MatMul Tutorial: Quantization

C++ API example demonstrating how one can perform reduced precision matrix-matrix multiplication using MatMul and the accuracy of the result compared to the floating point computations.


The example is focused around the following computation:

\[C = A \times B\]

First, we produce the reference result, having the original matrices \(A\) and \(B\) be in dnnl::memory::data_type::f32 data type.

For reduced precision computations, the matrices \(A\) and \(C\) will use dnnl::memory::data_type::u8 data type and would have the appropriate zero points. For the matrix \(B\), we will use the dnnl::memory::data_type::s8 data type, assuming that the data is centered around zero (hence, the zero point would be simply 0).

The quantization formula is:

\[X_{f32}(:) := scale\_X \cdot (X_{int8}(:) - zp\_X),\]


  • \(X_{f32}(:)\) original matrix;

  • \(X_{int8}(:)\) quantized matrix, where int8 is either u8 (uint8_t) for the matrices \(A\) and \(C\), or s8 (int8_t) for the matrix \(B\);

  • \(scale\_X\) f32 scaling factor. For simplicity we will use a single scale factor for each matrix, though for better accuracy it might be a good idea to use per-N-dimension scaling factor for the matrix B.

  • \(zp\_X\) integer quantization parameter “zero point” (essentially, the representation of the real 0 in the quantized data type).

For a given matrix \(X_{f32}\) and int8 data type (u8 or s8), the process of finding the proper \(scale\_X\) and \(zp\_X\) is a research problem and can be different depending on the domain. For example purposes, we will use the simplest approach by mapping the maximum (minimum) \(X_{f32}\) elements to the maximum (minimum) number in the corresponding integer data type, using the following formulas:

  1. Since:

    • \(max(X_{f32}(:)) = scale\_X \cdot (max_{int8} - zp\_X)\)

    • \(min(X_{f32}(:)) = scale\_X \cdot (min_{int8} - zp\_X)\)

  2. Hence:

    • \(scale\_X = \frac{max(X_{f32}(:)) - min(X_{f32}(:))}{max_{int8} - min_{int8}}\)

    • \(zp\_X = max_{int8} - \frac{max(X_{f32}(:))}{scale\_X}\)

It is worth noting that quantization parameters are not always computed at actual run-time. For example, if we perform MatMul operation for similar matrices (in a sense that data distribution is similar between the runs) we can simply guess the proper quantization parameters by collecting some statistics during the early runs. This approach is called static quantization. It gives good performance (since no cycles are spent on computing those parameters) and is typically used in reduced precision CNN inference. However, the static quantization has an obvious disadvantage the guessed parameters might not work well for some particular matrices. For example, that would most likely be the case if we could not guarantee the similarity of the input matrices. In this case, the dynamic quantization would be used, i.e. the parameters (re-)computed at runtime. This gives slightly worse performance, but that might be inevitable due to accuracy considerations.

Both approaches are demonstrated in this example.

Other details:

  • For simplicity all matrices will be stored in Row-Major format.

  • The shapes of the matrices are assumed to be known at creation time. However, for dynamic quantization we would consider q10n parameters (\(scale\_X\) and \(zp\_X\)) to be known at run-time only. On the contrary, for the static quantization these parameters are known at creation time as well.

* Copyright 2019-2020 Intel Corporation
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* See the License for the specific language governing permissions and
* limitations under the License.

#include <cassert>
#include <cctype>
#include <cmath>
#include <cstdio>
#include <iostream>
#include <random>
#include <stdexcept>
#include <vector>
#include <type_traits>

#include "oneapi/dnnl/dnnl.hpp"

#include "example_utils.hpp"

using namespace dnnl;

enum class q10n_scheme_t { DYNAMIC, STATIC };

namespace {

void init_vector(std::vector<float> &v, float min_value, float max_value) {
    std::mt19937 gen;
    std::uniform_real_distribution<float> u(min_value, max_value);

    for (auto &e : v)
        e = u(gen);

template <typename T>
void find_min_max(const std::vector<T> &v, float &min_value, float &max_value) {
    min_value = max_value = v[0];
    for (auto &e : v) {
        min_value = std::min<float>(min_value, e);
        max_value = std::max<float>(max_value, e);

template <typename T>
void compute_q10n_params(const char *message, const std::vector<float> &v,
        float &scale, int32_t &zp) {
    // Find property of T integer type
    // Simple trick to improve accuracy: shrink the range a little bit
    float max_int = (float)std::numeric_limits<T>::max() - 1;
    float min_int = (float)std::numeric_limits<T>::lowest() + 1;

