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Finding the maximum value in a large array is a fundamental operation in machine learning (softmax, attention), image processing (contrast detection), and scientific computing. While the algorithm parallels sum reduction, max reduction has unique considerations around identity values, argmax tracking, and numerical stability. This guide covers optimized max reduction patterns, including finding both the maximum value and its index (argmax), handling special cases like NaN and infinity, and integrating max reduction into larger GPU workflows.
Use -INFINITY (negative infinity) as the identity value for max reduction. This ensures correct results regardless of input values.
#include <limits>
#include <cuda_fp16.h>
// For float
__device__ const float FLOAT_NEG_INF = -INFINITY;
// Or: -std::numeric_limits<float>::infinity()
// For double
__device__ const double DOUBLE_NEG_INF = -INFINITY;
// For half precision
__device__ const __half HALF_NEG_INF = __float2half(-INFINITY);
// Kernel initialization
__global__ void reduce_max(float* input, float* output, int n) {
float max_val = -INFINITY; // Correct identity
for (int i = tid; i < n; i += gridSize) {
max_val = fmaxf(max_val, input[i]);
}
// ... reduction continues
}Track both maximum value and its index in a single pass. Use a struct or pair to carry both through the reduction tree.
struct MaxWithIndex {
float val;
int idx;
};
__device__ MaxWithIndex warpReduceMaxIdx(MaxWithIndex data) {
for (int offset = 16; offset > 0; offset /= 2) {
float other_val = __shfl_down_sync(0xffffffff, data.val, offset);
int other_idx = __shfl_down_sync(0xffffffff, data.idx, offset);
if (other_val > data.val ||
(other_val == data.val && other_idx < data.idx)) {
data.val = other_val;
data.idx = other_idx;
}
}
return data;
}
__global__ void argmax(float* input, MaxWithIndex* output, int n) {
MaxWithIndex local = {-INFINITY, -1};
int tid = blockIdx.x * blockDim.x + threadIdx.x;
for (int i = tid; i < n; i += blockDim.x * gridDim.x) {
if (input[i] > local.val) {
local.val = input[i];
local.idx = i;
}
}
// Block-level reduction
local = blockReduceMaxIdx(local);
if (threadIdx.x == 0) output[blockIdx.x] = local;
}Handle NaN values explicitly. IEEE 754 comparisons with NaN return false, which can cause incorrect results if not handled.
// fmaxf handles NaN correctly - returns the non-NaN value
// But be explicit about NaN propagation policy
// Option 1: Propagate NaN (if any input is NaN, output is NaN)
__device__ float nan_propagate_max(float a, float b) {
if (isnan(a)) return a;
if (isnan(b)) return b;
return fmaxf(a, b);
}
// Option 2: Ignore NaN (only return NaN if ALL inputs are NaN)
__device__ float nan_ignore_max(float a, float b) {
if (isnan(a)) return b;
if (isnan(b)) return a;
return fmaxf(a, b);
}
// Option 3: Use fmaxf directly (returns non-NaN, or NaN if both NaN)
__device__ float max_val = fmaxf(a, b); // Standard behavior
// Reduction with NaN handling
for (int i = tid; i < n; i += gridSize) {
float val = input[i];
if (!isnan(val)) {
max_val = fmaxf(max_val, val);
}
}Reduce multiple rows simultaneously for batched operations. Each block handles one row, maximizing parallelism for batch operations.
// Each block reduces one row
__global__ void rowwise_max(float* matrix, float* row_max,
int rows, int cols) {
int row = blockIdx.x;
if (row >= rows) return;
float* row_ptr = matrix + row * cols;
float max_val = -INFINITY;
// Each thread handles multiple columns
for (int col = threadIdx.x; col < cols; col += blockDim.x) {
max_val = fmaxf(max_val, row_ptr[col]);
}
// Warp reduction
max_val = warpReduceMax(max_val);
// Block reduction for blocks > 32 threads
__shared__ float shared[32];
int lane = threadIdx.x % 32;
int wid = threadIdx.x / 32;
if (lane == 0) shared[wid] = max_val;
__syncthreads();
if (wid == 0) {
max_val = (lane < blockDim.x / 32) ? shared[lane] : -INFINITY;
max_val = warpReduceMax(max_val);
if (lane == 0) row_max[row] = max_val;
}
}
// Launch: one block per row
rowwise_max<<<batch_size, 256>>>(data, output, batch_size, seq_len);Combine max finding with softmax computation. First find max, then compute exp(x - max) for numerical stability.
// Two-pass approach for numerically stable softmax
// Pass 1: Find row max
// Pass 2: Compute exp(x - max) and sum, then normalize
// Or fused single-pass with online algorithm
__global__ void online_softmax(float* input, float* output, int n) {
float max_val = -INFINITY;
float sum = 0.0f;
// Online max and sum computation
for (int i = threadIdx.x; i < n; i += blockDim.x) {
float val = input[i];
if (val > max_val) {
sum = sum * expf(max_val - val) + 1.0f;
max_val = val;
} else {
sum += expf(val - max_val);
}
}
// Reduce max across threads
max_val = blockReduceMax(max_val);
// Adjust sum for new global max, then reduce
sum = sum * expf(local_max - max_val);
sum = blockReduceSum(sum);
// Final normalization pass
for (int i = threadIdx.x; i < n; i += blockDim.x) {
output[i] = expf(input[i] - max_val) / sum;
}
}This naive implementation uses wrong identity values, lacks argmax, and handles NaN incorrectly.
