文章目錄

作業1：建立你的回圈神經網路
- 1. RNN 前向傳播
- - 1.1 RNN 單元
  - 1.2 RNN 前向傳播
- 2. LSTM 網路
- - 2.1 LSTM 單元
  - 2.2 LSTM 前向傳播
- 3. RNN 反向傳播
- - 3.1 基礎 RNN 反向傳播
  - 3.2 LSTM 反向傳播
  - 3.3 LSTM RNN網路反向傳播
作業2：字符級語言模型：恐龍島
- 1. 問題陳述
- - 1.1 資料集和預處理
  - 1.2 模型預覽
- 2. 構建模塊
- - 2.1 在優化回圈中進行梯度修剪
  - 2.2 采樣
- 3. 建立語言模型
- - 3.1 梯度下降
  - 3.2 訓練模型
- 4. 創作莎士比亞詩歌
作業3：用LSTM網路即興演奏爵士樂獨奏
- 1. 問題陳述
- - 1.1 資料集
  - 1.2 模型預覽

測驗題：參考博文

筆記：05.序列模型 W1.回圈序列模型

作業1：建立你的回圈神經網路

RNN 模型對序列問題（如NLP）非常有效，因為它有記憶，能記住一些資訊，并傳遞至后面的時間步當中

匯入一些包

import numpy as np
from rnn_utils import *

1. RNN 前向傳播

這是一個基本的RNN模型，其輸入輸出等長

1.1 RNN 單元

# GRADED FUNCTION: rnn_cell_forward

def rnn_cell_forward(xt, a_prev, parameters):
    """
    Implements a single forward step of the RNN-cell as described in Figure (2)

    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)
    parameters -- python dictionary containing:
                        Wax -- Weight matrix multiplying the input, numpy array of shape (n_a, n_x)
                        Waa -- Weight matrix multiplying the hidden state, numpy array of shape (n_a, n_a)
                        Wya -- Weight matrix relating the hidden-state to the output, numpy array of shape (n_y, n_a)
                        ba --  Bias, numpy array of shape (n_a, 1)
                        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)
    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, a_prev, xt, parameters)
    """
    
    # Retrieve parameters from "parameters"
    Wax = parameters["Wax"]
    Waa = parameters["Waa"]
    Wya = parameters["Wya"]
    ba = parameters["ba"]
    by = parameters["by"]
    
    ### START CODE HERE ### (≈2 lines)
    # compute next activation state using the formula given above
    # 按公式寫即可
    a_next = np.tanh(np.dot(Wax, xt)+np.dot(Waa, a_prev)+ba)
    # compute output of the current cell using the formula given above
    yt_pred = softmax(np.dot(Wya, a_next)+by) 
    ### END CODE HERE ###
    
    # store values you need for backward propagation in cache
    cache = (a_next, a_prev, xt, parameters)
    
    return a_next, yt_pred, cache

1.2 RNN 前向傳播

把上面的單元重復n次，前一個輸出，作為下一個單元的輸入

# GRADED FUNCTION: rnn_forward

def rnn_forward(x, a0, parameters):
    """
    Implement the forward propagation of the recurrent neural network described in Figure (3).

    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:
                        Waa -- Weight matrix multiplying the hidden state, numpy array of shape (n_a, n_a)
                        Wax -- Weight matrix multiplying the input, numpy array of shape (n_a, n_x)
                        Wya -- Weight matrix relating the hidden-state to the output, numpy array of shape (n_y, n_a)
                        ba --  Bias numpy array of shape (n_a, 1)
                        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_pred -- 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 caches, x)
    """
    
    # Initialize "caches" which will contain the list of all caches
    caches = []
    
    # Retrieve dimensions from shapes of x and Wy
    n_x, m, T_x = x.shape
    n_y, n_a = parameters["Wya"].shape
    
    ### START CODE HERE ###
    
    # initialize "a" and "y" with zeros (≈2 lines)
    a = np.zeros((n_a, m, T_x))
    y_pred = np.zeros((n_y, m, T_x))
    
    # Initialize a_next (≈1 line)
    a_next = a0
    
    # loop over all time-steps
    for t in range(T_x):
        # Update next hidden state, compute the prediction, get the cache (≈1 line)
        a_next, yt_pred, cache = rnn_cell_forward(x[:,:,t], a_next, parameters)
        # Save the value of the new "next" hidden state in a (≈1 line)
        a[:,:,t] = a_next
        # Save the value of the prediction in y (≈1 line)
        y_pred[:,:,t] = yt_pred
        # Append "cache" to "caches" (≈1 line)
        caches.append(cache)
        
    ### END CODE HERE ###
    
    # store values needed for backward propagation in cache
    caches = (caches, x)
    
    return a, y_pred, caches

上面的模型存在梯度消失的問題，預測值是根據區域資訊來預測的

下面我們建立更復雜的 LSTM 模型，它可以更好的解決梯度消失問題，它可以記住一些資訊，并在后序很多步中保留

2. LSTM 網路

forget 門， Γ f < t > ∈ ( 0 , 1 ) \Gamma_f^{<t>} \in (0,1) Γf<t>?∈(0,1)，等于0，就是忘記該資訊
Γ f ? t ? = σ ( W f [ a ? t ? 1 ? , x ? t ? ] + b f ) \Gamma_f^{\langle t \rangle} = \sigma(W_f[a^{\langle t-1 \rangle}, x^{\langle t \rangle}] + b_f) Γf?t??=σ(Wf?[a?t?1?,x?t?]+bf?)
update 門，是否更新當前的資訊至記憶
Γ u ? t ? = σ ( W u [ a ? t ? 1 ? , x { t } ] + b u ) \Gamma_u^{\langle t \rangle} = \sigma(W_u[a^{\langle t-1 \rangle}, x^{\{t\}}] + b_u) Γu?t??=σ(Wu?[a?t?1?,x{t}]+bu?)
更新單元
c ~ ? t ? = tanh ? ( W c [ a ? t ? 1 ? , x ? t ? ] + b c ) \tilde{c}^{\langle t \rangle} = \tanh(W_c[a^{\langle t-1 \rangle}, x^{\langle t \rangle}] + b_c) c~?t?=tanh(Wc?[a?t?1?,x?t?]+bc?)

c ? t ? = Γ f ? t ? ? c ? t ? 1 ? + Γ u ? t ? ? c ~ ? t ? c^{\langle t \rangle} = \Gamma_f^{\langle t \rangle}* c^{\langle t-1 \rangle} + \Gamma_u^{\langle t \rangle} *\tilde{c}^{\langle t \rangle} c?t?=Γf?t???c?t?1?+Γu?t???c~?t?

output 門
Γ o ? t ? = σ ( W o [ a ? t ? 1 ? , x ? t ? ] + b o ) \Gamma_o^{\langle t \rangle}= \sigma(W_o[a^{\langle t-1 \rangle}, x^{\langle t \rangle}] + b_o) Γo?t??=σ(Wo?[a?t?1?,x?t?]+bo?)

a ? t ? = Γ o ? t ? ? tanh ? ( c ? t ? ) a^{\langle t \rangle} = \Gamma_o^{\langle t \rangle}* \tanh(c^{\langle t \rangle}) a?t?=Γo?t???tanh(c?t?)

