How to normalize matrix columns in CUDA with max performance?

Question

How to effectively normalize matrix columns in CUDA?

My matrix is stored in column-major, and the typical size is 2000x200.

The operation can be represented in the following matlab code.

A = rand(2000,200);

A = exp(A);
A = A./repmat(sum(A,1), [size(A,1) 1]);

Can this be done effectively by Thrust, cuBLAS and/or cuNPP?

A rapid implementation including 4 kernels is shown as follows.

Wondering if these can be done in 1 or 2 kernels to improve the performance, especially for the column summation step implemented by cublasDgemv().

#include <cuda.h>
#include <curand.h>
#include <cublas_v2.h>
#include <thrust/device_vector.h>
#include <thrust/device_ptr.h>
#include <thrust/transform.h>
#include <thrust/iterator/constant_iterator.h>
#include <math.h>

struct Exp
{
    __host__ __device__ void operator()(double& x)
    {
        x = exp(x);
    }
};

struct Inv
{
    __host__ __device__ void operator()(double& x)
    {
        x = (double) 1.0 / x;
    }
};

int main()
{
    cudaDeviceSetCacheConfig(cudaFuncCachePreferShared);
    cublasHandle_t hd;
    curandGenerator_t rng;
    cublasCreate(&hd);
    curandCreateGenerator(&rng, CURAND_RNG_PSEUDO_DEFAULT);

    const size_t m = 2000, n = 200;
    const double c1 = 1.0;
    const double c0 = 0.0;

    thrust::device_vector<double> A(m * n);
    thrust::device_vector<double> sum(1 * n);
    thrust::device_vector<double> one(m * n, 1.0);

    double* pA = thrust::raw_pointer_cast(&A[0]);
    double* pSum = thrust::raw_pointer_cast(&sum[0]);
    double* pOne = thrust::raw_pointer_cast(&one[0]);

    for (int i = 0; i < 100; i++)
    {
        curandGenerateUniformDouble(rng, pA, A.size());


        thrust::for_each(A.begin(), A.end(), Exp());

        cublasDgemv(hd, CUBLAS_OP_T, m, n,
                &c1, pA, m, pOne, 1, &c0, pSum, 1);

        thrust::for_each(sum.begin(), sum.end(), Inv());

        cublasDdgmm(hd, CUBLAS_SIDE_RIGHT, m, n, pA, m, pSum, 1, pA, m);
    }

    curandDestroyGenerator(rng);
    cublasDestroy(hd);

    return 0;
}

Answer 1

I compared the performance of 3 approaches on M2090 with CUDA 5.0.

1. [173.179 us] cublas implementation as shown in the question
2. [733.734 us] pure Thrust implementation with thrust::reduce_by_key from @talonmies
3. [1.508 ms] pure Thrust implementation with thrust::inclusive_scan_by_key

It can be seen that,

1. cublas has highest performance in this case;
2. both thrust::reduce_by_key & thrust::inclusive_scan_by_key launch multiple kernels, which lead to extra overhead;
3. thrust::inclusive_scan_by_key writes much more data to DRAM compared to thrust::reduce_by_key, which can be one of the reasons for longer kernel time;
4. the main performance difference between cublas and thrust approach is the matrix column summation. thrust is slower possibly because thrust::reduce_by_key is designed to do reduction on segments with variant length, but cublas_gemv() can only apply to fixed length segments (row/col).

When the matrix A is large enough to ignore the kernel launching overhead, the cublas appoach still performs best. The profiling result on A_{20,000 x 2,000} is shown as follows.

Fusing the first for_each operation with the cublasSgemv call as indicated by @talonmies may further improve the performance, but I think kernel written by hand should be used instead of thrust::reduce_by_key.

The code for the 3 approaches is shown as follows.

#include <cuda.h>
#include <curand.h>
#include <cublas_v2.h>
#include <thrust/device_vector.h>
#include <thrust/device_ptr.h>
#include <thrust/transform.h>
#include <thrust/reduce.h>
#include <thrust/scan.h>
#include <thrust/iterator/counting_iterator.h>
#include <thrust/iterator/transform_iterator.h>
#include <thrust/iterator/discard_iterator.h>
#include <thrust/iterator/permutation_iterator.h>
#include <math.h>

struct Exp: public thrust::unary_function<double, double>
{
    __host__ __device__ double operator()(double x)
    {
        return exp(x);
    }
};

struct Inv: public thrust::unary_function<double, double>
{
    __host__ __device__ double operator()(double x)
    {
        return (double) 1.0 / x;
    }
};

template<typename T>
struct MulC: public thrust::unary_function<T, T>
{
    T C;
    __host__ __device__ MulC(T c) :
        C(c)
    {
    }
    __host__ __device__ T operator()(T x)
    {
        return x * C;
    }
};

template<typename T>
struct line2col: public thrust::unary_function<T, T>
{
    T C;
    __host__ __device__ line2col(T C) :
            C(C)
    {
    }

