Assignment | 05-week1 -Building your Recurrent Neural Network - Step by Step

該系列僅在原課程基礎上課後作業部分添加個人學習筆記，如有錯誤，還請批評指教。- ZJ

Coursera 課程 |deeplearning.ai |網易雲課堂

CSDN：http://blog.csdn.net/JUNJUN_ZHAO/article/details/79400107

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Welcome to Course 5’s first assignment! In this assignment, you will implement your first Recurrent Neural Network in numpy.

Recurrent Neural Networks (RNN) are very effective for Natural Language Processing and other sequence tasks because they have “memory”. They can read inputs x⟨t⟩$x^{⟨t⟩}$ (such as words) one at a time, and remember some information/context through the hidden layer activations that get passed from one time-step to the next. This allows a uni-directional RNN to take information from the past to process later inputs. A bidirection RNN can take context from both the past and the future.

歡迎來到課程5的第一項任務！ 在這個任務中，你將使用 numpy 實現你的第一個循環神經網絡。

循環神經網絡（RNN）對自然語言處理和其他序列任務非常有效，因爲它們具有“記憶”細胞的作用。 他們可以一次讀取一個輸入x⟨t⟩$x^{⟨t⟩}$ （例如單詞），並通過從一個時間步到另一個時間步的隱藏層激活記住一些信息/上下文。 這允許單向 RNN 從過去獲取信息以處理稍後的輸入。 雙向 RNN 可以從過去和未來中獲取背景信息。

Notation:
- Superscript [l]$[l]$ denotes an object associated with the lth$l^{th}$ layer. 層
- Example: a[4]$a^{[4]}$ is the 4th$4^{th}$ layer activation. 第 4 層的激活值 W[5]$W^{[5]}$ and b[5]$b^{[5]}$ are the 5th$5^{th}$ layer parameters. 第 5 層的參數

• Superscript (i)$(i)$ denotes an object associated with the ith$i^{th}$ example. 樣本

  • Example: x(i)$x^{(i)}$ is the ith$i^{th}$ training example input. 第 i 個輸入 訓練樣本

• Superscript ⟨t⟩$⟨t⟩$ denotes an object at the tth$t^{th}$ time-step. 時間步

  • Example: x⟨t⟩$x^{⟨t⟩}$ is the input x at the tth$t^{th}$ time-step. x(i)⟨t⟩$x^{(i)⟨t⟩}$ is the input at the tth$t^{th}$ time-step of example i$i$ .

• Lowerscript i$i$ denotes the ith$i^{th}$ entry of a vector. 一個向量的 第 i 個值

  • Example: a[l]i$a_i^{[l]}$ denotes the ith$i^{th}$ entry of the activations in layer l$l$ .

We assume that you are already familiar with numpy and/or have completed the previous courses of the specialization. Let’s get started!

【知乎】Softmax 函數的特點和作用是什麼？ https://www.zhihu.com/question/23765351

Let’s first import all the packages that you will need during this assignment.

import numpy as np
from rnn_utils import *

'''
rnn_utils 中的代碼

'''

import numpy as np

# https://www.zhihu.com/question/23765351  Softmax 函數的特點和作用是什麼？

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

# softmax(x)
# 如 x = [1, 5, 3]
# x - np.max(x) 相當於 [1, 5, 3] - [5, 5, 5] = [-1, 0, -2] 避免值過大 所以都# 剪去最大值，按比例縮小
# 然後 np.exp(x - np.max(x)) 逐一計算，數組中每一個元素的 指數
# e_x / e_x.sum(axis=0) 得出最後結果
# b = np.array([2,3,4]) b.shape = (3,)
# c = np.mat([2,3,4])  c.shape = (1,3)
# d = np.mat([[1],[2],[3]]) d.shape = (3,1)

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


def initialize_adam(parameters) :
    """
    Initializes v and s as two python dictionaries with:
                - keys: "dW1", "db1", ..., "dWL", "dbL" 
                - values: numpy arrays of zeros of the same shape as the corresponding gradients/parameters.

    Arguments:
    parameters -- python dictionary containing your parameters.
                    parameters["W" + str(l)] = Wl
                    parameters["b" + str(l)] = bl

    Returns: 
    v -- python dictionary that will contain the exponentially weighted average of the gradient.
                    v["dW" + str(l)] = ...
                    v["db" + str(l)] = ...
    s -- python dictionary that will contain the exponentially weighted average of the squared gradient.
                    s["dW" + str(l)] = ...
                    s["db" + str(l)] = ...

    """
    # 由上面的Arguments 中的 對 parameters 描述可知，parameters 字典中包含所有參數，
    # 參數有 w and  b 兩種，所以 //2 取小的整數 （7 //2 = 3 , -3 //2 = -2） 
    L = len(parameters) // 2 # number of layers in the neural networks 神經網絡的層數
    v = {}
    s = {}

    # Initialize v, s. Input: "parameters". Outputs: "v, s". 初始化 要輸出的 "v, s"字典中要包含的參數 v["dW"] = 0 ,v["db"] = 0, s["dW"] = 0, s["db"] = 0 
    # 它們的 shape 與 W 和 b 相同
    for l in range(L):
    ### START CODE HERE ### (approx. 4 lines)
        v["dW" + str(l+1)] = np.zeros(parameters["W" + str(l+1)].shape)
        v["db" + str(l+1)] = np.zeros(parameters["b" + str(l+1)].shape)
        s["dW" + str(l+1)] = np.zeros(parameters["W" + str(l+1)].shape)
        s["db" + str(l+1)] = np.zeros(parameters["b" + str(l+1)].shape)
    ### END CODE HERE ###

    return v, s


def update_parameters_with_adam(parameters, grads, v, s, t, learning_rate = 0.01,
                                beta1 = 0.9, beta2 = 0.999,  epsilon = 1e-8):
    """
    Update parameters using Adam

    Arguments:
    parameters -- python dictionary containing your parameters:
                    parameters['W' + str(l)] = Wl
                    parameters['b' + str(l)] = bl
    grads -- python dictionary containing your gradients for each parameters:
                    grads['dW' + str(l)] = dWl
                    grads['db' + str(l)] = dbl
    v -- Adam variable, moving average of the first gradient, python dictionary
    s -- Adam variable, moving average of the squared gradient, python dictionary
    learning_rate -- the learning rate, scalar.
    beta1 -- Exponential decay hyperparameter for the first moment estimates 0.9 
    beta2 -- Exponential decay hyperparameter for the second moment estimates 0.999
    epsilon -- hyperparameter preventing division by zero in Adam updates 1e-8