// Naive max reduction with common mistakes
__global__ void max_naive(float* g_in, float* g_out, int n) {
extern __shared__ float sdata[];
int tid = threadIdx.x;
int i = blockIdx.x * blockDim.x + threadIdx.x;
// PROBLEM 1: Using 0 as identity - wrong for negative arrays!
sdata[tid] = (i < n) ? g_in[i] : 0; // Should be -INFINITY
__syncthreads();
// PROBLEM 2: Using if/else instead of fmaxf (no NaN handling)
for (int s = blockDim.x / 2; s > 0; s >>= 1) {
if (tid < s) {
if (sdata[tid + s] > sdata[tid]) { // NaN unsafe!
sdata[tid] = sdata[tid + s];
}
}
__syncthreads();
}
if (tid == 0) g_out[blockIdx.x] = sdata[0];
}
// PROBLEM 3: No argmax - need second kernel to find index
// PROBLEM 4: Not handling ties - which index if multiple max?This optimized version tracks both max value and index, uses warp shuffles, handles ties consistently, and works with -INFINITY identity.
// Optimized max reduction with argmax support
struct ValIdx { float val; int idx; };
__device__ ValIdx warpReduceMax(ValIdx vi) {
#pragma unroll
for (int offset = 16; offset > 0; offset /= 2) {
float o_val = __shfl_down_sync(0xffffffff, vi.val, offset);
int o_idx = __shfl_down_sync(0xffffffff, vi.idx, offset);
if (o_val > vi.val || (o_val == vi.val && o_idx < vi.idx)) {
vi.val = o_val;
vi.idx = o_idx;
}
}
return vi;
}
__device__ ValIdx blockReduceMax(ValIdx vi) {
__shared__ ValIdx shared[32];
int lane = threadIdx.x % 32;
int wid = threadIdx.x / 32;
vi = warpReduceMax(vi);
if (lane == 0) shared[wid] = vi;
__syncthreads();
vi = (threadIdx.x < blockDim.x / 32) ?
shared[lane] : ValIdx{-INFINITY, -1};
if (wid == 0) vi = warpReduceMax(vi);
return vi;
}
template<int BLOCK_SIZE>
__global__ void argmax_optimized(float* input, ValIdx* output, int n) {
ValIdx local = {-INFINITY, -1};
int tid = blockIdx.x * blockDim.x + threadIdx.x;
int gridSize = BLOCK_SIZE * gridDim.x;
// Grid-stride loop
for (int i = tid; i < n; i += gridSize) {
float val = input[i];
if (val > local.val) {
local.val = val;
local.idx = i;
}
}
// Block reduction
local = blockReduceMax(local);
if (threadIdx.x == 0) {
output[blockIdx.x] = local;
}
}
// Final reduction of block results (small array)
__global__ void argmax_final(ValIdx* block_results, ValIdx* output,
int num_blocks) {
ValIdx local = {-INFINITY, -1};
for (int i = threadIdx.x; i < num_blocks; i += blockDim.x) {
ValIdx br = block_results[i];
if (br.val > local.val ||
(br.val == local.val && br.idx < local.idx)) {
local = br;
}
}
local = blockReduceMax(local);
if (threadIdx.x == 0) *output = local;
}| Metric | Naive | Optimized | Improvement |
|---|---|---|---|
| Execution Time (16M elements) | 1.9 ms | 0.11 ms | 17x faster |
| Memory Bandwidth (% peak) | 42% | 91% | 2.2x better |
| With Argmax Overhead | 3.8 ms (2 kernels) | 0.11 ms (fused) | 35x faster |
| Batched (1024 rows x 4096 cols) | 12.4 ms | 0.8 ms | 15.5x faster |
Decide on a tie-breaking policy: first occurrence (smallest index), last occurrence (largest index), or any (undefined behavior). The examples use first occurrence by preferring smaller indices when values are equal. For stable sorting in ML, first occurrence is typically preferred.
Use fmaxf() when possible. It handles NaN correctly (returns non-NaN value), compiles to a single instruction (FMNMX), and handles +0/-0 correctly. Use explicit comparisons only when you need custom NaN behavior or are tracking argmax.
For small K (< 32), maintain a sorted buffer in registers and insert each element. For larger K, use a heap-based approach or radix select algorithm. NVIDIA CUB provides cub::DeviceRadixSort for efficient top-K. For ML applications, consider approximate methods like random sampling.
Yes! Common fused patterns include: max + exp (softmax), max + L2 norm, max + min (range finding), and max + index for attention. Fusing saves memory bandwidth by avoiding intermediate storage. The online softmax algorithm is a great example of fusing max and sum.
Same pattern with different operator
Generalization of reduction pattern
Alternative for top-K finding
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