NNDL 作业10：第六章课后题（LSTM | GRU）
目录
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.
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
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
附加题 6-1P 什么时候应该用GRU? 什么时候用LSTM?
附加题 6-2P LSTM BP推导，并用Numpy实现
总结
参考
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：
令 
为在第k时刻函数g(·)的输入，在计算公式6.34中的误差项 
时，梯度可能过大，从而导致梯度过大问题。
解决方法：使用长短期记忆神经网络。
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
LSTM结构图:
- 遗忘门
：控制上一个隐藏状态要遗忘多少信息 - 输入门
：当前状态的候选状态有多少信息需要保存 - 输出门
：当前隐藏状态有多少需要输出给外部状态
LSTM 中梯度的传播有很多条路径，但
=
⨀+
⨀这条路径上只有逐元素相乘和相加的操作，梯度流最稳定；但是其他路径上梯度流与普通 RNN 类似，照样会发生相同的权重矩阵反复连乘。
由于总的远距离梯度 = 各条路径的远距离梯度之和，即便其他远距离路径梯度消失了，只要保证有一条远距离路径（就是上面说的那条高速公路）梯度不消失，总的远距离梯度就不会消失（正常梯度 + 消失梯度 = 正常梯度）。因此 LSTM 通过改善一条路径上的梯度问题拯救了总体的远距离梯度。
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结构图:
GRU的参数较少（U和W都较小），因此其训练速度更快，或需要归纳的数据更少。相对应的，如果有足够的训练数据，表达能力更强的LSTM或许效果更佳。GRU具有调节信息流动的门单元，但没有一个单独的记忆单元，GRU将输入门和遗忘门整合成一个升级门，通过门来控制梯度。
附加题 6-1P 什么时候应该用GRU? 什么时候用LSTM?
- 一般来说两者效果差不多，性能在很多任务上也不分伯仲。GRU参数更少，收敛更快；数据量很大时，LSTM效果会更好一些，因为LSTM参数也比GRU参数多一些。
- 产生新的state的方式不同，LSTM有两个不同的gate，分别是forget gate (f gate)和input gate(i gate)，而GRUs只有一种update gate(z gate)；
- GRU要在上一时刻的隐层信息的基础上乘上一个重置门，而LSTM无需门来对其控制，LSTM必须考虑上一时刻的隐层信息对当前隐层的影响，而GRU则可选择是否考虑上一时刻的隐层信息对当前时刻的影响。
附加题 6-2P LSTM BP推导，并用Numpy实现
首先求出它们在t时刻的梯度，然后再求出他们最终的梯度。
Numpy实现：
# coding=gbk
import os
os.environ['KMP_DUPLICATE_LIB_OK']='True'
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：
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]
Process finished with exit code 0
总结
这次作业的推导有点难，参考了网上的推导过程，收获了很多，对于LSTM网络和GRU网络有了更多的了解，还要继续努力啊！
参考
《神经网络的梯度推导与代码验证》之LSTM的前向传播和反向梯度推导 - SumwaiLiu - 博客园
GRU(Gated Recurrent Unit) 更新过程推导及简单代码实现 - 一只有恒心的小菜鸟 - 博客园
零基础入门深度学习(6) - 长短时记忆网络(LSTM) - 作业部落 Cmd Markdown 编辑阅读器