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一、RNN简介

RNN（Recurrent Neural Network）中文循环神经网络，用于处理序列数据。它与传统人工神经网络和卷积神经网络的输入和输出相互独立不同，依赖它独特的神经结构（循环核）获得“记忆能力”

注意与递归神经网络(Recursive Neural Network)RNN区分，同时循环神经网络为短期记忆，与（Long Short-Term Memory networks）LSTM的长期记忆不同

二、RNN关键结构

各参数含义：

• $x_t$：序列t的输入层的值，$s_t$：序列t的隐藏层的值 ，$o_t$：序列t的输出层的值
• $U$：输入层到隐藏层的权重矩阵 ，$V$：隐藏层到输出层的权重矩阵
• $W$：隐藏层上一次的值作为这一次输入的权重

注意事项：

• 同不同序列t时的W，V，U相同，即RNN的Weight sharing
• 结构图中每一步都会有输出，但实际中很可能只需最后一步的输出
• 为了降低网络复杂度，$s_t$只包含前面若干隐藏层的状态

三、RNN的训练方式

本质还是梯度下降的反向传播，由前向传播得到的预测值与真实值构建损失函数，更新W、U、V求解最小值：
$$\begin{gathered} S_t=f(U\cdot X_t+W\cdot S_{t-1}+b) \\ {O}_t=g(V\cdot S_t) \\ {L}_t=\frac{1}{2}(Y_t-O_t{)}^2 \end{gathered}$$
如果对$t_3$的U、V、W求偏导如下：

$$\begin{array}{rl}& \frac{\partial L_3}{\partial V}=\frac{\partial L_3}{\partial O_3}\frac{\partial O_3}{\partial V}\\ & \frac{\partial L_3}{\partial U}=\frac{\partial L_3}{\partial O_3}\frac{\partial O_3}{\partial S_3}\frac{\partial S_3}{\partial U}+\frac{\partial L_3}{\partial O_3}\frac{\partial O_3}{\partial S_3}\frac{\partial S_3}{\partial S_2}\frac{\partial S_2}{\partial U}+\frac{\partial L_3}{\partial O_3}\frac{\partial O_3}{\partial S_3}\frac{\partial S_3}{\partial S_2}\frac{\partial S_2}{\partial S_1}\frac{\partial S_1}{\partial U}\\ & \frac{\partial L_3}{\partial W}=\frac{\partial L_3}{\partial O_3}\frac{\partial O_3}{\partial S_3}\frac{\partial S_3}{\partial W}+\frac{\partial L_3}{\partial O_3}\frac{\partial O_3}{\partial S_3}\frac{\partial S_3}{\partial S_2}\frac{\partial S_2}{\partial W}+\frac{\partial L_3}{\partial O_3}\frac{\partial O_3}{\partial S_3}\frac{\partial S_3}{\partial S_2}\frac{\partial S_2}{\partial S_1}\frac{\partial S_1}{\partial W}\\ & 因为有：\\ & O_3=V{S}_3+b_2\\ & S_3=U{X}_3+W{S}_2+b_1\\ & S_2=U{X}_2+W{S}_1+b_1\\ & S_1=U{X}_1+W{S}_0+b_1\end{array}$$
可以看到U和W对于序列产生了依赖，并且可以得到：
$$\begin{array}{rl}& \frac{\partial L_t}{\partial U}=\sum _{k=0}^t\frac{\partial L_t}{\partial O_t}\frac{\partial O_t}{\partial S_t}(\prod _{j=k+1}^t\frac{\partial S_j}{\partial S_{j-1}})\frac{\partial S_k}{\partial U}\\ & \frac{\partial L_t}{\partial W}=\sum _{k=0}^t\frac{\partial L_t}{\partial O_t}\frac{\partial O_t}{\partial S_t}(\prod _{j=k+1}^t\frac{\partial S_j}{\partial S_{j-1}})\frac{\partial S_k}{\partial W}\end{array}$$
最后将结果放入激活函数即可

四、时间序列预测

预测一个正弦函数的走势
第一部分：构建样本数据

start = np.random.randint(3, size=1)[0]
time_steps = np.linspace(start, start + 10, num_time_steps)
data = np.sin(time_steps)
data = data.reshape(num_time_steps, 1)
x = torch.tensor(data[:-1]).float().view(1, num_time_steps - 1, 1)
y = torch.tensor(data[1:]).float().view(1, num_time_steps - 1, 1)

第二部分：构建循环神经网络结构

class Net(nn.Module):def __init__(self, ):super(Net, self).__init__()self.rnn = nn.RNN(input_size=input_size,hidden_size=hidden_size,num_layers=1,batch_first=True,)for p in self.rnn.parameters():nn.init.normal_(p, mean=0.0, std=0.001)self.linear = nn.Linear(hidden_size, output_size)def forward(self, x, hidden_prev):out, hidden_prev = self.rnn(x, hidden_prev)# [b, seq, h]out = out.view(-1, hidden_size)out = self.linear(out) # [seq,h] => [seq,1]out = out.unsqueeze(dim=0)# [1,seq,1]return out, hidden_prev

