Gemmlowp

Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference

Quantization scheme

Equation(1): \[ r=S(q-Z)\\ where\,\bold{S}\,is\,scale,\,\bold{Z}\,is\,zero-point\\ where\,\bold{r}\,is\,real\,value,\,\bold{q}\,is\,quantization\,value. \] Consider the multiplication of two square N X N matrices of real numbers, \(r_{1}\) and \(r_2\), with their product represented by \(r_3=r_1*r_2\).We denote the entries of each of these matrices \(r_{\alpha}(\alpha=1, 2, or\,3)\) as \(r_{\alpha}^{(i,j)}\) for \(1\le i,j\le N\), and the quantization parameters with which they are quantized as \((S_{\alpha}, Z_{\alpha})\).We denote the quantized entries by \(q_{\alpha}^{(i,j)}\). so the equation is follow:

Equation(2): \[ r_{\alpha}^{(i,j)}=S_{\alpha}(q_{\alpha}^{(i,j)}-Z_{\alpha}) \] From the definition of matrix multiplication, we have the follow equation:

Equation(3): \[ S_3(q_{3}^{(i,k)}-Z_3)=\sum_{j=1}^{N}S_{1}(q_1^{(i,j)}-Z_{1})S_{2}(q_2^{(j,k)}-Z_2) \] So, we can be rewritten as:

Equation(4): \[ q_3^{i,k}=Z_3+M\sum_{j=1}^{N}(q_1^{(i,j)}-Z_1)(q_2^{(j,k)}-Z_2)\\ where\,as\,\,M =\,\frac{S_1S_2}{S_3} \] In Equation (4), the only non-integer is the multiplier M. As a constant depending only on the quantization scales \(S1, S2, S3\), it can be computed offline. We empirically find it to always be in the interval (0, 1), and can therefore express it in the normalized form: \[ M=2^{-n}M_0\\ where\,\,M0\,\,is\,\,in\,\,the\,\,interval\,\,[0.5, 1) \] The following is the example code from the Google gemmlowp’s quantization example code, the example code include the scale and zero-point calculation and convert the float to uint8 data. and run the uint8 in the inference stage. Here the code by gemmlowp: if you feel like test this code, you can clone the gemmlowp repository and add the follow code to {gemm_dir}/contrib/CMakeList.txt: then you can mkdir build; pushd build; cmake ../contrib/; make you’ll get the quantization_example executed file.

add_execuble(quantization_example
    "${gemmlowp_src}/doc/quantization_example.cc" ${gemmlowp_public_headers})
target_link_libraries(quantization_example ${EXTERNAL_LIBRARIES})

// Example code illustrating the theory exposed in doc/quantization.md

/* Command line to build and run on x86:
c++ doc/quantization_example.cc -I . --std=c++11 -msse4.1 -lpthread \
  -o /tmp/quantization_example && \
/tmp/quantization_example
*/

#include <algorithm>
#include <cassert>
#include <cmath>
#include <cstdint>
#include <iostream>
#include <random>
#include <vector>
#include "../public/gemmlowp.h"
#include "../public/output_stages.h"

// We will handle both float and quantized matrices, which we will
// represent as gemmlowp::MatrixMap.
// We will need to be able to print them.

// Output a matrix to a std::ostream
template <typename tScalar, gemmlowp::MapOrder tOrder>
std::ostream& operator<<(std::ostream& s,
                         const gemmlowp::MatrixMap<tScalar, tOrder>& m) {
  for (int i = 0; i < m.rows(); i++) {
    for (int j = 0; j < m.cols(); j++) {
      if (j) {
        s << '\t';
      }
      s << static_cast<float>(m(i, j));
    }
    s << '\n';
  }
  return s;
}

// Find the min and max value in a float matrix.
template <gemmlowp::MapOrder tOrder>
void FindMinMax(const gemmlowp::MatrixMap<float, tOrder>& m, float* min,
                float* max) {
  *min = *max = m(0, 0);
  for (int i = 0; i < m.rows(); i++) {
    for (int j = 0; j < m.cols(); j++) {
      const float val = m(i, j);
      *min = std::min(*min, val);
      *max = std::max(*max, val);
    }
  }
}

// A structure to hold quantization parameters 'scale' and 'zero_point'
// as discussed in doc/quantization.md. As explained there, the meaning
// of these values is as the constants in the quantization equation
//
//   real_value = scale * (quantized_value - zero_point)
//
// In other words, 'zero_point' is the quantized value that corresponds
// to the real value 0, and 'scale' is the difference of real values
// corresponding to consecutive quantized values.
struct QuantizationParams {
  float scale;
  std::uint8_t zero_point;
};

