NNDL 作业10:第六章课后题(LSTM | GRU)

习题6-3 当使用公式(6.50)作为循环神经网络得状态更新公式时,分析其可能存在梯度爆炸的原因并给出解决办法.

公式(6.50)为:h_{t}=h_{t-1}+g(x_{t},h_{t-1};\Theta )

在计算误差项时,可能会出现梯度过大的情况,解决办法为:使用长短期神经网络。

习题6-4 推导LSTM网络中参数的梯度,并分析其避免梯度消失的效果 

NNDL 作业10:第六章课后题(LSTM | GRU)_第1张图片 图1

NNDL 作业10:第六章课后题(LSTM | GRU)_第2张图片NNDL 作业10:第六章课后题(LSTM | GRU)_第3张图片 3.Cell状态(长时记忆) 

NNDL 作业10:第六章课后题(LSTM | GRU)_第4张图片

4.Cell输出(短时记忆)

NNDL 作业10:第六章课后题(LSTM | GRU)_第5张图片 5.遗忘门

NNDL 作业10:第六章课后题(LSTM | GRU)_第6张图片 6.输入门

NNDL 作业10:第六章课后题(LSTM | GRU)_第7张图片效果:LSTM把原本RNN的单元改造成一个叫做CEC的部件,这个部件保证了误差将以常数的形式在网络中流动 ,并在此基础上添加输入门和输出门使得模型变成非线性的,并可以调整不同时序的输出对模型后续动作的影响。 

习题6-5 推导GRU网络中参数的梯度,并分析其避免梯度消失的效果

GRU神经网络的整体结构与RNN神经网络的整体结构相同,区别在于GRU神经网络中的隐藏层单元引入了门控单元,其具体的结构如下图所示:

相比于RNN神经元,GRU的神经元略微的有些复杂,下面,逐个的对上面的计算进行说明。

NNDL 作业10:第六章课后题(LSTM | GRU)_第8张图片 NNDL 作业10:第六章课后题(LSTM | GRU)_第9张图片 NNDL 作业10:第六章课后题(LSTM | GRU)_第10张图片 NNDL 作业10:第六章课后题(LSTM | GRU)_第11张图片 NNDL 作业10:第六章课后题(LSTM | GRU)_第12张图片 NNDL 作业10:第六章课后题(LSTM | GRU)_第13张图片 NNDL 作业10:第六章课后题(LSTM | GRU)_第14张图片效果: GRU它引⼊了重置⻔(reset gate)和更新⻔(update gate) 的概念,从而修改了循环神经⽹络中隐藏状态的计算⽅式。

附加题 6-1P 什么时候应该用GRU? 什么时候用LSTM?

        LSTM与GRU的一个不同就是LSTM使用记忆单元来更新隐状态,而GRU使用的是候选隐状态。但本质上都是对原有信息和现有信息的保留与丢弃的选择,最后的目的也都是隐状态的更新。实际应用中LSTM与GRU效果并没有大的差距,但GRU更加简便。经过发展,时序数据的双向的顺序,后面也出现了双向LSTM与双向GRU。GRU参数比LSTM少,所以容易收敛。 数据集大的情况下,LSTM的表达性能还是比GRU好。在一般数据集上 GRU和LSTM的性能差不多。

附加题 6-2P LSTM BP推导,并用Numpy实现

代码如下:

import numpy as np
import torch
 
def sigmoid(x):
    return 1 / (1 + np.exp(-x))
 
class LSTMCell:
    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.dc_prev = None
        self.dh_prev = None
 
        self.weight_ih_grad_stack = []
        self.weight_hh_grad_stack = []
        self.bias_ih_grad_stack = []
        self.bias_hh_grad_stack = []
 
        self.x_stack = []
        self.dx_list = []
        self.dh_prev_stack = []
 
        self.h_prev_stack = []
        self.c_prev_stack = []
 
        self.h_next_stack = []
        self.c_next_stack = []
 
        self.input_gate_stack = []
        self.forget_gate_stack = []
        self.output_gate_stack = []
        self.cell_memory_stack = []
    def __call__(self, x, h_prev, c_prev):
        a_vector = np.dot(x, self.weight_ih.T) + np.dot(h_prev, self.weight_hh.T)
        a_vector += self.bias_ih + self.bias_hh
 
        h_size = np.shape(h_prev)[1]
        a_i = a_vector[:, h_size * 0:h_size * 1]
        a_f = a_vector[:, h_size * 1:h_size * 2]
        a_c = a_vector[:, h_size * 2:h_size * 3]
        a_o = a_vector[:, h_size * 3:]
 
        input_gate = sigmoid(a_i)
        forget_gate = sigmoid(a_f)
        cell_memory = np.tanh(a_c)
        output_gate = sigmoid(a_o)
 
        c_next = (forget_gate * c_prev) + (input_gate * cell_memory)
        h_next = output_gate * np.tanh(c_next)
 
        self.x_stack.append(x)
 
        self.h_prev_stack.append(h_prev)
        self.c_prev_stack.append(c_prev)
 
