From 12e86bb63b4bde510f281581268b95cba5b71bcc Mon Sep 17 00:00:00 2001 From: xeon27 Date: Fri, 28 Feb 2025 14:52:11 -0500 Subject: [PATCH 01/12] Add placeholder books for 6 cuda kernels --- books/compute/src/cuda/kernels/layernorm_forward.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/books/compute/src/cuda/kernels/layernorm_forward.md b/books/compute/src/cuda/kernels/layernorm_forward.md index dc78369..7aee937 100644 --- a/books/compute/src/cuda/kernels/layernorm_forward.md +++ b/books/compute/src/cuda/kernels/layernorm_forward.md @@ -3,6 +3,8 @@ # Kernels for LayerNorm forward pass + + {{ #aipr_header }} From 634379098fde60cf89a1f1dc3f2a5e1d18cd8907 Mon Sep 17 00:00:00 2001 From: xeon27 Date: Wed, 5 Mar 2025 15:41:52 -0500 Subject: [PATCH 02/12] Add section headers --- books/compute/src/cuda/kernels/layernorm_forward.md | 2 -- 1 file changed, 2 deletions(-) diff --git a/books/compute/src/cuda/kernels/layernorm_forward.md b/books/compute/src/cuda/kernels/layernorm_forward.md index 7aee937..dc78369 100644 --- a/books/compute/src/cuda/kernels/layernorm_forward.md +++ b/books/compute/src/cuda/kernels/layernorm_forward.md @@ -3,8 +3,6 @@ # Kernels for LayerNorm forward pass - - {{ #aipr_header }} From f2e48ccd3fbb62550caa4bc892786cd820ca42ef Mon Sep 17 00:00:00 2001 From: xeon27 Date: Wed, 5 Mar 2025 23:20:45 -0500 Subject: [PATCH 03/12] Add figures --- .../cuda/kernels/imgs/layernorm_kernel1.png | Bin 0 -> 129672 bytes .../cuda/kernels/imgs/layernorm_kernel2.png | Bin 0 -> 553203 bytes .../cuda/kernels/imgs/layernorm_kernel3.png | Bin 0 -> 137975 bytes .../cuda/kernels/imgs/layernorm_kernel5.png | Bin 0 -> 446412 bytes 4 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 books/compute/src/cuda/kernels/imgs/layernorm_kernel1.png create mode 100644 books/compute/src/cuda/kernels/imgs/layernorm_kernel2.png create mode 100644 books/compute/src/cuda/kernels/imgs/layernorm_kernel3.png create mode 100644 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z=234sgcB#zew(*y7xlSlQA5QpL&I=J6R=M;vLpE~c*F79fg&~-t}%1%FQbMmr;aDl z8;LPiqWupJwbE8CblQx==!fH^zWmJ>@=EMX)}d>+iSdRAPzCgQB{nt_+Z#EXf&$8V zH2@iAovDo-@}tunG|A$}UFJU;g`d0oi2nay|H}FQgYW5o{FkvT|M9i|Yrpyb9h322 fxAyPZsVA4Nzu~H8UeNf0Uaq&pV0*^4BWM2~Z0qQ3 literal 0 HcmV?d00001 From e2c6fd86ac1b37d88057209eebdd5fc5abe8e1ad Mon Sep 17 00:00:00 2001 From: xeon27 Date: Fri, 7 Mar 2025 00:28:07 -0500 Subject: [PATCH 05/12] Fix latex bug --- books/compute/src/cuda/kernels/layernorm_forward.md | 11 +++++++++++ 1 file changed, 11 insertions(+) diff --git a/books/compute/src/cuda/kernels/layernorm_forward.md b/books/compute/src/cuda/kernels/layernorm_forward.md index dc78369..0471e29 100644 --- a/books/compute/src/cuda/kernels/layernorm_forward.md +++ b/books/compute/src/cuda/kernels/layernorm_forward.md @@ -14,7 +14,11 @@ last D dimensions of the activation tensor as described in [this foundational paper](https://arxiv.org/abs/1607.06450) by Ba et al. (2016). The normalization equation is given below: +<<<<<<< HEAD $$y = \frac{x - \mathbb{E}[x]}{\sqrt{Var[x] + \epsilon}} \times \gamma + \beta$$ +======= +$$y = \frac{x - \mathbb{E}[x]}{\sqrt{Var[x] + \epsilon}} * \gamma + \beta$$ +>>>>>>> 522c130 (Fix latex bug) where, \\(\mathbb{E}[z]\\) and \\(Var[z]\\) are the expectation and variance of random variable \\(z\\), respectively. Note that in the above \\(\epsilon\\) @@ -66,6 +70,7 @@ in that segment. The kernel code is broken down into 4 steps: $$\mathbb{E}[x] = \frac{1}{C} \sum_{i=1}^{C} x_i$$ 2. Variance and reciprocal of standard deviation (rstd) calculation +<<<<<<< HEAD $$Var[x] = \frac{1}{C} \sum_{i=1}^{C} (x_i - \mathbb{E}[x])^2$$ @@ -76,6 +81,12 @@ in that segment. The kernel code is broken down into 4 steps: $$y_i = ((x_i - \mathbb{E}[x]) * rstd[x]) * \gamma_i + \beta_i$$ +======= +$$Var[x] = \frac{1}{C} \sum_{i=1}^{C} (x_i - \mathbb{E}[x])^2$$ +$$rstd[x] = \frac{1}{\sqrt{Var[x] + \epsilon}}$$ +3. Apply mean and variance normalization and then scale and shift with the learnable weight and bias parameters +$$y_i = ((x_i - \mathbb{E}[x]) * rstd[x]) * \gamma_i + \beta_i$$ +>>>>>>> 522c130 (Fix latex bug) 4. Store mean and rstd for backward pass The kernel uses a 1D grid and block as shown in Figure-1a. From 5eceddcfa06a647a07d5fefa4f4774fb45321049 Mon Sep 17 00:00:00 2001 From: xeon27 Date: Fri, 7 Mar 2025 00:31:31 -0500 Subject: [PATCH 06/12] Space out equations --- books/compute/src/cuda/kernels/layernorm_forward.md | 11 ----------- 1 file changed, 11 deletions(-) diff --git a/books/compute/src/cuda/kernels/layernorm_forward.md