// Given the min and max values of a float array, return
// reasonable quantization parameters to use for this array.
QuantizationParams ChooseQuantizationParams(float min, float max) {
  // We extend the [min, max] interval to ensure that it contains 0.
  // Otherwise, we would not meet the requirement that 0 be an exactly
  // representable value.
  min = std::min(min, 0.f);
  max = std::max(max, 0.f);

  // the min and max quantized values, as floating-point values
  const float qmin = 0;
  const float qmax = 255;

  // First determine the scale.
  const double scale = (max - min) / (qmax - qmin);

  // Zero-point computation.
  // First the initial floating-point computation. The zero-point can be
  // determined from solving an affine equation for any known pair
  // (real value, corresponding quantized value).
  // We know two such pairs: (rmin, qmin) and (rmax, qmax).
  // Let's use the first one here.
  const double initial_zero_point = qmin - min / scale;

  // Now we need to nudge the zero point to be an integer
  // (our zero points are integer, and this is motivated by the requirement
  // to be able to represent the real value "0" exactly as a quantized value,
  // which is required in multiple places, for example in Im2col with SAME
  // padding).
  std::uint8_t nudged_zero_point = 0;
  if (initial_zero_point < qmin) {
    nudged_zero_point = qmin;
  } else if (initial_zero_point > qmax) {
    nudged_zero_point = qmax;
  } else {
    nudged_zero_point =
        static_cast<std::uint8_t>(std::round(initial_zero_point));
  }

  QuantizationParams result;
  result.scale = scale;
  result.zero_point = nudged_zero_point;
  return result;
}

template <gemmlowp::MapOrder tLhsOrder, gemmlowp::MapOrder tRhsOrder,
          gemmlowp::MapOrder tResultOrder>
void FloatMatrixMultiplication(
    const gemmlowp::MatrixMap<const float, tLhsOrder>& lhs,
    const gemmlowp::MatrixMap<const float, tRhsOrder>& rhs,
    gemmlowp::MatrixMap<float, tResultOrder>* result) {
  assert(lhs.cols() == rhs.rows());
  assert(lhs.rows() == result->rows());
  assert(rhs.cols() == result->cols());
  for (int i = 0; i < lhs.rows(); i++) {
    for (int k = 0; k < rhs.cols(); k++) {
      (*result)(i, k) = 0;
      for (int j = 0; j < lhs.cols(); j++) {
        (*result)(i, k) += lhs(i, j) * rhs(j, k);
      }
    }
  }
}

void Quantize(const QuantizationParams& qparams, const std::vector<float>& src,
              std::vector<std::uint8_t>* dst) {
  assert(src.size() == dst->size());
  for (std::size_t i = 0; i < src.size(); i++) {
    const float real_val = src[i];
    const float transformed_val = qparams.zero_point + real_val / qparams.scale;
    const float clamped_val = std::max(0.f, std::min(255.f, transformed_val));
    (*dst)[i] = static_cast<std::uint8_t>(std::round(clamped_val));
  }
}

void Dequantize(const QuantizationParams& qparams,
                const std::vector<std::uint8_t>& src, std::vector<float>* dst) {
  assert(src.size() == dst->size());
  for (std::size_t i = 0; i < src.size(); i++) {
    const std::uint8_t quantized_val = src[i];
    (*dst)[i] = qparams.scale * (quantized_val - qparams.zero_point);
  }
}

template <typename tScalar, gemmlowp::MapOrder tOrder>
class MatrixWithStorage {
 public:
  MatrixWithStorage(int rows, int cols)
      : storage(rows * cols), matrix_map(storage.data(), rows, cols) {}
  void MakeRandom() {
    static std::mt19937 random_engine;
    std::uniform_real_distribution<float> distribution(-1, 1);
    for (auto& x : storage) {
      x = static_cast<tScalar>(distribution(random_engine));
    }
  }
  gemmlowp::MatrixMap<const tScalar, tOrder> ConstMap() const {
    return gemmlowp::MatrixMap<const tScalar, tOrder>(
        storage.data(), matrix_map.rows(), matrix_map.cols());
  }
  gemmlowp::MatrixMap<tScalar, tOrder> Map() {
    return gemmlowp::MatrixMap<tScalar, tOrder>(
        storage.data(), matrix_map.rows(), matrix_map.cols());
  }
  const std::vector<tScalar>& Storage() const { return storage; }
  std::vector<tScalar>& Storage() { return storage; }

 private:
  std::vector<tScalar> storage;
  gemmlowp::MatrixMap<tScalar, tOrder> matrix_map;
};

template <typename tScalar, gemmlowp::MapOrder tOrder>
std::ostream& operator<<(std::ostream& s,
                         const MatrixWithStorage<tScalar, tOrder>& m) {
  return s << m.ConstMap();
}