    Returns:
    parameters -- python dictionary containing your updated parameters 字典包含你要更新的參數
    v -- Adam variable, moving average of the first gradient, python dictionary
    s -- Adam variable, moving average of the squared gradient, python dictionary
    """

    L = len(parameters) // 2                 # number of layers in the neural networks
    v_corrected = {}                         # Initializing first moment estimate, python dictionary
    s_corrected = {}                         # Initializing second moment estimate, python dictionary

    # Perform Adam update on all parameters 更新所有參數
    for l in range(L):
        # Moving average of the gradients. Inputs: "v, grads, beta1". Output: "v".
        ### START CODE HERE ### (approx. 2 lines)
        v["dW" + str(l+1)] = beta1 * v["dW" + str(l+1)] + (1 - beta1) * grads["dW" + str(l+1)] 
        v["db" + str(l+1)] = beta1 * v["db" + str(l+1)] + (1 - beta1) * grads["db" + str(l+1)] 
        ### END CODE HERE ###

        # Compute bias-corrected first moment estimate. Inputs: "v, beta1, t". Output: "v_corrected".
        ### START CODE HERE ### (approx. 2 lines)
        v_corrected["dW" + str(l+1)] = v["dW" + str(l+1)] / (1 - beta1**t)
        v_corrected["db" + str(l+1)] = v["db" + str(l+1)] / (1 - beta1**t)
        ### END CODE HERE ###

        # Moving average of the squared gradients. Inputs: "s, grads, beta2". Output: "s".
        ### START CODE HERE ### (approx. 2 lines)
        s["dW" + str(l+1)] = beta2 * s["dW" + str(l+1)] + (1 - beta2) * (grads["dW" + str(l+1)] ** 2)
        s["db" + str(l+1)] = beta2 * s["db" + str(l+1)] + (1 - beta2) * (grads["db" + str(l+1)] ** 2)
        ### END CODE HERE ###

        # Compute bias-corrected second raw moment estimate. Inputs: "s, beta2, t". Output: "s_corrected".
        ### START CODE HERE ### (approx. 2 lines)
        s_corrected["dW" + str(l+1)] = s["dW" + str(l+1)] / (1 - beta2 ** t)
        s_corrected["db" + str(l+1)] = s["db" + str(l+1)] / (1 - beta2 ** t)
        ### END CODE HERE ###

        # Update parameters. Inputs: "parameters, learning_rate, v_corrected, s_corrected, epsilon". Output: "parameters".
        ### START CODE HERE ### (approx. 2 lines) 
        # 對於 後面 np.sqrt 那有些疑問哈，我看論文中原本的算法，以及吳恩達老師講解的算法 epsilon 應該是在 np.sqrt 之後再加上的 
       # parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - learning_rate * v_corrected["dW" + str(l+1)] / np.sqrt(s_corrected["dW" + str(l+1)] + epsilon)
       # parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - learning_rate * v_corrected["db" + str(l+1)] / np.sqrt(s_corrected["db" + str(l+1)] + epsilon)

        parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - learning_rate * v_corrected["dW" + str(l+1)]/(np.sqrt(s_corrected["dW" + str(l+1)])+epsilon)
        parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - learning_rate * v_corrected["db" + str(l+1)]/(np.sqrt(s_corrected["db" + str(l+1)])+epsilon)

        ### END CODE HERE ###

    return parameters, v, s

1 - Forward propagation for the basic Recurrent Neural Network

Later this week, you will generate music using an RNN. The basic RNN that you will implement has the structure below. In this example, Tx=Ty$T_x=T_y$ .

Figure 1: Basic RNN model

Here’s how you can implement an RNN:

Steps:
1. Implement the calculations needed for one time-step of the RNN.
2. Implement a loop over Tx$T_x$ time-steps in order to process all the inputs, one at a time.

Let’s go!

1.1 - RNN cell

A Recurrent neural network can be seen as the repetition 重複 of a single cell. You are first going to implement the computations for a single time-step. The following figure describes the operations for a single time-step of an RNN cell.

循環神經網絡可以被看作是單個細胞的重複。 你首先要實現一個時間步驟的計算。 下圖描述了 RNN 單元的單個時間步驟的操作。

Figure 2: Basic RNN cell. Takes as input x⟨t⟩$x^{⟨t⟩}$ (current input) and a⟨t−1⟩$a^{⟨t-1⟩}$ (previous hidden state containing information from the past), and outputs a⟨t⟩$a^{⟨t⟩}$ which is given to the next RNN cell and also used to predict y⟨t⟩$y^{⟨t⟩}$

Exercise: Implement the RNN-cell described in Figure (2).

Instructions:
1. Compute the hidden state with tanh activation: a⟨t⟩=tanh(Waaa⟨t−1⟩+Waxx⟨t⟩+ba)$a^{⟨t⟩}=\tanh (W_{aa}{a}^{⟨t-1⟩}+W_{ax}{x}^{⟨t⟩}+b_a)$ .
2. Using your new hidden state a⟨t⟩$a^{⟨t⟩}$ , compute the prediction y^⟨t⟩=softmax(Wyaa⟨t⟩+by)${\hat{y}}^{⟨t⟩}=softmax(W_{ya}{a}^{⟨t⟩}+b_y)$ . We provided you a function: softmax.
3. Store (a⟨t⟩,a⟨t−1⟩,x⟨t⟩,parameters)$(a^{⟨t⟩},a^{⟨t-1⟩},x^{⟨t⟩},parameters)$ in cache
4. Return a⟨t⟩$a^{⟨t⟩}$ , y⟨t⟩$y^{⟨t⟩}$ and cache

We will vectorize over m$m$ examples. Thus, x⟨t⟩$x^{⟨t⟩}$ will have dimension (nx,m)$(n_x,m)$ , and a⟨t⟩$a^{⟨t⟩}$ will have dimension (na,m)$(n_a,m)$ .