b/books/compute/src/cuda/kernels/layernorm_forward.md index 0471e29..dc78369 100644 --- a/books/compute/src/cuda/kernels/layernorm_forward.md +++ b/books/compute/src/cuda/kernels/layernorm_forward.md @@ -14,11 +14,7 @@ last D dimensions of the activation tensor as described in [this foundational paper](https://arxiv.org/abs/1607.06450) by Ba et al. (2016). The normalization equation is given below: -<<<<<<< HEAD $$y = \frac{x - \mathbb{E}[x]}{\sqrt{Var[x] + \epsilon}} \times \gamma + \beta$$ -======= -$$y = \frac{x - \mathbb{E}[x]}{\sqrt{Var[x] + \epsilon}} * \gamma + \beta$$ ->>>>>>> 522c130 (Fix latex bug) where, \\(\mathbb{E}[z]\\) and \\(Var[z]\\) are the expectation and variance of random variable \\(z\\), respectively. Note that in the above \\(\epsilon\\) @@ -70,7 +66,6 @@ in that segment. The kernel code is broken down into 4 steps: $$\mathbb{E}[x] = \frac{1}{C} \sum_{i=1}^{C} x_i$$ 2. Variance and reciprocal of standard deviation (rstd) calculation -<<<<<<< HEAD $$Var[x] = \frac{1}{C} \sum_{i=1}^{C} (x_i - \mathbb{E}[x])^2$$ @@ -81,12 +76,6 @@ in that segment. The kernel code is broken down into 4 steps: $$y_i = ((x_i - \mathbb{E}[x]) * rstd[x]) * \gamma_i + \beta_i$$ -======= -$$Var[x] = \frac{1}{C} \sum_{i=1}^{C} (x_i - \mathbb{E}[x])^2$$ -$$rstd[x] = \frac{1}{\sqrt{Var[x] + \epsilon}}$$ -3. Apply mean and variance normalization and then scale and shift with the learnable weight and bias parameters -$$y_i = ((x_i - \mathbb{E}[x]) * rstd[x]) * \gamma_i + \beta_i$$ ->>>>>>> 522c130 (Fix latex bug) 4. Store mean and rstd for backward pass The kernel uses a 1D grid and block as shown in Figure-1a. From aed7590f28f9da28be474202b291272389c63040 Mon Sep 17 00:00:00 2001 From: xeon27 Date: Tue, 18 Mar 2025 14:56:32 -0400 Subject: [PATCH 07/12] Add references --- .../src/cuda/kernels/layernorm_forward.md | 146 ++++++++++++++++-- 1 file changed, 134 insertions(+), 12 deletions(-) diff --git a/books/compute/src/cuda/kernels/layernorm_forward.md b/books/compute/src/cuda/kernels/layernorm_forward.md index dc78369..f7b2075 100644 --- a/books/compute/src/cuda/kernels/layernorm_forward.md +++ b/books/compute/src/cuda/kernels/layernorm_forward.md @@ -11,11 +11,18 @@ The Layer Normalization (LayerNorm) operation applies normalization across the last D dimensions of the activation tensor as described in +<<<<<<< HEAD [this foundational paper](https://arxiv.org/abs/1607.06450) by Ba et al. (2016). The normalization equation is given below: +======= +[this paper](https://arxiv.org/abs/1607.06450). The normalization equation is +given below: + +> > > > > > > cf1c71d (Add references) $$y = \frac{x - \mathbb{E}[x]}{\sqrt{Var[x] + \epsilon}} \times \gamma + \beta$$ +<<<<<<< HEAD where, \\(\mathbb{E}[z]\\) and \\(Var[z]\\) are the expectation and variance of random variable \\(z\\), respectively. Note that in the above \\(\epsilon\\) is a small term for to avoid division by zero errors, whereas \\(\gamma\\) and \\(\beta\\) @@ -42,28 +49,59 @@ For benchmarking purposes, we use the following configuration: - \\(T = 1024\\) - \\(C = 768\\) -The following table shows memory bandwidth for each kernel on a **A40 GPU for -block size 512**. The last column shows improvement **over the first kernel**: +======= +This book outlines and provides a detailed explanation of a series of CUDA +kernel implementations of LayerNorm forward pass based on the +[llm.c](https://github.com/karpathy/llm.c/tree/master/dev/cuda) repository. +Please refer to the [Layer Normalization book](../../../../fundamentals/src/normalizations/layernorm.md) +for conceptual understanding and other details about the operation. For the +purpose of this book, lets implement kernels for LayerNorm in the +[Transformer](../../../../nlp/src/llms/architecture/transformer.md) architecture +for language modeling which expects **a tensor of shape $(B, T, C)$ as input**, +where $B$ is the batch size, $T$ is the sequence length and $C$ is the hidden +dimension size. LayerNorm is applied to the last dimension. $B = 8$, $T = 1024$ +and $C = 768$ is set for benchmarking. + +> > > > > > > cf1c71d (Add references) +> > > > > > > The following table shows memory bandwidth for each kernel on a **A40 GPU for +> > > > > > > block size 512**. The last column shows improvement **over the first kernel**: | Kernel # | Bandwidth (GB/s) | Improvement | +<<<<<<< HEAD | :------- | ---------------: | :---------- | -| 1 | 41.43 | - | -| 2 | 201.25 | 4.9x | -| 3 | 362.10 | 8.7x | -| 4 | 432.03 | 10.4x | -| 5 | 538.88 | 13x | +| 1 | 41.43 | - | +| 2 | 201.25 | 4.9x | +| 3 | 362.10 | 8.7x | +| 