第三部分：迭代训练并计算loss

model = Net()
criterion = nn.MSELoss()
optimizer = optim.Adam(model.parameters(), lr)hidden_prev = torch.zeros(1, 1, hidden_size)for iter in range(6000):start = np.random.randint(10, size=1)[0]time_steps = np.linspace(start, start + 10, num_time_steps)data = np.sin(time_steps)data = data.reshape(num_time_steps, 1)x = torch.tensor(data[:-1]).float().view(1, num_time_steps - 1, 1)y = torch.tensor(data[1:]).float().view(1, num_time_steps - 1, 1)output, hidden_prev = model(x, hidden_prev)hidden_prev = hidden_prev.detach() #不会具有梯度loss = criterion(output, y)model.zero_grad()loss.backward()optimizer.step()if iter % 100 == 0:print("Iteration: {} loss {}".format(iter, loss.item()))

第四部分：绘制预测值并比较

predictions = []
input = x[:, 0, :]
for _ in range(x.shape[1]):input = input.view(1, 1, 1)(pred, hidden_prev) = model(input, hidden_prev)input = predpredictions.append(pred.detach().numpy().ravel()[0])x = x.data.numpy().ravel()
y = y.data.numpy()
plt.scatter(time_steps[:-1], x, s=90)
plt.plot(time_steps[:-1], x)plt.scatter(time_steps[1:], predictions)
plt.show()

迭代200次的图像：

迭代6000次的图像：

完整代码：

import numpy as np
import torch
import torch.nn as nn
import torch.optim as optim
from matplotlib import pyplot as plt
num_time_steps = 50
input_size = 1
hidden_size = 16
output_size = 1
lr=0.01class Net(nn.Module):def __init__(self, ):super(Net, self).__init__()self.rnn = nn.RNN(input_size=input_size,hidden_size=hidden_size,num_layers=1,batch_first=True,)for p in self.rnn.parameters():nn.init.normal_(p, mean=0.0, std=0.001)self.linear = nn.Linear(hidden_size, output_size)def forward(self, x, hidden_prev):out, hidden_prev = self.rnn(x, hidden_prev)# [b, seq, h]out = out.view(-1, hidden_size)out = self.linear(out) # [seq,h] => [seq,1]out = out.unsqueeze(dim=0)# [1,seq,1]return out, hidden_prevmodel = Net()
criterion = nn.MSELoss()
optimizer = optim.Adam(model.parameters(), lr)hidden_prev = torch.zeros(1, 1, hidden_size)for iter in range(200):start = np.random.randint(10, size=1)[0]time_steps = np.linspace(start, start + 10, num_time_steps)data = np.sin(time_steps)data = data.reshape(num_time_steps, 1)x = torch.tensor(data[:-1]).float().view(1, num_time_steps - 1, 1)y = torch.tensor(data[1:]).float().view(1, num_time_steps - 1, 1)output, hidden_prev = model(x, hidden_prev)hidden_prev = hidden_prev.detach() #不会具有梯度loss = criterion(output, y)model.zero_grad()loss.backward()optimizer.step()if iter % 100 == 0:print("Iteration: {} loss {}".format(iter, loss.item()))start = np.random.randint(3, size=1)[0]
time_steps = np.linspace(start, start + 10, num_time_steps)
data = np.sin(time_steps)
data = data.reshape(num_time_steps, 1)
x = torch.tensor(data[:-1]).float().view(1, num_time_steps - 1, 1)
y = torch.tensor(data[1:]).float().view(1, num_time_steps - 1, 1)predictions = []
input = x[:, 0, :]
for _ in range(x.shape[1]):input = input.view(1, 1, 1)(pred, hidden_prev) = model(input, hidden_prev)input = predpredictions.append(pred.detach().numpy().ravel()[0])x = x.data.numpy().ravel()
y = y.data.numpy()
plt.scatter(time_steps[:-1], x, s=90)
plt.plot(time_steps[:-1], x)plt.scatter(time_steps[1:], predictions)
plt.show()

五、梯度弥散和梯度爆炸问题

• 梯度弥散(消失）：由于导数的链式法则，连续多层小于1的梯度相乘会使梯度越来越小，最终导致某层梯度为0。梯度被近距离梯度主导，导致模型难以学到远距离的依赖关系
• 梯度爆炸：初始化权值过大，梯度更新量是会成指数级增长的，前面层会比后面层变化的更快，就会导致权值越来越大

上面两个问题都是RNN训练时的难题，解决它们需要不断的实操经验和更加升入的理解