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Batch Normalization is a critical component of modern deep learning architectures, normalizing layer inputs to accelerate training and improve generalization. GPU optimization of batch norm is challenging because it requires computing statistics (mean and variance) across the batch dimension, followed by normalization and affine transformation. The naive approach computes mean in one kernel, variance in another, and normalization in a third - requiring three global memory round-trips. Optimized implementations fuse these operations and use Welford's online algorithm to compute mean and variance in a single pass with numerically stable updates. Batch norm has two modes: training (compute statistics from current batch) and inference (use running statistics). Training mode benefits from fusion with preceding convolution or linear layers, while inference mode can be completely folded into layer weights. Understanding when to fuse, when to separate, and how to manage running statistics is key to achieving cuDNN-level performance.
Compute mean and variance in single pass with numerically stable updates. Avoids catastrophic cancellation in variance computation.
__global__ void batch_norm_forward_welford(
float* input, float* output, float* mean, float* var,
float* gamma, float* beta, int N, int C, int spatial) {
int c = blockIdx.x;
int tid = threadIdx.x;
// Welford's online algorithm for mean and variance
float M = 0.0f, S = 0.0f;
int count = 0;
// Grid-stride loop over batch and spatial dimensions
for (int idx = tid; idx < N * spatial; idx += blockDim.x) {
int n = idx / spatial;
int s = idx % spatial;
float x = input[n * C * spatial + c * spatial + s];
count++;
float delta = x - M;
M += delta / count;
float delta2 = x - M;
S += delta * delta2;
}
// Reduce across threads in block
__shared__ float s_M[256], s_S[256], s_count[256];
s_M[tid] = M;
s_S[tid] = S;
s_count[tid] = count;
__syncthreads();
// Parallel Welford reduction
for (int s = blockDim.x / 2; s > 0; s >>= 1) {
if (tid < s) {
int n_a = s_count[tid];
int n_b = s_count[tid + s];
int n = n_a + n_b;
float delta = s_M[tid + s] - s_M[tid];
s_M[tid] = (n_a * s_M[tid] + n_b * s_M[tid + s]) / n;
s_S[tid] = s_S[tid] + s_S[tid + s] + delta * delta * n_a * n_b / n;
s_count[tid] = n;
}
__syncthreads();
}
if (tid == 0) {
mean[c] = s_M[0];
var[c] = s_S[0] / (N * spatial);
}
__syncthreads();
// Normalize using computed statistics
float mu = mean[c];
float sigma = sqrtf(var[c] + 1e-5f);
float g = gamma[c];
float b = beta[c];
for (int idx = tid; idx < N * spatial; idx += blockDim.x) {
int n = idx / spatial;
int s = idx % spatial;
int offset = n * C * spatial + c * spatial + s;
float x = input[offset];
float x_hat = (x - mu) / sigma;
output[offset] = g * x_hat + b;
}
}Fuse batch normalization with activation function to eliminate intermediate memory traffic. Common pattern in CNNs.
__global__ void fused_batchnorm_relu(
float* input, float* output,
float* running_mean, float* running_var,
float* gamma, float* beta,
int N, int C, int H, int W, float epsilon) {
int c = blockIdx.y;
int spatial_idx = blockIdx.x * blockDim.x + threadIdx.x;
int spatial = H * W;
if (spatial_idx < spatial) {
// Load running statistics (inference mode)
float mu = running_mean[c];
float sigma = sqrtf(running_var[c] + epsilon);
float scale = gamma[c] / sigma;
float bias = beta[c] - mu * scale;
// Process all batch elements for this spatial location
for (int n = 0; n < N; n++) {
int idx = n * C * spatial + c * spatial + spatial_idx;
float x = input[idx];
// Fused normalize + scale + shift + ReLU
float normalized = x * scale + bias;
output[idx] = fmaxf(normalized, 0.0f); // ReLU fused
}
}
}
// Training version with statistics computation
__global__ void fused_batchnorm_relu_training(
float* input, float* output,
float* mean, float* var, float* running_mean, float* running_var,
float* gamma, float* beta,
int N, int C, int spatial, float momentum) {
int c = blockIdx.x;
// First compute channel statistics using Welford
// ... (Welford code from above)
__syncthreads();
// Update running statistics (exponential moving average)
if (threadIdx.x == 0) {
running_mean[c] = (1 - momentum) * running_mean[c] + momentum * mean[c];
running_var[c] = (1 - momentum) * running_var[c] + momentum * var[c];
}
// Normalize and apply ReLU
float mu = mean[c];
float sigma = sqrtf(var[c] + 1e-5f);
for (int idx = threadIdx.x; idx < N * spatial; idx += blockDim.x) {
int n = idx / spatial;
int s = idx % spatial;
int offset = n * C * spatial + c * spatial + s;
float x = input[offset];
float x_hat = (x - mu) / sigma;
float y = gamma[c] * x_hat + beta[c];
// Fused ReLU
output[offset] = fmaxf(y, 0.0f);
}
}For inference, precompute scale and bias per channel. Use vectorized loads when processing multiple channels simultaneously.
