HBU-NNDL 作业9：分别使用numpy和pytorch实现BPTT

目录

6-1P：推导RNN反向传播算法BPTT.

RNN前向传播

沿时反向传播BPTT(Backpropagation Through Time)

设计简单RNN模型，分别用Numpy、Pytorch实现反向传播算子，并代入数值测试. 

心得体会

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6-1P：推导RNN反向传播算法BPTT.

参考：学习笔记-循环神经网络(RNN)及沿时反向传播BPTT - 知乎

RNN前向传播

假设有一个时间序列，在每一时刻我们有：

这就是RNN的结构。可以看到，每一时刻t tt的隐含状态都是由当前时刻的输入和上一时刻的隐含状态共同得到的。下面是详细的符号定义：

符号	含义	维度
	第t时刻的输入
	第t时刻隐层的带权输入
	第t时刻的隐含状态
	第t时刻输出层的带权输入
	第t时刻的输出
	第t时刻的损失	标量
	隐层对输入的参数，整个模型共享
	隐层对状态的参数，整个模型共享
	输出层参数，整个模型共享
	隐层的偏置，整个模型共享
	输出层偏置，整个模型共享
	输出层激活函数
	隐层激活函数

沿时反向传播BPTT(Backpropagation Through Time)

计算出后，并不能立即对模型参数进行更新。需要沿着时间t tt不断给出输入，计算出所有时刻的损失。模型总损失为

1、求

由公式 和 ，很容易有:

推广到矩阵形式，即：

2、求

细心的人会发现，与之前 不同，这次的 时间上标加在了括号外面。简单说一下原因：由于V在输出层，所以它在每一时刻的梯度只与当前时刻的损失有关。但U和W在隐藏层，参与到了下一时刻的运算。在求它们每一时刻的梯度时，要使用总损失E来表示。

观察公式和，有：

计算这一项时，由于RNN的特性：计算时，同时需要和。所以不仅会对当前时刻的输出造成影响，也会影响到下一时刻的输出，变量间具体的依赖关系如下图所示：

 所以，应该包含两部分：

 前半部分：

 后半部分：

带入原式，得到：

 引入误差符号，记。

上式可改写为：

 推广到矩阵形式，即：

3、求

 观察公式，有：

可发现与形式基本相同。所以很容易直接得出的矩阵形式：

 

通过上式比较，我们可以找到的计算方式

 

设计简单RNN模型，分别用Numpy、Pytorch实现反向传播算子，并代入数值测试. 

参考链接：L5W1作业1 手把手实现循环神经网络_追寻远方的人的博客-CSDN博客

import torch
import numpy as np


class RNNCell:
    def __init__(self, weight_ih, weight_hh,
                 bias_ih, bias_hh):
        self.weight_ih = weight_ih
        self.weight_hh = weight_hh
        self.bias_ih = bias_ih
        self.bias_hh = bias_hh

        self.x_stack = []
        self.dx_list = []
        self.dw_ih_stack = []
        self.dw_hh_stack = []
        self.db_ih_stack = []
        self.db_hh_stack = []

        self.prev_hidden_stack = []
        self.next_hidden_stack = []

        # temporary cache
        self.prev_dh = None

    def __call__(self, x, prev_hidden):
        self.x_stack.append(x)

        next_h = np.tanh(
            np.dot(x, self.weight_ih.T)
            + np.dot(prev_hidden, self.weight_hh.T)
            + self.bias_ih + self.bias_hh)

        self.prev_hidden_stack.append(prev_hidden)
        self.next_hidden_stack.append(next_h)
        # clean cache
        self.prev_dh = np.zeros(next_h.shape)
        return next_h

    def backward(self, dh):
        x = self.x_stack.pop()
        prev_hidden = self.prev_hidden_stack.pop()
        next_hidden = self.next_hidden_stack.pop()

        d_tanh = (dh + self.prev_dh) * (1 - next_hidden ** 2)
        self.prev_dh = np.dot(d_tanh, self.weight_hh)

        dx = np.dot(d_tanh, self.weight_ih)
        self.dx_list.insert(0, dx)