# 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 np.tanh 
    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 注意使用上面計算出來的 a_next 值，進行下一步計算 softmax 在rnn_utils 中
    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

np.random.seed(1)
#  randn 標準正太分佈中 返回值
xt = np.random.randn(3,10)
a_prev = np.random.randn(5,10)
Waa = np.random.randn(5,5)
Wax = np.random.randn(5,3)
Wya = np.random.randn(2,5)
ba = np.random.randn(5,1)
by = np.random.randn(2,1)
parameters = {"Waa": Waa, "Wax": Wax, "Wya": Wya, "ba": ba, "by": by}

a_next, yt_pred, cache = rnn_cell_forward(xt, a_prev, parameters)
print("a_next[4] = ", a_next[4])
print("a_next.shape = ", a_next.shape)
print("yt_pred[1] =", yt_pred[1])
print("yt_pred.shape = ", yt_pred.shape)

a_next[4] =  [ 0.59584544  0.18141802  0.61311866  0.99808218  0.85016201  0.99980978
 -0.18887155  0.99815551  0.6531151   0.82872037]
a_next.shape =  (5, 10)
yt_pred[1] = [0.9888161  0.01682021 0.21140899 0.36817467 0.98988387 0.88945212
 0.36920224 0.9966312  0.9982559  0.17746526]
yt_pred.shape =  (2, 10)

Expected Output:

**a_next[4]**:	[ 0.59584544 0.18141802 0.61311866 0.99808218 0.85016201 0.99980978 -0.18887155 0.99815551 0.6531151 0.82872037]
**a_next.shape**:	(5, 10)
**yt[1]**:	[ 0.9888161 0.01682021 0.21140899 0.36817467 0.98988387 0.88945212 0.36920224 0.9966312 0.9982559 0.17746526]
**yt.shape**:	(2, 10)

1.2 - RNN forward pass

You can see an RNN as the repetition of the cell you’ve just built. If your input sequence of data is carried over 10 time steps, then you will copy the RNN cell 10 times. Each cell takes as input the hidden state from the previous cell (a⟨t−1⟩$a^{⟨t-1⟩}$ ) and the current time-step’s input data (x⟨t⟩$x^{⟨t⟩}$ ). It outputs a hidden state (a⟨t⟩$a^{⟨t⟩}$ ) and a prediction (y⟨t⟩$y^{⟨t⟩}$ ) for this time-step.

Figure 3: Basic RNN. The input sequence x=(x⟨1⟩,x⟨2⟩,...,x⟨Tx⟩)$x=(x^{⟨1⟩},x^{⟨2⟩},...,x^{⟨T_x⟩})$ is carried over Tx$T_x$ time steps. The network outputs y=(y⟨1⟩,y⟨2⟩,...,y⟨Tx⟩)$y=(y^{⟨1⟩},y^{⟨2⟩},...,y^{⟨T_x⟩})$ .

Exercise: Code the forward propagation of the RNN described in Figure (3).

Instructions:
1. Create a vector of zeros (a$a$ ) that will store all the hidden states computed by the RNN.
2. Initialize the “next” hidden state as a0$a_0$ (initial hidden state).
3. Start looping over each time step, your incremental index is t$t$ :
- Update the “next” hidden state and the cache by running rnn_cell_forward
- Store the “next” hidden state in a$a$ (tth$t^{th}$ position)
- Store the prediction in y
- Add the cache to the list of caches
4. Return a$a$ , y$y$ and caches

1.創建一個零向量（a$a$ ），它將存儲由 RNN 計算的所有隱藏狀態。
2.將“下一個”隱藏狀態初始化爲a0$a_0$ （初始隱藏狀態）。
3.開始循環遍歷每個時間步，您的增量索引是t$t$ ：
- 通過運行rnn_cell_forward來更新“下一個”隱藏狀態和緩存
- 將“下一個”隱藏狀態存儲在a$a$ （tth$t^{th}$ 位置）
- 將預測存儲在 y 中
- 將緩存添加到緩存列表
4.返回a$a$ ，y$y$ 並緩存

# 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 Wya 
    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

# 錯誤點： x.shape 大小這麼表示

np.random.seed(1)
x = np.random.randn(3,10,4)
a0 = np.random.randn(5,10)
Waa = np.random.randn(5,5)
Wax = np.random.randn(5,3)
Wya = np.random.randn(2,5)
ba = np.random.randn(5,1)
by = np.random.randn(2,1)
parameters = {"Waa": Waa, "Wax": Wax, "Wya": Wya, "ba": ba, "by": by}

a, y_pred, caches = rnn_forward(x, a0, parameters)
print("a[4][1] = ", a[4][1])
print("a.shape = ", a.shape)
print("y_pred[1][3] =", y_pred[1][3])
print("y_pred.shape = ", y_pred.shape)
print("caches[1][1][3] =", caches[1][1][3])
print("len(caches) = ", len(caches))

a[4][1] =  [-0.99999375  0.77911235 -0.99861469 -0.99833267]
a.shape =  (5, 10, 4)
y_pred[1][3] = [0.79560373 0.86224861 0.11118257 0.81515947]
y_pred.shape =  (2, 10, 4)
caches[1][1][3] = [-1.1425182  -0.34934272 -0.20889423  0.58662319]
len(caches) =  2

Expected Output:

**a[4][1]**:	[-0.99999375 0.77911235 -0.99861469 -0.99833267]
**a.shape**:	(5, 10, 4)
**y[1][3]**:	[ 0.79560373 0.86224861 0.11118257 0.81515947]
**y.shape**:	(2, 10, 4)
**cache[1][1][3]**:	[-1.1425182 -0.34934272 -0.20889423 0.58662319]
**len(cache)**:	2

Congratulations! You’ve successfully built the forward propagation of a recurrent neural network from scratch. This will work well enough for some applications, but it suffers from vanishing gradient problems. So it works best when each output y⟨t⟩$y^{⟨t⟩}$ can be estimated using mainly “local” context (meaning information from inputs x⟨t′⟩$x^{⟨t^{\prime }⟩}$ where t′$t^{\prime }$ is not too far from t$t$ ).

In the next part, you will build a more complex LSTM model, which is better at addressing vanishing gradients. The LSTM will be better able to remember a piece of information and keep it saved for many timesteps.

恭喜！ 您已經成功構建了從頭開始的循環神經網絡的向前傳播。 這對於某些應用程序來說可以工作得很好，但是會遇到逐漸消失的梯度問題。 因此，當每個輸出 y⟨t⟩$y^{⟨t⟩}$ 可以使用主要的“local”上下文（意思是來自輸入信息 x⟨t′⟩$x^{⟨t^{\prime }⟩}$ ，其中 t′$t^{\prime }$ 是 離t$t$ 不太遠）。

在接下來的部分中，您將構建一個更復雜的 LSTM 模型，該模型更適合處理消失漸變。 LSTM 將能夠更好地記住一段信息，並將其保存多個時間步。

2 - Long Short-Term Memory (LSTM) network 長短時記憶網絡

This following figure shows the operations of an LSTM-cell.