        self.c_next_stack.append(c_next)
        self.h_next_stack.append(h_next)
 
        self.input_gate_stack.append(input_gate)
        self.forget_gate_stack.append(forget_gate)
        self.output_gate_stack.append(output_gate)
        self.cell_memory_stack.append(cell_memory)
 
        self.dc_prev = np.zeros_like(c_next)
        self.dh_prev = np.zeros_like(h_next)
 
        return h_next, c_next
 
    def backward(self, dh_next):
        x_stack = self.x_stack.pop()
 
        h_prev = self.h_prev_stack.pop()
        c_prev = self.c_prev_stack.pop()
 
        c_next = self.c_next_stack.pop()
 
        input_gate = self.input_gate_stack.pop()
        forget_gate = self.forget_gate_stack.pop()
        output_gate = self.output_gate_stack.pop()
        cell_memory = self.cell_memory_stack.pop()
 
        dh = dh_next + self.dh_prev
 
        d_tanh_c = dh * output_gate * (1 - np.square(np.tanh(c_next)))
        dc = d_tanh_c + self.dc_prev
 
        dc_prev = dc * forget_gate
        self.dc_prev = dc_prev
 
        d_input_gate = dc * cell_memory
        d_forget_gate = dc * c_prev
        d_cell_memory = dc * input_gate
 
        d_output_gate = dh * np.tanh(c_next)
 
        d_ai = d_input_gate * input_gate * (1 - input_gate)
        d_af = d_forget_gate * forget_gate * (1 - forget_gate)
        d_ao = d_output_gate * output_gate * (1 - output_gate)
        d_ac = d_cell_memory * (1 - np.square(cell_memory))
 
        da = np.concatenate((d_ai, d_af, d_ac, d_ao), axis=1)
 
        dx = np.dot(da, self.weight_ih)
        dh_prev = np.dot(da, self.weight_hh)
        self.dh_prev = dh_prev
 
        self.dx_list.insert(0, dx)
        self.dh_prev_stack.append(dh_prev)
 
        self.weight_ih_grad_stack.append(np.dot(da.T, x_stack))
        self.weight_hh_grad_stack.append(np.dot(da.T, h_prev))
 
        db = np.sum(da, axis=0)
        self.bias_ih_grad_stack.append(db)
        self.bias_hh_grad_stack.append(db)
 
        return dh_prev
 
 
np.random.seed(123)
torch.random.manual_seed(123)
np.set_printoptions(precision=6, suppress=True)
 
lstm_torch = torch.nn.LSTMCell(2, 3).double()
lstm_numpy = LSTMCell(lstm_torch.weight_ih.data.numpy(),
                      lstm_torch.weight_hh.data.numpy(),
                      lstm_torch.bias_ih.data.numpy(),
                      lstm_torch.bias_hh.data.numpy())
 
x_numpy = np.random.random((4, 2))
x_torch = torch.tensor(x_numpy, requires_grad=True)
 
h_numpy = np.random.random((4, 3))
h_torch = torch.tensor(h_numpy, requires_grad=True)
 
c_numpy = np.random.random((4, 3))
c_torch = torch.tensor(c_numpy, requires_grad=True)
 
dh_numpy = np.random.random((4, 3))
dh_torch = torch.tensor(dh_numpy, requires_grad=True)
 
h_numpy, c_numpy = lstm_numpy(x_numpy, h_numpy, c_numpy)
h_torch, c_torch = lstm_torch(x_torch, (h_torch, c_torch))
h_torch.backward(dh_torch)
 
dh_numpy = lstm_numpy.backward(dh_numpy)
 
print("h_numpy :\n", h_numpy)
print("h_torch :\n", h_torch.data.numpy())
 
print("---------------------------------")
print("c_numpy :\n", c_numpy)
print("c_torch :\n", c_torch.data.numpy())
 
print("---------------------------------")
print("dx_numpy :\n", np.sum(lstm_numpy.dx_list, axis=0))
print("dx_torch :\n", x_torch.grad.data.numpy())
 
print("---------------------------------")
print("w_ih_grad_numpy :\n",
      np.sum(lstm_numpy.weight_ih_grad_stack, axis=0))
print("w_ih_grad_torch :\n",
      lstm_torch.weight_ih.grad.data.numpy())
 
print("---------------------------------")
print("w_hh_grad_numpy :\n",
      np.sum(lstm_numpy.weight_hh_grad_stack, axis=0))
print("w_hh_grad_torch :\n",
      lstm_torch.weight_hh.grad.data.numpy())
 
print("---------------------------------")
print("b_ih_grad_numpy :\n",
      np.sum(lstm_numpy.bias_ih_grad_stack, axis=0))
print("b_ih_grad_torch :\n",
      lstm_torch.bias_ih.grad.data.numpy())
 
print("---------------------------------")
print("b_hh_grad_numpy :\n",
      np.sum(lstm_numpy.bias_hh_grad_stack, axis=0))
print("b_hh_grad_torch :\n",
      lstm_torch.bias_hh.grad.data.numpy())