        dw_ih = np.dot(d_tanh.T, x)
        self.dw_ih_stack.append(dw_ih)

        dw_hh = np.dot(d_tanh.T, prev_hidden)
        self.dw_hh_stack.append(dw_hh)

        self.db_ih_stack.append(d_tanh)
        self.db_hh_stack.append(d_tanh)

        return self.dx_list


if __name__ == '__main__':
    np.random.seed(123)
    torch.random.manual_seed(123)
    np.set_printoptions(precision=6, suppress=True)

    rnn_PyTorch = torch.nn.RNN(4, 5).double()
    rnn_numpy = RNNCell(rnn_PyTorch.all_weights[0][0].data.numpy(),
                        rnn_PyTorch.all_weights[0][1].data.numpy(),
                        rnn_PyTorch.all_weights[0][2].data.numpy(),
                        rnn_PyTorch.all_weights[0][3].data.numpy())

    nums = 3
    x3_numpy = np.random.random((nums, 3, 4))
    x3_tensor = torch.tensor(x3_numpy, requires_grad=True)

    h3_numpy = np.random.random((1, 3, 5))
    h3_tensor = torch.tensor(h3_numpy, requires_grad=True)

    dh_numpy = np.random.random((nums, 3, 5))
    dh_tensor = torch.tensor(dh_numpy, requires_grad=True)

    h3_tensor = rnn_PyTorch(x3_tensor, h3_tensor)
    h_numpy_list = []

    h_numpy = h3_numpy[0]
    for i in range(nums):
        h_numpy = rnn_numpy(x3_numpy[i], h_numpy)
        h_numpy_list.append(h_numpy)

    h3_tensor[0].backward(dh_tensor)
    for i in reversed(range(nums)):
        rnn_numpy.backward(dh_numpy[i])

    print("numpy_hidden :\n", np.array(h_numpy_list))
    print("torch_hidden :\n", h3_tensor[0].data.numpy())
    print("-----------------------------------------------")

    print("dx_numpy :\n", np.array(rnn_numpy.dx_list))
    print("dx_torch :\n", x3_tensor.grad.data.numpy())
    print("------------------------------------------------")

    print("dw_ih_numpy :\n",
          np.sum(rnn_numpy.dw_ih_stack, axis=0))
    print("dw_ih_torch :\n",
          rnn_PyTorch.all_weights[0][0].grad.data.numpy())
    print("------------------------------------------------")

    print("dw_hh_numpy :\n",
          np.sum(rnn_numpy.dw_hh_stack, axis=0))
    print("dw_hh_torch :\n",
          rnn_PyTorch.all_weights[0][1].grad.data.numpy())
    print("------------------------------------------------")

    print("db_ih_numpy :\n",
          np.sum(rnn_numpy.db_ih_stack, axis=(0, 1)))
    print("db_ih_torch :\n",
          rnn_PyTorch.all_weights[0][2].grad.data.numpy())
    print("-----------------------------------------------")
    print("db_hh_numpy :\n",
          np.sum(rnn_numpy.db_hh_stack, axis=(0, 1)))
    print("db_hh_torch :\n",
          rnn_PyTorch.all_weights[0][3].grad.data.numpy())

numpy_hidden :
 [[[ 0.4686   -0.298203  0.741399 -0.446474  0.019391]
  [ 0.365172 -0.361254  0.426838 -0.448951  0.331553]
  [ 0.589187 -0.188248  0.684941 -0.45859   0.190099]]

 [[ 0.146213 -0.306517  0.297109  0.370957 -0.040084]
  [-0.009201 -0.365735  0.333659  0.486789  0.061897]
  [ 0.030064 -0.282985  0.42643   0.025871  0.026388]]

 [[ 0.225432 -0.015057  0.116555  0.080901  0.260097]
  [ 0.368327  0.258664  0.357446  0.177961  0.55928 ]
  [ 0.103317 -0.029123  0.182535  0.216085  0.264766]]]
torch_hidden :
 [[[ 0.4686   -0.298203  0.741399 -0.446474  0.019391]
  [ 0.365172 -0.361254  0.426838 -0.448951  0.331553]
  [ 0.589187 -0.188248  0.684941 -0.45859   0.190099]]

