BatchNorm backward implementation (#461)

* Implemented batchnorm-backward Blockwise and Multiblock kernels

* Add batchnorm-backward device op

* Add batchnorm-backward host-reference op

* Add batchnorm-backward example

* Parameters renaming in batchnorm backward kernels and device op

* Change in the example to loose the threshold for ScaleDiff checking

* Add comments to explain the implementation of batchnorm-backward

* Parameters renaming again in batchnorm backward kernels

* Improve the expression calculation for performance

* Add batchnorm backward to README

* Add comments to explain inv-variance in batchnorm forward and backward

* Renaming the batchnorm forward training and inferring examples

* Add/update the comments for batchnorm-backward kernels

* Renaming again

* Add block_sync_lds between two consecutive blockwise reductions

* Move common expression 1/N out of the static_for loops

* Add dy_elementwise_op

* Renaming in backward example again

* Add checking for reduceDims in reference_batchnorm_backward

* Update to comments and codes format

* Rename in the comments

* Remove common expression out of the loop in reference_batchnorm_backward_nhwc_c

* Add block_sync_lds() between blockwise reduction again

* Fix comments again

* Remove int8 from batchnorm-forward instances since it is not needed for forward training and could fail test

example/34_batchnorm/CMakeLists.txt

-add_example_executable(example_batchnorm_forward batchnorm_forward_nhwc.cpp)
-add_example_executable(example_batchnorm_infer batchnorm_infer_nhwc.cpp)
+add_example_executable(example_batchnorm_forward_training batchnorm_forward_training_nhwc.cpp)
+add_example_executable(example_batchnorm_forward_inferring batchnorm_forward_inferring_nhwc.cpp)
+add_example_executable(example_batchnorm_backward batchnorm_backward_nhwc.cpp)

