k-means Clustering (cudaFlow)
Contents
Following up on k-means Clustering, this page studies how to accelerate a k-means workload on a GPU using tf::
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:
- for every data point, find the nearest centroid (L2 distance or other measurements) and assign the point to it
- 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"); // allocate GPU memory tf::Task allocate_px = taskflow.emplace([&](){ cudaMalloc(&d_px, N*sizeof(float)); }).name("allocate_px"); tf::Task allocate_py = taskflow.emplace([&](){ cudaMalloc(&d_py, N*sizeof(float)); }).name("allocate_py"); tf::Task allocate_mx = taskflow.emplace([&](){ cudaMalloc(&d_mx, K*sizeof(float)); } ).name("allocate_mx"); tf::Task allocate_my = taskflow.emplace([&](){ cudaMalloc(&d_my, K*sizeof(float)); }).name("allocate_my"); tf::Task allocate_sy = taskflow.emplace([&](){ cudaMalloc(&d_sy, K*sizeof(float)); }).name("allocate_sy"); tf::Task allocate_c = taskflow.emplace([&](){ cudaMalloc(&d_c, K*sizeof(float)); }).name("allocate_c"); // transfer data from the host to the GPU tf::Task h2d = taskflow.emplace([&](tf::cudaFlow& cf){ cf.copy(d_px, h_px.data(), N).name("h2d_px"); cf.copy(d_py, h_py.data(), N).name("h2d_py"); cf.copy(d_mx, h_mx.data(), K).name("h2d_mx"); cf.copy(d_my, h_my.data(), K).name("h2d_my"); }).name("h2d"); // GPU task graph of the main k-means body tf::Task kmeans = taskflow.emplace([&](tf::cudaFlow& cf){ tf::cudaTask zero_c = cf.zero(d_c, K).name("zero_c"); tf::cudaTask zero_sx = cf.zero(d_sx, K).name("zero_sx"); tf::cudaTask zero_sy = cf.zero(d_sy, K).name("zero_sy"); tf::cudaTask cluster = cf.kernel( (N+1024-1) / 1024, 1024, 0, assign_clusters, d_px, d_py, N, d_mx, d_my, d_sx, d_sy, K, d_c ).name("cluster"); tf::cudaTask 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); }).name("update_means"); // condition task to check convergence tf::Task condition = taskflow.emplace([i=0, M] () mutable { return i++ < M ? 0 : 1; }).name("converged?"); // transfer the result of clusters from GPU to host tf::Task stop = taskflow.emplace([&](tf::cudaFlow& cf){ cf.copy(h_mx.data(), d_mx, K).name("d2h_mx"); cf.copy(h_my.data(), d_my, K).name("d2h_my"); }).name("d2h"); // deallocated GPU memory tf::Task 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(condition); condition.precede(kmeans, stop); stop.precede(free); // dump the taskflow without expanding GPU task graphs taskflow.dump(std::cout); // run the taskflow executor.run(taskflow).wait(); // dump the entire taskflow taskflow.dump(std::cout); }
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 second dump after executing the taskflow produces the following diagram, with all cudaFlows expanded:
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.
Repeat the Execution of the k-means cudaFlow
We observe the GPU task graph parameters remain unchanged across all k-means iterations. In this case, we can leverage tf::
tf::Task kmeans = taskflow.emplace([&](tf::cudaFlow& cf){ tf::cudaTask zero_c = cf.zero(d_c, K).name("zero_c"); tf::cudaTask zero_sx = cf.zero(d_sx, K).name("zero_sx"); tf::cudaTask zero_sy = cf.zero(d_sy, K).name("zero_sy"); tf::cudaTask cluster = cf.kernel( (N+1024-1) / 1024, 1024, 0, assign_clusters, d_px, d_py, N, d_mx, d_my, d_sx, d_sy, K, d_c ).name("cluster"); tf::cudaTask 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); // we ask the executor to launch the cudaFlow by M times cf.offload_n(M); }).name("update_means"); // ... // 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);
At the last line of the cudaFlow closure, we call cf.offload_n(M)
to ask the executor to repeatedly run the cudaFlow by M
times. Compared with the version using conditional tasking, the cudaFlow here is created only one time and thus the overhead is reduced.
We can see from the above taskflow the condition task is removed.
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.
N | K | M | CPU Sequential | CPU Parallel | GPU (conditional taksing) | GPU (using offload_n) |
---|---|---|---|---|---|---|
10 | 5 | 10 | 0.14 ms | 77 ms | 1 ms | 1 ms |
100 | 10 | 100 | 0.56 ms | 86 ms | 7 ms | 1 ms |
1000 | 10 | 1000 | 10 ms | 98 ms | 55 ms | 13 ms |
10000 | 10 | 10000 | 1006 ms | 713 ms | 458 ms | 183 ms |
100000 | 10 | 100000 | 102483 ms | 49966 ms | 7952 ms | 4725 ms |
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::