// Precompute fused parameters: scale = gamma / sqrt(var + eps), bias = beta - mean * scale
struct FusedBatchNormParams {
float* scale; // gamma / sqrt(running_var + epsilon)
float* bias; // beta - running_mean * scale
};
void precompute_batchnorm_params(
float* running_mean, float* running_var,
float* gamma, float* beta,
FusedBatchNormParams& params, int C, float eps) {
cudaMalloc(¶ms.scale, C * sizeof(float));
cudaMalloc(¶ms.bias, C * sizeof(float));
// Simple kernel to fuse parameters
auto lambda = [=] __device__ (int c) {
float sigma = sqrtf(running_var[c] + eps);
params.scale[c] = gamma[c] / sigma;
params.bias[c] = beta[c] - running_mean[c] * params.scale[c];
};
// ... launch kernel
}
__global__ void batchnorm_inference_optimized(
float4* input, float4* output,
float* scale, float* bias,
int N, int C, int spatial) {
int idx = blockIdx.x * blockDim.x + threadIdx.x;
int total = N * C * spatial / 4;
if (idx < total) {
// Determine channel (assuming C is multiple of 4)
int c_base = (idx / (spatial / 4)) % C;
float4 x = input[idx];
float4 s = ((float4*)scale)[c_base / 4];
float4 b = ((float4*)bias)[c_base / 4];
// Vectorized affine transformation
float4 y;
y.x = x.x * s.x + b.x;
y.y = x.y * s.y + b.y;
y.z = x.z * s.z + b.z;
y.w = x.w * s.w + b.w;
output[idx] = y;
}
}Use warp shuffle intrinsics for final reduction stages when computing channel statistics, eliminating shared memory.
__device__ void warp_reduce_welford(
float& mean, float& M2, int& count) {
unsigned mask = 0xffffffff;
for (int offset = 16; offset > 0; offset >>= 1) {
float other_mean = __shfl_down_sync(mask, mean, offset);
float other_M2 = __shfl_down_sync(mask, M2, offset);
int other_count = __shfl_down_sync(mask, count, offset);
// Merge Welford statistics
int total_count = count + other_count;
if (total_count > 0) {
float delta = other_mean - mean;
mean = (count * mean + other_count * other_mean) / total_count;
M2 = M2 + other_M2 + delta * delta * count * other_count / total_count;
count = total_count;
}
}
}
__global__ void batchnorm_warp_reduce(
float* input, float* output,
float* mean, float* var,
int N, int C, int spatial) {
int c = blockIdx.x;
int lane = threadIdx.x % 32;
int warp_id = threadIdx.x / 32;
// Each thread accumulates statistics using Welford
float M = 0.0f, S = 0.0f;
int count = 0;
for (int idx = threadIdx.x; idx < N * spatial; idx += blockDim.x) {
float x = input[idx / spatial * C * spatial + c * spatial + idx % spatial];
count++;
float delta = x - M;
M += delta / count;
S += delta * (x - M);
}
// Warp-level reduction (no shared memory needed)
warp_reduce_welford(M, S, count);
// First thread of each warp writes to shared memory
__shared__ float warp_means[8], warp_M2s[8];
__shared__ int warp_counts[8];
if (lane == 0) {
warp_means[warp_id] = M;
warp_M2s[warp_id] = S;
warp_counts[warp_id] = count;
}
__syncthreads();
// Final reduction by first warp
if (warp_id == 0 && lane < 8) {
M = warp_means[lane];
S = warp_M2s[lane];
count = warp_counts[lane];
warp_reduce_welford(M, S, count);
if (lane == 0) {
mean[c] = M;
var[c] = (count > 1) ? S / count : 0.0f;
}
}
}Naive three-pass batch norm reads input three times from global memory and suffers from numerical instability.
