Learning from Examples » k-means Clustering (cudaFlow)

Define the k-means Kernels

Recall that the k-means algorithm has the following steps:

• Step 1: initialize k random centroids
• Step 2: for every data point, find the nearest centroid (L2 distance or other measurements) and assign the point to it
• Step 3: for every centroid, move the centroid to the average of the points assigned to that centroid
• Step 4: go to Step 2 until converged (no more changes in the last few iterations) or maximum iterations reached

We observe Step 2 and Step 3 of the algorithm are parallelizable across individual points for use to harness the power of GPU:

1. for every data point, find the nearest centroid (L2 distance or other measurements) and assign the point to it
2. for every centroid, move the centroid to the average of the points assigned to that centroid.

At a fine-grained level, we request one GPU thread to work on one point for Step 2 and one GPU thread to work on one centroid for Step 3.

// px/py: 2D points
// N: number of points
// mx/my: centroids
// K: number of clusters
// sx/sy/c: storage to compute the average
__global__ void assign_clusters(
  float* px, float* py, int N, 
  float* mx, float* my, float* sx, float* sy, int K, int* c
) {
  const int index = blockIdx.x * blockDim.x + threadIdx.x;

  if (index >= N) {
    return;
  }

  // Make global loads once.
  float x = px[index];
  float y = py[index];

  float best_dance = FLT_MAX;
  int best_k = 0;
  for (int k = 0; k < K; ++k) {
    float d = L2(x, y, mx[k], my[k]);
    if (d < best_d) {
      best_d = d;
      best_k = k;
    }   
  }

  atomicAdd(&sx[best_k], x); 
  atomicAdd(&sy[best_k], y); 
  atomicAdd(&c [best_k], 1); 
}

// mx/my: centroids, sx/sy/c: storage to compute the average
__global__ void compute_new_means(
  float* mx, float* my, float* sx, float* sy, int* c
) {
  int k = threadIdx.x;
  int count = max(1, c[k]);  // turn 0/0 to 0/1
  mx[k] = sx[k] / count;
  my[k] = sy[k] / count;
}

When we recompute the cluster centroids to be the mean of all points assigned to a particular centroid, multiple GPU threads may access the sum arrays, sx and sy, and the count array, c. To avoid data race, we use a simple atomicAdd method.

Define the k-means cudaFlow

Based on the two kernels, we can define the cudaFlow for the k-means workload below:

// N: number of points
// K: number of clusters
// M: number of iterations
// px/py: 2D point vector 
void kmeans_gpu(
  int N, int K, int M, cconst std::vector<float>& px, const std::vector<float>& py
) {
  std::vector<float> h_mx, h_my;
  float *d_px, *d_py, *d_mx, *d_my, *d_sx, *d_sy, *d_c;

  for(int i=0; i<K; ++i) {
    h_mx.push_back(h_px[i]);
    h_my.push_back(h_py[i]);
  }

  // create a taskflow graph
  tf::Executor executor;
  tf::Taskflow taskflow("K-Means");

  auto allocate_px = taskflow.emplace([&](){
    TF_CHECK_CUDA(cudaMalloc(&d_px, N*sizeof(float)), "failed to allocate d_px");
  }).name("allocate_px");

  auto allocate_py = taskflow.emplace([&](){
    TF_CHECK_CUDA(cudaMalloc(&d_py, N*sizeof(float)), "failed to allocate d_py");
  }).name("allocate_py");

  auto allocate_mx = taskflow.emplace([&](){
    TF_CHECK_CUDA(cudaMalloc(&d_mx, K*sizeof(float)), "failed to allocate d_mx");
  }).name("allocate_mx");

  auto allocate_my = taskflow.emplace([&](){
    TF_CHECK_CUDA(cudaMalloc(&d_my, K*sizeof(float)), "failed to allocate d_my");
  }).name("allocate_my");

  auto allocate_sx = taskflow.emplace([&](){
    TF_CHECK_CUDA(cudaMalloc(&d_sx, K*sizeof(float)), "failed to allocate d_sx");
  }).name("allocate_sx");