Figure 4: LSTM-cell. This tracks and updates a “cell state” or memory variable c⟨t⟩$c^{⟨t⟩}$ at every time-step, which can be different from a⟨t⟩$a^{⟨t⟩}$ .

Similar to the RNN example above, you will start by implementing the LSTM cell for a single time-step. Then you can iteratively call it from inside a for-loop to have it process an input with Tx$T_x$ time-steps.

與上面的 RNN 例子相似，首先先實現一個時間步的 LSTM 單元，然後再迭代循環調用。

About the gates

- Forget gate

For the sake of this illustration, lets assume we are reading words in a piece of text, and want use an LSTM to keep track of grammatical structures 語法結構, such as whether the subject 主語 is singular or plural單數還是複數. If the subject changes from a singular word to a plural word, we need to find a way to get rid of our previously stored memory value of the singular/plural state. In an LSTM, the forget gate lets us do this: 如果主語從單數變爲複數，那麼需要去掉之前存儲的記憶值， LSTM 中的遺忘門 就是做這件事的。

Γ⟨t⟩f=σ(Wf[a⟨t−1⟩,x⟨t⟩]+bf)(1)$$\begin{array}{}\text{(1)}& {\mathrm{\Gamma }}_f^{⟨t⟩}=\sigma (W_f[a^{⟨t-1⟩},x^{⟨t⟩}]+b_f)\end{array}$$

Here, Wf$W_f$ are weights that govern the forget gate’s behavior. We concatenate 連接 [a⟨t−1⟩,x⟨t⟩]$[a^{⟨t-1⟩},x^{⟨t⟩}]$ and multiply by Wf$W_f$ . The equation above results in a vector Γ⟨t⟩f${\mathrm{\Gamma }}_f^{⟨t⟩}$ with values between 0 and 1. This forget gate vector will be multiplied element-wise by the previous cell state c⟨t−1⟩$c^{⟨t-1⟩}$ . So if one of the values of Γ⟨t⟩f${\mathrm{\Gamma }}_f^{⟨t⟩}$ is 0 (or close to 0) then it means that the LSTM should remove that piece of information (e.g. the singular subject) in the corresponding component of c⟨t−1⟩$c^{⟨t-1⟩}$ . If one of the values is 1, then it will keep the information. 若遺忘門的值是 0 或接近 0 那麼 移除掉 c⟨t−1⟩$c^{⟨t-1⟩}$ 的信息

- Update gate

Once we forget that the subject being discussed is singular, we need to find a way to update it to reflect that the new subject is now plural. Here is the formulat for the update gate: 找到一個方法去更新它，現在的主語是複數

Γ⟨t⟩u=σ(Wu[a⟨t−1⟩,x{t}]+bu)(2)$$\begin{array}{}\text{(2)}& {\mathrm{\Gamma }}_u^{⟨t⟩}=\sigma (W_u[a^{⟨t-1⟩},x^{\{t\}}]+b_u)\end{array}$$

Similar to the forget gate, here Γ⟨t⟩u${\mathrm{\Gamma }}_u^{⟨t⟩}$ is again a vector of values between 0 and 1. This will be multiplied element-wise with c~⟨t⟩${\tilde{c}}^{⟨t⟩}$ , in order to compute c⟨t⟩$c^{⟨t⟩}$ .

- Updating the cell

To update the new subject we need to create a new vector of numbers that we can add to our previous cell state. The equation we use is:

c~⟨t⟩=tanh(Wc[a⟨t−1⟩,x⟨t⟩]+bc)(3)$$\begin{array}{}\text{(3)}& {\tilde{c}}^{⟨t⟩}=\tanh (W_c[a^{⟨t-1⟩},x^{⟨t⟩}]+b_c)\end{array}$$

Finally, the new cell state is:

c⟨t⟩=Γ⟨t⟩f∗c⟨t−1⟩+Γ⟨t⟩u∗c~⟨t⟩(4)$$\begin{array}{}\text{(4)}& c^{⟨t⟩}={\mathrm{\Gamma }}_f^{⟨t⟩}\ast c^{⟨t-1⟩}+{\mathrm{\Gamma }}_u^{⟨t⟩}\ast {\tilde{c}}^{⟨t⟩}\end{array}$$

- Output gate

To decide which outputs we will use, we will use the following two formulas:

Γ⟨t⟩o=σ(Wo[a⟨t−1⟩,x⟨t⟩]+bo)(5)$$\begin{array}{}\text{(5)}& {\mathrm{\Gamma }}_o^{⟨t⟩}=\sigma (W_o[a^{⟨t-1⟩},x^{⟨t⟩}]+b_o)\end{array}$$

a⟨t⟩=Γ⟨t⟩o∗tanh(c⟨t⟩)(6)$$\begin{array}{}\text{(6)}& a^{⟨t⟩}={\mathrm{\Gamma }}_o^{⟨t⟩}\ast \tanh (c^{⟨t⟩})\end{array}$$

Where in equation 5 you decide what to output using a sigmoid function and in equation 6 you multiply that by the tanh$\tanh$ of the previous state.

2.1 - LSTM cell

Exercise: Implement the LSTM cell described in the Figure (3).

Instructions:
1. Concatenate a⟨t−1⟩$a^{⟨t-1⟩}$ and x⟨t⟩$x^{⟨t⟩}$ in a single matrix: concat=[a⟨t−1⟩x⟨t⟩]$concat=\left[\begin{array}{c}{a}^{⟨t-1⟩}\\ x^{⟨t⟩}\end{array}\right]$
2. Compute all the formulas 1-6. You can use sigmoid() (provided) and np.tanh().
3. Compute the prediction y⟨t⟩$y^{⟨t⟩}$ . You can use softmax() (provided).