运行结果:

C:\ProgramData\Anaconda3\envs\pytorch\python.exe D:/刘子豪/应用程序/pycharm/python项目/liu/亚太杯.py
h_numpy :
 [[ 0.055856  0.234159  0.138457]
 [ 0.094461  0.245843  0.224411]
 [ 0.020396  0.086745  0.082545]
 [-0.003794  0.040677  0.063094]]
h_torch :
 [[ 0.055856  0.234159  0.138457]
 [ 0.094461  0.245843  0.224411]
 [ 0.020396  0.086745  0.082545]
 [-0.003794  0.040677  0.063094]]
---------------------------------
c_numpy :
 [[ 0.092093  0.384992  0.213364]
 [ 0.151362  0.424671  0.318313]
 [ 0.033245  0.141979  0.120822]
 [-0.0061    0.062946  0.094999]]
c_torch :
 [[ 0.092093  0.384992  0.213364]
 [ 0.151362  0.424671  0.318313]
 [ 0.033245  0.141979  0.120822]
 [-0.0061    0.062946  0.094999]]
---------------------------------
dx_numpy :
 [[-0.144016  0.029775]
 [-0.229789  0.140921]
 [-0.246041 -0.009354]
 [-0.088844  0.036652]]
dx_torch :
 [[-0.144016  0.029775]
 [-0.229789  0.140921]
 [-0.246041 -0.009354]
 [-0.088844  0.036652]]
---------------------------------
w_ih_grad_numpy :
 [[-0.056788 -0.036448]
 [ 0.018742  0.014428]
 [ 0.007827  0.024828]
 [ 0.07856   0.05437 ]
 [ 0.061267  0.045952]
 [ 0.083886  0.0655  ]
 [ 0.229755  0.156008]
 [ 0.345218  0.251984]
 [ 0.430385  0.376664]
 [ 0.014239  0.011767]
 [ 0.054866  0.044531]
 [ 0.04654   0.048565]]
w_ih_grad_torch :
 [[-0.056788 -0.036448]
 [ 0.018742  0.014428]
 [ 0.007827  0.024828]
 [ 0.07856   0.05437 ]
 [ 0.061267  0.045952]
 [ 0.083886  0.0655  ]
 [ 0.229755  0.156008]
 [ 0.345218  0.251984]
 [ 0.430385  0.376664]
 [ 0.014239  0.011767]
 [ 0.054866  0.044531]
 [ 0.04654   0.048565]]
---------------------------------
w_hh_grad_numpy :
 [[-0.037698 -0.048568 -0.021069]
 [ 0.016749  0.016277  0.007556]
 [ 0.035743  0.02156   0.000111]
 [ 0.060824  0.069505  0.029101]
 [ 0.060402  0.051634  0.025643]
 [ 0.068116  0.06966   0.035544]
 [ 0.168965  0.217076  0.075904]
 [ 0.248277  0.290927  0.138279]
 [ 0.384974  0.401949  0.167006]
 [ 0.015448  0.0139    0.005158]
 [ 0.057147  0.048975  0.022261]
 [ 0.057297  0.048308  0.017745]]
w_hh_grad_torch :
 [[-0.037698 -0.048568 -0.021069]
 [ 0.016749  0.016277  0.007556]
 [ 0.035743  0.02156   0.000111]
 [ 0.060824  0.069505  0.029101]
 [ 0.060402  0.051634  0.025643]
 [ 0.068116  0.06966   0.035544]
 [ 0.168965  0.217076  0.075904]
 [ 0.248277  0.290927  0.138279]
 [ 0.384974  0.401949  0.167006]
 [ 0.015448  0.0139    0.005158]
 [ 0.057147  0.048975  0.022261]
 [ 0.057297  0.048308  0.017745]]
---------------------------------
b_ih_grad_numpy :
 [-0.084682  0.032588  0.046412  0.126449  0.111421  0.139337  0.361956
  0.539519  0.761838  0.027649  0.103695  0.099405]
b_ih_grad_torch :
 [-0.084682  0.032588  0.046412  0.126449  0.111421  0.139337  0.361956
  0.539519  0.761838  0.027649  0.103695  0.099405]
---------------------------------
b_hh_grad_numpy :
 [-0.084682  0.032588  0.046412  0.126449  0.111421  0.139337  0.361956
  0.539519  0.761838  0.027649  0.103695  0.099405]
b_hh_grad_torch :
 [-0.084682  0.032588  0.046412  0.126449  0.111421  0.139337  0.361956
  0.539519  0.761838  0.027649  0.103695  0.099405]

进程已结束,退出代码为 0

心得体会:

        这次实验的推导感觉挺有难度,在网上找到了一些大佬的推导过程,对于LSTM网络和GRU网络有了深刻的了解。

参考博客:

1. LSTM公式详解&推导

2. 深度学习——循环神经网络GRU公式推导

3. 深度学习笔记-目录 (ai-start.com)

4. DeepLearning.AI: Start or Advance Your Career in AI

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