4 | 432.03 | 10.4x | +| 5 | 538.88 | 13x | +======= +|:---------|-----------------:|:------------| +| 1 | 41.43 | - | +| 2 | 201.25 | 4.9x | +| 3 | 362.10 | 8.7x | +| 4 | 432.03 | 10.4x | +| 5 | 538.88 | 13x | + +> > > > > > > cf1c71d (Add references) ## Kernel 1 The first kernel is a copy of the CPU implementation. It parallelizes +<<<<<<< HEAD over the first 2 dimensions, \\(B\\) and \\(T\\), where \\(N = B\*T\\). **A single thread (see Figure-1a) is responsible for normalizing** -**one segment of size C**, hence it loops over all elements -in that segment. The kernel code is broken down into 4 steps: +======= +over the first 2 dimensions, $B$ and $T$, where $N = B*T$. +**A single thread (see Figure-1) is responsible for normalizing** + +> > > > > > > cf1c71d (Add references) +> > > > > > > **one segment of size C**, hence it loops over all elements +> > > > > > > in that segment. The kernel code is broken down into 4 steps: 1. Mean calculation - $$\mathbb{E}[x] = \frac{1}{C} \sum_{i=1}^{C} x_i$$ +<<<<<<< HEAD +$$\mathbb{E}[x] = \frac{1}{C} \sum_{i=1}^{C} x_i$$ 2. Variance and reciprocal of standard deviation (rstd) calculation @@ -78,8 +116,27 @@ in that segment. The kernel code is broken down into 4 steps: 4. Store mean and rstd for backward pass -The kernel uses a 1D grid and block as shown in Figure-1a. -Also note that all operations are implemented in a single kernel. +# The kernel uses a 1D grid and block as shown in Figure-1a. + + $$\mathbb{E}[x] = \frac{1}{C} \sum_{i=1}^{C} x_i$$ + +2. Variance and reciprocal of standard deviation (rstd) calculation + + $$Var[x] = \frac{1}{C} \sum_{i=1}^{C} (x_i - \mathbb{E}[x])^2$$ + + $$rstd[x] = \frac{1}{\sqrt{Var[x] + \epsilon}}$$ + +3. Apply mean and variance normalization and then scale and + shift with the learnable weight and bias parameters + + $$y_i = ((x_i - \mathbb{E}[x]) * rstd[x]) * \gamma_i + \beta_i$$ + +4. Store mean and rstd for backward pass + +The kernel uses a 1D grid and block as shown in Figure-1. + +> > > > > > > cf1c71d (Add references) +> > > > > > > Also note that all operations are implemented in a single kernel.

layernorm_kernel1 @@ -92,6 +149,8 @@ Also note that all operations are implemented in a single kernel. Figure-1a: Kernel 1 Illustration. +<<<<<<< HEAD +
layernorm_kernel1_code
@@ -117,6 +176,69 @@ thread sums corresponding elements and stores it in a _shared memory array_ array_ are iteratively reduced to obtain the final sum. For more details, see the [thread coarsening](../concepts/thread_coarsening.md) and [reduction](../concepts/reduction.md) pocket references. +======= + +```cpp +__global__ void layernorm_forward_kernel1( + float* out, float* mean, float* rstd, + const float* inp, const float* weight, const float* bias, + int N, int C +) { + int idx = blockIdx.x * blockDim.x + threadIdx.x; + float eps = 1.0e-5f; + + if (idx < N) { + // Start index of the input segment for this thread, inp[idx, :] + const float* x = inp + idx * C; + + // Compute mean + float m = 0.0f; + for (int i = 0; i < C; i++) { + m += x[i]; + } + m /= C; + + // Compute variance (without any bias correction) + float v = 0.0f; + for (int i = 0; i < C; i++) { + float diff = x[i] - m; + v += diff * diff; + } + v /= C; + + // Compute rstd + float r = 1.0f / sqrt(v + eps); + + // Compute output + // Start index of the output segment for this thread, out[idx, :] + float* y = out + idx * C; + for (int i = 0; i < C; i++) { + float o_prime = (x[i] - m) * r; // normalized output + float o = o_prime * weight[i] + bias[i]; // scale and shift + y[i] = o; + } + + // Store mean and rstd for backward pass + mean[idx] = m; + rstd[idx] = r; + } +} +``` + +## Kernel 2 + +In Kernel 2, steps 1, 2 and 3 are implemented as separate kernels. For the _mean_ +and _rstd_ kernels, **each block is responsible for one segment of C** instead of +each thread (see Figure-2) which allows for further parallelization. Whereas for +the _normalization_ kernel (step 3), each thread calculates one output element. + +Since both the _mean_ and _rstd_ calculations involve the sum operation, they +make use of _thread coarsening_ and _reduction_. In _thread coarsening_, each +thread sums corresponding elements and stores it in a _shared memory array_ +(same size as the thread block). In _reduction_, the elements in the _shared +array_ are iteratively reduced to obtain the final sum. + +> > > > > > > cf1c71d (Add references) These optimizations lead to an improvement of **~5x over Kernel 1** (for block size 512). From c87a1661c25d13983bdfbfc90cda7f29ef015810 Mon Sep 17 00:00:00 