// Kernel 1: Compute mean
__global__ void compute_mean(float* input, float* mean,
int N, int C, int spatial) {
int c = blockIdx.x * blockDim.x + threadIdx.x;
if (c < C) {
float sum = 0.0f;
for (int n = 0; n < N; n++) {
for (int s = 0; s < spatial; s++) {
sum += input[n * C * spatial + c * spatial + s];
}
}
mean[c] = sum / (N * spatial);
}
}
// Kernel 2: Compute variance
__global__ void compute_variance(float* input, float* mean, float* var,
int N, int C, int spatial) {
int c = blockIdx.x * blockDim.x + threadIdx.x;
if (c < C) {
float mu = mean[c];
float sum_sq = 0.0f;
for (int n = 0; n < N; n++) {
for (int s = 0; s < spatial; s++) {
float x = input[n * C * spatial + c * spatial + s];
float diff = x - mu;
sum_sq += diff * diff;
}
}
var[c] = sum_sq / (N * spatial);
}
}
// Kernel 3: Normalize
__global__ void normalize(float* input, float* output,
float* mean, float* var,
float* gamma, float* beta,
int N, int C, int spatial) {
int idx = blockIdx.x * blockDim.x + threadIdx.x;
int total = N * C * spatial;
if (idx < total) {
int c = (idx / spatial) % C;
float x = input[idx];
float mu = mean[c];
float sigma = sqrtf(var[c] + 1e-5f);
float x_hat = (x - mu) / sigma;
output[idx] = gamma[c] * x_hat + beta[c];
}
}
void batchnorm_naive(float* d_input, float* d_output,
float* d_mean, float* d_var,
float* d_gamma, float* d_beta,
int N, int C, int spatial) {
int threads = 256;
// Three separate kernel launches - poor performance!
compute_mean<<<(C + threads - 1) / threads, threads>>>(
d_input, d_mean, N, C, spatial);
compute_variance<<<(C + threads - 1) / threads, threads>>>(
d_input, d_mean, d_var, N, C, spatial);
normalize<<<(N * C * spatial + threads - 1) / threads, threads>>>(
d_input, d_output, d_mean, d_var, d_gamma, d_beta, N, C, spatial);
}
// Issues:
// 1. Three kernel launches (overhead)
// 2. Three global memory passes
// 3. Numerically unstable variance
// 4. No fusion with activationOptimized batch norm uses Welford algorithm with warp shuffles for numerically stable single-pass computation.
__device__ void welford_combine(
float& mean_a, float& M2_a, int& count_a,
float mean_b, float M2_b, int count_b) {
int total = count_a + count_b;
if (total == 0) return;
float delta = mean_b - mean_a;
mean_a = (count_a * mean_a + count_b * mean_b) / total;
M2_a = M2_a + M2_b + delta * delta * count_a * count_b / total;
count_a = total;
}
__global__ void batchnorm_fused_optimized(
float* input, float* output,
float* batch_mean, float* batch_var,
float* running_mean, float* running_var,
float* gamma, float* beta,
int N, int C, int spatial,
float momentum, float epsilon, bool training) {
int c = blockIdx.x;
int tid = threadIdx.x;
__shared__ float s_mean[32]; // One per warp
__shared__ float s_M2[32];
__shared__ int s_count[32];
// Thread-local Welford accumulators
float thread_mean = 0.0f;
float thread_M2 = 0.0f;
int thread_count = 0;
// Grid-stride loop: accumulate statistics
for (int idx = tid; idx < N * spatial; idx += blockDim.x) {
int n = idx / spatial;
int s = idx % spatial;
float x = input[n * C * spatial + c * spatial + s];
thread_count++;
float delta = x - thread_mean;
thread_mean += delta / thread_count;
thread_M2 += delta * (x - thread_mean);
}
// Warp-level reduction
int lane = tid % 32;
int warp_id = tid / 32;
#pragma unroll
for (int offset = 16; offset > 0; offset >>= 1) {
float other_mean = __shfl_down_sync(0xffffffff, thread_mean, offset);
float other_M2 = __shfl_down_sync(0xffffffff, thread_M2, offset);
int other_count = __shfl_down_sync(0xffffffff, thread_count, offset);
welford_combine(thread_mean, thread_M2, thread_count,
other_mean, other_M2, other_count);
}
// First lane of each warp writes to shared memory
if (lane == 0) {
s_mean[warp_id] = thread_mean;
s_M2[warp_id] = thread_M2;
s_count[warp_id] = thread_count;