# 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 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)

    Note: ft/it/ot stand for the forget/update/output gates, cct stands for the candidate value (c tilde),
          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) 連接矩陣時，先計算好，矩陣的形狀大小，初始化 zeros 然後再對應部分再 添加對應矩陣
    concat = np.zeros((n_x + n_a, m))
    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.matmul(Wf, concat) + bf)
    it = sigmoid(np.matmul(Wi, concat) + bi)
    cct = np.tanh(np.matmul(Wc, concat) + bc)
    c_next = it * cct + ft * c_prev
    ot = sigmoid(np.matmul(Wo, concat) + bo)
    a_next = ot * np.tanh(c_next)

    # Compute prediction of the LSTM cell (≈1 line)
    yt_pred = softmax(np.matmul(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

np.random.seed(1)
xt = np.random.randn(3,10)
a_prev = np.random.randn(5,10)
c_prev = np.random.randn(5,10)
Wf = np.random.randn(5, 5+3)
bf = np.random.randn(5,1)
Wi = np.random.randn(5, 5+3)
bi = np.random.randn(5,1)
Wo = np.random.randn(5, 5+3)
bo = np.random.randn(5,1)
Wc = np.random.randn(5, 5+3)
bc = np.random.randn(5,1)
Wy = np.random.randn(2,5)
by = np.random.randn(2,1)

parameters = {"Wf": Wf, "Wi": Wi, "Wo": Wo, "Wc": Wc, "Wy": Wy, "bf": bf, "bi": bi, "bo": bo, "bc": bc, "by": by}

a_next, c_next, yt, cache = lstm_cell_forward(xt, a_prev, c_prev, parameters)
print("a_next[4] = ", a_next[4])
print("a_next.shape = ", c_next.shape)
print("c_next[2] = ", c_next[2])
print("c_next.shape = ", c_next.shape)
print("yt[1] =", yt[1])
print("yt.shape = ", yt.shape)
print("cache[1][3] =", cache[1][3])
print("len(cache) = ", len(cache))

a_next[4] =  [-0.66408471  0.0036921   0.02088357  0.22834167 -0.85575339  0.00138482
  0.76566531  0.34631421 -0.00215674  0.43827275]
a_next.shape =  (5, 10)
c_next[2] =  [ 0.63267805  1.00570849  0.35504474  0.20690913 -1.64566718  0.11832942
  0.76449811 -0.0981561  -0.74348425 -0.26810932]
c_next.shape =  (5, 10)
yt[1] = [0.79913913 0.15986619 0.22412122 0.15606108 0.97057211 0.31146381
 0.00943007 0.12666353 0.39380172 0.07828381]
yt.shape =  (2, 10)
cache[1][3] = [-0.16263996  1.03729328  0.72938082 -0.54101719  0.02752074 -0.30821874
  0.07651101 -1.03752894  1.41219977 -0.37647422]
len(cache) =  10

Expected Output:

**a_next[4]**:	[-0.66408471 0.0036921 0.02088357 0.22834167 -0.85575339 0.00138482 0.76566531 0.34631421 -0.00215674 0.43827275]
**a_next.shape**:	(5, 10)
**c_next[2]**:	[ 0.63267805 1.00570849 0.35504474 0.20690913 -1.64566718 0.11832942 0.76449811 -0.0981561 -0.74348425 -0.26810932]
**c_next.shape**:	(5, 10)
**yt[1]**:	[ 0.79913913 0.15986619 0.22412122 0.15606108 0.97057211 0.31146381 0.00943007 0.12666353 0.39380172 0.07828381]
**yt.shape**:	(2, 10)
**cache[1][3]**:	[-0.16263996 1.03729328 0.72938082 -0.54101719 0.02752074 -0.30821874 0.07651101 -1.03752894 1.41219977 -0.37647422]
**len(cache)**:	10

2.2 - Forward pass for LSTM

Now that you have implemented one step of an LSTM, you can now iterate this over this using a for-loop to process a sequence of Tx$T_x$ inputs.

Figure 4: LSTM over multiple time-steps.

Exercise: Implement lstm_forward() to run an LSTM over Tx$T_x$ time-steps.

Note: c⟨0⟩$c^{⟨0⟩}$ is initialized with zeros.

# 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 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)
    """

    # Initialize "caches", which will track the list of all the caches
    caches = []

    ### START CODE HERE ###
    # Retrieve dimensions from shapes of x 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(a_next.shape)

    # 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

np.random.seed(1)
x = np.random.randn(3,10,7)
a0 = np.random.randn(5,10)
Wf = np.random.randn(5, 5+3)
bf = np.random.randn(5,1)
Wi = np.random.randn(5, 5+3)
bi = np.random.randn(5,1)
Wo = np.random.randn(5, 5+3)
bo = np.random.randn(5,1)
Wc = np.random.randn(5, 5+3)
bc = np.random.randn(5,1)
Wy = np.random.randn(2,5)
by = np.random.randn(2,1)

parameters = {"Wf": Wf, "Wi": Wi, "Wo": Wo, "Wc": Wc, "Wy": Wy, "bf": bf, "bi": bi, "bo": bo, "bc": bc, "by": by}

a, y, c, caches = lstm_forward(x, a0, parameters)
print("a[4][3][6] = ", a[4][3][6])
print("a.shape = ", a.shape)
print("y[1][4][3] =", y[1][4][3])
print("y.shape = ", y.shape)
print("caches[1][1[1]] =", caches[1][1][1])
print("c[1][2][1]", c[1][2][1])
print("len(caches) = ", len(caches))

a[4][3][6] =  0.17211776753291672
a.shape =  (5, 10, 7)
y[1][4][3] = 0.9508734618501101
y.shape =  (2, 10, 7)
caches[1][1[1]] = [ 0.82797464  0.23009474  0.76201118 -0.22232814 -0.20075807  0.18656139
  0.41005165]
c[1][2][1] -0.8555449167181981
len(caches) =  2

Expected Output:

**a[4][3][6]** =	0.172117767533
**a.shape** =	(5, 10, 7)
**y[1][4][3]** =	0.95087346185
**y.shape** =	(2, 10, 7)
**caches[1][1][1]** =	[ 0.82797464 0.23009474 0.76201118 -0.22232814 -0.20075807 0.18656139 0.41005165]
**c[1][2][1]** =	-0.855544916718
**len(caches)** =	2

Congratulations! You have now implemented the forward passes for the basic RNN and the LSTM. When using a deep learning framework, implementing the forward pass is sufficient to build systems that achieve great performance.

The rest of this notebook is optional, and will not be graded.

3 - Backpropagation in recurrent neural networks (OPTIONAL / UNGRADED)

In modern deep learning frameworks, you only have to implement the forward pass, and the framework takes care of the backward pass, so most deep learning engineers do not need to bother with the details of the backward pass. If however you are an expert in calculus and want to see the details of backprop in RNNs, you can work through this optional portion of the notebook.