2001 From: jacobthebanana Date: Fri, 21 Mar 2025 18:35:42 +0000 Subject: [PATCH 08/12] Added gpt2 encoder kernel references. --- .../src/cuda/kernels/encoder_forward.md | 43 +++++++++++++++++++ .../src/cuda/kernels/imgs/encoder_kernel1.svg | 4 ++ .../src/cuda/kernels/imgs/encoder_kernel2.svg | 4 ++ .../src/cuda/kernels/imgs/encoder_kernel3.svg | 4 ++ .../cuda/kernels/imgs/encoder_overview.svg | 4 ++ 5 files changed, 59 insertions(+) create mode 100644 books/compute/src/cuda/kernels/imgs/encoder_kernel1.svg create mode 100644 books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg create mode 100644 books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg create mode 100644 books/compute/src/cuda/kernels/imgs/encoder_overview.svg diff --git a/books/compute/src/cuda/kernels/encoder_forward.md b/books/compute/src/cuda/kernels/encoder_forward.md index b26e78a..0fda63f 100644 --- a/books/compute/src/cuda/kernels/encoder_forward.md +++ b/books/compute/src/cuda/kernels/encoder_forward.md @@ -6,6 +6,49 @@ +## Introduction + +After tokenization, batches of input texts would be turned into arrays of token ids . The GPT-2 Encoder will turn these input ids into hidden representations that can be proceessed with the remainder of the transformer blocks: + +- Input: + - "input" `inp`, produced by the tokenizer- An integer array of shape $(B,\, T)$, where $B$ denotes the batch size (a total of $B$ sequences in the batch) while $T$ denotes the tokens in each sequence (each sequence contains a maximum of $T$ tokens.) + - token embeddings `wte`, from the model parameters- a float array of shape (vocab\_size, $C$), where $C$ is the latent dimension of this transformer model. The $n$th row of this matrix is the embedding vector for the $n$th token. + - position embeddings `wpe`, provided- a float array of shape (max\_length, $C$), where $C$ is the latent dimension of the transformer model. The $n$th role of this matrix represents the position embedding vector for the a token at position $n$ . +- Output: float tensor of shape $(B, T, C)$, where in addition to the batch ($B$) and sequence length ($T$) dimension, a "channel" $C$ is added ($C$ is the same as the latent embedding dimension of the transformer.) + +![GPT-2 Encoder Input and Output Format](./imgs/encoder_overview.svg) + +Unlike most other parts of a transformer, the embedding kernel contains notably few calculations- for the forward pass, both the word embeddings and the position embeddings are **pre-computed** and available as arrays in memory. The only job of the encoder kernel is to load the appropriate rows from both matrices, add these togehter, and write the sum to the appropriate location in memory. + +## Kernel 1 + +![GPT-2 Encoder Kernel 1](./imgs/encoder_kernel1.svg) + +THis kernel uses a total of $B \times T$ threads, where each thread handles a token in the $(B,\, T)$ input array. Within each thread, a for-loop iterates over the hidden dimensions, reading one float from `wte` and from `wpe`, take the sum, and write the sum to the output tensor. + +While this approach might work fine on CPU because of the predictable memory access pattern, the for-loop approach does not fully unleash the multiprocessing potentials of the GPU. + +## Kernel 2 + +![GPT-2 Encoder Kernel 2](./imgs/encoder_kernel2.svg) + +This kernel uses substantially more threads than kernel 1, a total of $B \times T \times C$, one for each element in the output. (Recall that in modern GPTs, the hidden dimension $C$ is usually in the thousands if not larger.) Each thread handles one float in the output tensor- the thread would read one float from the `wte` token embedding array, one float from the `wpe` position embedding array, sum the two floating point numbers, and write the result to the appropriate location in the output tensor. + +This appraoch improves over kernel 1 by using many more threads in parallel. However, since each thread needs to do its own memory read/write, the memory access pattern becomes a lot less predictable. Random memory access might quickly become a bottleneck for this approach. + +## Kernel 3 + +![GPT-2 Encoder Kernel 3](./imgs/encoder_kernel3.svg) + +On NVIDIA GPUs, reading and writing consecutive chunks of memory at a time can be significantly more efficient than doing so one byte at a time. This kernel leverages this feature to make processing a lot more efficient. + + +While in the kernel, the additions are defined in a for-loop across the $\texttt{x128::size}$, one floating point number at a time, the kernel uses `#pragma unroll` to automatically optimizes this part of the code during compilation. + +## References + +1. Code for encoder forward kernels from [llm.c](https://github.com/karpathy/llm.c/blob/master/dev/cuda/encoder_forward.cu) + {{#author VectorInstitute}} diff --git a/books/compute/src/cuda/kernels/imgs/encoder_kernel1.svg b/books/compute/src/cuda/kernels/imgs/encoder_kernel1.svg new file mode 100644 index 0000000..586f859 --- /dev/null +++ b/books/compute/src/cuda/kernels/imgs/encoder_kernel1.svg @@ -0,0 +1,4 @@ + + + +