    __host__ __device__ T operator()(T i)
    {
        return i / C;
    }
};

int main()
{
    cudaDeviceSetCacheConfig(cudaFuncCachePreferShared);
    cublasHandle_t hd;
    curandGenerator_t rng;
    cublasCreate(&hd);
    curandCreateGenerator(&rng, CURAND_RNG_PSEUDO_DEFAULT);

    const size_t m = 2000, n = 200;
    const double c1 = 1.0;
    const double c0 = 0.0;

    thrust::device_vector<double> A(m * n);
    thrust::device_vector<double> B(m * n);
    thrust::device_vector<double> C(m * n);
    thrust::device_vector<double> sum1(1 * n);
    thrust::device_vector<double> sum2(1 * n);
    thrust::device_vector<double> one(m * n, 1);

    double* pA = thrust::raw_pointer_cast(&A[0]);
    double* pB = thrust::raw_pointer_cast(&B[0]);
    double* pSum1 = thrust::raw_pointer_cast(&sum1[0]);
    double* pSum2 = thrust::raw_pointer_cast(&sum2[0]);
    double* pOne = thrust::raw_pointer_cast(&one[0]);

    curandGenerateUniformDouble(rng, pA, A.size());

    const int count = 2;

    for (int i = 0; i < count; i++)
    {
        thrust::transform(A.begin(), A.end(), B.begin(), Exp());
        cublasDgemv(hd, CUBLAS_OP_T, m, n, &c1, pB, m, pOne, 1, &c0, pSum1, 1);
        thrust::transform(sum1.begin(), sum1.end(), sum1.begin(), Inv());
        cublasDdgmm(hd, CUBLAS_SIDE_RIGHT, m, n, pB, m, pSum2, 1, pB, m);
    }

    for (int i = 0; i < count; i++)
    {
        thrust::reduce_by_key(
                thrust::make_transform_iterator(thrust::make_counting_iterator(0), line2col<int>(m)),
                thrust::make_transform_iterator(thrust::make_counting_iterator(0), line2col<int>(m)) + A.size(),
                thrust::make_transform_iterator(A.begin(), Exp()),
                thrust::make_discard_iterator(),
                sum2.begin());
        thrust::transform(
                A.begin(), A.end(),
                thrust::make_permutation_iterator(
                        sum2.begin(),
                        thrust::make_transform_iterator(thrust::make_counting_iterator(0), line2col<int>(m))),
                C.begin(),
                thrust::divides<double>());
    }

    for (int i = 0; i < count; i++)
    {
        thrust::inclusive_scan_by_key(
                thrust::make_transform_iterator(thrust::make_counting_iterator(0), line2col<int>(m)),
                thrust::make_transform_iterator(thrust::make_counting_iterator(0), line2col<int>(m)) + A.size(),
                thrust::make_transform_iterator(A.begin(), Exp()),
                C.begin());
        thrust::copy(
                thrust::make_permutation_iterator(
                        C.begin() + m - 1,
                        thrust::make_transform_iterator(thrust::make_counting_iterator(0), MulC<int>(m))),
                thrust::make_permutation_iterator(
                        C.begin() + m - 1,
                        thrust::make_transform_iterator(thrust::make_counting_iterator(0), MulC<int>(m))) + n,
                sum2.begin());
        thrust::transform(
                A.begin(), A.end(),
                thrust::make_permutation_iterator(
                        sum2.begin(),
                        thrust::make_transform_iterator(thrust::make_counting_iterator(0), line2col<int>(m))),
                C.begin(),
                thrust::divides<double>());
    }

    curandDestroyGenerator(rng);
    cublasDestroy(hd);

    return 0;
}

Answer 2

You should be able to fuse the first for_each operation with the cublasSgemv call into a single reduce_by_key call. If you define/redefine functors as:

struct Accessor : public thrust::unary_function<int,int>
{
    int lda;
    __host__ __device__ Accessor(int _lda) : lda(_lda) {};
    __host__ __device__ int operator()(const int& idx)
    {
        return idx/lda;
    }
};

struct Exp : public thrust::unary_function<double,double>
{
    __host__ __device__ double operator()(const double& x)
    {
        return exp(x);
    }
};

struct Inv : public thrust::unary_function<double,double>
{
    __host__ __device__ double operator()(const double& x)
    {
        return double(1.0) / x;
    }
};

You can then calculate the normalised output as

Accessor columns(m);
thrust::reduce_by_key(
        thrust::make_transform_iterator(thrust::make_counting_iterator(int(0)), columns),
        thrust::make_transform_iterator(thrust::make_counting_iterator(int(m*n)), columns),
        thrust::make_transform_iterator(A.begin(), Exp()),
        thrust::make_discard_iterator(),
        sum.begin());

thrust::for_each(sum.begin(), sum.end(), Inv());

cublasDdgmm(hd, CUBLAS_SIDE_RIGHT, m, n, pA, m, pSum, 1, pA, m);

[disclaimer: all code written in browser and is untested, use at own risk]

Apart from reducing the number of kernel calls, using fancy iterators eliminates the need for the large unit matrix which should reduce memory footprint and total number of memory transactions to do the summation and exponentiation operations.

Answer 3

You could use ArrayFire in the following manner

array A = randu(2000, 2000);
A = exp(A);
A /= tile(sum(A, 0), A.dims(0), 1);

You could do this in thrust as well. But if you are going to work with matrices (as opposed to plain vectors), you'd have to do it in a for loop which would not be as efficient.

DISCLAIMER I am a developer at Accelereyes, working on arrayfire.

EDIT I am working on generating new benchmarks as requested.

EDIT We found performance bugs for exp in our code because of this benchmark. We are reviewing and fixing it.