When in an earlier course you implemented a simple (fully connected) neural network, you used backpropagation to compute the derivatives with respect to the cost to update the parameters. Similarly, in recurrent neural networks you can to calculate the derivatives with respect to the cost in order to update the parameters. The backprop equations are quite complicated and we did not derive them in lecture. However, we will briefly present them below. 求導過程，這裏不會進行詳細推導，只是簡單的展示結果。

在現代的深度學習框架中，您只需要實現前向傳播，並且框架負責關注反向傳播，因此大多數深度學習工程師不需要爲反向傳播的細節而煩惱。 但是，如果您是微積分專家，並希望查看 RNN 中 backprop 的詳細信息，則可以通過筆記的這一可選部分進行操作。

在之前的課程中，您實現了一個簡單的（完全連接的）神經網絡，您使用反向傳播來計算更新參數的費用方面的導數。 同樣，在循環神經網絡中，您可以根據成本計算導數以更新參數。 backprop方程相當複雜，我們在講座中沒有推導出它們。 但是，我們將在下面簡要介紹它們。

3.1 - Basic RNN backward pass

We will start by computing the backward pass for the basic RNN-cell.

Figure 5: RNN-cell’s backward pass. Just like in a fully-connected neural network, the derivative of the cost function J$J$ backpropagates through the RNN by following the chain-rule from calculas. The chain-rule is also used to calculate (∂J∂Wax,∂J∂Waa,∂J∂b)$(\frac{\mathrm{\partial }J}{\mathrm{\partial }{W}_{ax}},\frac{\mathrm{\partial }J}{\mathrm{\partial }{W}_{aa}},\frac{\mathrm{\partial }J}{\mathrm{\partial }b})$ to update the parameters (Wax,Waa,ba)$(W_{ax},W_{aa},b_a)$ .

Deriving the one step backward functions:

To compute the rnn_cell_backward you need to compute the following equations. It is a good exercise to derive them by hand.

The derivative of tanh$\tanh$ is 1−tanh(x)2$1-\tanh (x{)}^2$ . You can find the complete proof here. Note that: sec(x)2=1−tanh(x)2$\sec (x{)}^2=1-\tanh (x{)}^2$

Similarly for ∂a⟨t⟩∂Wax,∂a⟨t⟩∂Waa,∂a⟨t⟩∂b$\frac{\mathrm{\partial }{a}^{⟨t⟩}}{\mathrm{\partial }{W}_{ax}},\frac{\mathrm{\partial }{a}^{⟨t⟩}}{\mathrm{\partial }{W}_{aa}},\frac{\mathrm{\partial }{a}^{⟨t⟩}}{\mathrm{\partial }b}$ , the derivative of tanh(u)$\tanh (u)$ is (1−tanh(u)2)du$(1-\tanh (u{)}^2)du$ .

The final two equations also follow same rule and are derived using the tanh$\tanh$ derivative. Note that the arrangement is done in a way to get the same dimensions to match.

def rnn_cell_backward(da_next, cache):
    """
    Implements the backward pass for the RNN-cell (single time-step). 注意這裏是 基本的 RNN 單元 網絡的反向傳播

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

    Returns:
    gradients -- python dictionary containing:
                        dxt -- 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 先把之前存的參數提取出來
    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)
    # 根據  提供的參數 da_next -- Gradient of loss with respect to next hidden state 
    # 以及 上面提到的公式 tanh(u) = (1- tanh(u)**2)*du ，這裏的 du 就是 da_next  tanh(u) 是 a_next   
    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)
    #  axis=0 列方向上操作 axis=1 行方向上操作  keepdims=True 矩陣的二維特性
    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

np.random.seed(1)
xt = np.random.randn(3,10)
a_prev = np.random.randn(5,10)
Wax = np.random.randn(5,3)
Waa = np.random.randn(5,5)
Wya = np.random.randn(2,5)
b = np.random.randn(5,1)
by = np.random.randn(2,1)
parameters = {"Wax": Wax, "Waa": Waa, "Wya": Wya, "ba": ba, "by": by}

a_next, yt, cache = rnn_cell_forward(xt, a_prev, parameters)

da_next = np.random.randn(5,10)
gradients = rnn_cell_backward(da_next, cache)
print("gradients[\"dxt\"][1][2] =", gradients["dxt"][1][2])
print("gradients[\"dxt\"].shape =", gradients["dxt"].shape)
print("gradients[\"da_prev\"][2][3] =", gradients["da_prev"][2][3])
print("gradients[\"da_prev\"].shape =", gradients["da_prev"].shape)
print("gradients[\"dWax\"][3][1] =", gradients["dWax"][3][1])
print("gradients[\"dWax\"].shape =", gradients["dWax"].shape)
print("gradients[\"dWaa\"][1][2] =", gradients["dWaa"][1][2])
print("gradients[\"dWaa\"].shape =", gradients["dWaa"].shape)
print("gradients[\"dba\"][4] =", gradients["dba"][4])
print("gradients[\"dba\"].shape =", gradients["dba"].shape)

gradients["dxt"][1][2] = -0.4605641030588796
gradients["dxt"].shape = (3, 10)
gradients["da_prev"][2][3] = 0.08429686538067671
gradients["da_prev"].shape = (5, 10)
gradients["dWax"][3][1] = 0.3930818739219304
gradients["dWax"].shape = (5, 3)
gradients["dWaa"][1][2] = -0.2848395578696066
gradients["dWaa"].shape = (5, 5)
gradients["dba"][4] = [0.80517166]
gradients["dba"].shape = (5, 1)

Expected Output:

**gradients[“dxt”][1][2]** =	-0.460564103059
**gradients[“dxt”].shape** =	(3, 10)
**gradients[“da_prev”][2][3]** =	0.0842968653807
**gradients[“da_prev”].shape** =	(5, 10)
**gradients[“dWax”][3][1]** =	0.393081873922
**gradients[“dWax”].shape** =	(5, 3)
**gradients[“dWaa”][1][2]** =	-0.28483955787
**gradients[“dWaa”].shape** =	(5, 5)
**gradients[“dba”][4]** =	[ 0.80517166]
**gradients[“dba”].shape** =	(5, 1)

Backward pass through the RNN

Computing the gradients of the cost with respect to a⟨t⟩$a^{⟨t⟩}$ at every time-step t$t$ is useful because it is what helps the gradient backpropagate to the previous RNN-cell. To do so, you need to iterate through all the time steps starting at the end, and at each step, you increment the overall dba$d{b}_a$ , dWaa$d{W}_{aa}$ , dWax$d{W}_{ax}$ and you store dx$dx$ .

Instructions:

Implement the rnn_backward function. Initialize the return variables with zeros first and then loop through all the time steps while calling the rnn_cell_backward at each time timestep, update the other variables accordingly.

