# LayerNorm2D Forward with CK Tile

This example demonstrates efficient 2D layer normalization using the CK Tile programming model, leveraging tile-based parallelism and advanced fusion for transformer and LLM workloads.

---

## Algorithm and Math

LayerNorm computes, for each row $x$:
$$
\mu = \frac{1}{N} \sum_{i=1}^N x_i,\quad \sigma^2 = \frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2
$$
$$
\hat{x}_i = \frac{x_i - \mu}{\sqrt{\sigma^2 + \epsilon}},\quad y_i = \gamma \hat{x}_i + \beta
$$

- **Welford's Algorithm**: Used for numerically stable, blockwise mean/variance computation. For $N \leq 4096$, a one-pass algorithm is used; for large $N$, a two-pass approach is adopted.

--

## Features

- **Prenorm/Postnorm Fusion**: Fused residual addition before/after normalization for transformer blocks.
- **Smooth/Dynamic Quantization**: Rowwise int8 quantization with per-token scale, supporting smoothquant for LLMs.
- **Flexible Precision**: Supports fp16, bf16, int8 output.
- **Efficient for Large N**: Two-pass pipeline for $N > 4096$.
- **Highly Modular**: Easily extendable for new fusion or quantization strategies.

---

## Build & Run

```
# in the root of ck_tile
mkdir build && cd build
../script/cmake-ck-dev.sh  ../ <arch>  # you can replace this <arch> to gfx90a, gfx942...
make tile_example_layernorm2d_fwd -j
```
This will result in an executable `build/bin/tile_example_layernorm2d_fwd`

## Example
```
args:
          -m    m dimension (default:3328)
          -n    n dimension (default:4096)
     -stride    stride per row, if -1 then equal to n (default:-1)
          -e    epsilon (default:1e-5)
    -save_mv    save mean/variance(invstd) or not. set to 1 in training case (default:0)
          -v    cpu validation or not (default:1)
      -kname    print kernel name or not (default:1)
     -prec_i    input precision (default:fp16)
     -prec_o    output precision, set auto will be the same as input (default:auto)
    -prec_sm    output quant scale type, set auto will be the same as input. used when fquant=1 (default:auto)
    -prec_sy    output quant scale type, set auto will be the same as input. used when fquant=1 or 2 (default:auto)
       -fadd    fused-add, 0:no fused add, 1:preadd+store, 2:preadd only (default:0)
     -fquant    fused-quant, 0:no, 1:smooth-dynamic-quant, 2:dynamic-quant (default:0)
     -warmup    cold iter (default:5)
     -repeat    hot iter (default:20)
       -json    0: No Json, 1: Dump Results in Json format (default:0)
   -jsonfile    json file name to dump results (default:layernorm2d_fwd.json)

```
---

## Technical Details

## Welford online algorithm
We use welfold algorithm to update `mean`/`variance` block by block. For `N <=4096` case we can compute `mean`/`var`/`normalization` within one loop, we call it `one-pass`. For large N case, it is hard to keep `mean`/`var` inside register/LDS and then computation `normalization`, so we need to load input twice, first time to compute `mean`/`var` block-by-block, then load input another time to compute the `normalization`. We call it `two-pass`.

## mean/variance save
In training case the mean/variance need to store out (TBD, not supported yet).

## prenorm/postnorm

![](misc/pnorm.png)

Since [prenorm/postnorm](https://arxiv.org/pdf/1906.01787) is quite common in LLM blocks, this example boosts this feature by kernel fusion. Note that `prenorm`/`postnorm` always need to do elementwise-add a `shortcut` before the actual layernorm computation, and optionally store out the result to global. You can use `-fadd=1` to test `pre-add+store`, or `-fadd=2` to test `pre-add` without store out (not codegen by default).

## smooth-quant/dynamic-quant
We support smooth/dynamic quantization for `int8` output, by setting `-fquant=1` and `-prec_o=int8`. In this case the output will doing a rowwise dynamic quantization like below. Note that smooth-quant require input a `(1*N)` size per-channel scale(in fp32 in our example, though this is customizable), then elememt-wise multiply the tensor for each row, then compute the rowwise dynamic quant. if set `-fquant=2` will have the input per-channel scale stage, only the dynamic quant. This case is supported in our kernel but by default not generated (TBD: add some filter in generate.py support on-demand codegen)
![](misc/dquant.png)

```
# assume output int8, hidden_states is [m, n] shape and in fp16/bf16
# [m, 1]
per_token_amax, _ = torch.max(
     input=torch.abs(hidden_states), 
     dim=-1, 
     keepdim=True
)
per_token_scale = per_token_amax.to(dtype=torch.float32) / 127.0

# quant hidden_states
hidden_states = (hidden_states / per_token_scale).to(dtype=torch.int8)

return hidden_states, per_token_scale
# hidden_states now is int8 will feed to next layer as intput
# per_token_scale will be used as dequant factor later layer
```
## limitations
Note that `fquant=2`, `fadd=2`, `prec_sm/prec_sy` other than `fp32` are not by default generated. Though our kernel template suppor this. (TBD: add some flag in generate.py) to generate those instance on demand. Beside, `N>8192` case will by default using two-pass pipeline, and `-fquant=1/2` are not supported yet. If need suport `N>8192` and `fused+residual+store`, you can use this example together with `12_smoothquant`, to construct layernorm+residual, and smoothquant, 2 kernels for this purpose.

```
# some case
# standard fp16 layernorm 2d, m=10. n=1024
./build/bin/tile_example_layernorm2d_fwd  -m=10 -n=1024

# standard fp16 layernorm 2d, m=10. n=1024, fused-smooth-quant, output in int8
./build/bin/tile_example_layernorm2d_fwd  -m=10 -n=1024 -prec_o=int8 -fquant=1

# standard fp16 layernorm 2d, m=10. n=1024, fused-smooth-quant+fused-add-store, output in int8
./build/bin/tile_example_layernorm2d_fwd  -m=10 -n=1024 -prec_o=int8 -fquant=1 -fadd=1
```
---

## Source Structure

- **Kernel**: `layernorm2d_fwd.hpp` (tile-programming kernel template)
- **Executable**: `layernorm2d_fwd.cpp` (argument parsing, kernel launch)
- **Codegen**: `generate.py` (instantiates kernels for different configs)
- **Misc**: `misc/` (algorithm diagrams, e.g., prenorm/postnorm, quantization)

---

## Related CK Tile Examples

- [01_fmha](../01_fmha/README.md): Fused multi-head attention (FMHA)
- [03_gemm](../03_gemm/README.md): Tile-programming GEMM
- [12_smoothquant](../12_smoothquant/README.md): Standalone smoothquant kernel

For and distribution, see `include/ck_tile/tile_program/tile_distribution/`.

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[Back to CK Tile Examples](../README.md)