    // Read more in CPU / Section 1 here:
    if (std::is_same<T, uint8_t>::value) max_int /= 2;

    // Find min and max value in array
    float min_val = v[0], max_val = v[0];
    find_min_max(v, min_val, max_val);

    // Compute appropriate scale
    scale = (max_val - min_val) / (max_int - min_int);

    // Compute appropriate offset
    if (std::is_same<T, int8_t>::value)
        zp = 0;
        zp = (int32_t)(max_int - max_val / scale);
    printf("\tComputing q10n params for %s\n"
           "\t\tData type: %s\n"
           "\t\tScale:%.3g (inverse scale:%.3g)\n"
           "\t\tZero point:%d\n\n",
            message, std::is_same<T, int8_t>::value ? "int8_t" : "uint8_t",
            scale, 1 / scale, zp);

int compare_vectors(const std::vector<float> &v1,
        const std::vector<uint8_t> &v2, float scale_v2, int32_t zp_v2,
        float threshold) {
    double v1_l2 = 0, diff_l2 = 0;
    for (size_t n = 0; n < v1.size(); ++n) {
        float v2_n = scale_v2 * (v2[n] - zp_v2); // deq10n v2
        float diff = v1[n] - v2_n;
        v1_l2 += v1[n] * v1[n];
        diff_l2 += diff * diff;

    v1_l2 = std::sqrt(v1_l2);
    diff_l2 = std::sqrt(diff_l2);
    bool ok = diff_l2 <= threshold * v1_l2;

    printf("\tComparison (using l2-norms)\n"
           "\t\tReference matrix:%g\n\t\tError:%g\n\t\tRelative error:%g\n"
           "\nAccuracy check: %s\n\n",
            v1_l2, diff_l2, diff_l2 / v1_l2, ok ? "OK" : "FAILED");

    return ok ? 0 : 1;

} // namespace

engine eng(engine::kind::cpu, 0); // We create a global engine for simplicity

// Quantize float data into X_int_m oneDNN memory using the q10n parameters
// Inputs:
// - X_f32 -- source f32 matrix
// - scale_X, zp_X -- quantization parameters
// - q10n_scheme -- dynamic or static, to mimic real-world applications wrt to
//                  how the q10n parameters are passed to reorders
// Outputs:
// - X_int_m -- prepared oneDNN memory that would hold quantized values
void quantize(q10n_scheme_t q10n_scheme, const std::vector<float> &X_f32,
        float scale_X, int32_t zp_X, memory &X_int_m) {
    using dt = memory::data_type;

    // Depending on `q10n_scheme` pretend the values come at run-time (dynamic)
    // or were known at creation time (static).
    float inv_scale_X = 1.f / scale_X;

    const bool is_dynamic_q10n = q10n_scheme == q10n_scheme_t::DYNAMIC;

    stream s(eng);

    memory::desc x_int_md = X_int_m.get_desc();
    const auto &dims =;

    memory::desc x_f32_md({dims[0], dims[1]}, dt::f32, {dims[1], 1});
    memory X_f32_m(x_f32_md, eng, (void *);

    primitive_attr q10n_attr;
    q10n_attr.set_output_scales(/* mask */ 0,
            {is_dynamic_q10n ? DNNL_RUNTIME_F32_VAL : inv_scale_X});
    q10n_attr.set_zero_points(DNNL_ARG_DST, /* mask */ 0,
            {is_dynamic_q10n ? DNNL_RUNTIME_S32_VAL : zp_X});

    reorder::primitive_desc q10n_pd(eng, x_f32_md, eng, x_int_md, q10n_attr);
    if (is_dynamic_q10n) {
        memory scale_X_m({{1}, dt::f32, {1}}, eng, &inv_scale_X);
        memory zp_X_m({{1}, dt::s32, {1}}, eng, &zp_X);
                {{DNNL_ARG_SRC, X_f32_m}, {DNNL_ARG_DST, X_int_m},
                        {DNNL_ARG_ATTR_OUTPUT_SCALES, scale_X_m},
                        {DNNL_ARG_ATTR_ZERO_POINTS | DNNL_ARG_DST, zp_X_m}});
    } else {
                s, {{DNNL_ARG_SRC, X_f32_m}, {DNNL_ARG_DST, X_int_m}});