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(parameters["Wax"].shape)
    dWaa = np.zeros(parameters["Waa"].shape)
    dba = np.zeros(parameters["ba"].shape)
    da0 = np.zeros(a0.shape)
    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 += dWaxt
        dWaa += dWaat
        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

錯誤記錄：
gradients = rnn_cell_backward(da[:,:,t] + da_prevt, caches[t])
第一個參數是： da[:,:,t] + da_prevt, 這裏是每一次訓練後的更新迭代

np.random.seed(1)
x = np.random.randn(3,10,4)
a0 = np.random.randn(5,10)
Wax = np.random.randn(5,3)
Waa = np.random.randn(5,5)
Wya = np.random.randn(2,5)
ba = np.random.randn(5,1)
by = np.random.randn(2,1)
parameters = {"Wax": Wax, "Waa": Waa, "Wya": Wya, "ba": ba, "by": by}
a, y, caches = rnn_forward(x, a0, parameters)
da = np.random.randn(5, 10, 4)
gradients = rnn_backward(da, caches)

print("gradients[\"dx\"][1][2] =", gradients["dx"][1][2])
print("gradients[\"dx\"].shape =", gradients["dx"].shape)
print("gradients[\"da0\"][2][3] =", gradients["da0"][2][3])
print("gradients[\"da0\"].shape =", gradients["da0"].shape)
print("gradients[\"dWax\"][3][1] =", gradients["dWax"][3][1])
print("gradients[\"dWax\"].shape =", gradients["dWax"].shape)
print("gradients[\"dWaa\"][1][2] =", gradients["dWaa"][1][2])
print("gradients[\"dWaa\"].shape =", gradients["dWaa"].shape)
print("gradients[\"dba\"][4] =", gradients["dba"][4])
print("gradients[\"dba\"].shape =", gradients["dba"].shape)

gradients["dx"][1][2] = [-2.07101689 -0.59255627  0.02466855  0.01483317]
gradients["dx"].shape = (3, 10, 4)
gradients["da0"][2][3] = -0.3149423751266499
gradients["da0"].shape = (5, 10)
gradients["dWax"][3][1] = 11.264104496527777
gradients["dWax"].shape = (5, 3)
gradients["dWaa"][1][2] = 2.3033331265798926
gradients["dWaa"].shape = (5, 5)
gradients["dba"][4] = [-0.74747722]
gradients["dba"].shape = (5, 1)

Expected Output:

**gradients[“dx”][1][2]** =	[-2.07101689 -0.59255627 0.02466855 0.01483317]
**gradients[“dx”].shape** =	(3, 10, 4)
**gradients[“da0”][2][3]** =	-0.314942375127
**gradients[“da0”].shape** =	(5, 10)
**gradients[“dWax”][3][1]** =	11.2641044965
**gradients[“dWax”].shape** =	(5, 3)
**gradients[“dWaa”][1][2]** =	2.30333312658
**gradients[“dWaa”].shape** =	(5, 5)
**gradients[“dba”][4]** =	[-0.74747722]
**gradients[“dba”].shape** =	(5, 1)

3.2 - LSTM backward pass

3.2.1 One Step backward

The LSTM backward pass is slighltly more complicated than the forward one. We have provided you with all the equations for the LSTM backward pass below. (If you enjoy calculus exercises feel free to try deriving these from scratch yourself.)

3.2.2 gate derivatives

dΓ⟨t⟩o=danext∗tanh(cnext)∗Γ⟨t⟩o∗(1−Γ⟨t⟩o)(7)$$\begin{array}{}\text{(7)}& d{\mathrm{\Gamma }}_o^{⟨t⟩}=d{a}_{next}\ast \tanh (c_{next})\ast {\mathrm{\Gamma }}_o^{⟨t⟩}\ast (1-{\mathrm{\Gamma }}_o^{⟨t⟩})\end{array}$$

dc~⟨t⟩=dcnext∗Γ⟨t⟩i+Γ⟨t⟩o(1−tanh(cnext)2)∗it∗danext∗c~⟨t⟩∗(1−tanh(c~)2)(8)$$\begin{array}{}\text{(8)}& d{\tilde{c}}^{⟨t⟩}=d{c}_{next}\ast {\mathrm{\Gamma }}_i^{⟨t⟩}+{\mathrm{\Gamma }}_o^{⟨t⟩}(1-\tanh (c_{next}{)}^2)\ast i_t\ast d{a}_{next}\ast {\tilde{c}}^{⟨t⟩}\ast (1-\tanh (\tilde{c}{)}^2)\end{array}$$

dΓ⟨t⟩u=dcnext∗c~⟨t⟩+Γ⟨t⟩o(1−tanh(cnext)2)∗c~⟨t⟩∗danext∗Γ⟨t⟩u∗(1−Γ⟨t⟩u)(9)$$\begin{array}{}\text{(9)}& d{\mathrm{\Gamma }}_u^{⟨t⟩}=d{c}_{next}\ast {\tilde{c}}^{⟨t⟩}+{\mathrm{\Gamma }}_o^{⟨t⟩}(1-\tanh (c_{next}{)}^2)\ast {\tilde{c}}^{⟨t⟩}\ast d{a}_{next}\ast {\mathrm{\Gamma }}_u^{⟨t⟩}\ast (1-{\mathrm{\Gamma }}_u^{⟨t⟩})\end{array}$$

dΓ⟨t⟩f=dcnext∗c~prev+Γ⟨t⟩o(1−tanh(cnext)2)∗cprev∗danext∗Γ⟨t⟩f∗(1−Γ⟨t⟩f)(10)$$\begin{array}{}\text{(10)}& d{\mathrm{\Gamma }}_f^{⟨t⟩}=d{c}_{next}\ast {\tilde{c}}_{prev}+{\mathrm{\Gamma }}_o^{⟨t⟩}(1-\tanh (c_{next}{)}^2)\ast c_{prev}\ast d{a}_{next}\ast {\mathrm{\Gamma }}_f^{⟨t⟩}\ast (1-{\mathrm{\Gamma }}_f^{⟨t⟩})\end{array}$$