example/34_batchnorm/README.md

@@ -53,4 +53,29 @@ Start running 10 times...
 Perf: 1.28235 ms, 523.329 GB/s
 ```
 
+## Run ```batchnorm backward nhwc```
+```bash
+# -D <xxx> : input 4-d tensor lengths
+# -v <x> :   verification (0=no, 1=yes)
+Arg1: data type (0: fp16, 1: fp32, 3: int8, 5: bp16, 6: fp64)
+Arg2 -- 1/0 to indicate whether to use saved mean and invVariance
+Arg3 -- init method used for dy and bnScale (0=no init, 1=single integer value, 2=scope integer value, 3=decimal value)
+Arg4 -- time kernel (0=no, 1=yes)
+Arg5: use multi-block welford (0=n0, 1=yes)
+./bin/example_batchnorm_backward -D 128,16,3,1024 -v 1 0 0 3 1 1
+```
 
+Result 
+```
+./bin/example_batchnorm_backward -D 128,16,3,1024 -v 1 0 0 3 1 1
+launch_and_time_kernel: grid_dim {6144, 1, 1}, block_dim {256, 1, 1} 
+Warm up 1 time
+Start running 10 times...
+launch_and_time_kernel: grid_dim {6144, 1, 1}, block_dim {256, 1, 1} 
+Warm up 1 time
+Start running 10 times...
+launch_and_time_kernel: grid_dim {6144, 1, 1}, block_dim {256, 1, 1} 
+Warm up 1 time
+Start running 10 times...
+Perf: 0.411026 ms, 91.8702 GB/s
+```

include/ck/tensor_operation/gpu/device/device_batchnorm_backward.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2022, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include <array>
+#include <memory>
+
+#include "ck/ck.hpp"
+#include "ck/tensor_operation/gpu/device/device_base.hpp"
+
+namespace ck {
+namespace tensor_operation {
+namespace device {
+
+template <index_t Rank, index_t NumBatchNormReduceDim, typename DyElementwiseOp>
+struct DeviceBatchNormBwd : public BaseOperator
+{
+    static constexpr index_t NumInvariantDim = Rank - NumBatchNormReduceDim;
+
+    virtual std::unique_ptr<BaseArgument>
+    MakeArgumentPointer(const std::array<index_t, Rank> xyLengths,
+                        const std::array<index_t, Rank> xStrides,
+                        const std::array<index_t, Rank> dyStrides,
+                        const std::array<index_t, Rank> dxStrides,
+                        const std::array<int, NumBatchNormReduceDim> reduceDims,
+                        const std::array<ck::index_t, NumInvariantDim> bnScaleBiasMeanVarLengths,
+                        const std::array<ck::index_t, NumInvariantDim> bnScaleStrides,
+                        const std::array<ck::index_t, NumInvariantDim> bnBiasStrides,
+                        const std::array<ck::index_t, NumInvariantDim> bnMeanVarStrides,
+                        const void* p_x,
+                        const void* p_dy,
+                        const void* p_scale,
+                        const void* p_savedMean,
+                        const void* p_savedInvVar,
+                        double epsilon,
+                        const DyElementwiseOp dy_elementwise_op,
+                        void* p_dx,
+                        void* p_dscale,
+                        void* p_dbias) = 0;
+
+    virtual std::unique_ptr<BaseInvoker> MakeInvokerPointer() = 0;
+};
+
+template <index_t Rank, index_t NumBatchNormReduceDim, typename DyElementwiseOp>
+using DeviceBatchNormBwdPtr =
+    std::unique_ptr<DeviceBatchNormBwd<Rank, NumBatchNormReduceDim, DyElementwiseOp>>;
+
+} // namespace device
+} // namespace tensor_operation
+} // namespace ck

include/ck/tensor_operation/gpu/grid/batchnorm_multiblock/gridwise_multiblock_welford_first_half.hpp

@@ -93,6 +93,9 @@ struct GridwiseMultiblockWelfordFirstHalf
     static constexpr index_t M_BlockTileSize = MThreadClusterSize * MThreadSliceSize;
     static constexpr index_t K_BlockTileSize = KThreadClusterSize * KThreadSliceSize;
 
+    // clang-format off
+    // First half of the Multiblock Welford method to calculate mean and variance, used by both batchnorm-forward and batchnorm-backward.
+    // clang-format on
     __device__ static void Run(const XGridDesc_M_K& x_grid_desc_m_k,
                                const MeanVarCountGridDesc_M_G& mean_var_count_grid_desc_m_g,
                                const GetReduceCountPerThreadFunctor& get_reduce_count_per_thread,

include/ck/tensor_operation/gpu/grid/batchnorm_multiblock/gridwise_multiblock_welford_second_half_batchnorm_forward_final.hpp

@@ -529,6 +529,7 @@ struct GridwiseWelfordSecondHalfBatchNormForwardFinal
             auto result_inv_var_global_buf = make_dynamic_buffer<AddressSpaceEnum::Global>(
                 resultSaveInvVariance, mean_var_grid_desc_m.GetElementSpaceSize());
 
+            // calculate inv-variance as 1/sqrt(epsilon+variance)