2.1 LSTM 單元

# GRADED FUNCTION: lstm_cell_forward

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 save gate, numpy array of shape (n_a, n_a + n_x)
                        bi -- Bias of the save 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 focus gate, numpy array of shape (n_a, n_a + n_x)
                        bo --  Bias of the focus 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)
    
    Note: ft/it/ot stand for the forget/update/output gates, cct stands for the candidate value (c tilda),
          c stands for the memory value
    """

    # Retrieve parameters from "parameters"
    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"]
    
    # Retrieve dimensions from shapes of xt and Wy
    n_x, m = xt.shape
    n_y, n_a = Wy.shape

    ### START CODE HERE ###
    # Concatenate a_prev and xt (≈3 lines)
    concat = np.concatenate((a_prev, xt), axis=0)
    concat[: n_a, :] = a_prev
    concat[n_a :, :] = xt

    # Compute values for ft, it, cct, c_next, ot, a_next using the formulas given figure (4) (≈6 lines)
    ft = sigmoid(np.dot(Wf, concat)+bf) # forget 門
    it = sigmoid(np.dot(Wi, concat)+bi) # update 門
    cct = np.tanh(np.dot(Wc, concat)+bc)
    c_next = ft*c_prev + it*cct
    ot = sigmoid(np.dot(Wo, concat)+bo) # output 門
    a_next = ot*np.tanh(c_next)
    
    # Compute prediction of the LSTM cell (≈1 line)
    yt_pred = softmax(np.dot(Wy, a_next)+by)
    ### END CODE HERE ###

    # store values needed for backward propagation in cache
    cache = (a_next, c_next, a_prev, c_prev, ft, it, cct, ot, xt, parameters)

    return a_next, c_next, yt_pred, cache

2.2 LSTM 前向傳播

# GRADED FUNCTION: lstm_forward

def lstm_forward(x, a0, parameters):
    """
    Implement the forward propagation of the recurrent neural network using an LSTM-cell described in Figure (3).

    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 save gate, numpy array of shape (n_a, n_a + n_x)
                        bi -- Bias of the save 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 focus gate, numpy array of shape (n_a, n_a + n_x)
                        bo -- Bias of the focus 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)
    """

    # Initialize "caches", which will track the list of all the caches
    caches = []
    
    ### START CODE HERE ###
    # Retrieve dimensions from shapes of xt and Wy (≈2 lines)
    n_x, m, T_x = x.shape
    n_y, n_a = parameters['Wy'].shape
    
    # initialize "a", "c" and "y" with zeros (≈3 lines)
    a = np.zeros((n_a, m, T_x))
    c = np.zeros((n_a, m, T_x))
    y = np.zeros((n_y, m, T_x))
    
    # Initialize a_next and c_next (≈2 lines)
    a_next = a0
    c_next = np.zeros((n_a, m))
    
    # loop over all time-steps
    for t in range(T_x):
        # Update next hidden state, next memory state, compute the prediction, get the cache (≈1 line)
        a_next, c_next, yt, cache = lstm_cell_forward(x[:,:,t], a_next, c_next, parameters)
        # Save the value of the new "next" hidden state in a (≈1 line)
        a[:,:,t] = a_next
        # Save the value of the prediction in y (≈1 line)
        y[:,:,t] = yt
        # Save the value of the next cell state (≈1 line)
        c[:,:,t]  = c_next
        # Append the cache into caches (≈1 line)
        caches.append(cache)
        
    ### END CODE HERE ###
    
    # store values needed for backward propagation in cache
    caches = (caches, x)

    return a, y, c, caches

3. RNN 反向傳播

深度學習框架一般都會幫你自動實作反向傳播，下面我們來簡要看看

3.1 基礎 RNN 反向傳播

def rnn_cell_backward(da_next, cache):
    """
    Implements the backward pass for the RNN-cell (single time-step).

    Arguments:
    da_next -- Gradient of loss with respect to next hidden state
    cache -- python dictionary containing useful values (output of rnn_step_forward())

    Returns:
    gradients -- python dictionary containing:
                        dx -- Gradients of input data, of shape (n_x, m)
                        da_prev -- Gradients of previous hidden state, of shape (n_a, m)
                        dWax -- Gradients of input-to-hidden weights, of shape (n_a, n_x)
                        dWaa -- Gradients of hidden-to-hidden weights, of shape (n_a, n_a)
                        dba -- Gradients of bias vector, of shape (n_a, 1)
    """
    
    # Retrieve values from cache
    (a_next, a_prev, xt, parameters) = cache
    
    # Retrieve values from parameters
    Wax = parameters["Wax"]
    Waa = parameters["Waa"]
    Wya = parameters["Wya"]
    ba = parameters["ba"]
    by = parameters["by"]

    ### START CODE HERE ###
    # compute the gradient of tanh with respect to a_next (≈1 line)
    dtanh = (1-a_next**2)*da_next

    # compute the gradient of the loss with respect to Wax (≈2 lines)
    dxt = np.dot(Wax.T, dtanh)
    dWax = np.dot(dtanh, xt.T)

    # compute the gradient with respect to Waa (≈2 lines)
    da_prev = np.dot(Waa.T, dtanh)
    dWaa = np.dot(dtanh, a_prev.T)

    # compute the gradient with respect to b (≈1 line)
    dba = np.sum(dtanh, axis=1, keepdims=True)

    ### END CODE HERE ###
    
    # Store the gradients in a python dictionary
    gradients = {"dxt": dxt, "da_prev": da_prev, "dWax": dWax, "dWaa": dWaa, "dba": dba}
    
    return gradients

在整個 RNN 網路上實作反向傳播

def rnn_backward(da, caches):
    """
    Implement the backward pass for a RNN over an entire sequence of input data.