// Floating point MatMul
// Inputs:
// - Shape: M, N, K
// - Matrices A and B
// Outputs:
// - Matrix C
void f32_matmul_compute(int64_t M, int64_t N, int64_t K,
        const std::vector<float> &A_f32, const std::vector<float> &B_f32,
        std::vector<float> &C_f32) {
    // Initialize memory descriptors that describes matrices in Row-Major format
    memory::desc a_md({M, K}, memory::data_type::f32, {K, 1});
    memory::desc b_md({K, N}, memory::data_type::f32, {N, 1});
    memory::desc c_md({M, N}, memory::data_type::f32, {N, 1});

    // Wrap raw pointers into oneDNN memory objects
    memory A_f32_m(a_md, eng, (void *);
    memory B_f32_m(b_md, eng, (void *);
    memory C_f32_m(c_md, eng, (void *);

    // Create a MatMul primitive
    matmul::desc matmul_d(a_md, b_md, c_md);
    matmul::primitive_desc matmul_pd(matmul_d, eng);
    matmul matmul_p(matmul_pd);

    stream s(eng);
            {{DNNL_ARG_SRC, A_f32_m}, {DNNL_ARG_WEIGHTS, B_f32_m},
                    {DNNL_ARG_DST, C_f32_m}});

// Reduced precision MatMul with **dynamic** quantization
// Inputs:
// - Shape: M, N, K
// - Matrices A and B in float (would be quantized inside the function)
// Outputs:
// - Matrix C in uint8_t
// - Quantization parameters: scale_C and zp_C
void dynamic_q10n_matmul(int64_t M, int64_t N, int64_t K,
        const std::vector<float> &A_f32, const std::vector<float> &B_f32,
        std::vector<uint8_t> &C_u8, float &scale_C, int32_t &zp_C) {
    stream s(eng);

    float scale_A, scale_B;
    int32_t zp_A, zp_B;

    // We compute q10n parameters here, but in the real world applications for
    // inputs these parameters are transferred from the previous layers
    compute_q10n_params<uint8_t>("A", A_f32, scale_A, zp_A);
    compute_q10n_params<int8_t>("B", B_f32, scale_B, zp_B);
    assert(zp_B == 0 && "for int8 q10n we assume zero point = 0");

    // Quantize matrix A_u8 using reorder primitive
    std::vector<uint8_t> A_u8(M * K, 0);
    memory::desc a_u8_md({M, K}, memory::data_type::u8, {K, 1});
    memory A_u8_m(a_u8_md, eng, (void *);
    quantize(q10n_scheme_t::DYNAMIC, A_f32, scale_A, zp_A, A_u8_m);

    // Quantize matrix B_s8 using reorder primitive
    std::vector<uint8_t> B_s8(K * N, 0);
    memory::desc b_s8_md({K, N}, memory::data_type::s8, {N, 1});
    memory B_s8_m(b_s8_md, eng, (void *);
    quantize(q10n_scheme_t::DYNAMIC, B_f32, scale_B, 0, B_s8_m);

    // Compute C_f32. We cannot directly compute C_u8 since we don't know the
    // appropriate quantization parameters.
    // Note: typically the computed data type in this case is int32_t and not
    //       float. But for brevity we are going to embed the scale_A and
    //       scale_B directly in this quantized MatMul, and hence will get the
    //       intermediate computation in floating point anyways, so there is
    //       no sense to convert the result to int32_t.
    //       In theory, we could postpone using the scale_A and scale_B, compute
    //       the exact C_s32 := (A_u8 - zp_A) * B_s8, and then find the
    //       appropriate quantization parameters for matrix C.
    //       Let it be an exercise :)

    std::vector<float> C_f32(M * N, 0);
    memory::desc c_f32_md({M, N}, memory::data_type::f32, {N, 1});
    memory C_f32_m(c_f32_md, eng, (void *);