Writes one dimension at a time, in a for-loop
Number of programs:
batch size * max length
Sequence
Batch
\ No newline at end of file diff --git a/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg b/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg new file mode 100644 index 0000000..d7cefe9 --- /dev/null +++ b/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg @@ -0,0 +1,4 @@ + + + +
Number of programs:
batch size * max length * hidden dim
Each program writes only one float
Sequence
Batch
Hidden Dim
\ No newline at end of file diff --git a/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg b/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg new file mode 100644 index 0000000..2539db5 --- /dev/null +++ b/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg @@ -0,0 +1,4 @@ + + + +
Number of programs:
batch size * max length * hidden dim //  
Each program writes up to floats along the embedding dimension
Hidden Dim
Batch
Sequence
\ No newline at end of file diff --git a/books/compute/src/cuda/kernels/imgs/encoder_overview.svg b/books/compute/src/cuda/kernels/imgs/encoder_overview.svg new file mode 100644 index 0000000..065b8f8 --- /dev/null +++ b/books/compute/src/cuda/kernels/imgs/encoder_overview.svg @@ -0,0 +1,4 @@ + + + +
Token Embeddings
(vocab size, 
hidden dim)
Position Embeddings
(max length, hidden dim)
Token IDs of input
(batch size, 
max length)
Output
(batch size,
max length, 
hidden dim)
\ No newline at end of file From 4c570071a3503e3f70cf8171f187f25ee5009d43 Mon Sep 17 00:00:00 2001 From: jacobthebanana Date: Fri, 21 Mar 2025 19:30:30 +0000 Subject: [PATCH 09/12] Updated diagrams for encoder kernel. --- books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg | 2 +- books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg | 2 +- books/compute/src/cuda/kernels/imgs/encoder_overview.svg | 2 +- 3 files changed, 3 insertions(+), 3 deletions(-) diff --git a/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg b/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg index d7cefe9..4dba26f 100644 --- a/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg +++ b/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg @@ -1,4 +1,4 @@ -
Number of programs:
batch size * max length * hidden dim
Each program writes only one float
Sequence
Batch
Hidden Dim
\ No newline at end of file +
Number of programs:
batch size * max length * hidden dim
Each program writes only one float
Batch
Hidden Dim
\ No newline at end of file diff --git a/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg b/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg index 2539db5..101191d 100644 --- a/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg +++ b/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg @@ -1,4 +1,4 @@ -
Number of programs:
batch size * max length * hidden dim //  
Each program writes up to floats along the embedding dimension
Hidden Dim
Batch
Sequence
\ No newline at end of file +
Number of programs:
batch size * max length * hidden dim //  
Each program writes up to floats along the embedding dimension
Hidden Dim
Batch
Sequence
\ No newline at end of file diff --git a/books/compute/src/cuda/kernels/imgs/encoder_overview.svg b/books/compute/src/cuda/kernels/imgs/encoder_overview.svg index 065b8f8..7c1ae16 100644 --- a/books/compute/src/cuda/kernels/imgs/encoder_overview.svg +++ b/books/compute/src/cuda/kernels/imgs/encoder_overview.svg @@ -1,4 +1,4 @@ -
Token Embeddings
(vocab size, 
hidden dim)
Position Embeddings
(max length, hidden dim)
Token IDs of input
(batch size, 
max length)
Output
(batch size,
max length, 
hidden dim)
\ No newline at end of file +
Token Embeddings
(vocab size, 
hidden dim)
Position Embeddings
(max length, hidden dim)
Token IDs of input
(batch size, 
max length)
token id
position
Output
(batch size,
max length, 
hidden dim)