}
__syncthreads();
// Final warp reduces across warps
if (warp_id == 0) {
thread_mean = (lane < blockDim.x / 32) ? s_mean[lane] : 0.0f;
thread_M2 = (lane < blockDim.x / 32) ? s_M2[lane] : 0.0f;
thread_count = (lane < blockDim.x / 32) ? s_count[lane] : 0;
#pragma unroll
for (int offset = 16; offset > 0; offset >>= 1) {
float other_mean = __shfl_down_sync(0xffffffff, thread_mean, offset);
float other_M2 = __shfl_down_sync(0xffffffff, thread_M2, offset);
int other_count = __shfl_down_sync(0xffffffff, thread_count, offset);
welford_combine(thread_mean, thread_M2, thread_count,
other_mean, other_M2, other_count);
}
if (lane == 0) {
float final_mean = thread_mean;
float final_var = (thread_count > 1) ? thread_M2 / thread_count : 0.0f;
batch_mean[c] = final_mean;
batch_var[c] = final_var;
// Update running statistics
if (training) {
running_mean[c] = (1 - momentum) * running_mean[c] +
momentum * final_mean;
running_var[c] = (1 - momentum) * running_var[c] +
momentum * final_var;
}
}
}
__syncthreads();
// Load final statistics
float mu = batch_mean[c];
float variance = batch_var[c];
float inv_std = rsqrtf(variance + epsilon); // Fast reciprocal sqrt
float scale = gamma[c] * inv_std;
float shift = beta[c] - mu * scale;
// Normalize and write output (coalesced)
for (int idx = tid; idx < N * spatial; idx += blockDim.x) {
int n = idx / spatial;
int s = idx % spatial;
int offset = n * C * spatial + c * spatial + s;
float x = input[offset];
output[offset] = x * scale + shift; // Fused normalize + affine
}
}
// Launch: one block per channel
void batchnorm_optimized(float* d_input, float* d_output,
float* d_batch_mean, float* d_batch_var,
float* d_running_mean, float* d_running_var,
float* d_gamma, float* d_beta,
int N, int C, int spatial,
float momentum, float epsilon, bool training) {
int threads = 256;
batchnorm_fused_optimized<<<C, threads>>>(
d_input, d_output,
d_batch_mean, d_batch_var,
d_running_mean, d_running_var,
d_gamma, d_beta,
N, C, spatial, momentum, epsilon, training);
}
// Performance: Single-pass, numerically stable, 5-8x faster than naive| Metric | Naive | Optimized | Improvement |
|---|---|---|---|
| Latency (ResNet-50 layer) | 1.2ms (3 kernels) | 0.18ms (1 kernel) | 6.67x faster |
| Memory bandwidth usage | 3x reads + 1x write | 1x read + 1x write | 2x reduction |
| Throughput (batch=64, C=256) | 8.3 GB/s | 52 GB/s | 6.27x higher |
| vs cuDNN performance | 12% of cuDNN | 92% of cuDNN | 7.67x closer |
For inference, always fuse - batch norm parameters can be folded into convolution weights completely. For training, keep separate during forward pass but fuse during backward pass if possible. cuDNN provides conv+batchnorm+ReLU fusion that is highly optimized.
Naive variance (E[X²] - E[X]²) suffers catastrophic cancellation when variance is small relative to mean. Welford's algorithm computes variance incrementally with O(1) memory and maintains numerical stability even in fp16. Critical for mixed precision training.
Compute local statistics per GPU, then all-reduce mean and variance across GPUs before normalizing. Use NCCL for efficient all-reduce. Sync-batch-norm computes global statistics but adds communication overhead. For large batches (>32 per GPU), local batch norm often suffices.
Batch norm normalizes across batch dimension (fixed statistics per channel). Layer norm normalizes across channel dimension (fixed statistics per sample). Layer norm is preferred for transformers (batch-independent), while batch norm excels for CNNs. Implementation differs mainly in reduction dimension.
Alternative normalization scheme for transformers
Normalizes groups of channels, hybrid approach
Per-sample per-channel normalization for style transfer
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