    Arguments:
    da -- Upstream gradients of all hidden states, of shape (n_a, m, T_x)
    caches -- tuple containing information from the forward pass (rnn_forward)
    
    Returns:
    gradients -- python dictionary containing:
                        dx -- Gradient w.r.t. the input data, numpy-array of shape (n_x, m, T_x)
                        da0 -- Gradient w.r.t the initial hidden state, numpy-array of shape (n_a, m)
                        dWax -- Gradient w.r.t the input's weight matrix, numpy-array of shape (n_a, n_x)
                        dWaa -- Gradient w.r.t the hidden state's weight matrix, numpy-arrayof shape (n_a, n_a)
                        dba -- Gradient w.r.t the bias, of shape (n_a, 1)
    """
        
    ### START CODE HERE ###
    
    # Retrieve values from the first cache (t=1) of caches (≈2 lines)
    (caches, x) = caches
    (a1, a0, x1, parameters) = caches[0]
    
    # Retrieve dimensions from da's and x1's shapes (≈2 lines)
    n_a, m, T_x = da.shape
    n_x, m = x1.shape
    
    # initialize the gradients with the right sizes (≈6 lines)
    dx = np.zeros((n_x, m, T_x))
    dWax = np.zeros((n_a, n_x))
    dWaa = np.zeros((n_a, n_a))
    dba = np.zeros((n_a, 1))
    da0 = np.zeros((n_a, m))
    da_prevt = np.zeros((n_a, m))
    
    # Loop through all the time steps
    for t in reversed(range(T_x)):
        # Compute gradients at time step t. Choose wisely the "da_next" and the "cache" to use in the backward propagation step. (≈1 line)
        gradients = rnn_cell_backward(da[:,:,t]+da_prevt, caches[t])
        # Retrieve derivatives from gradients (≈ 1 line)
        dxt, da_prevt, dWaxt, dWaat, dbat = gradients['dxt'],gradients['da_prev'],gradients['dWax'],gradients['dWaa'],gradients['dba']
        # Increment global derivatives w.r.t parameters by adding their derivative at time-step t (≈4 lines)
        dx[:, :, t] = dxt
        dWax = dWax + dWaxt
        dWaa = dWaa + dWaat
        dba = dba + dbat
        
    # Set da0 to the gradient of a which has been backpropagated through all time-steps (≈1 line) 
    da0 = da_prevt
    ### END CODE HERE ###

    # Store the gradients in a python dictionary
    gradients = {"dx": dx, "da0": da0, "dWax": dWax, "dWaa": dWaa,"dba": dba}
    
    return gradients

3.2 LSTM 反向傳播

gate 導數：

d Γ o ? t ? = d a n e x t ? tanh ? ( c n e x t ) ? Γ o ? t ? ? ( 1 ? Γ o ? t ? ) d \Gamma_o^{\langle t \rangle} = da_{next}*\tanh(c_{next}) * \Gamma_o^{\langle t \rangle}*(1-\Gamma_o^{\langle t \rangle}) dΓo?t??=danext??tanh(cnext?)?Γo?t???(1?Γo?t??)

d c ~ ? t ? = d c n e x t ? Γ i ? t ? + Γ o ? t ? ( 1 ? tanh ? ( c n e x t ) 2 ) ? i t ? d a n e x t ? c ~ ? t ? ? ( 1 ? tanh ? ( c ~ ) 2 ) d\tilde c^{\langle t \rangle} = dc_{next}*\Gamma_i^{\langle t \rangle}+ \Gamma_o^{\langle t \rangle} (1-\tanh(c_{next})^2) * i_t * da_{next} * \tilde c^{\langle t \rangle} * (1-\tanh(\tilde c)^2) dc~?t?=dcnext??Γi?t??+Γo?t??(1?tanh(cnext?)2)?it??danext??c~?t??(1?tanh(c~)2)

d Γ u ? t ? = d c n e x t ? c ~ ? t ? + Γ o ? t ? ( 1 ? tanh ? ( c n e x t ) 2 ) ? c ~ ? t ? ? d a n e x t ? Γ u ? t ? ? ( 1 ? Γ u ? t ? ) d\Gamma_u^{\langle t \rangle} = dc_{next}*\tilde c^{\langle t \rangle} + \Gamma_o^{\langle t \rangle} (1-\tanh(c_{next})^2) * \tilde c^{\langle t \rangle} * da_{next}*\Gamma_u^{\langle t \rangle}*(1-\Gamma_u^{\langle t \rangle}) dΓu?t??=dcnext??c~?t?+Γo?t??(1?tanh(cnext?)2)?c~?t??danext??Γu?t???(1?Γu?t??)

d Γ f ? t ? = d c n e x t ? c ~ p r e v + Γ o ? t ? ( 1 ? tanh ? ( c n e x t ) 2 ) ? c p r e v ? d a n e x t ? Γ f ? t ? ? ( 1 ? Γ f ? t ? ) d\Gamma_f^{\langle t \rangle} = dc_{next}*\tilde c_{prev} + \Gamma_o^{\langle t \rangle} (1-\tanh(c_{next})^2) * c_{prev} * da_{next}*\Gamma_f^{\langle t \rangle}*(1-\Gamma_f^{\langle t \rangle}) dΓf?t??=dcnext??c~prev?+Γo?t??(1?tanh(cnext?)2)?cprev??danext??Γf?t???(1?Γf?t??)

parameter 導數

d W f = d Γ f ? t ? ? ( a p r e v x t ) T dW_f = d\Gamma_f^{\langle t \rangle} * \begin{pmatrix} a_{prev} \\ x_t\end{pmatrix}^T dWf?=dΓf?t???(aprev?xt??)T

d W u = d Γ u ? t ? ? ( a p r e v x t ) T dW_u = d\Gamma_u^{\langle t \rangle} * \begin{pmatrix} a_{prev} \\ x_t\end{pmatrix}^T dWu?=dΓu?t???(aprev?xt??)T

d W c = d c ~ ? t ? ? ( a p r e v x t ) T dW_c = d\tilde c^{\langle t \rangle} * \begin{pmatrix} a_{prev} \\ x_t\end{pmatrix}^T dWc?=dc~?t??(aprev?xt??)T

d W o = d Γ o ? t ? ? ( a p r e v x t ) T dW_o = d\Gamma_o^{\langle t \rangle} * \begin{pmatrix} a_{prev} \\ x_t\end{pmatrix}^T dWo?=dΓo?t???(aprev?xt??)T

為了計算 d b f , d b u , d b c , d b o db_f, db_u, db_c, db_o dbf?,dbu?,dbc?,dbo? ，在 d Γ f ? t ? , d Γ u ? t ? , d c ~ ? t ? , d Γ o ? t ? d\Gamma_f^{\langle t \rangle}, d\Gamma_u^{\langle t \rangle}, d\tilde c^{\langle t \rangle}, d\Gamma_o^{\langle t \rangle} dΓf?t??,dΓu?t??,dc~?t?,dΓo?t?? 水平軸 (axis= 1)上求和，注意keep_dims = True

d a p r e v = W f T ? d Γ f ? t ? + W u T ? d Γ u ? t ? + W c T ? d c ~ ? t ? + W o T ? d Γ o ? t ? da_{prev} = W_f^T*d\Gamma_f^{\langle t \rangle} + W_u^T * d\Gamma_u^{\langle t \rangle}+ W_c^T * d\tilde c^{\langle t \rangle} + W_o^T * d\Gamma_o^{\langle t \rangle} daprev?=WfT??dΓf?t??+WuT??dΓu?t??+WcT??dc~?t?+WoT??dΓo?t??