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b/books/compute/src/cuda/kernels/layernorm_forward.md index f7b2075..dc78369 100644 --- a/books/compute/src/cuda/kernels/layernorm_forward.md +++ b/books/compute/src/cuda/kernels/layernorm_forward.md @@ -11,18 +11,11 @@ The Layer Normalization (LayerNorm) operation applies normalization across the last D dimensions of the activation tensor as described in -<<<<<<< HEAD [this foundational paper](https://arxiv.org/abs/1607.06450) by Ba et al. (2016). The normalization equation is given below: -======= -[this paper](https://arxiv.org/abs/1607.06450). The normalization equation is -given below: - -> > > > > > > cf1c71d (Add references) $$y = \frac{x - \mathbb{E}[x]}{\sqrt{Var[x] + \epsilon}} \times \gamma + \beta$$ -<<<<<<< HEAD where, \\(\mathbb{E}[z]\\) and \\(Var[z]\\) are the expectation and variance of random variable \\(z\\), respectively. Note that in the above \\(\epsilon\\) is a small term for to avoid division by zero errors, whereas \\(\gamma\\) and \\(\beta\\) @@ -49,59 +42,28 @@ For benchmarking purposes, we use the following configuration: - \\(T = 1024\\) - \\(C = 768\\) -======= -This book outlines and provides a detailed explanation of a series of CUDA -kernel implementations of LayerNorm forward pass based on the -[llm.c](https://github.com/karpathy/llm.c/tree/master/dev/cuda) repository. -Please refer to the [Layer Normalization book](../../../../fundamentals/src/normalizations/layernorm.md) -for conceptual understanding and other details about the operation. For the -purpose of this book, lets implement kernels for LayerNorm in the -[Transformer](../../../../nlp/src/llms/architecture/transformer.md) architecture -for language modeling which expects **a tensor of shape $(B, T, C)$ as input**, -where $B$ is the batch size, $T$ is the sequence length and $C$ is the hidden -dimension size. LayerNorm is applied to the last dimension. $B = 8$, $T = 1024$ -and $C = 768$ is set for benchmarking. - -> > > > > > > cf1c71d (Add references) -> > > > > > > The following table shows memory bandwidth for each kernel on a **A40 GPU for -> > > > > > > block size 512**. The last column shows improvement **over the first kernel**: +The following table shows memory bandwidth for each kernel on a **A40 GPU for +block size 512**. The last column shows improvement **over the first kernel**: | Kernel # | Bandwidth (GB/s) | Improvement | -<<<<<<< HEAD | :------- | ---------------: | :---------- | -| 1 | 41.43 | - | -| 2 | 201.25 | 4.9x | -| 3 | 362.10 | 8.7x | -| 4 | 432.03 | 10.4x | -| 5 | 538.88 | 13x | -======= -|:---------|-----------------:|:------------| -| 1 | 41.43 | - | -| 2 | 201.25 | 4.9x | -| 3 | 362.10 | 8.7x | -| 4 | 432.03 | 10.4x | -| 5 | 538.88 | 13x | - -> > > > > > > cf1c71d (Add references) +| 1 | 41.43 | - | +| 2 | 201.25 | 4.9x | +| 3 | 362.10 | 8.7x | +| 4 | 432.03 | 10.4x | +| 5 | 538.88 | 13x | ## Kernel 1 The first kernel is a copy of the CPU implementation. It parallelizes -<<<<<<< HEAD over the first 2 dimensions, \\(B\\) and \\(T\\), where \\(N = B\*T\\). **A single thread (see Figure-1a) is responsible for normalizing** -======= -over the first 2 dimensions, $B$ and $T$, where $N = B*T$. -**A single thread (see Figure-1) is responsible for normalizing** - -> > > > > > > cf1c71d (Add references) -> > > > > > > **one segment of size C**, hence it loops over all elements -> > > > > > > in that segment. The kernel code is broken down into 4 steps: +**one segment of size C**, hence it loops over all elements +in that segment. The kernel code is broken down into 4 steps: 1. Mean calculation -<<<<<<< HEAD -$$\mathbb{E}[x] = \frac{1}{C} \sum_{i=1}^{C} x_i$$ + $$\mathbb{E}[x] = \frac{1}{C} \sum_{i=1}^{C} x_i$$ 2. Variance and reciprocal of standard deviation (rstd) calculation @@ -116,27 +78,8 @@ $$\mathbb{E}[x] = \frac{1}{C} \sum_{i=1}^{C} x_i$$ 4. Store mean and rstd for backward pass -# The kernel uses a 1D grid and block as shown in Figure-1a. - - $$\mathbb{E}[x] = \frac{1}{C} \sum_{i=1}^{C} x_i$$ - -2. Variance and reciprocal of standard deviation (rstd) calculation - - $$Var[x] = \frac{1}{C} \sum_{i=1}^{C} (x_i - \mathbb{E}[x])^2$$ - - $$rstd[x] = \frac{1}{\sqrt{Var[x] + \epsilon}}$$ - -3. Apply mean and variance normalization and then scale and - shift with the learnable weight and bias parameters - - $$y_i = ((x_i - \mathbb{E}[x]) * rstd[x]) * \gamma_i + \beta_i$$ - -4. Store mean and rstd for backward pass - -The kernel uses a 1D grid and block as shown in Figure-1. - -> > > > > > > cf1c71d (Add references) -> > > > > > > Also note that all operations are implemented in a single kernel. +The kernel uses a 1D grid and block as shown in Figure-1a. +Also note that all operations are implemented in a single kernel.