             static_for<0, MThreadSliceSize, 1>{}([&](auto I) {
                 welford_var_thread_buf(I) =
                     type_convert<AccDataType>(1.0f) / sqrt(epsilon + welford_var_thread_buf[I]);

include/ck/tensor_operation/gpu/grid/gridwise_batchnorm_forward_blockwise_welford.hpp

@@ -441,6 +441,7 @@ struct GridwiseBatchNormForwardWithBlockwiseWelford
             auto result_inv_var_global_buf = make_dynamic_buffer<AddressSpaceEnum::Global>(
                 resultSaveInvVariance, mean_var_grid_desc_m.GetElementSpaceSize());
 
+            // calculate inv-variance as 1/sqrt(epsilon+variance), stored in place of variance
             static_for<0, MThreadSliceSize, 1>{}([&](auto I) {
                 var_thread_buf(I) =
                     type_convert<AccDataType>(1.0f) / sqrt(epsilon + var_thread_buf[I]);

include/ck/tensor_operation/gpu/grid/gridwise_multiblock_welford_first_half.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2022, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include "ck/utility/data_type.hpp"
+#include "ck/utility/math.hpp"
+#include "ck/tensor_operation/gpu/block/blockwise_welford.hpp"
+#include "ck/tensor_operation/gpu/thread/threadwise_welford.hpp"
+#include "ck/tensor_operation/gpu/thread/threadwise_tensor_slice_transfer.hpp"
+#include "ck/tensor_operation/gpu/element/element_wise_operation.hpp"
+
+namespace ck {
+
+template <typename GridwiseMultiblockWelfordFirstHalf_,
+          typename XDataType,
+          typename MeanVarDataType,
+          typename XGridDesc_M_K,
+          typename MeanVarCountGridDesc_M_G,
+          typename GetReduceCountPerThreadFunctor>
+__global__ void kernel_multiblock_welford_first_half(
+    const XGridDesc_M_K x_grid_desc_m_k,
+    const MeanVarCountGridDesc_M_G mean_var_count_grid_desc_m_g,
+    const GetReduceCountPerThreadFunctor get_reduce_count_per_thread,
+    index_t num_k_block_tile_iteration,
+    const XDataType* const __restrict__ p_x,
+    MeanVarDataType* const p_welford_mean,
+    MeanVarDataType* const p_welford_variance,
+    int32_t* const p_welford_count)
+{
+    GridwiseMultiblockWelfordFirstHalf_::Run(x_grid_desc_m_k,
+                                             mean_var_count_grid_desc_m_g,
+                                             get_reduce_count_per_thread,
+                                             num_k_block_tile_iteration,
+                                             p_x,
+                                             p_welford_mean,
+                                             p_welford_variance,
+                                             p_welford_count);
+};
+
+template <typename XDataType,
+          typename AccDataType,
+          typename MeanVarDataType,
+          typename XGridDesc_M_K,
+          typename MeanVarCountGridDesc_M_G,
+          typename GetReduceCountPerThreadFunctor,
+          index_t BlockSize,
+          index_t MThreadClusterSize,
+          index_t KThreadClusterSize,
+          index_t MThreadSliceSize,
+          index_t KThreadSliceSize,
+          index_t XSrcCountSrcVectorDim,
+          index_t XSrcCountSrcVectorSize>
+struct GridwiseMultiblockWelfordFirstHalf
+{
+    static_assert((XSrcCountSrcVectorDim == 0 && MThreadSliceSize % XSrcCountSrcVectorSize == 0) ||
+                      (XSrcCountSrcVectorDim == 1 &&
+                       KThreadSliceSize % XSrcCountSrcVectorSize == 0),
+                  "Invalid thread slice sizes and/or vector sizes configuration, please check!");
+
+    static constexpr bool reorder_thread_cluster = (XSrcCountSrcVectorDim == 0);
+
+    using ThreadClusterLengths_M_K = Sequence<MThreadClusterSize, KThreadClusterSize>;
+
+    using ThreadBufferDimAccessOrder =
+        typename conditional<reorder_thread_cluster, Sequence<1, 0>, Sequence<0, 1>>::type;
+
+    using ThreadClusterArrangeOrder =
+        typename conditional<reorder_thread_cluster, Sequence<1, 0>, Sequence<0, 1>>::type;
+
+    static constexpr auto thread_cluster_desc =
+        make_cluster_descriptor(ThreadClusterLengths_M_K{}, ThreadClusterArrangeOrder{});
+
+    using ThreadReduceSrcDesc_M_K = decltype(make_naive_tensor_descriptor_packed(
+        make_tuple(Number<MThreadSliceSize>{}, Number<KThreadSliceSize>{})));
+    using ThreadReduceDstDesc_M =
+        decltype(make_naive_tensor_descriptor_packed(make_tuple(Number<MThreadSliceSize>{})));
+
+    using ThreadwiseWelford =
+        ThreadwiseWelford<AccDataType, ThreadReduceSrcDesc_M_K, ThreadReduceDstDesc_M>;
+
+    using BlockwiseWelford = BlockwiseWelford<AccDataType,
