You write custom CUDA kernels to replace the pytorch operators in the given GeGLU architecture to get speedups.

You have complete freedom to choose the set of operators you want to replace. You may make the decision to replace some operators with custom CUDA kernels and leave others unchanged. You may replace multiple operators with custom implementations, consider operator fusion opportunities (combining multiple operators into a single kernel, for example, combining chunk+gelu+elementwise_mul), or algorithmic changes (such as optimized memory access patterns). You are only limited by your imagination.

This code implements Bregman divergence + GELU activation with CUDA optimizations:

Bregman divergence computation - For squared norm generator f(x)=x²: D(x,t) = x² - t² - 2t(x-t).

Single parallel reduction - Warp shuffle for sum reduction of divergence terms.

Shared memory reduction - Standard warp/block reduction pattern.

Mathematical simplification - Computes divergence efficiently using pre-computed squared terms.

Fused activation - Applies GELU to the total divergence sum.

Grid-stride loop - Threads process multiple elements for load balancing.

Memory coalescing - Contiguous tensor access patterns.

Batch parallelism - One CUDA block per input row.

CUDA math function - Uses erff() for GELU activation.

Single-pass computation - Computes all divergence components in one memory traversal.



Here's an example to show you the syntax of inline embedding custom CUDA operators in torch: The example given architecture is:
import torch
import torch.nn as nn
import torch.nn.functional as F


class Model(nn.Module):
    def __init__(self, target):
        super(Model, self).__init__()
        self.target = nn.Parameter(target)

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        fx = x * x
        ft = self.target * self.target
        grad_ft = 2.0 * self.target

        bregman = fx - ft - grad_ft * (x - self.target)
        dist = torch.sum(bregman, dim=-1)
        return F.gelu(dist)


batch_size = 128
input_dim = 1024


def get_inputs():
    x = torch.randn(batch_size, input_dim)
    return [x]


def get_init_inputs():
    target = torch.randn(input_dim)
    return [target]