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编辑
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

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.

梯度爆炸问题：在公式 $Z_k=Uh_{k-1}+Wx_k +b$ 为在第k kk时刻函数 $g (*)$ 的输入，在计算公式(6.34)中的误差项 $δ_{t,k} = \frac{\partial L_{t}}{\partial z_{k}}$ ，梯度可能会过大，从而导致梯度爆炸问题。

解决办法：增加门控装置。

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编辑

手动推导一下

分析避免梯度消失的效果：简单的来讲，RNN的输出由连乘决定，连乘可能会出现梯度消失。LSTM的输出由累加决定，累加的方式克服了上述的问题。

梯度更新时，梯度的大小主要取决于前后状态导数再连乘关系，即： $\prod_{j}^{t}\frac{\partial c_{j}}{\partial c_{j-1}}$ 。

RNN两个状态求导满足（tanh激活函数类似）： $\sigma\left( 1-\sigma \right)W$ ，求导绝对值范围为： $[0\sim\frac{1}{4}W]$ 。

而LSTM输入门状态更新时，两个状态求导则中有一部分直接取决于forget gate的输出值： $\sigma\left( Wx+b \right)$ ，求导绝对值范围为 $\left[ 0\sim1 \right]$ 。

由于RNN很大程度上取决于W 的取值（与4的大小关系），若认定最终所期望学得的系数W 是绝对值稀疏的，即其要么大要么小，对于RNN而言是容易产生梯度消失和梯度爆炸的。而LSTM相比于RNN更稳定，稀疏的话其对应的求导范围看极端一点也是0和1，是不是就类似Relu的作用了呢？因此，在通过几次连乘操作时LSTM相比RNN而言能缓解梯度的消失问题。

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

推导一下：

分析：LSTM和GRU里面都存储了一个中间状态，并把中间状态持续从前往后传，和当前步的特征合并来做预测，合并过程用的是加法模型，这点其实和ResNet的残差模块有点类似。

附加题 6-1P 什么时候应该用GRU? 什么时候用LSTM?

先说明一下GRU和LSTM的区别：GRU作为LSTM的一种变体，将忘记门和输入门合成了一个单一的更新门。同样还混合了细胞状态和隐藏状态，加诸其他一些改动。最终的模型比标准的 LSTM 模型要简单，也是非常流行的变体。GRU 输入输出的结构与普通的 RNN 相似，其中的内部思想与 LSTM 相似。与 LSTM 相比，GRU 内部少了一个”门控“，参数比LSTM 少，但是却也能够达到与 LSTM 相当的功能。

对于 LSTM 与 GRU 而言，由于 GRU 参数更少，收敛速度更快，因此其实际花费时间要少很多，这可以大大加速了我们的迭代过程。而从表现上讲，二者之间孰优孰劣并没有定论，这要依据具体的任务和数据集而定，而实际上，二者之间的 performance 差距往往并不大，远没有调参所带来的效果明显

附加题 6-2P LSTM BP推导

import numpy as np

def sigmoid(x):
    return 1/(1+np.exp(-x))

def softmax(x):
    e_x = np.exp(x-np.max(x))# 防溢出
    return e_x/e_x.sum(axis=0)


def lstm_cell_forward(xt, a_prev, c_prev, parameters):    
    """
    Implement a single forward step of the LSTM-cell as described in Figure (4)
    Arguments:
    xt -- your input data at timestep "t", numpy array of shape (n_x, m).
    a_prev -- Hidden state at timestep "t-1", numpy array of shape (n_a, m)
    c_prev -- Memory state at timestep "t-1", numpy array of shape (n_a, m)
    parameters -- python dictionary containing:
    Wf -- Weight matrix of the forget gate, numpy array of shape (n_a, n_a + n_x)
    bf -- Bias of the forget gate, numpy array of shape (n_a, 1)
    Wi -- Weight matrix of the update gate, numpy array of shape (n_a, n_a + n_x)
    bi -- Bias of the update gate, numpy array of shape (n_a, 1)
    Wc -- Weight matrix of the first "tanh", numpy array of shape (n_a, n_a + n_x)
    bc --  Bias of the first "tanh", numpy array of shape (n_a, 1)
    Wo -- Weight matrix of the output gate, numpy array of shape (n_a, n_a + n_x)
    bo --  Bias of the output gate, numpy array of shape (n_a, 1)Wy -- Weight matrix relating the hidden-state to the output, numpy array of shape (n_y, n_a)
    by -- Bias relating the hidden-state to the output, numpy array of shape (n_y, 1)
    Returns:
    a_next -- next hidden state, of shape (n_a, m)
    c_next -- next memory state, of shape (n_a, m)
    yt_pred -- prediction at timestep "t", numpy array of shape (n_y, m)
    cache -- tuple of values needed for the backward pass, contains (a_next, c_next, a_prev, c_prev, xt, parameters)
    """
 