d c p r e v = d c n e x t Γ f ? t ? + Γ o ? t ? ? ( 1 ? tanh ? ( c n e x t ) 2 ) ? Γ f ? t ? ? d a n e x t dc_{prev} = dc_{next}\Gamma_f^{\langle t \rangle} + \Gamma_o^{\langle t \rangle} * (1- \tanh(c_{next})^2)*\Gamma_f^{\langle t \rangle}*da_{next} dcprev?=dcnext?Γf?t??+Γo?t???(1?tanh(cnext?)2)?Γf?t???danext?

d x ? t ? = W f T ? d Γ f ? t ? + W u T ? d Γ u ? t ? + W c T ? d c ~ t + W o T ? d Γ o ? t ? dx^{\langle t \rangle} = W_f^T*d\Gamma_f^{\langle t \rangle} + W_u^T * d\Gamma_u^{\langle t \rangle}+ W_c^T * d\tilde c_t + W_o^T * d\Gamma_o^{\langle t \rangle} dx?t?=WfT??dΓf?t??+WuT??dΓu?t??+WcT??dc~t?+WoT??dΓo?t??
注：感覺上面的公式跟正確答案的代碼有點對不上，

def lstm_cell_backward(da_next, dc_next, cache):
    """
    Implement the backward pass for the LSTM-cell (single time-step).

    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 input 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)
    """

    # Retrieve information from "cache"
    (a_next, c_next, a_prev, c_prev, ft, it, cct, ot, xt, parameters) = cache
    
    ### START CODE HERE ###
    # Retrieve dimensions from xt's and a_next's shape (≈2 lines)
    n_x, m = xt.shape
    n_a, m = a_next.shape
    
    # Compute gates related derivatives, you can find their values can be found by looking carefully at equations (7) to (10) (≈4 lines)
    dot = da_next*np.tanh(c_next)*ot*(1-ot)
    dcct = (dc_next*it+ot*(1-np.tanh(c_next)**2)*it*da_next)*(1-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)
    
    # Compute parameters related derivatives. Use equations (11)-(14) (≈8 lines)
    concat = np.concatenate((a_prev, xt), axis=0)
    dWf = np.dot(dft,concat.T)
    dWi = np.dot(dit,concat.T)
    dWc = np.dot(dcct,concat.T)
    dWo = np.dot(dot,concat.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)

    # Compute derivatives w.r.t previous hidden state, previous memory state and input. Use equations (15)-(17). (≈3 lines)
    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.tanh(c_next)**2)*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)
    ### END CODE HERE ###
    
    # Save gradients in dictionary
    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

3.3 LSTM RNN網路反向傳播

def lstm_backward(da, caches):
    
    """
    Implement the backward pass for the RNN with LSTM-cell (over a whole sequence).

    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)
    """

    # Retrieve values from the first cache (t=1) of caches.
    (caches, x) = caches
    (a1, c1, a0, c0, f1, i1, cc1, o1, x1, parameters) = caches[0]
    
    ### START CODE HERE ###
    # Retrieve dimensions from da's and x1's shapes (≈2 lines)
    n_a, m, T_x = da.shape
    n_x, m = x1.shape
    
    # initialize the gradients with the right sizes (≈12 lines)
    dx = np.zeros([n_x, m, T_x])
    da0 = np.zeros([n_a, m])
    da_prevt = np.zeros([n_a, 1])
    dc_prevt = np.zeros([n_a, 1])
    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])
    
    # loop back over the whole sequence
    for t in reversed(range(T_x)):
        # Compute all gradients using lstm_cell_backward
        gradients = lstm_cell_backward(da[:,:,t], dc_prevt, caches[t])
        # da_prevt, dc_prevt = gradients['da_prev'], gradients["dc_prev"]
        # Store or add the gradient to the parameters' previous step's gradient
        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']
    # Set the first activation's gradient to the backpropagated gradient da_prev.
    da0 = gradients['da_prev']
    
    ### END CODE HERE ###

    # Store the gradients in a python dictionary
    gradients = {"dx": dx, "da0": da0, "dWf": dWf,"dbf": dbf, "dWi": dWi,"dbi": dbi,
                "dWc": dWc,"dbc": dbc, "dWo": dWo,"dbo": dbo}
    
    return gradients

作業2：字符級語言模型：恐龍島

恐龍回歸了，你要給恐龍命名，你的助手收集了他們能找到的所有恐龍名稱的串列，并將它們編譯到這個資料集中，

要創建新的恐龍名稱，您將構建一個字符級語言模型來生成新名稱，您的演算法將學習不同的名稱模式，并隨機生成新的名稱，

通過完成這項作業，你將學到：

如何存盤文本資料以使用RNN進行處理
如何合成資料，通過在每個時間步采樣預測并將其傳遞給下一個RNN單元
如何建立字符級文本生成RNN網路
為什么梯度修剪很重要

加載一些包

import numpy as np
from utils import *
import random
from random import shuffle

1. 問題陳述

1.1 資料集和預處理

data = open('dinos.txt', 'r').read()
data= data.lower()
chars = list(set(data))
data_size, vocab_size = len(data), len(chars)
print('There are %d total characters and %d unique characters in your data.' % (data_size, vocab_size))

輸出：

There are 19909 total characters and 27 unique characters in your data.

所有恐龍的名字有 26個唯一的字母，還有\n

建立 字符：數字 哈希映射關系

char_to_ix = { ch:i for i,ch in enumerate(sorted(chars)) }
ix_to_char = { i:ch for i,ch in enumerate(sorted(chars)) }
print(ix_to_char)
print(char_to_ix)

輸出：

{0: '\n', 1: 'a', 2: 'b', 3: 'c', 4: 'd', 5: 'e', 6: 'f', 7: 'g', 8: 'h', 9: 'i', 10: 'j', 11: 'k', 12: 'l', 13: 'm', 14: 'n', 15: 'o', 16: 'p', 17: 'q', 18: 'r', 19: 's', 20: 't', 21: 'u', 22: 'v', 23: 'w', 24: 'x', 25: 'y', 26: 'z'}

{'\n': 0, 'a': 1, 'b': 2, 'c': 3, 'd': 4, 'e': 5, 'f': 6, 'g': 7, 'h': 8, 'i': 9, 'j': 10, 'k': 11, 'l': 12, 'm': 13, 'n': 14, 'o': 15, 'p': 16, 'q': 17, 'r': 18, 's': 19, 't': 20, 'u': 21, 'v': 22, 'w': 23, 'x': 24, 'y': 25, 'z': 26}

1.2 模型預覽

模型結構：

初始化引數
運行優化回圈
1.前向傳播計算損失
2.反向傳播計算對應的梯度
3.梯度修剪，防止梯度爆炸
4.使用梯度更新引數
回傳學習到的引數

2. 構建模塊

模塊1：梯度修剪，防止梯度爆炸
模塊2：采樣，生成字符

2.1 在優化回圈中進行梯度修剪

在更新引數之前，先對梯度進行修剪，限制在一定的大小范圍內，對不在范圍內的取最近的區間端點值

numpy.clip(a, a_min, a_max, out=None)https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.clip.html

### GRADED FUNCTION: clip

def clip(gradients, maxValue):
    '''
    Clips the gradients' values between minimum and maximum.
    