layernorm_kernel1 @@ -149,8 +92,6 @@ The kernel uses a 1D grid and block as shown in Figure-1. Figure-1a: Kernel 1 Illustration. -<<<<<<< HEAD -
layernorm_kernel1_code
@@ -176,69 +117,6 @@ thread sums corresponding elements and stores it in a _shared memory array_ array_ are iteratively reduced to obtain the final sum. For more details, see the [thread coarsening](../concepts/thread_coarsening.md) and [reduction](../concepts/reduction.md) pocket references. -======= - -```cpp -__global__ void layernorm_forward_kernel1( - float* out, float* mean, float* rstd, - const float* inp, const float* weight, const float* bias, - int N, int C -) { - int idx = blockIdx.x * blockDim.x + threadIdx.x; - float eps = 1.0e-5f; - - if (idx < N) { - // Start index of the input segment for this thread, inp[idx, :] - const float* x = inp + idx * C; - - // Compute mean - float m = 0.0f; - for (int i = 0; i < C; i++) { - m += x[i]; - } - m /= C; - - // Compute variance (without any bias correction) - float v = 0.0f; - for (int i = 0; i < C; i++) { - float diff = x[i] - m; - v += diff * diff; - } - v /= C; - - // Compute rstd - float r = 1.0f / sqrt(v + eps); - - // Compute output - // Start index of the output segment for this thread, out[idx, :] - float* y = out + idx * C; - for (int i = 0; i < C; i++) { - float o_prime = (x[i] - m) * r; // normalized output - float o = o_prime * weight[i] + bias[i]; // scale and shift - y[i] = o; - } - - // Store mean and rstd for backward pass - mean[idx] = m; - rstd[idx] = r; - } -} -``` - -## Kernel 2 - -In Kernel 2, steps 1, 2 and 3 are implemented as separate kernels. For the _mean_ -and _rstd_ kernels, **each block is responsible for one segment of C** instead of -each thread (see Figure-2) which allows for further parallelization. Whereas for -the _normalization_ kernel (step 3), each thread calculates one output element. - -Since both the _mean_ and _rstd_ calculations involve the sum operation, they -make use of _thread coarsening_ and _reduction_. In _thread coarsening_, each -thread sums corresponding elements and stores it in a _shared memory array_ -(same size as the thread block). In _reduction_, the elements in the _shared -array_ are iteratively reduced to obtain the final sum. - -> > > > > > > cf1c71d (Add references) These optimizations lead to an improvement of **~5x over Kernel 1** (for block size 512). From 6c0018b471ae45095e9b9a468a9a41eae626c207 Mon Sep 17 00:00:00 2001 From: Val Andrei Fajardo Date: Thu, 10 Apr 2025 09:13:17 -0400 Subject: [PATCH 12/12] lint --- .../src/cuda/kernels/encoder_forward.md | 71 ++++++++++++++----- .../src/cuda/kernels/imgs/encoder_kernel1.svg | 2 +- .../src/cuda/kernels/imgs/encoder_kernel2.svg | 2 +- .../src/cuda/kernels/imgs/encoder_kernel3.svg | 2 +- .../cuda/kernels/imgs/encoder_overview.svg | 2 +- 5 files changed, 57 insertions(+), 22 deletions(-) diff --git a/books/compute/src/cuda/kernels/encoder_forward.md b/books/compute/src/cuda/kernels/encoder_forward.md index 0fda63f..dfaee84 100644 --- a/books/compute/src/cuda/kernels/encoder_forward.md +++ b/books/compute/src/cuda/kernels/encoder_forward.md @@ -1,3 +1,5 @@ + + # Kernels for positional encoder forward pass in GPT-2 @@ -8,47 +10,80 @@ ## Introduction -After tokenization, batches of input texts would be turned into arrays of token ids . The GPT-2 Encoder will turn these input ids into hidden representations that can be proceessed with the remainder of the transformer blocks: - -- Input: - - "input" `inp`, produced by the tokenizer- An integer array of shape $(B,\, T)$, where $B$ denotes the batch size (a total of $B$ sequences in the batch) while $T$ denotes the tokens in each sequence (each sequence contains a maximum of $T$ tokens.) - - token embeddings `wte`, from the model parameters- a float array of shape (vocab\_size, $C$), where $C$ is the latent dimension of this transformer model. The $n$th row of this matrix is the embedding vector for the $n$th token. - - position embeddings `wpe`, provided- a float array of shape (max\_length, $C$), where $C$ is the latent dimension of the transformer model. The $n$th role of this matrix represents the position embedding vector for the a token at position $n$ . -- Output: float tensor of shape $(B, T, C)$, where in addition to the batch ($B$) and sequence length ($T$) dimension, a "channel" $C$ is added ($C$ is the same as the latent embedding dimension of the transformer.) +After tokenization, batches of input texts would be turned into arrays of token +ids . The GPT-2 Encoder will turn these input ids into hidden representations that +can be proceessed with the remainder of the transformer blocks: + +- Input: + - "input" `inp`, produced by the tokenizer- An integer array of shape $(B,\, T)$, + where $B$ denotes the batch size (a total of $B$ sequences in the batch) while + $T$ denotes the tokens in each sequence (each sequence contains a maximum of + $T$ tokens.) + - token embeddings `wte`, from the model parameters- a float array of shape + (vocab_size, $C$), where $C$ is the latent dimension of this transformer model. + The $n$th row of this matrix is the embedding vector for the $n$th token. + - position embeddings `wpe`, provided- a float array of shape (max_length, $C$), + where $C$ is the latent dimension of the transformer model. The $n$th role of + this matrix represents the position embedding vector for the a token at position + $n$ . +- Output: float tensor of shape $(B, T, C)$, where in addition to the batch ($B$) + and sequence length ($T$) dimension, a "channel" $C$ is added ($C$ is the same + as the latent embedding dimension of the transformer.) ![GPT-2 Encoder Input and Output Format](./imgs/encoder_overview.svg) -Unlike most other parts of a transformer, the embedding kernel contains notably few calculations- for the forward pass, both the word embeddings and the position embeddings are **pre-computed** and available as arrays in memory. The only job of the encoder kernel is to load the appropriate rows from both matrices, add these togehter, and write the sum to the appropriate location in memory. +Unlike most other parts of a transformer, the embedding kernel contains notably +few calculations- for the forward pass, both the word embeddings and the position +embeddings are **pre-computed** and available as arrays in memory. The only job +of the encoder kernel is to load the appropriate rows from both matrices, add +these together, and write the sum to the appropriate location in memory. -## Kernel 1 +## Kernel 1 ![GPT-2 Encoder Kernel 1](./imgs/encoder_kernel1.svg) -THis kernel uses a total of $B \times T$ threads, where each thread handles a token in the $(B,\, T)$ input array. Within each thread, a for-loop iterates over the hidden dimensions, reading one float from `wte` and from `wpe`, take the sum, and write the sum to the output tensor. +THis kernel uses a total of $B \times T$ threads, where each thread handles a token +in the $(B,\, T)$ input array. Within each thread, a for-loop iterates over the +hidden dimensions, reading one float from `wte` and from `wpe`, take the sum, +and write the sum to the output tensor. -While this approach might work fine on CPU because of the predictable memory access pattern, the for-loop approach does not fully unleash the multiprocessing potentials of the GPU. +While this approach might work fine on CPU because of the predictable memory access +pattern, the for-loop approach does not fully unleash the multiprocessing potentials +of the GPU. ## Kernel 2 ![GPT-2 Encoder Kernel 2](./imgs/encoder_kernel2.svg) -This kernel uses substantially more threads than kernel 1, a total of $B \times T \times C$, one for each element in the output. (Recall that in modern GPTs, the hidden dimension $C$ is usually in the thousands if not larger.) Each thread handles one float in the output tensor- the thread would read one float from the `wte` token embedding array, one float from the `wpe` position embedding array, sum the two floating point numbers, and write the result to the appropriate location in the output tensor. +This kernel uses substantially more threads than kernel 1, a total of $B\timesT\timesC$, +one for each element in the output. (Recall that in modern GPTs, the hidden dimension +$C$ is usually in the thousands if not larger.) Each thread handles one float +in the output tensor- the thread would read one float from the `wte` token embedding +array, one float from the `wpe` position embedding array, sum the two floating +point numbers, and write the result to the appropriate location in the output +tensor. -This appraoch improves over kernel 1 by using many more threads in parallel. However, since each thread needs to do its own memory read/write, the memory access pattern becomes a lot less predictable. Random memory access might quickly become a bottleneck for this approach. +This approach improves over kernel 1 by using many more threads in parallel. However, +since each thread needs to do its own memory read/write, the memory access pattern +becomes a lot less predictable. Random memory access might quickly become a bottleneck +for this approach. ## Kernel 3 ![GPT-2 Encoder Kernel 3](./imgs/encoder_kernel3.svg) -On NVIDIA GPUs, reading and writing consecutive chunks of memory at a time can be significantly more efficient than doing so one byte at a time. This kernel leverages this feature to make processing a lot more efficient. - +On NVIDIA GPUs, reading and writing consecutive chunks of memory at a time can be +significantly more efficient than doing so one byte at a time. This kernel leverages +this feature to make processing a lot more efficient. -While in the kernel, the additions are defined in a for-loop across the $\texttt{x128::size}$, one floating point number at a time, the kernel uses `#pragma unroll` to automatically optimizes this part of the code during compilation. +While in the kernel, the additions are defined in a for-loop across the $\texttt{x128::size}$, +one floating point number at a time, the kernel uses `#pragma unroll` to automatically +optimizes this part of the code during compilation. -## References +#### References 1. Code for encoder forward kernels from [llm.c](https://github.com/karpathy/llm.c/blob/master/dev/cuda/encoder_forward.cu) -{{#author VectorInstitute}} +{{#author jacobthebanana}} diff --git a/books/compute/src/cuda/kernels/imgs/encoder_kernel1.svg b/books/compute/src/cuda/kernels/imgs/encoder_kernel1.svg index 586f859..a3b39e1 100644 --- a/books/compute/src/cuda/kernels/imgs/encoder_kernel1.svg +++ b/books/compute/src/cuda/kernels/imgs/encoder_kernel1.svg @@ -1,4 +1,4 @@ -
Writes one dimension at a time, in a for-loop
Number of programs:
batch size * max length
Sequence
Batch
\ No newline at end of file +
Writes one dimension at a time, in a for-loop
Number of programs:
batch size * max length
Sequence
Batch
diff --git a/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg b/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg index 4dba26f..9ebee47 100644 --- a/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg +++ b/books/compute/src/cuda/kernels/imgs/encoder_kernel2.svg @@ -1,4 +1,4 @@ -
Number of programs:
batch size * max length * hidden dim
Each program writes only one float
Batch
Hidden Dim
\ No newline at end of file +
Number of programs:
batch size * max length * hidden dim
Each program writes only one float
Batch
Hidden Dim
diff --git a/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg b/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg index 101191d..571d474 100644 --- a/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg +++ b/books/compute/src/cuda/kernels/imgs/encoder_kernel3.svg @@ -1,4 +1,4 @@ -
Number of programs:
batch size * max length * hidden dim //  
Each program writes up to floats along the embedding dimension
Hidden Dim
Batch
Sequence
\ No newline at end of file +
Number of programs:
batch size * max length * hidden dim //  
Each program writes up to floats along the embedding dimension
Hidden Dim
Batch
Sequence
diff --git a/books/compute/src/cuda/kernels/imgs/encoder_overview.svg b/books/compute/src/cuda/kernels/imgs/encoder_overview.svg index 7c1ae16..1a07189 100644 --- a/books/compute/src/cuda/kernels/imgs/encoder_overview.svg +++ b/books/compute/src/cuda/kernels/imgs/encoder_overview.svg @@ -1,4 +1,4 @@ -
Token Embeddings
(vocab size, 
hidden dim)
Position Embeddings
(max length, hidden dim)
Token IDs of input
(batch size, 
max length)
token id
position
Output
(batch size,
max length, 
hidden dim)
\ No newline at end of file +
Token Embeddings
(vocab size, 
hidden dim)
Position Embeddings
(max length, hidden dim)
Token IDs of input
(batch size, 
max length)
token id
position
Output
(batch size,
max length, 
hidden dim)

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