+                                              BlockSize,
+                                              ThreadClusterLengths_M_K,
+                                              ThreadClusterArrangeOrder,
+                                              false>;
+
+    using PassThroughOp = tensor_operation::element_wise::PassThrough;
+
+    static constexpr auto I0 = Number<0>{};
+    static constexpr auto I1 = Number<1>{};
+
+    static constexpr index_t M_BlockTileSize = MThreadClusterSize * MThreadSliceSize;
+    static constexpr index_t K_BlockTileSize = KThreadClusterSize * KThreadSliceSize;
+
+    __device__ static void Run(const XGridDesc_M_K& x_grid_desc_m_k,
+                               const MeanVarCountGridDesc_M_G& mean_var_count_grid_desc_m_g,
+                               const GetReduceCountPerThreadFunctor& get_reduce_count_per_thread,
+                               index_t num_k_block_tile_iteration,
+                               const XDataType* const __restrict__ p_x,
+                               MeanVarDataType* const p_welford_mean,
+                               MeanVarDataType* const p_welford_variance,
+                               int32_t* const p_welford_count)
+    {
+        StaticBuffer<AddressSpaceEnum::Vgpr, AccDataType, MThreadSliceSize * KThreadSliceSize, true>
+            x_thread_buf;
+
+        StaticBuffer<AddressSpaceEnum::Vgpr, AccDataType, MThreadSliceSize, true>
+            welford_mean_thread_buf;
+        StaticBuffer<AddressSpaceEnum::Vgpr, AccDataType, MThreadSliceSize, true>
+            welford_var_thread_buf;
+        StaticBuffer<AddressSpaceEnum::Vgpr, int32_t, MThreadSliceSize, true>
+            welford_count_thread_buf;
+
+        const index_t blkgroup_size = mean_var_count_grid_desc_m_g.GetLength(I1);
+
+        const index_t thread_local_id = get_thread_local_1d_id();
+        const index_t block_global_id = get_block_1d_id();
+        const index_t blkgroup_id     = block_global_id / blkgroup_size;
+        const index_t block_local_id  = block_global_id % blkgroup_size;
+
+        const auto thread_cluster_idx =
+            thread_cluster_desc.CalculateBottomIndex(make_multi_index(thread_local_id));
+
+        const auto thread_m_cluster_id = thread_cluster_idx[I0];
+        const auto thread_k_cluster_id = thread_cluster_idx[I1];
+
+        using ThreadBufferLengths_M_K = Sequence<MThreadSliceSize, KThreadSliceSize>;
+        using ThreadBufferLengths_M_1 = Sequence<MThreadSliceSize, 1>;
+
+        constexpr auto thread_buffer_desc_m_k = make_naive_tensor_descriptor_packed(
+            make_tuple(Number<MThreadSliceSize>{}, Number<KThreadSliceSize>{}));
+        constexpr auto thread_buffer_desc_m_1 = make_naive_tensor_descriptor_packed(
+            make_tuple(Number<MThreadSliceSize>{}, Number<1>{}));
+
+        const index_t reduceSizePerBlock = K_BlockTileSize * num_k_block_tile_iteration;
+
+        auto threadwise_x_load = ThreadwiseTensorSliceTransfer_v2<XDataType,
+                                                                  AccDataType,
+                                                                  XGridDesc_M_K,
+                                                                  decltype(thread_buffer_desc_m_k),
+                                                                  ThreadBufferLengths_M_K,
+                                                                  ThreadBufferDimAccessOrder,
+                                                                  XSrcCountSrcVectorDim,
+                                                                  XSrcCountSrcVectorSize,
+                                                                  1,
+                                                                  true>(
+            x_grid_desc_m_k,
+            make_multi_index(blkgroup_id * M_BlockTileSize + thread_m_cluster_id * MThreadSliceSize,
+                             block_local_id * reduceSizePerBlock +
+                                 thread_k_cluster_id * KThreadSliceSize));
+
+        auto threadwise_welford_mean_var_store =
+            ThreadwiseTensorSliceTransfer_v1r3<AccDataType,
+                                               MeanVarDataType,
+                                               decltype(thread_buffer_desc_m_1),
+                                               MeanVarCountGridDesc_M_G,
+                                               PassThroughOp,
+                                               ThreadBufferLengths_M_1,