    # 获取参数字典中各个参数
    Wf = parameters["Wf"]
    bf = parameters["bf"]
    Wi = parameters["Wi"]
    bi = parameters["bi"]
    Wc = parameters["Wc"]
    bc = parameters["bc"]
    Wo = parameters["Wo"]
    bo = parameters["bo"]
    Wy = parameters["Wy"]
    by = parameters["by"]    
    # 获取 xt 和 Wy 的维度参数
    n_x, m = xt.shape
    n_y, n_a = Wy.shape    
    # 拼接 a_prev 和 xt
    concat = np.zeros((n_a + n_x, m))
    concat[: n_a, :] = a_prev
    concat[n_a :, :] = xt    
    # 计算遗忘门、更新门、记忆细胞候选值、下一时间步的记忆细胞、输出门和下一时间步的隐状态值
    ft = sigmoid(np.matmul(Wf, concat) + bf)
    it = sigmoid(np.matmul(Wi, concat) + bi)
    cct = np.tanh(np.matmul(Wc, concat) + bc)
    c_next = ft*c_prev + it*cct
    ot = sigmoid(np.matmul(Wo, concat) + bo)
    a_next = ot*np.tanh(c_next)    
    # 计算 LSTM 的预测输出
    yt_pred = softmax(np.matmul(Wy, a_next) + by)    
    # 保存各计算结果值
    cache = (a_next, c_next, a_prev, c_prev, ft, it, cct, ot, xt, parameters)    
    return a_next, c_next, yt_pred, cache

def lstm_forward(x, a0, parameters):    
    """
    Arguments:
    x -- Input data for every time-step, of shape (n_x, m, T_x).
    a0 -- Initial hidden state, of shape (n_a, m)
    parameters -- python dictionary containing:
                 Wf -- Weight matrix of the forget gate, numpy array of shape (n_a, n_a + n_x)
                 bf -- Bias of the forget gate, numpy array of shape (n_a, 1)
                 Wi -- Weight matrix of the update gate, numpy array of shape (n_a, n_a + n_x)
                 bi -- Bias of the update gate, numpy array of shape (n_a, 1)
                 Wc -- Weight matrix of the first "tanh", numpy array of shape (n_a, n_a + n_x)
                 bc -- Bias of the first "tanh", numpy array of shape (n_a, 1)
                 Wo -- Weight matrix of the output gate, numpy array of shape (n_a, n_a + n_x)
                 bo -- Bias of the output gate, numpy array of shape (n_a, 1)
                 Wy -- Weight matrix relating the hidden-state to the output, numpy array of shape (n_y, n_a)
                 by -- Bias relating the hidden-state to the output, numpy array of shape (n_y, 1)
    Returns:
    a -- Hidden states for every time-step, numpy array of shape (n_a, m, T_x)
    y -- Predictions for every time-step, numpy array of shape (n_y, m, T_x)
    caches -- tuple of values needed for the backward pass, contains (list of all the caches, x)
    """
    # 初始化缓存列表
    caches = []    
    # 获取 x 和 参数 Wy 的维度大小
    n_x, m, T_x = x.shape
    n_y, n_a = parameters['Wy'].shape    
    # 初始化 a, c 和 y 的值
    a = np.zeros((n_a, m, T_x))
    c = np.zeros((n_a, m, T_x))
    y = np.zeros((n_y, m, T_x))    
    # 初始化 a_next 和 c_next
    a_next = a0
    c_next = np.zeros(a_next.shape)    
    # 循环所有时间步
    for t in range(T_x):        
    # 更新下一时间步隐状态值、记忆值并计算预测 
        a_next, c_next, yt, cache = lstm_cell_forward(x[:,:,t], a_next, c_next, parameters)        
        # 在 a 中保存新的激活值 
        a[:,:,t] = a_next        
        # 在 a 中保存预测值
        y[:,:,t] = yt        
        # 在 c 中保存记忆值
        c[:,:,t]  = c_next        
        # 添加到缓存列表
        caches.append(cache)    
        # 保存各计算值供反向传播调用
    caches = (caches, x)    
    return a, y, c, caches