 [[ 0.146213 -0.306517  0.297109  0.370957 -0.040084]
  [-0.009201 -0.365735  0.333659  0.486789  0.061897]
  [ 0.030064 -0.282985  0.42643   0.025871  0.026388]]

 [[ 0.225432 -0.015057  0.116555  0.080901  0.260097]
  [ 0.368327  0.258664  0.357446  0.177961  0.55928 ]
  [ 0.103317 -0.029123  0.182535  0.216085  0.264766]]]
-----------------------------------------------
dx_numpy :
 [[[-0.643965  0.215931 -0.476378  0.072387]
  [-1.221727  0.221325 -0.757251  0.092991]
  [-0.59872  -0.065826 -0.390795  0.037424]]

 [[-0.537631 -0.303022 -0.364839  0.214627]
  [-0.815198  0.392338 -0.564135  0.217464]
  [-0.931365 -0.254144 -0.561227  0.164795]]

 [[-1.055966  0.249554 -0.623127  0.009784]
  [-0.45858   0.108994 -0.240168  0.117779]
  [-0.957469  0.315386 -0.616814  0.205634]]]
dx_torch :
 [[[-0.643965  0.215931 -0.476378  0.072387]
  [-1.221727  0.221325 -0.757251  0.092991]
  [-0.59872  -0.065826 -0.390795  0.037424]]

 [[-0.537631 -0.303022 -0.364839  0.214627]
  [-0.815198  0.392338 -0.564135  0.217464]
  [-0.931365 -0.254144 -0.561227  0.164795]]

 [[-1.055966  0.249554 -0.623127  0.009784]
  [-0.45858   0.108994 -0.240168  0.117779]
  [-0.957469  0.315386 -0.616814  0.205634]]]
------------------------------------------------
dw_ih_numpy :
 [[3.918335 2.958509 3.725173 4.157478]
 [1.261197 0.812825 1.10621  0.97753 ]
 [2.216469 1.718251 2.366936 2.324907]
 [3.85458  3.052212 3.643157 3.845696]
 [1.806807 1.50062  1.615917 1.521762]]
dw_ih_torch :
 [[3.918335 2.958509 3.725173 4.157478]
 [1.261197 0.812825 1.10621  0.97753 ]
 [2.216469 1.718251 2.366936 2.324907]
 [3.85458  3.052212 3.643157 3.845696]
 [1.806807 1.50062  1.615917 1.521762]]
------------------------------------------------
dw_hh_numpy :
 [[ 2.450078  0.243735  4.269672  0.577224  1.46911 ]
 [ 0.421015  0.372353  0.994656  0.962406  0.518992]
 [ 1.079054  0.042843  2.12169   0.863083  0.757618]
 [ 2.225794  0.188735  3.682347  0.934932  0.955984]
 [ 0.660546 -0.321076  1.554888  0.833449  0.605201]]
dw_hh_torch :
 [[ 2.450078  0.243735  4.269672  0.577224  1.46911 ]
 [ 0.421015  0.372353  0.994656  0.962406  0.518992]
 [ 1.079054  0.042843  2.12169   0.863083  0.757618]
 [ 2.225794  0.188735  3.682347  0.934932  0.955984]
 [ 0.660546 -0.321076  1.554888  0.833449  0.605201]]
------------------------------------------------
db_ih_numpy :
 [7.568411 2.175445 4.335336 6.820628 3.51003 ]
db_ih_torch :
 [7.568411 2.175445 4.335336 6.820628 3.51003 ]
-----------------------------------------------
db_hh_numpy :
 [7.568411 2.175445 4.335336 6.820628 3.51003 ]
db_hh_torch :
 [7.568411 2.175445 4.335336 6.820628 3.51003 ]

 

心得体会

这次作业手推了一遍BPTT，因为网课注意力不太集中，所以这次作业做起来还是有点费劲。

在我参考的文章学习笔记-循环神经网络(RNN)及沿时反向传播BPTT - 知乎中，作者提到了将BPTT的误差分为了网络方向上的误差与时间上的误差，我认为这样更易于理解BPTT。