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Softmax is ubiquitous in deep learning - from attention mechanisms in transformers to classification layers. A naive implementation requires three passes over data (max, sum, normalize), but optimized versions compute it in a single pass using the online softmax algorithm. This guide covers numerical stability, warp-level reductions, memory access patterns, and fusion strategies that can achieve 5x+ speedup over naive implementations.
Compute max, sum, and normalize in a single pass by maintaining running statistics.
Use __shfl_down_sync for fast warp-level max and sum reductions.
Use float4 loads for 4x memory throughput on aligned data.
Combine softmax with Q*K^T multiply and V multiplication for FlashAttention.
Naive three-pass implementation reads data 3 times from global memory.
__global__ void softmax_naive(float* input, float* output, int N) {
int row = blockIdx.x;
float* in_row = input + row * N;
float* out_row = output + row * N;
// Pass 1: find max
float max_val = -INFINITY;
for (int i = 0; i < N; i++) max_val = fmaxf(max_val, in_row[i]);
// Pass 2: compute sum of exp
float sum = 0.0f;
for (int i = 0; i < N; i++) sum += expf(in_row[i] - max_val);
// Pass 3: normalize
for (int i = 0; i < N; i++) out_row[i] = expf(in_row[i] - max_val) / sum;
}Online algorithm with warp shuffles reduces memory reads by 60%.
__global__ void softmax_online(float* input, float* output, int N) {
int row = blockIdx.x;
int tid = threadIdx.x;
float* in_row = input + row * N;
float* out_row = output + row * N;
// Online computation: track max and sum simultaneously
float thread_max = -INFINITY;
float thread_sum = 0.0f;
for (int i = tid; i < N; i += blockDim.x) {
float val = in_row[i];
float new_max = fmaxf(thread_max, val);
thread_sum = thread_sum * expf(thread_max - new_max) + expf(val - new_max);
thread_max = new_max;
}
// Warp reduction for max and sum
for (int offset = 16; offset > 0; offset /= 2) {
float other_max = __shfl_down_sync(0xffffffff, thread_max, offset);
float other_sum = __shfl_down_sync(0xffffffff, thread_sum, offset);
float new_max = fmaxf(thread_max, other_max);
thread_sum = thread_sum * expf(thread_max - new_max) + other_sum * expf(other_max - new_max);
thread_max = new_max;
}
// Broadcast final values
float final_max = __shfl_sync(0xffffffff, thread_max, 0);
float final_sum = __shfl_sync(0xffffffff, thread_sum, 0);
// Single pass output
for (int i = tid; i < N; i += blockDim.x) {
out_row[i] = expf(in_row[i] - final_max) / final_sum;
}
}| Metric | Naive | Optimized | Improvement |
|---|---|---|---|
| Throughput (GB/s) | 180 | 720 | 4x |
| Latency (μs) | 45 | 12 | 3.8x |
| Memory Reads | 3N | 2N | 33% reduction |
Subtracting max prevents numerical overflow. Without it, exp(x) can overflow to infinity for large x values. The math is unchanged because exp(x-max)/sum(exp(xi-max)) = exp(x)/sum(exp(xi)).
Online softmax computes max and sum in a single pass by updating running statistics. When a new max is found, the running sum is rescaled by exp(old_max - new_max) to maintain correctness.
Softmax uses sum reduction
Softmax uses max reduction
Softmax is core of attention
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