Write CUDA LayerNorm Kernel

A numerically stable CUDA LayerNorm kernel uses a single-pass Welford reduction for mean and variance, combined with FP32 accumulation to prevent overflow when processing FP16 or BF16 inputs.

Yes, the CUDA LayerNorm implementation handles arbitrary hidden dimensions, including non-power-of-two sizes, by performing proper per-row normalization across the normalization dimension for N x S x D tensor shapes.

Yes, the kernel includes an optional RMSNorm variant that normalizes by root mean square without mean subtraction, and supports a backward pass with saved intermediates for gradient computation.

Welford online accumulation computes mean and variance in a single pass, ensuring numerical stability and reducing memory bandwidth usage compared to two-pass variance methods in GPU LayerNorm kernels.

Epsilon should be placed inside the rsqrtf function as rsqrtf(var + epsilon), ensuring the CUDA LayerNorm kernel applies normalization correctly and prevents division by zero during variance scaling.

The best way to optimize LayerNorm for FP16 inputs is to use FP32 accumulation for all variance and mean computations, ensuring numerical stability while maintaining high throughput in your CUDA kernel.