// Given a real_multiplier in the interval (0, 1),
// produces a pair (quantized_multiplier, right_shift) where
// quantized_multiplier is an int32 representing a fixed-point value
// in the interval [-1, 1)  (in practice we only produce positive values)
// and right_shift is an amount to shift right by, so that the
// floating-point multiplication of some int32 input value by real_multiplier,
//
//   return static_cast<int32>(int32_value * real_multiplier);
//
// is best approximated by the integer-arithmetic-only code
//
//   return RoundingRightShift(
//       FixedPointMultiplication(int32_value, quantized_multiplier),
//       right_shift);
//
// This is how to obtain the fixed-point multiplier and right shift
// parameters to pass to
// OutputStageQuantizeDownInt32ByFixedPoint.
//
// Note: all this code only needs to run offline to generate the quantized
// neural network workload, not at runtime on the
// device on which quantized neural networks need to run. So it's not
// performance-critical at all.
void QuantizeMultiplierSmallerThanOne(float real_multiplier,
                                      std::int32_t* quantized_multiplier,
                                      int* right_shift) {
  assert(real_multiplier > 0.f);
  assert(real_multiplier < 1.f);
  int s = 0;
  // We want to bring the real multiplier into the interval [1/2, 1).
  // We can do so by multiplying it by two, and recording how many times
  // we multiplied by two so that we can compensate that by a right
  // shift by the same amount.
  while (real_multiplier < 0.5f) {
    real_multiplier *= 2.0f;
    s++;
  }
  // Now that the real multiplier is in [1/2, 1), we convert it
  // into a fixed-point number.
  std::int64_t q =
      static_cast<std::int64_t>(std::round(real_multiplier * (1ll << 31)));
  assert(q <= (1ll << 31));
  // Handle the special case when the real multiplier was so close to 1
  // that its fixed-point approximation was undistinguishable from 1.
  // We handle this by dividing it by two, and remembering to decrement
  // the right shift amount.
  if (q == (1ll << 31)) {
    q /= 2;
    s--;
  }
  assert(s >= 0);
  assert(q <= std::numeric_limits<std::int32_t>::max());
  *quantized_multiplier = static_cast<std::int32_t>(q);
  *right_shift = s;
}

int main() {
  std::cout.precision(3);

  const int rows = 2;
  const int depth = 4;
  const int cols = 3;
  const auto kOrder = gemmlowp::MapOrder::ColMajor;

  std::cout << "First, let us make some float matrices LHS and RHS, "
            << "and compute their product.\n"
            << std::endl;
  MatrixWithStorage<float, kOrder> float_lhs(rows, depth);
  float_lhs.MakeRandom();
  MatrixWithStorage<float, kOrder> float_rhs(depth, cols);
  float_rhs.MakeRandom();
  MatrixWithStorage<float, kOrder> reference_float_result(rows, cols);
  auto reference_float_result_map = reference_float_result.Map();
  FloatMatrixMultiplication(float_lhs.ConstMap(), float_rhs.ConstMap(),
                            &reference_float_result_map);
  std::cout << "Here is the float LHS matrix:\n" << float_lhs << std::endl;
  std::cout << "Here is the float RHS matrix:\n" << float_rhs << std::endl;
  std::cout << "Here is the float product (LHS * RHS) matrix obtained by "
            << "ordinary float matrix multiplication, i.e. as far as we are "
            << "concerned, the REFERENCE RESULT:\n"
            << reference_float_result << std::endl;

  std::cout
      << "Now we embark on reproducing this result using "
      << "quantized arithmetic. The code below splits into two parts: "
      << "quantization code that only needs to run offline (e.g. to "
      << "generate a quantized neural network workload), and actual "
      << "runtime quantized code, which is typically performance-critical "
      << "and where we typically do not want to use any floating-point "
      << "arithmetic. We want to clearly distinguish between the two.\n"
      << std::endl;

  std::cout << "The below is OFFLINE QUANTIZATION CODE. We still use some "
            << "floating-point arithmetic in the process of generating the "
            << "quantized workload to be run on-device.\n"
            << std::endl;

  std::cout
      << "Now, let us choose quantization parameters for these matrices. "
      << "You might ask, what good is quantization if we need to pick "
      << "quantization parameters for the result before we can run the "
      << "quantized computation to obtain the result? The idea is that we "
      << "target applications such as neural networks, where unknown results "
      << "are only allowed to vary within preexisting bounds. In practice, the "
      << "bounds for the results are typically learned during the neural "
         "network "
      << "training process. The min and max of the result do not have to be "
      << "exact. If they are too broad, we just get lower quantization "
         "accuracy. "
      << "If they are too narrow, we just get clamping at the bounds.\n"
      << std::endl;