    Arguments:
    gradients -- a dictionary containing the gradients "dWaa", "dWax", "dWya", "db", "dby"
    maxValue -- everything above this number is set to this number, and everything less than -maxValue is set to -maxValue
    
    Returns: 
    gradients -- a dictionary with the clipped gradients.
    '''
    
    dWaa, dWax, dWya, db, dby = gradients['dWaa'], gradients['dWax'], gradients['dWya'], gradients['db'], gradients['dby']
   
    ### START CODE HERE ###
    # clip to mitigate exploding gradients, loop over [dWax, dWaa, dWya, db, dby]. (≈2 lines)
    for gradient in [dWax, dWaa, dWya, db, dby]:
        np.clip(gradient, -maxValue, maxValue, out=gradient)
    ### END CODE HERE ###
    
    gradients = {"dWaa": dWaa, "dWax": dWax, "dWya": dWya, "db": db, "dby": dby}
    
    return gradients

2.2 采樣

假設你的模型已經訓練好了，你要生成新的文本（字符）

步驟：

給模型一個虛擬的輸入 x ? 1 ? = 0 ? x^{\langle 1 \rangle} = \vec{0} x?1?=0 ， a ? 0 ? = 0 ? a^{\langle 0 \rangle} = \vec{0} a?0?=0
運行一次前向傳播，得到 a ? 1 ? a^{\langle 1 \rangle} a?1? 和 y ^ ? 1 ? \hat{y}^{\langle 1 \rangle} y^??1?

a ? t + 1 ? = tanh ? ( W a x x ? t ? + W a a a ? t ? + b ) a^{\langle t+1 \rangle} = \tanh(W_{ax} x^{\langle t \rangle } + W_{aa} a^{\langle t \rangle } + b) a?t+1?=tanh(Wax?x?t?+Waa?a?t?+b)

z ? t + 1 ? = W y a a ? t + 1 ? + b y z^{\langle t + 1 \rangle } = W_{ya} a^{\langle t + 1 \rangle } + b_y z?t+1?=Wya?a?t+1?+by?

y ^ ? t + 1 ? = s o f t m a x ( z ? t + 1 ? ) \hat{y}^{\langle t+1 \rangle } = softmax(z^{\langle t + 1 \rangle }) y^??t+1?=softmax(z?t+1?)

根據 y ^ ? t + 1 ? \hat{y}^{\langle t+1 \rangle } y^??t+1? 的概率分布選擇一個字符，可以使用 np.random.choice

一個例子：

np.random.seed(0)
p = np.array([0.1, 0.0, 0.7, 0.2])
index = np.random.choice([0, 1, 2, 3], p = p.ravel())

使用 ont-hot 編碼后的 x ? t + 1 ? x^{\langle t + 1 \rangle } x?t+1? 寫入 x x x ，前向傳播 x ? t + 1 ? x^{\langle t + 1 \rangle } x?t+1? 直到遇見\n（EOS 結束標志）

# GRADED FUNCTION: sample

def sample(parameters, char_to_ix, seed):
    """
    Sample a sequence of characters according to a sequence of probability distributions output of the RNN

    Arguments:
    parameters -- python dictionary containing the parameters Waa, Wax, Wya, by, and b. 
    char_to_ix -- python dictionary mapping each character to an index.
    seed -- used for grading purposes. Do not worry about it.

    Returns:
    indices -- a list of length n containing the indices of the sampled characters.
    """
    
    # Retrieve parameters and relevant shapes from "parameters" dictionary
    Waa, Wax, Wya, by, b = parameters['Waa'], parameters['Wax'], parameters['Wya'], parameters['by'], parameters['b']
    vocab_size = by.shape[0]
    n_a = Waa.shape[1]
    
    ### START CODE HERE ###
    # Step 1: Create the one-hot vector x for the first character (initializing the sequence generation). (≈1 line)
    x = np.zeros((vocab_size, 1))
    # Step 1': Initialize a_prev as zeros (≈1 line)
    a_prev = np.zeros((n_a, 1))
    
    # Create an empty list of indices, this is the list which will contain the list of indices of the characters to generate (≈1 line)
    indices = []
    
    # Idx is a flag to detect a newline character, we initialize it to -1
    idx = -1 
    
    # Loop over time-steps t. At each time-step, sample a character from a probability distribution and append 
    # its index to "indices". We'll stop if we reach 50 characters (which should be very unlikely with a well 
    # trained model), which helps debugging and prevents entering an infinite loop. 
    counter = 0
    newline_character = char_to_ix['\n']
    
    while (idx != newline_character and counter != 50):
        
        # Step 2: Forward propagate x using the equations (1), (2) and (3)
        a = np.tanh(np.dot(Wax, x)+np.dot(Waa, a_prev)+b)
        z = np.dot(Wya, a)+by
        y = softmax(z)
        
        # for grading purposes
        np.random.seed(counter+seed) 
        
        # Step 3: Sample the index of a character within the vocabulary from the probability distribution y
        idx = np.random.choice(list(range(vocab_size)), p = y.ravel())

        # Append the index to "indices"
        indices.append(idx)
        
        # Step 4: Overwrite the input character as the one corresponding to the sampled index.
        x = np.zeros((vocab_size, 1))
        x[idx] = 1
        
        # Update "a_prev" to be "a"
        a_prev = a
        
        # for grading purposes
        seed += 1
        counter +=1
        
    ### END CODE HERE ###

    if (counter == 50):
        indices.append(char_to_ix['\n'])
    
    return indices

3. 建立語言模型

3.1 梯度下降

已經寫好的函式：

def rnn_forward(X, Y, a_prev, parameters):
    """ Performs the forward propagation through the RNN and computes the cross-entropy loss.
    It returns the loss' value as well as a "cache" storing values to be used in the backpropagation."""
    ....
    return loss, cache

def rnn_backward(X, Y, parameters, cache):
    """ Performs the backward propagation through time to compute the gradients of the loss with respect
    to the parameters. It returns also all the hidden states."""
    ...
    return gradients, a

def update_parameters(parameters, gradients, learning_rate):
    """ Updates parameters using the Gradient Descent Update Rule."""
    ...
    return parameters

優化程序如下：

# GRADED FUNCTION: optimize

def optimize(X, Y, a_prev, parameters, learning_rate = 0.01):
    """
    Execute one step of the optimization to train the model.
    
