You write custom CUDA kernels to replace the PyTorch operators in the given EvoNorm 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 normalization+affine_transform+nonlinear_gating), or algorithmic changes (such as optimized memory access patterns). You are only limited by your imagination.


Overview
This implementation provides a highly optimized CUDA kernel for computing the Log-Cosh loss function, which is a smooth alternative to Mean Absolute Error (MAE) that is less sensitive to outliers than Mean Squared Error (MSE).

Mathematical Formulation
The Log-Cosh loss is defined as:
L(x, y) = log(cosh(x - y)) = |x-y| + log(1 + exp(-2|x-y|)) - log(2)

key Optimizations
1. Vectorized Memory Access
Uses float4 data type for coalesced memory operations

Processes 4 elements per thread simultaneously

Reduces memory transaction overhead by 75%

2. Numerical Stability
Implements the stable formulation: |diff| + log1p(exp(-2*|diff|)) - log(2)

Avoids numerical overflow in cosh() calculation

Uses log1p() for accurate logarithm of (1 + x)

3. Parallel Reduction Strategy
Warp-level reduction: 32-thread warp shuffle operations

Block-level reduction: Shared memory for intra-block reduction

Global reduction: Atomic operations for cross-block summation

4. Flexible Reduction Modes
reduction=0: Element-wise output (no reduction)

reduction=1: Mean reduction (sum / n_elements)

reduction=2: Sum reduction


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 math

N, C, H, W = 32, 64, 56, 56


class LogCoshLoss(nn.Module):
    def __init__(self, reduction='mean'):
        super().__init__()
        self.reduction = reduction

    def forward(self, input, target):
        diff = input - target
        loss = torch.abs(diff) + torch.nn.functional.softplus(-2. * torch.abs(diff)) - math.log(2.0)

        if self.reduction == 'mean':
            return loss.mean()
        elif self.reduction == 'sum':
            return loss.sum()
        return loss


class Model(nn.Module):
    def __init__(self, reduction='mean'):
        super().__init__()
        self.op = LogCoshLoss(reduction)

    def forward(self, input, target):
        return self.op(input, target)


def get_inputs():
    input = torch.randn(N, C, H, W, dtype=torch.float32)
    target = torch.randn(N, C, H, W, dtype=torch.float32)
    return [input, target]


def get_init_inputs():
    return ['mean']