  auto allocate_sy = taskflow.emplace([&](){
    TF_CHECK_CUDA(cudaMalloc(&d_sy, K*sizeof(float)), "failed to allocate d_sy");
  }).name("allocate_sy");
  auto allocate_c = taskflow.emplace([&](){
    TF_CHECK_CUDA(cudaMalloc(&d_c, K*sizeof(float)), "failed to allocate dc");
  }).name("allocate_c");

  auto h2d = taskflow.emplace([&](){
    cudaMemcpy(d_px, h_px.data(), N*sizeof(float), cudaMemcpyDefault);
    cudaMemcpy(d_py, h_py.data(), N*sizeof(float), cudaMemcpyDefault);
    cudaMemcpy(d_mx, h_mx.data(), K*sizeof(float), cudaMemcpyDefault);
    cudaMemcpy(d_my, h_my.data(), K*sizeof(float), cudaMemcpyDefault);
  }).name("h2d");

  auto kmeans = taskflow.emplace([&](){

    tf::cudaFlow cf;

    auto zero_c = cf.zero(d_c, K).name("zero_c");
    auto zero_sx = cf.zero(d_sx, K).name("zero_sx");
    auto zero_sy = cf.zero(d_sy, K).name("zero_sy");

    auto cluster = cf.kernel(
      (N+512-1) / 512, 512, 0,
      assign_clusters, d_px, d_py, N, d_mx, d_my, d_sx, d_sy, K, d_c
    ).name("cluster");

    auto new_centroid = cf.kernel(
      1, K, 0,
      compute_new_means, d_mx, d_my, d_sx, d_sy, d_c
    ).name("new_centroid");

    cluster.precede(new_centroid)
           .succeed(zero_c, zero_sx, zero_sy);

    // Repeat the execution for M times
    tf::cudaStream stream;
    for(int i=0; i<M; i++) {
      cf.run(stream);
    }
    stream.synchronize();
  }).name("update_means");

  auto stop = taskflow.emplace([&](){
    cudaMemcpy(h_mx.data(), d_mx, K*sizeof(float), cudaMemcpyDefault);
    cudaMemcpy(h_my.data(), d_my, K*sizeof(float), cudaMemcpyDefault);
  }).name("d2h");

  auto free = taskflow.emplace([&](){
    cudaFree(d_px);
    cudaFree(d_py);
    cudaFree(d_mx);
    cudaFree(d_my);
    cudaFree(d_sx);
    cudaFree(d_sy);
    cudaFree(d_c);
  }).name("free");

  // build up the dependency
  h2d.succeed(allocate_px, allocate_py, allocate_mx, allocate_my);

  kmeans.succeed(allocate_sx, allocate_sy, allocate_c, h2d)
        .precede(stop);

  stop.precede(free);

  // run the taskflow
  executor.run(taskflow).wait();

  //std::cout << "dumping kmeans graph ...\n";
  taskflow.dump(std::cout);
  return {h_mx, h_my};
}

The first dump before executing the taskflow produces the following diagram. The condition tasks introduces a cycle between itself and update_means. Each time it goes back to update_means, the cudaFlow is reconstructed with captured parameters in the closure and offloaded to the GPU.

The main cudaFlow task, update_means, must not run before all required data has settled down. It precedes a condition task that circles back to itself until we reach M iterations. When iteration completes, the condition task directs the execution path to the cudaFlow, h2d, to copy the results of clusters to h_mx and h_my and then deallocate all GPU memory.

Benchmarking

We run three versions of k-means, sequential CPU, parallel CPUs, and one GPU, on a machine of 12 Intel i7-8700 CPUs at 3.20 GHz and a Nvidia RTX 2080 GPU using various numbers of 2D point counts and iterations.

When the number of points is larger than 10K, both parallel CPU and GPU implementations start to pick up the speed over than the sequential version. We can see that using the built-in predicate, tf::cudaFlow::offload_n, can avoid repetitively creating the graph over and over, resulting in two times faster than conditional tasking.