+                                               Sequence<0, 1>,
+                                               1,
+                                               1,
+                                               InMemoryDataOperationEnum::Set,
+                                               1,
+                                               true>(
+                mean_var_count_grid_desc_m_g,
+                make_multi_index(blkgroup_id * M_BlockTileSize +
+                                     thread_m_cluster_id * MThreadSliceSize,
+                                 block_local_id),
+                PassThroughOp{});
+
+        auto threadwise_welford_count_store =
+            ThreadwiseTensorSliceTransfer_v1r3<int32_t,
+                                               int32_t,
+                                               decltype(thread_buffer_desc_m_1),
+                                               MeanVarCountGridDesc_M_G,
+                                               PassThroughOp,
+                                               ThreadBufferLengths_M_1,
+                                               Sequence<0, 1>,
+                                               1,
+                                               1,
+                                               InMemoryDataOperationEnum::Set,
+                                               1,
+                                               true>(
+                mean_var_count_grid_desc_m_g,
+                make_multi_index(blkgroup_id * M_BlockTileSize +
+                                     thread_m_cluster_id * MThreadSliceSize,
+                                 block_local_id),
+                PassThroughOp{});
+
+        constexpr auto thread_copy_fwd_step_m_k = make_multi_index(0, K_BlockTileSize);
+
+        const auto x_global_val_buf = make_dynamic_buffer<AddressSpaceEnum::Global>(
+            p_x, x_grid_desc_m_k.GetElementSpaceSize());
+
+        auto welford_mean_global_val_buf = make_dynamic_buffer<AddressSpaceEnum::Global>(
+            p_welford_mean, mean_var_count_grid_desc_m_g.GetElementSpaceSize());
+
+        auto welford_var_global_val_buf = make_dynamic_buffer<AddressSpaceEnum::Global>(
+            p_welford_variance, mean_var_count_grid_desc_m_g.GetElementSpaceSize());
+
+        auto welford_count_global_val_buf = make_dynamic_buffer<AddressSpaceEnum::Global>(
+            p_welford_count, mean_var_count_grid_desc_m_g.GetElementSpaceSize());
+
+        auto threadwise_welford = ThreadwiseWelford();
+        threadwise_welford.max_count_ =
+            get_reduce_count_per_thread(block_local_id, thread_k_cluster_id);
+
+        static_for<0, MThreadSliceSize, 1>{}([&](auto I) {
+            welford_mean_thread_buf(I) = type_convert<AccDataType>(0.0f);
+            welford_var_thread_buf(I)  = type_convert<AccDataType>(0.0f);
+        });
+
+        for(index_t reducedTiles = 0; reducedTiles < num_k_block_tile_iteration; ++reducedTiles)
+        {
+            threadwise_x_load.Run(x_grid_desc_m_k,
+                                  x_global_val_buf,
+                                  thread_buffer_desc_m_k,
+                                  make_tuple(I0, I0),
+                                  x_thread_buf);
+
+            threadwise_x_load.MoveSrcSliceWindow(x_grid_desc_m_k, thread_copy_fwd_step_m_k);
+            threadwise_welford.Run(x_thread_buf, welford_mean_thread_buf, welford_var_thread_buf);
+        }
+
+        static_for<0, MThreadSliceSize, 1>{}([&](auto I) {
+            if constexpr(I > 0)
+                block_sync_lds();
+
+            welford_count_thread_buf(I) = threadwise_welford.cur_count_;
+            BlockwiseWelford::Run(
+                welford_mean_thread_buf(I), welford_var_thread_buf(I), welford_count_thread_buf(I));
+        });
+
+        if(thread_k_cluster_id == 0)
+        {
+            threadwise_welford_mean_var_store.Run(thread_buffer_desc_m_1,
+                                                  make_tuple(I0, I0),
+                                                  welford_mean_thread_buf,
+                                                  mean_var_count_grid_desc_m_g,
+                                                  welford_mean_global_val_buf);
+
+            threadwise_welford_mean_var_store.Run(thread_buffer_desc_m_1,
+                                                  make_tuple(I0, I0),
+                                                  welford_var_thread_buf,