    Arguments:
    X -- list of integers, where each integer is a number that maps to a character in the vocabulary.
    Y -- list of integers, exactly the same as X but shifted one index to the left.
    a_prev -- previous hidden state.
    parameters -- python dictionary containing:
                        Wax -- Weight matrix multiplying the input, numpy array of shape (n_a, n_x)
                        Waa -- Weight matrix multiplying the hidden state, numpy array of shape (n_a, n_a)
                        Wya -- Weight matrix relating the hidden-state to the output, numpy array of shape (n_y, n_a)
                        b --  Bias, numpy array of shape (n_a, 1)
                        by -- Bias relating the hidden-state to the output, numpy array of shape (n_y, 1)
    learning_rate -- learning rate for the model.
    
    Returns:
    loss -- value of the loss function (cross-entropy)
    gradients -- python dictionary containing:
                        dWax -- Gradients of input-to-hidden weights, of shape (n_a, n_x)
                        dWaa -- Gradients of hidden-to-hidden weights, of shape (n_a, n_a)
                        dWya -- Gradients of hidden-to-output weights, of shape (n_y, n_a)
                        db -- Gradients of bias vector, of shape (n_a, 1)
                        dby -- Gradients of output bias vector, of shape (n_y, 1)
    a[len(X)-1] -- the last hidden state, of shape (n_a, 1)
    """
    
    ### START CODE HERE ###
    
    # Forward propagate through time (≈1 line)
    loss, cache = rnn_forward(X,Y,a_prev,parameters)
    
    # Backpropagate through time (≈1 line)
    gradients, a = rnn_backward(X,Y,parameters,cache)
    
    # Clip your gradients between -5 (min) and 5 (max) (≈1 line)
    gradients = clip(gradients, maxValue=5)
    
    # Update parameters (≈1 line)
    parameters = update_parameters(parameters, gradients, learning_rate)
    
    ### END CODE HERE ###
    
    return loss, gradients, a[len(X)-1]

3.2 訓練模型

給定恐龍名稱的資料集，使用資料集的每一行（一個名稱）作為一個訓練樣本，
每100步隨機梯度下降，抽樣10個隨機選擇的名字，看看演算法是如何做的，記住隨機打亂資料集

當樣本包含一個恐龍的名字時，創建訓練樣本 ( X , Y ) (X,Y) (X,Y) ：

index = j % len(examples)
X = [None] + [char_to_ix[ch] for ch in examples[index]] 
Y = X[1:] + [char_to_ix["\n"]]

Y 跟 X 一樣，但是往左偏移了 1 位，最后加了一個結束符\n

# GRADED FUNCTION: model

def model(data, ix_to_char, char_to_ix, num_iterations = 35000, n_a = 50, dino_names = 7, vocab_size = 27):
    """
    Trains the model and generates dinosaur names. 
    
    Arguments:
    data -- text corpus
    ix_to_char -- dictionary that maps the index to a character
    char_to_ix -- dictionary that maps a character to an index
    num_iterations -- number of iterations to train the model for
    n_a -- number of units of the RNN cell
    dino_names -- number of dinosaur names you want to sample at each iteration. 
    vocab_size -- number of unique characters found in the text, size of the vocabulary
    
    Returns:
    parameters -- learned parameters
    """
    
    # Retrieve n_x and n_y from vocab_size
    n_x, n_y = vocab_size, vocab_size
    
    # Initialize parameters
    parameters = initialize_parameters(n_a, n_x, n_y)
    
    # Initialize loss (this is required because we want to smooth our loss, don't worry about it)
    loss = get_initial_loss(vocab_size, dino_names)
    
    # Build list of all dinosaur names (training examples).
    with open("dinos.txt") as f:
        examples = f.readlines()
    examples = [x.lower().strip() for x in examples]
    
    # Shuffle list of all dinosaur names
    shuffle(examples)
    
    # Initialize the hidden state of your LSTM
    a_prev = np.zeros((n_a, 1))
    
    # Optimization loop
    for j in range(num_iterations):
        
        ### START CODE HERE ###
        
        # Use the hint above to define one training example (X,Y) (≈ 2 lines)
        index = j%len(examples)
        X = [None]+[char_to_ix[ch] for ch in examples[index]]
        Y = X[1:] + [char_to_ix['\n']]
        
        # Perform one optimization step: Forward-prop -> Backward-prop -> Clip -> Update parameters
        # Choose a learning rate of 0.01
        curr_loss, gradients, a_prev = optimize(X,Y,a_prev,parameters,learning_rate=0.01)
        
        ### END CODE HERE ###
        
        # Use a latency trick to keep the loss smooth. It happens here to accelerate the training.
        loss = smooth(loss, curr_loss)

        # Every 2000 Iteration, generate "n" characters thanks to sample() to check if the model is learning properly
        if j % 2000 == 0:
            
            print('Iteration: %d, Loss: %f' % (j, loss) + '\n')
            
            # The number of dinosaur names to print
            seed = 0
            for name in range(dino_names):
                
                # Sample indices and print them
                sampled_indices = sample(parameters, char_to_ix, seed)
                print_sample(sampled_indices, ix_to_char)
                
                seed += 1  # To get the same result for grading purposed, increment the seed by one. 
      
            print('\n')
        
    return parameters

運行模型

parameters = model(data, ix_to_char, char_to_ix)

您應該觀察模型在第一次迭代中輸出隨機字符，
在幾千次迭代之后，模型應該學會生成看起來合理的名稱，

后續生成的大部分帶有osaurus后綴（拉丁詞根，蜥蜴類的）

Iteration: 0, Loss: 23.093929

Nkzxwtdmfqoeyhsqwasjjjvu
Kneb
Kzxwtdmfqoeyhsqwasjjjvu
Neb
Zxwtdmfqoeyhsqwasjjjvu
Eb
Xwtdmfqoeyhsqwasjjjvu


Iteration: 2000, Loss: 27.865115

Livtos
Hnba
Iwtos
Lca
Xuscandorawhus
Ba
Tos


Iteration: 4000, Loss: 25.632137

Livosaqrasaurus
Imacaipqia
Iwtosaurus
Lebagosan
Xusiangopdtipos
Acaipon
Torangosaurus


Iteration: 6000, Loss: 24.694657

Mhytosaurus
Imacaesaurus
Iustolmascatarosaurus
Macagptoia
Wustandosaurus
Baaerpe
Stoimatonyirosaurus


Iteration: 8000, Loss: 24.138770

Nhyusicheoravfpsadrenitochustelanfetalkang
Klecalosaurus
Lyusodomophxgshuaomimus
Ngaagosaurus
Xutognatoptkoroclingos
Eeahosaurus
Troenatoptloroclingos