    // Create and compute a reduced precision MatMul primitive
        primitive_attr matmul_attr;
        matmul_attr.set_output_scales(/* mask */ 0, {DNNL_RUNTIME_F32_VAL});
                DNNL_ARG_SRC, /* mask */ 0, {DNNL_RUNTIME_S32_VAL});

        matmul::desc matmul_d(a_u8_md, b_s8_md, c_f32_md);
        matmul::primitive_desc matmul_pd(matmul_d, matmul_attr, eng);
        matmul matmul_p(matmul_pd);

        // Pretend the values come at run-time
        float output_scale = scale_A * scale_B;

        memory output_scales_m(
                {{1}, memory::data_type::f32, {1}}, eng, &output_scale);
        memory zp_A_m({{1}, memory::data_type::s32, {1}}, eng, &zp_A);

                {{DNNL_ARG_SRC, A_u8_m}, {DNNL_ARG_WEIGHTS, B_s8_m},
                        {DNNL_ARG_DST, C_f32_m},
                        {DNNL_ARG_ATTR_OUTPUT_SCALES, output_scales_m},
                        {DNNL_ARG_ATTR_ZERO_POINTS | DNNL_ARG_SRC, zp_A_m}});

    // Find quantization parameters for matrix C
    compute_q10n_params<uint8_t>("C", C_f32, scale_C, zp_C);

    // Finally quantize the matrix C
    memory::desc c_u8_md({M, N}, memory::data_type::u8, {N, 1});
    memory C_u8_m(c_u8_md, eng, (void *);
    quantize(q10n_scheme_t::DYNAMIC, C_f32, scale_C, zp_C, C_u8_m);

// Reduced precision MatMul with **static** quantization
// Inputs:
// - Shape: M, N, K
// - Matrices A and B in float (would be quantized inside the function using
//   given q10n parameters)
// - Quantization parameters for all 3 matrices:
//   - scale_A, zp_A
//   - scale_B
//   - scale_C, zp_C
// Outputs:
// - Matrix C in uint8_t
void static_q10n_matmul(int64_t M, int64_t N, int64_t K,
        const std::vector<float> &A_f32, const std::vector<float> &B_f32,
        float scale_A, int32_t zp_A, float scale_B, float scale_C, int32_t zp_C,
        std::vector<uint8_t> &C_u8) {
    stream s(eng);

    // Quantize matrix A_u8 using reorder primitive
    std::vector<uint8_t> A_u8(M * K, 0);
    memory::desc a_u8_md({M, K}, memory::data_type::u8, {K, 1});
    memory A_u8_m(a_u8_md, eng, (void *);
    quantize(q10n_scheme_t::STATIC, A_f32, scale_A, zp_A, A_u8_m);

    // Quantize matrix B_s8 using reorder primitive
    std::vector<uint8_t> B_s8(K * N, 0);
    memory::desc b_s8_md({K, N}, memory::data_type::s8, {N, 1});
    memory B_s8_m(b_s8_md, eng, (void *);
    quantize(q10n_scheme_t::STATIC, B_f32, scale_B, 0, B_s8_m);

    // Directly compute C_u8, since we know quantization parameters for the
    // matrix C. This is the key difference compare to **dynamic** quantization.
        memory::desc c_u8_md({M, N}, memory::data_type::u8, {N, 1});
        memory C_u8_m(c_u8_md, eng, (void *);

        primitive_attr matmul_attr;
                /* mask */ 0, {scale_A * scale_B / scale_C});
        matmul_attr.set_zero_points(DNNL_ARG_SRC, /* mask */ 0, {zp_A});
        matmul_attr.set_zero_points(DNNL_ARG_DST, /* mask */ 0, {zp_C});

        matmul::desc matmul_d(a_u8_md, b_s8_md, c_u8_md);
        matmul::primitive_desc matmul_pd(matmul_d, matmul_attr, eng);
        matmul matmul_p(matmul_pd);