+                                                  mean_var_count_grid_desc_m_g,
+                                                  welford_var_global_val_buf);
+
+            threadwise_welford_count_store.Run(thread_buffer_desc_m_1,
+                                               make_tuple(I0, I0),
+                                               welford_count_thread_buf,
+                                               mean_var_count_grid_desc_m_g,
+                                               welford_count_global_val_buf);
+        };
+    }
+};
+
+} // namespace ck

library/include/ck/library/reference_tensor_operation/cpu/reference_batchnorm_backward_nhwc_c.hpp

+// SPDX-License-Identifier: MIT
+// Copyright (c) 2018-2022, Advanced Micro Devices, Inc. All rights reserved.
+
+#pragma once
+
+#include <iostream>
+#include <sstream>
+#include <algorithm>
+
+#include "ck/tensor_operation/gpu/device/device_batchnorm_backward.hpp"
+
+namespace ck {
+namespace tensor_operation {
+namespace host {
+
+template <typename XDataType,
+          typename DyDataType,
+          typename DxDataType,
+          typename AccDataType,
+          typename ScaleDataType,
+          typename BiasDataType,
+          typename MeanVarDataType,
+          typename DyElementwiseOp>
+struct ReferenceBatchNormBwd_Input_N_H_W_C_Output_C
+    : public device::DeviceBatchNormBwd<4, 3, DyElementwiseOp>
+{
+    struct Argument : public device::BaseArgument
+    {
+        Argument(const std::array<index_t, 4> xyLengths,
+                 const std::array<index_t, 4> xStrides,
+                 const std::array<index_t, 4> dyStrides,
+                 const std::array<index_t, 4> dxStrides,
+                 const std::array<int, 3> reduceDims,
+                 const std::array<ck::index_t, 1> bnScaleBiasMeanVarLengths,
+                 const std::array<ck::index_t, 1> bnScaleStrides,
+                 const std::array<ck::index_t, 1> bnBiasStrides,
+                 const std::array<ck::index_t, 1> bnMeanVarStrides,
+                 const XDataType* p_x,
+                 const DyDataType* p_dy,
+                 const ScaleDataType* p_scale,
+                 const MeanVarDataType* p_savedMean,
+                 const MeanVarDataType* p_savedInvVar,
+                 double epsilon,
+                 const DyElementwiseOp dy_elementwise_op,
+                 DxDataType* p_dx,
+                 ScaleDataType* p_dscale,
+                 BiasDataType* p_dbias)
+            : p_x_(p_x),
+              p_dy_(p_dy),
+              p_scale_(p_scale),
+              p_savedMean_(p_savedMean),
+              p_savedInvVar_(p_savedInvVar),
+              epsilon_(epsilon),
+              dy_elementwise_op_(dy_elementwise_op),
+              p_dx_(p_dx),
+              p_dscale_(p_dscale),
+              p_dbias_(p_dbias)
+        {
+            ignore = xStrides;
+            ignore = dyStrides;
+            ignore = dxStrides;
+            ignore = bnScaleStrides;
+            ignore = bnBiasStrides;
+            ignore = bnMeanVarStrides;
+
+            if(xyLengths.size() != 4 || bnScaleBiasMeanVarLengths.size() != 1 ||
+               bnScaleBiasMeanVarLengths[0] != xyLengths[3])
+                throw std::runtime_error("Invalid tensor dimensions!");
+
+            if(reduceDims[0] != 0 || reduceDims[1] != 1 || reduceDims[2] != 2)
+                throw std::runtime_error("Invalid reduce dimensions!");
+
+            n_ = xyLengths[0];
+            h_ = xyLengths[1];
+            w_ = xyLengths[2];
+            c_ = xyLengths[3];
+
+            haveSavedMeanInvVar_ = (p_savedMean != nullptr && p_savedInvVar != nullptr);
+        }
+
+        const XDataType* p_x_;
+        const DyDataType* p_dy_;
+        const ScaleDataType* p_scale_;
+        const MeanVarDataType* p_savedMean_;
+        const MeanVarDataType* p_savedInvVar_;
+
+        double epsilon_;
+        const DyElementwiseOp dy_elementwise_op_;
+
+        DxDataType* p_dx_;
+        ScaleDataType* p_dscale_;
+        BiasDataType* p_dbias_;
+
+        bool haveSavedMeanInvVar_;
+
+        index_t n_, h_, w_, c_;
+    };
+
+    struct Invoker : public device::BaseInvoker
+    {
+        float Run(const Argument& arg)
+        {
+            auto thread_reduce_func = [&](auto iC) {
+                AccDataType reduceSize = type_convert<AccDataType>(arg.n_) *
+                                         type_convert<AccDataType>(arg.h_) *
+                                         type_convert<AccDataType>(arg.w_);
+                index_t offset_C = iC;