Iteration: 10000, Loss: 23.604738

Ngyusichaosaurus
Inecamosaurus
Kytrodoninaweosanqosaurosaurus
Ncaadosaurus
Xustangosaurus
Caadosaurus
Trocheosaurus


Iteration: 12000, Loss: 23.576294

Mivustandosaurus
Inceaeus
Jyustandorix
Macacitadantithinviceyalosaurus
Xustanesaurus
Cabarsan
Trrangosaurus


Iteration: 14000, Loss: 23.446216

Ngyrosaurus
Kiecanosaurus
Lyuroknesaurus
Nebairopadrus
Xusrangpreusaurus
Daahosaurus
Torangosaurus


Iteration: 16000, Loss: 23.113554

Mewtosaurus
Inedahosaurus
Iwtroceplocuriosaurus
Macamosaurus
Xustangriasaurus
Cabarpelarops
Troceratosaurus


Iteration: 18000, Loss: 23.254092

Mevutoneosaurus
Inecaltona
Kyutollessaurus
Macaisteialus
Xustarchulultitan
Caaerta
Trodicticurotoknathus


Iteration: 20000, Loss: 23.110590

Onwutonganmaurosaurus
Lkehalosaurus
Lyutolidon
Omaakrong
Xwuterasaurus
Daakosaurus
Trokianlaus


Iteration: 22000, Loss: 22.879895

Lixsopelisaurus
Indaaerosaurus
Iwuskanesaurus
Lecacosaurus
Yuusangosaurus
Ccacosaurus
Trochenoguchosaurus


Iteration: 24000, Loss: 22.836100

Miwtosaurus
Kidiabrong
Lyuspangtomuqusgarihisialopupia
Macalosaurus
Ywurophosaurus
Edalosaurus
Tyrhimosaurus


Iteration: 26000, Loss: 22.734218

Levotolia
Ilaca
Kyusolegosaurus
Lacacisaurus
Wstrasaurus
Caaeosaurus
Surapignaveratapaldys


Iteration: 28000, Loss: 22.750129

Piwustaorathus
Ligabiskia
Lyvusaurus
Pecalosaurus
Xutolomisaurus
Egaiskia
Trocibisaurus


Iteration: 30000, Loss: 22.524480

Lixusaurus
Hicaaeros
Ivrpolopopaudus
Lebairus
Xuromelosaurus
Baaishaecitaurus
Surciinidus


Iteration: 32000, Loss: 22.514697

Mgxusoconltfus
Kiceadosaurus
Lyusteodon
Ngaberopa
Wusteodon
Cabbqukaclus
Surangosaurus


Iteration: 34000, Loss: 22.639142

Llytrodon
Ingaaeropechus
Ivstonnatopulorocophisairus
Lecagosaurus
Xusudolosaurus
Caadosaurus
Surangosaurus

結論：

可以看到，演算法已經開始產生可信的恐龍名字接近訓練結束
起初，它是生成隨機字符，但到最后你可以看到恐龍的名字有很酷的結尾
模型也了解到恐龍的名字往往以saurus(蜥蜴)、don、aura、tor等結尾

4. 創作莎士比亞詩歌

你可以使用莎士比亞詩集，而不是從恐龍名字的資料集中學習，使用 LSTM 單元，你可以學習更長的依賴關系跨越很多字符

from __future__ import print_function
from keras.callbacks import LambdaCallback
from keras.models import Model, load_model, Sequential
from keras.layers import Dense, Activation, Dropout, Input, Masking
from keras.layers import LSTM
from keras.utils.data_utils import get_file
from keras.preprocessing.sequence import pad_sequences
from shakespeare_utils import *
import sys
import io

模型已經是訓練好的，把這個模型再訓練一個時代，當它完成時，可以運行generate_output，它將提示您輸入（<40個字符），這首詩將從你的句子開始，模型將為你完成這首詩的剩余部分！

print_callback = LambdaCallback(on_epoch_end=on_epoch_end)

model.fit(x, y, batch_size=128, epochs=1, callbacks=[print_callback])

# Run this cell to try with different inputs without having to re-train the model 
generate_output()

輸出：
Write the beginning of your poem, the Shakespeare machine will complete it. Your input is:

我輸入love is forever

我輸入love is forever （加一個空格）

Keras Team’s text generation https://github.com/keras-team/keras/blob/master/examples/lstm_text_generation.py

作業3：用LSTM網路即興演奏爵士樂獨奏

注意：pip install music21 安裝這個包

from __future__ import print_function
import IPython
import sys
from music21 import *
import numpy as np
from grammar import *
from qa import *
from preprocess import * 
from music_utils import *
from data_utils import *
from keras.models import load_model, Model
from keras.layers import Dense, Activation, Dropout, Input, LSTM, Reshape, Lambda, RepeatVector
from keras.initializers import glorot_uniform
from keras.utils import to_categorical
from keras.optimizers import Adam
from keras import backend as K

1. 問題陳述

你要給朋友過生日，你想創作一段音樂，但是你不懂音樂，你要使用 LSTM RNN 生成音樂

1.1 資料集

聽一下這段音樂

IPython.display.Audio('./data/30s_seq.mp3')

我們的音樂生成系統將使用 78 個獨特的值(聲調)，運行以下代碼來加載原始音樂資料并將其預處理為數字

X, Y, n_values, indices_values = load_music_utils()
print('shape of X:', X.shape)
print('number of training examples:', X.shape[0])
print('Tx (length of sequence):', X.shape[1])
print('total # of unique values:', n_values)
print('Shape of Y:', Y.shape)

輸出：

shape of X: (60, 30, 78)
number of training examples: 60
Tx (length of sequence): 30
total # of unique values: 78
Shape of Y: (30, 60, 78)

X：維度 ( m , T x , 78 ) (m, T_x,78) (m,Tx?,78) m 個樣本，每個樣本有30個音樂值，每個值用 78 維的 one-hot 編碼表示
Y：跟 X 一樣，向左移動了一步，維度重塑為 ( T y , m , 78 ) (T_y, m, 78) (Ty?,m,78) ， T y = T x T_y=T_x Ty?=Tx?，方便給 LSTM 喂資料
n_values：資料集里獨立的編碼個數：78
indices_values：編碼字典映射序號，0-77

1.2 模型預覽

使用 64 維隱藏狀態的 LSTM

n_a = 64

LSTM 參考 https://keras.io/zh/layers/recurrent/#lstm

Dense 參考 https://keras.io/zh/layers/core/#dense

reshapor = Reshape((1, 78))                        # Used in Step 2.B of djmodel(), below
LSTM_cell = LSTM(n_a, return_state = True)         # Used in Step 2.C
densor = Dense(n_values, activation='softmax')     # Used in Step 2.D