def lstm_cell_backward(da_next, dc_next, cache):    
    """
    Arguments:
    da_next -- Gradients of next hidden state, of shape (n_a, m)
    dc_next -- Gradients of next cell state, of shape (n_a, m)
    cache -- cache storing information from the forward pass
    Returns:
    gradients -- python dictionary containing:
     dxt -- Gradient of input data at time-step t, of shape (n_x, m)
     da_prev -- Gradient w.r.t. the previous hidden state, numpy array of shape (n_a, m)
     dc_prev -- Gradient w.r.t. the previous memory state, of shape (n_a, m, T_x)
     dWf -- Gradient w.r.t. the weight matrix of the forget gate, numpy array of shape (n_a, n_a + n_x)
     dWi -- Gradient w.r.t. the weight matrix of the update gate, numpy array of shape (n_a, n_a + n_x)
     dWc -- Gradient w.r.t. the weight matrix of the memory gate, numpy array of shape (n_a, n_a + n_x)
     dWo -- Gradient w.r.t. the weight matrix of the output gate, numpy array of shape (n_a, n_a + n_x)
     dbf -- Gradient w.r.t. biases of the forget gate, of shape (n_a, 1)
     dbi -- Gradient w.r.t. biases of the update gate, of shape (n_a, 1)
     dbc -- Gradient w.r.t. biases of the memory gate, of shape (n_a, 1)
     dbo -- Gradient w.r.t. biases of the output gate, of shape (n_a, 1)
    """
 
    # 获取缓存值
    (a_next, c_next, a_prev, c_prev, ft, it, cct, ot, xt, parameters) = cache    # 获取 xt 和 a_next 的维度大小
    n_x, m = xt.shape
    n_a, m = a_next.shape    
    # 计算各种门的梯度
    dot = da_next * np.tanh(c_next) * ot * (1 - ot)
    dcct = dc_next * it + ot * (1 - np.tanh(c_next) ** 2) * it * da_next * cct * (1 - np.tanh(cct) ** 2)
    dit = dc_next * cct + ot * (1 - np.tanh(c_next) ** 2) * cct * da_next * it * (1 - it)
    dft = dc_next * c_prev + ot * (1 - np.tanh(c_next) ** 2) * c_prev * da_next * ft * (1 - ft)    # 计算各参数的梯度 
    dWf = np.dot(dft, np.concatenate((a_prev, xt), axis=0).T)
    dWi = np.dot(dit, np.concatenate((a_prev, xt), axis=0).T)
    dWc = np.dot(dcct, np.concatenate((a_prev, xt), axis=0).T)
    dWo = np.dot(dot, np.concatenate((a_prev, xt), axis=0).T)
    dbf = np.sum(dft, axis=1, keepdims=True)
    dbi = np.sum(dit, axis=1, keepdims=True)
    dbc = np.sum(dcct, axis=1, keepdims=True)
    dbo = np.sum(dot, axis=1, keepdims=True)
 
    da_prev = np.dot(parameters['Wf'][:,:n_a].T, dft) + np.dot(parameters['Wi'][:,:n_a].T, dit) + np.dot(parameters['Wc'][:,:n_a].T, dcct) + np.dot(parameters['Wo'][:,:n_a].T, dot)
    dc_prev = dc_next*ft + ot*(1-np.square(np.tanh(c_next)))*ft*da_next
    dxt = np.dot(parameters['Wf'][:,n_a:].T,dft)+np.dot(parameters['Wi'][:,n_a:].T,dit)+np.dot(parameters['Wc'][:,n_a:].T,dcct)+np.dot(parameters['Wo'][:,n_a:].T,dot) 
 