+                AccDataType mean;
+                AccDataType invVar;
+
+                if(arg.haveSavedMeanInvVar_)
+                {
+                    mean   = arg.p_savedMean_[offset_C];
+                    invVar = arg.p_savedInvVar_[offset_C];
+                }
+                else
+                {
+                    AccDataType meansquare;
+
+                    meansquare = type_convert<AccDataType>(0.0f);
+                    mean       = type_convert<AccDataType>(0.0f);
+
+                    // compute mean, meanquare, variance, inv-variance
+                    for(index_t iN = 0; iN < arg.n_; iN++)
+                    {
+                        index_t offset_N = iN * arg.h_ * arg.w_ * arg.c_;
+                        for(index_t iH = 0; iH < arg.h_; iH++)
+                        {
+                            index_t offset_H = iH * arg.w_ * arg.c_;
+                            for(index_t iW = 0; iW < arg.w_; iW++)
+                            {
+                                index_t offset_W = iW * arg.c_;
+
+                                auto offset = offset_N + offset_H + offset_W + offset_C;
+
+                                AccDataType x = type_convert<AccDataType>(arg.p_x_[offset]);
+
+                                mean += x;
+                                meansquare += x * x;
+                            };
+                        }
+                    };
+
+                    mean       = mean / reduceSize;
+                    meansquare = meansquare / reduceSize;
+
+                    AccDataType variance = meansquare - mean * mean;
+                    invVar               = type_convert<AccDataType>(1.0f) /
+                             std::sqrt(type_convert<AccDataType>(arg.epsilon_) + variance);
+                };
+
+                AccDataType dbias  = type_convert<AccDataType>(0.0f); // Sum on NHW of dy
+                AccDataType dscale = type_convert<AccDataType>(0.0f); // Sum on NHW of dy * norm_x
+
+                // 1) calculate dy * (x - mean) * inv-variance
+                // 2) calculate sum(dy) on NHW dimensions
+                // 3) calculate sum(dy * norm_x) on NHW dimensions
+                for(index_t iN = 0; iN < arg.n_; iN++)
+                {
+                    index_t offset_N = iN * arg.h_ * arg.w_ * arg.c_;
+                    for(index_t iH = 0; iH < arg.h_; iH++)
+                    {
+                        index_t offset_H = iH * arg.w_ * arg.c_;
+                        for(index_t iW = 0; iW < arg.w_; iW++)
+                        {
+                            index_t offset_W = iW * arg.c_;
+
+                            auto offset = offset_N + offset_H + offset_W + offset_C;
+
+                            AccDataType x = type_convert<AccDataType>(arg.p_x_[offset]);
+
+                            AccDataType norm_x = (x - mean) * invVar;
+                            AccDataType dy     = type_convert<AccDataType>(arg.p_dy_[offset]);
+
+                            arg.dy_elementwise_op_(dy, dy);
+
+                            dbias += dy;
+                            dscale += norm_x * dy;
+                        };
+                    }
+                };
+
+                arg.p_dscale_[offset_C] = type_convert<ScaleDataType>(dscale);
+                arg.p_dbias_[offset_C]  = type_convert<BiasDataType>(dbias);
+
+                AccDataType scale = type_convert<AccDataType>(arg.p_scale_[offset_C]);
+                AccDataType multiplier =
+                    type_convert<AccDataType>(1.0f) / reduceSize * invVar * scale;
+
+                // 1) calculate tmp = dscale * (x - mean) * inv-variance
+                // 2) calculate dx = 1/nhw * inv-variance * scale * (nhw * dy - dbias - tmp)
+                for(index_t iN = 0; iN < arg.n_; iN++)
+                {
+                    index_t offset_N = iN * arg.h_ * arg.w_ * arg.c_;
+                    for(index_t iH = 0; iH < arg.h_; iH++)
+                    {
+                        index_t offset_H = iH * arg.w_ * arg.c_;
+                        for(index_t iW = 0; iW < arg.w_; iW++)
+                        {
+                            index_t offset_W = iW * arg.c_;
+
+                            auto offset = offset_N + offset_H + offset_W + offset_C;
+
+                            AccDataType x = type_convert<AccDataType>(arg.p_x_[offset]);
+
+                            AccDataType norm_x = (x - mean) * invVar;
+                            AccDataType dy     = type_convert<AccDataType>(arg.p_dy_[offset]);
