substantial gains with minimal effort. If further speed is needed, explore batching, quantization (especially for inference), and ensure fused attention is active via SDPA. For cutting-edge performance or unique model components, delving into Triton or leveraging libraries like Liger may be necessary.

The following table summarizes the key techniques discussed:

Table: LLM GEMM Optimization Techniques in PyTorch

Technique	Key PyTorch/Library Tool	Impact (Speed/Memory)	Ease of Use	When to Apply	Key Considerations/Trade-offs
Lower Precision (FP16/BF16)	`model.half()`, `.to(torch.bfloat16)`, `set_default_dtype`	++ Speed, + Memory	★★★★★	Almost always (Training & Inference)	FP16: Range issues (needs scaling). BF16: Ampere+. Accuracy.
Enable TF32	`torch.backends.cuda.matmul.allow_tf32 = True`	+ Speed (for FP32)	★★★★★	FP32 inputs on Ampere+ GPUs	Minimal accuracy loss vs FP32. No memory saving.
Align Dimensions	Model architecture design (multiples of 8/16/64)	+ Speed	★★★☆☆	During model design/selection	Requires architectural changes or padding.
Automatic Compilation	`torch.compile(model)`	++ Speed, +/- Memory	★★★★☆	PyTorch 2.0+, Training & Inference	Compile time overhead. Potential graph breaks.
Batched GEMM	`torch.bmm()`	++ Speed (for many small GEMMs)	★★★★☆	Batches of small matrices	Requires specific input shape `(B, M, K)`.
Quantization (INT8/INT4)	`bitsandbytes` (`load_in_8bit`, `Linear8bitLt`)	+ Speed, ++ Memory	★★★★☆	Primarily Inference (or QAT)	Accuracy loss possible. Careful setup.
Fused Attention	`torch.nn.functional.scaled_dot_product_attention` (SDPA)	++ Speed, ++ Memory (Seq Len)	★★★★☆	PyTorch 2.0+, Attention layers	Auto-uses FlashAttention/xFormers if avail.
Custom Kernels (Triton)	`@triton.jit`, `tl.*` functions	+++ Speed, +++ Memory (Custom)	★★☆☆☆	Specific bottlenecks, novel fusions	Kernel development effort & tuning required.
Pre-built Kernels (Liger)	`liger_kernel` patching APIs / AutoModel	++ Speed, ++ Memory	★★★★☆	Supported LLM layers (RMSNorm, etc.)	Library dependency. Model architecture support needed.

By systematically applying these GEMM optimization techniques, developers can significantly enhance the performance of their LLMs in PyTorch, making them faster, more efficient, and capable of tackling larger and more complex tasks.

Things about ML, DL, Data Science, AI

Tuesday, April 22, 2025

Accelerating Large Language Models with GEMM Optimization in PyTorch

1. Introduction: The Need for Speed in LLMs

2. What is GEMM? The Computational Core of LLMs

3. Why GPUs Excel at GEMM

4. LLMs as "GEMM Farms": The Scale of the Challenge

5. Level 1 Optimizations: The Low-Hanging Fruit in PyTorch

Technique 1: Leverage Lower Precision (FP16, BF16, TF32)

Technique 2: Aligning with Hardware (Dimension Multiples)

Technique 3: Automatic Acceleration with `torch.compile`

6. Level 2 Optimizations: Getting More Advanced

Technique 4: Batching Small Matmuls (`torch.bmm`)

Technique 5: Quantization (INT8/INT4) with `bitsandbytes`

Technique 6: Fused Operations (FlashAttention & xFormers via SDPA)

7. Level 3: Unleashing Custom Power with Triton

8. Leveraging Pre-Optimized Kernels: Introduction to Liger

9. When GEMM Isn't the Whole Story

10. Conclusion: Your GEMM Optimization Toolkit

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Tuesday, April 22, 2025

Accelerating Large Language Models with GEMM Optimization in PyTorch

1. Introduction: The Need for Speed in LLMs

2. What is GEMM? The Computational Core of LLMs

3. Why GPUs Excel at GEMM

4. LLMs as "GEMM Farms": The Scale of the Challenge

5. Level 1 Optimizations: The Low-Hanging Fruit in PyTorch

Technique 1: Leverage Lower Precision (FP16, BF16, TF32)

Technique 2: Aligning with Hardware (Dimension Multiples)

Technique 3: Automatic Acceleration with torch.compile

6. Level 2 Optimizations: Getting More Advanced

Technique 4: Batching Small Matmuls (torch.bmm)

Technique 5: Quantization (INT8/INT4) with bitsandbytes

Technique 6: Fused Operations (FlashAttention & xFormers via SDPA)

7. Level 3: Unleashing Custom Power with Triton

8. Leveraging Pre-Optimized Kernels: Introduction to Liger

9. When GEMM Isn't the Whole Story

10. Conclusion: Your GEMM Optimization Toolkit

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Post a Comment

Technique 3: Automatic Acceleration with `torch.compile`

Technique 4: Batching Small Matmuls (`torch.bmm`)

Technique 5: Quantization (INT8/INT4) with `bitsandbytes`