                {{DNNL_ARG_SRC, A_u8_m}, {DNNL_ARG_WEIGHTS, B_s8_m},
                        {DNNL_ARG_DST, C_u8_m}});

void compare_f32_and_quantized_matmuls() {
    // MatMul parameters
    const int64_t M = 10, N = 20, K = 30;

    // Data distribution for matrices A and B
    const float param_A_min_val = -2.f;
    const float param_A_max_val = 1.4f;

    const float param_B_min_val = -1.f;
    const float param_B_max_val = -param_B_min_val; // B is centered around 0

    // Thresholds
    // Ideally the threshold for static quantization should be a little higher
    // than for dynamic quantization. However, we will slightly cheat on the
    // guessed q10n parameters of matrix C (see below), so we will get pretty
    // good accuracy as well.
    const float threshold_dynamic_q10n = 3 * 1e-2f;
    const float threshold_static_q10n = 4 * 1e-2f;

    // Prepare matrices
    std::vector<float> A_f32(M * K), B_f32(K * N), C_f32(M * N, 0);
    init_vector(A_f32, param_A_min_val, param_A_max_val);
    init_vector(B_f32, param_B_min_val, param_B_max_val);

    // Compute _true_ f32 result
    f32_matmul_compute(M, N, K, A_f32, B_f32, C_f32);

    // Compute quantized variant (dynamic)
        printf("# DYNAMIC quantization\n\n");

        std::vector<uint8_t> C_u8_dynamic_q10n(M * N, 0);

        float scale_C_dynamic_q10n; // Q10n parameters we don't know yet
        int zp_C_dynamic_q10n;

        dynamic_q10n_matmul(M, N, K, A_f32, B_f32, C_u8_dynamic_q10n,
                scale_C_dynamic_q10n, zp_C_dynamic_q10n);

        // Compare _true_ f32 result with dynamic q10n
        int rc = compare_vectors(C_f32, C_u8_dynamic_q10n, scale_C_dynamic_q10n,
                zp_C_dynamic_q10n, threshold_dynamic_q10n);
        if (rc) throw std::logic_error("Dynamic quantization accuracy failed.");

    // Compute quantized variant (static)
        printf("# STATIC quantization\n\n");

        std::vector<uint8_t> C_u8_static_q10n(M * N, 0);

        // Let's pretend we know the appropriate q10n parameters (by gathering
        // some statistic or whatnot). For matrix C we will slightly _cheat_
        // and get the appropriate q10n from the actual C_f32 result that we
        // computed earlier. Of course, it is not what one would do in the
        // **static** q10n scheme (just by the definition of the **static**
        // q10n), but solely for the purpose of this example print "passed" in
        // the end :)
        const float scale_A_static_q10n
                = (param_A_max_val - param_A_min_val) / 128;
        const int zp_A_static_q10n
                = (int)(128 - param_A_max_val / scale_A_static_q10n);
        const float scale_B_static_q10n
                = (param_B_max_val - param_B_min_val) / 256;

        float scale_C_static_q10n;
        int zp_C_static_q10n;
        const char *warn_message
                = "C"
                  "\n\t* NOTE: These computation do not happen in real world *"
                  "\n\t*       applications and used here solely to simplify *"
                  "\n\t*       the example.                                  *"
                  "\n\t*       Please refer to the example source code for   *"
                  "\n\t*       more information.                             *"

                warn_message, C_f32, scale_C_static_q10n, zp_C_static_q10n);
        // !!! CHEATING ENDS HERE

        static_q10n_matmul(M, N, K, A_f32, B_f32, scale_A_static_q10n,
                zp_A_static_q10n, scale_B_static_q10n, scale_C_static_q10n,
                zp_C_static_q10n, C_u8_static_q10n);

        // Compare _true_ f32 result with static q10n
        int rc = compare_vectors(C_f32, C_u8_static_q10n, scale_C_static_q10n,
                zp_C_static_q10n, threshold_static_q10n);
        if (rc) throw std::logic_error("Static quantization accuracy failed.");

int main(int argc, char **argv) {
    return handle_example_errors(
            {engine::kind::cpu}, compare_f32_and_quantized_matmuls);