+
+                            arg.dy_elementwise_op_(dy, dy);
+
+                            AccDataType tmpVal = norm_x * dscale;
+
+                            AccDataType dx = multiplier * (reduceSize * dy - dbias - tmpVal);
+
+                            arg.p_dx_[offset] = type_convert<XDataType>(dx);
+                        };
+                    }
+                };
+            };
+
+            std::size_t num_thread      = std::thread::hardware_concurrency();
+            std::size_t work_per_thread = (arg.c_ + num_thread - 1) / num_thread;
+
+            std::vector<joinable_thread> threads(num_thread);
+
+            for(std::size_t it = 0; it < num_thread; ++it)
+            {
+                std::size_t ic_begin = it * work_per_thread;
+                std::size_t ic_end = std::min(static_cast<int>((it + 1) * work_per_thread), arg.c_);
+
+                auto f = [=] {
+                    for(std::size_t ic = ic_begin; ic < ic_end; ++ic)
+                    {
+                        thread_reduce_func(ic);
+                    }
+                };
+
+                threads[it] = joinable_thread(f);
+            }
+
+            return (0.0f);
+        };
+
+        float Run(const device::BaseArgument* p_arg,
+                  const StreamConfig& /*stream_config*/ = StreamConfig{}) override
+        {
+            return Run(*dynamic_cast<const Argument*>(p_arg));
+        };
+    };
+
+    bool IsSupportedArgument(const device::BaseArgument* p_arg) override
+    {
+        (void)p_arg;
+
+        return (true);
+    };
+
+    std::unique_ptr<device::BaseArgument>
+    MakeArgumentPointer(const std::array<index_t, 4> xyLengths,
+                        const std::array<index_t, 4> xStrides,
+                        const std::array<index_t, 4> dyStrides,
+                        const std::array<index_t, 4> dxStrides,
+                        const std::array<int, 3> reduceDims,
+                        const std::array<ck::index_t, 1> bnScaleBiasMeanVarLengths,
+                        const std::array<ck::index_t, 1> bnScaleStrides,
+                        const std::array<ck::index_t, 1> bnBiasStrides,
+                        const std::array<ck::index_t, 1> bnMeanVarStrides,
+                        const void* p_x,
+                        const void* p_dy,
+                        const void* p_scale,
+                        const void* p_savedMean,
+                        const void* p_savedInvVar,
+                        double epsilon,
+                        const DyElementwiseOp dy_elementwise_op,
+                        void* p_dx,
+                        void* p_dscale,
+                        void* p_dbias) override
+    {
+        return std::make_unique<Argument>(xyLengths,
+                                          xStrides,
+                                          dyStrides,
+                                          dxStrides,
+                                          reduceDims,
+                                          bnScaleBiasMeanVarLengths,
+                                          bnScaleStrides,
+                                          bnBiasStrides,
+                                          bnMeanVarStrides,
+                                          static_cast<const XDataType*>(p_x),
+                                          static_cast<const DyDataType*>(p_dy),
+                                          static_cast<const ScaleDataType*>(p_scale),
+                                          static_cast<const MeanVarDataType*>(p_savedMean),
+                                          static_cast<const MeanVarDataType*>(p_savedInvVar),
+                                          epsilon,
+                                          dy_elementwise_op,
+                                          static_cast<DxDataType*>(p_dx),
+                                          static_cast<ScaleDataType*>(p_dscale),
+                                          static_cast<BiasDataType*>(p_dbias));
+    };
+
+    std::unique_ptr<device::BaseInvoker> MakeInvokerPointer() override
+    {
+        return std::make_unique<Invoker>();
+    };
+
+    std::string GetTypeString() const override
+    {
+        auto str = std::stringstream();
+
+        // clang-format off
+        str << "Reference_BatchNorm_Backward_NHWC_C<" << std::endl;
+        // clang-format on
+
+        return str.str();
+    }
+};
+
+} // namespace host
+} // namespace tensor_operation
+} // namespace ck