實作djmodel()步驟：

創建空的 list output 存盤每個時間步的 LSTM 單元
for 回圈 t ∈ [ 1 , T x ] t \in [1,T_x] t∈[1,Tx?]
A. 從X里選擇第 i 個時間步向量，x = Lambda(lambda x: x[:,t,:])(X)
B. reshape x 為（1，78）使用 layer 物件 reshapor = Reshape((1, 78))
C. 運行 x 經過一步LSTM單元，記住用前一步的隱藏層狀態 a 和 cell 狀態 c 初始化 LSTM單元：a, _, c = LSTM_cell(input_x, initial_state=[previous hidden state, previous cell state])
D. 使用 dense + softmax 得到激活輸出
E. 記錄預測值到outputs

# GRADED FUNCTION: djmodel
 
def djmodel(Tx, n_a, n_values):
    """
    Implement the model
    
    Arguments:
    Tx -- length of the sequence in a corpus
    n_a -- the number of activations used in our model
    n_values -- number of unique values in the music data 
    
    Returns:
    model -- a keras model with the 
    """
    
    # Define the input of your model with a shape 
    X = Input(shape=(Tx, n_values))
    
    # Define s0, initial hidden state for the decoder LSTM
    a0 = Input(shape=(n_a,), name='a0')
    c0 = Input(shape=(n_a,), name='c0')
    a = a0
    c = c0
    
    ### START CODE HERE ### 
    # Step 1: Create empty list to append the outputs while you iterate (≈1 line)
    outputs = []
    
    # Step 2: Loop
    for t in range(Tx):
        
        # Step 2.A: select the "t"th time step vector from X. 
        x = Lambda(lambda x: x[:,t,:])(X)
        # Step 2.B: Use reshapor to reshape x to be (1, n_values) (≈1 line)
        x = reshapor(x)
        # Step 2.C: Perform one step of the LSTM_cell
        a, _, c = LSTM_cell(x, initial_state=[a, c])
        # Step 2.D: Apply densor to the hidden state output of LSTM_Cell
        out = densor(a)
        # Step 2.E: add the output to "outputs"
        outputs.append(out)
        
    # Step 3: Create model instance
    model = Model(inputs=[X, a0, c0], outputs=outputs)
    
    ### END CODE HERE ###
    
    return model

這段測驗，一直報錯，過不去，也找不到原因，，，

model = djmodel(Tx = 30 , n_a = 64, n_values = 78)

報錯：

LinAlgError                               Traceback (most recent call last)
<ipython-input-7-57eb2d19469c> in <module>
----> 1 model = djmodel(Tx = 30 , n_a = 64, n_values = 78)

<ipython-input-6-7a17ca9b5b35> in djmodel(Tx, n_a, n_values)
     35         x = reshapor(x)
     36         # Step 2.C: Perform one step of the LSTM_cell
---> 37         a, _, c = LSTM_cell(x, initial_state=[a, c])
     38         # Step 2.D: Apply densor to the hidden state output of LSTM_Cell
     39         out = densor(a)

c:\program files\python37\lib\site-packages\keras\layers\recurrent.py in __call__(self, inputs, initial_state, constants, **kwargs)
    582             if 'constants' in kwargs:
    583                 kwargs.pop('constants')
--> 584             output = super(RNN, self).__call__(full_input, **kwargs)
    585             self.input_spec = original_input_spec
    586             return output

c:\program files\python37\lib\site-packages\keras\engine\base_layer.py in __call__(self, inputs, **kwargs)
    461                                          'You can build it manually via: '
    462                                          '`layer.build(batch_input_shape)`')
--> 463                 self.build(unpack_singleton(input_shapes))
    464                 self.built = True
    465 

c:\program files\python37\lib\site-packages\keras\layers\recurrent.py in build(self, input_shape)
    500                 self.cell.build([step_input_shape] + constants_shape)
    501             else:
--> 502                 self.cell.build(step_input_shape)
    503 
    504         # set or validate state_spec

c:\program files\python37\lib\site-packages\keras\layers\recurrent.py in build(self, input_shape)
   1923             initializer=self.recurrent_initializer,
   1924             regularizer=self.recurrent_regularizer,
-> 1925             constraint=self.recurrent_constraint)
   1926 
   1927         if self.use_bias:

c:\program files\python37\lib\site-packages\keras\engine\base_layer.py in add_weight(self, name, shape, dtype, initializer, regularizer, trainable, constraint)
    277         if dtype is None:
    278             dtype = self.dtype
--> 279         weight = K.variable(initializer(shape, dtype=dtype),
    280                             dtype=dtype,
    281                             name=name,

c:\program files\python37\lib\site-packages\keras\initializers.py in __call__(self, shape, dtype)
    266             self.seed += 1
    267         a = rng.normal(0.0, 1.0, flat_shape)
--> 268         u, _, v = np.linalg.svd(a, full_matrices=False)
    269         # Pick the one with the correct shape.
    270         q = u if u.shape == flat_shape else v

<__array_function__ internals> in svd(*args, **kwargs)

c:\program files\python37\lib\site-packages\numpy\linalg\linalg.py in svd(a, full_matrices, compute_uv, hermitian)
   1624 
   1625         signature = 'D->DdD' if isComplexType(t) else 'd->ddd'
-> 1626         u, s, vh = gufunc(a, signature=signature, extobj=extobj)
   1627         u = u.astype(result_t, copy=False)
   1628         s = s.astype(_realType(result_t), copy=False)

c:\program files\python37\lib\site-packages\numpy\linalg\linalg.py in _raise_linalgerror_svd_nonconvergence(err, flag)
    104 
    105 def _raise_linalgerror_svd_nonconvergence(err, flag):
--> 106     raise LinAlgError("SVD did not converge")
    107 
    108 def _raise_linalgerror_lstsq(err, flag):

LinAlgError: SVD did not converge

這個生成音樂先不做了，繼續學習，如果有相同錯誤的小伙伴解決了，記得在下面留言告知方法，多謝了！

我的CSDN博客地址 https://michael.blog.csdn.net/

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05.序列模型 W1.回圈序列模型（作業：手寫RNN+恐龍名字生成）

文章目錄

作業1：建立你的回圈神經網路

1. RNN 前向傳播

1.1 RNN 單元

1.2 RNN 前向傳播

2. LSTM 網路

2.1 LSTM 單元

2.2 LSTM 前向傳播

3. RNN 反向傳播

3.1 基礎 RNN 反向傳播

3.2 LSTM 反向傳播

3.3 LSTM RNN網路反向傳播

作業2：字符級語言模型：恐龍島

1. 問題陳述

1.1 資料集和預處理

1.2 模型預覽

2. 構建模塊

2.1 在優化回圈中進行梯度修剪

2.2 采樣

3. 建立語言模型

3.1 梯度下降

3.2 訓練模型

4. 創作莎士比亞詩歌

作業3：用LSTM網路即興演奏爵士樂獨奏

1. 問題陳述

1.1 資料集

1.2 模型預覽