  float lhs_min, lhs_max, rhs_min, rhs_max, result_min, result_max;
  FindMinMax(float_lhs.Map(), &lhs_min, &lhs_max);
  FindMinMax(float_rhs.Map(), &rhs_min, &rhs_max);
  FindMinMax(reference_float_result.Map(), &result_min, &result_max);
  const auto lhs_qparams = ChooseQuantizationParams(lhs_min, lhs_max);
  const auto rhs_qparams = ChooseQuantizationParams(rhs_min, rhs_max);
  const auto result_qparams = ChooseQuantizationParams(result_min, result_max);

  std::cout << "For LHS, we have min = " << lhs_min << ", max = " << lhs_max
            << ", scale = " << lhs_qparams.scale
            << ", zero_point = " << static_cast<float>(lhs_qparams.zero_point)
            << std::endl;
  std::cout << "For RHS, we have min = " << rhs_min << ", max = " << rhs_max
            << ", scale = " << rhs_qparams.scale
            << ", zero_point = " << static_cast<float>(rhs_qparams.zero_point)
            << std::endl;
  std::cout << "For the result, we have min = " << result_min
            << ", max = " << result_max << ", scale = " << result_qparams.scale
            << ", zero_point = "
            << static_cast<float>(result_qparams.zero_point) << std::endl;

  std::cout << std::endl;

  MatrixWithStorage<std::uint8_t, kOrder> uint8_lhs(rows, depth);
  MatrixWithStorage<std::uint8_t, kOrder> uint8_rhs(depth, cols);
  MatrixWithStorage<std::uint8_t, kOrder> actual_uint8_result(rows, cols);

  Quantize(lhs_qparams, float_lhs.Storage(), &uint8_lhs.Storage());
  Quantize(rhs_qparams, float_rhs.Storage(), &uint8_rhs.Storage());

  std::cout << "Quantized uint8 LHS matrix:\n" << uint8_lhs << std::endl;
  std::cout << "Quantized uint8 RHS matrix:\n" << uint8_rhs << std::endl;

  const int lhs_offset = -lhs_qparams.zero_point;
  const int rhs_offset = -rhs_qparams.zero_point;
  const int result_offset = result_qparams.zero_point;

  const float real_multiplier =
      lhs_qparams.scale * rhs_qparams.scale / result_qparams.scale;
  std::int32_t quantized_multiplier;
  int right_shift;
  QuantizeMultiplierSmallerThanOne(real_multiplier, &quantized_multiplier,
                                   &right_shift);

  std::cout << "End of OFFLINE QUANTIZATION CODE.\n" << std::endl;

  std::cout << "The below is ON-DEVICE RUNTIME QUANTIZED CODE. "
            << "This is the part that is performance-critical and may only "
            << "use quantized arithmetic.\n"
            << std::endl;

  gemmlowp::OutputStageQuantizeDownInt32ByFixedPoint
      quantize_down_stage;
  quantize_down_stage.result_offset_after_shift = result_offset;
  quantize_down_stage.result_fixedpoint_multiplier = quantized_multiplier;
  quantize_down_stage.result_shift = right_shift;
  gemmlowp::OutputStageSaturatingCastToUint8 saturating_cast_stage;
  const auto& output_pipeline =
      std::make_tuple(quantize_down_stage, saturating_cast_stage);

  auto actual_uint8_result_map = actual_uint8_result.Map();
  gemmlowp::GemmContext gemm_context;
  gemmlowp::GemmWithOutputPipeline<std::uint8_t, std::uint8_t,
                                   gemmlowp::DefaultL8R8BitDepthParams>(
      &gemm_context, uint8_lhs.ConstMap(), uint8_rhs.ConstMap(),
      &actual_uint8_result_map, lhs_offset, rhs_offset, output_pipeline);

  std::cout << "Quantized uint8 result matrix obtained by quantized "
            << "multiplication:\n"
            << actual_uint8_result << std::endl;

  std::cout << "End of ON-DEVICE RUNTIME QUANTIZED CODE.\n" << std::endl;

  MatrixWithStorage<float, kOrder> actual_float_result(rows, cols);
  Dequantize(result_qparams, actual_uint8_result.Storage(),
             &actual_float_result.Storage());
  std::cout
      << "Here is the actual float product (LHS * RHS) matrix obtained by "
      << "dequantizing the above uint8 result, i.e. "
      << "as far as we are concerned, the ACTUAL RESULT:\n"
      << actual_float_result << std::endl;

  MatrixWithStorage<float, kOrder> diff_float_result(rows, cols);
  for (int i = 0; i < rows; i++) {
    for (int j = 0; j < cols; j++) {
      diff_float_result.Map()(i, j) =
          actual_float_result.Map()(i, j) - reference_float_result.Map()(i, j);
    }
  }

  std::cout << "Difference between ACTUAL and REFERENCE float results:\n"
            << diff_float_result << std::endl;
}

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