    # 将各梯度保存至字典
    gradients = {"dxt": dxt, "da_prev": da_prev, "dc_prev": dc_prev, "dWf": dWf,"dbf": dbf, "dWi": dWi,"dbi": dbi, 
                   "dWc": dWc,"dbc": dbc, "dWo": dWo,"dbo": dbo}    
    return gradients



def lstm_backward(da, caches):    
    """
    Arguments:
    da -- Gradients w.r.t the hidden states, numpy-array of shape (n_a, m, T_x)
    dc -- Gradients w.r.t the memory states, numpy-array of shape (n_a, m, T_x)
    caches -- cache storing information from the forward pass (lstm_forward)
    Returns:
    gradients -- python dictionary containing:
           dx -- Gradient of inputs, of shape (n_x, m, T_x)
           da0 -- Gradient w.r.t. the previous hidden state, numpy array of shape (n_a, m)
           dWf -- Gradient w.r.t. the weight matrix of the forget gate, numpy array of shape (n_a, n_a + n_x)
           dWi -- Gradient w.r.t. the weight matrix of the update gate, numpy array of shape (n_a, n_a + n_x)
           dWc -- Gradient w.r.t. the weight matrix of the memory gate, numpy array of shape (n_a, n_a + n_x)
           dWo -- Gradient w.r.t. the weight matrix of the save gate, numpy array of shape (n_a, n_a + n_x)
           dbf -- Gradient w.r.t. biases of the forget gate, of shape (n_a, 1)
           dbi -- Gradient w.r.t. biases of the update gate, of shape (n_a, 1)
           dbc -- Gradient w.r.t. biases of the memory gate, of shape (n_a, 1)
           dbo -- Gradient w.r.t. biases of the save gate, of shape (n_a, 1)
    """
 
    # 获取第一个缓存值
    (caches, x) = caches
    (a1, c1, a0, c0, f1, i1, cc1, o1, x1, parameters) = caches[0]    # 获取 da 和 x1 的形状大小
    n_a, m, T_x = da.shape
    n_x, m = x1.shape    
    # 初始化各梯度值
    dx = np.zeros((n_x, m, T_x))
    da0 = np.zeros((n_a, m))
    da_prevt = np.zeros((n_a, m))
    dc_prevt = np.zeros((n_a, m))
    dWf = np.zeros((n_a, n_a+n_x))
    dWi = np.zeros((n_a, n_a+n_x))
    dWc = np.zeros((n_a, n_a+n_x))
    dWo = np.zeros((n_a, n_a+n_x))
    dbf = np.zeros((n_a, 1))
    dbi = np.zeros((n_a, 1))
    dbc = np.zeros((n_a, 1))
    dbo = np.zeros((n_a, 1))    
    # 循环各时间步
    for t in reversed(range(T_x)):        
        # 使用 lstm 单元反向传播计算各梯度值
        gradients = lstm_cell_backward(da[:, :, t] + da_prevt, dc_prevt, caches[t])        
        # 保存各梯度值
        dx[:,:,t] = gradients['dxt']
        dWf = dWf + gradients['dWf']
        dWi = dWi + gradients['dWi']
        dWc = dWc + gradients['dWc']
        dWo = dWo + gradients['dWo']
        dbf = dbf + gradients['dbf']
        dbi = dbi + gradients['dbi']
        dbc = dbc + gradients['dbc']
        dbo = dbo + gradients['dbo']
 
    da0 = gradients['da_prev']
 
    gradients = {"dx": dx, "da0": da0, "dWf": dWf,"dbf": dbf, "dWi": dWi,"dbi": dbi,                
    "dWc": dWc,"dbc": dbc, "dWo": dWo,"dbo": dbo}    
    return gradients