3.2.3 parameter derivatives

dWf=dΓ⟨t⟩f∗(aprevxt)T(11)$$\begin{array}{}\text{(11)}& d{W}_f=d{\mathrm{\Gamma }}_f^{⟨t⟩}\ast {\left(\begin{array}{c}{a}_{prev}\\ x_t\end{array}\right)}^T\end{array}$$

dWu=dΓ⟨t⟩u∗(aprevxt)T(12)$$\begin{array}{}\text{(12)}& d{W}_u=d{\mathrm{\Gamma }}_u^{⟨t⟩}\ast {\left(\begin{array}{c}{a}_{prev}\\ x_t\end{array}\right)}^T\end{array}$$

dWc=dc~⟨t⟩∗(aprevxt)T(13)$$\begin{array}{}\text{(13)}& d{W}_c=d{\tilde{c}}^{⟨t⟩}\ast {\left(\begin{array}{c}{a}_{prev}\\ x_t\end{array}\right)}^T\end{array}$$

dWo=dΓ⟨t⟩o∗(aprevxt)T(14)$$\begin{array}{}\text{(14)}& d{W}_o=d{\mathrm{\Gamma }}_o^{⟨t⟩}\ast {\left(\begin{array}{c}{a}_{prev}\\ x_t\end{array}\right)}^T\end{array}$$

To calculate dbf,dbu,dbc,dbo$d{b}_f,d{b}_u,d{b}_c,d{b}_o$ you just need to sum across the horizontal (axis= 1) axis on dΓ⟨t⟩f,dΓ⟨t⟩u,dc~⟨t⟩,dΓ⟨t⟩o$d{\mathrm{\Gamma }}_f^{⟨t⟩},d{\mathrm{\Gamma }}_u^{⟨t⟩},d{\tilde{c}}^{⟨t⟩},d{\mathrm{\Gamma }}_o^{⟨t⟩}$ respectively. Note that you should have the keep_dims = True option.

Finally, you will compute the derivative with respect to the previous hidden state, previous memory state, and input.

daprev=WTf∗dΓ⟨t⟩f+WTu∗dΓ⟨t⟩u+WTc∗dc~⟨t⟩+WTo∗dΓ⟨t⟩o(15)$$\begin{array}{}\text{(15)}& d{a}_{prev}=W_f^T\ast d{\mathrm{\Gamma }}_f^{⟨t⟩}+W_u^T\ast d{\mathrm{\Gamma }}_u^{⟨t⟩}+W_c^T\ast d{\tilde{c}}^{⟨t⟩}+W_o^T\ast d{\mathrm{\Gamma }}_o^{⟨t⟩}\end{array}$$

Here, the weights for equations 13 are the first n_a, (i.e. Wf=Wf[:na,:]$W_f=W_f[:n_a,:]$ etc…)

dcprev=dcnextΓ⟨t⟩f+Γ⟨t⟩o∗(1−tanh(cnext)2)∗Γ⟨t⟩f∗danext(16)$$\begin{array}{}\text{(16)}& d{c}_{prev}=d{c}_{next}{\mathrm{\Gamma }}_f^{⟨t⟩}+{\mathrm{\Gamma }}_o^{⟨t⟩}\ast (1-\tanh (c_{next}{)}^2)\ast {\mathrm{\Gamma }}_f^{⟨t⟩}\ast d{a}_{next}\end{array}$$

dx⟨t⟩=WTf∗dΓ⟨t⟩f+WTu∗dΓ⟨t⟩u+WTc∗dc~t+WTo∗dΓ⟨t⟩o(17)$$\begin{array}{}\text{(17)}& d{x}^{⟨t⟩}=W_f^T\ast d{\mathrm{\Gamma }}_f^{⟨t⟩}+W_u^T\ast d{\mathrm{\Gamma }}_u^{⟨t⟩}+W_c^T\ast d{\tilde{c}}_t+W_o^T\ast d{\mathrm{\Gamma }}_o^{⟨t⟩}\end{array}$$

where the weights for equation 15 are from n_a to the end, (i.e. Wf=Wf[na:,:]$W_f=W_f[n_a:,:]$ etc…)

Exercise: Implement lstm_cell_backward by implementing equations 7−17$7-17$ below. Good luck! :)

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 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)
    """

    # 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.square(np.tanh(c_next))) * it * da_next) * (1 - np.square(cct))
    dit = (dc_next * cct + ot * (1 - np.square(np.tanh(c_next))) * cct * da_next) * it * (1 - it)
    dft = (dc_next * c_prev + ot *(1 - np.square(np.tanh(c_next))) * c_prev * da_next) * ft * (1 - ft)

    # Code equations (7) to (10) (≈4 lines)
    ##dit = None
    ##dft = None
    ##dot = None
    ##dcct = None
    concat = np.concatenate((a_prev, xt), axis=0)

    # Compute parameters related derivatives. Use equations (11)-(14) (≈8 lines)
    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.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)
    ### 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

錯誤記錄：

parameters['Wf'][:, :n_a].T 每一行的 第 0 到 n_a-1 列的數據取出來

parameters['Wf'][:, n_a:].T 每一行的 第 n_a 到最後列的數據 取出來

dcct = (dc_next * it + ot * (1 - np.square(np.tanh(c_next))) * it * da_next) * (1 - np.square(cct))

np.random.seed(1)
xt = np.random.randn(3,10)
a_prev = np.random.randn(5,10)
c_prev = np.random.randn(5,10)
Wf = np.random.randn(5, 5+3)
bf = np.random.randn(5,1)
Wi = np.random.randn(5, 5+3)
bi = np.random.randn(5,1)
Wo = np.random.randn(5, 5+3)
bo = np.random.randn(5,1)
Wc = np.random.randn(5, 5+3)
bc = np.random.randn(5,1)
Wy = np.random.randn(2,5)
by = np.random.randn(2,1)

parameters = {"Wf": Wf, "Wi": Wi, "Wo": Wo, "Wc": Wc, "Wy": Wy, "bf": bf, "bi": bi, "bo": bo, "bc": bc, "by": by}

a_next, c_next, yt, cache = lstm_cell_forward(xt, a_prev, c_prev, parameters)

da_next = np.random.randn(5,10)
dc_next = np.random.randn(5,10)
gradients = lstm_cell_backward(da_next, dc_next, cache)
print("gradients[\"dxt\"][1][2] =", gradients["dxt"][1][2])
print("gradients[\"dxt\"].shape =", gradients["dxt"].shape)
print("gradients[\"da_prev\"][2][3] =", gradients["da_prev"][2][3])
print("gradients[\"da_prev\"].shape =", gradients["da_prev"].shape)
print("gradients[\"dc_prev\"][2][3] =", gradients["dc_prev"][2][3])
print("gradients[\"dc_prev\"].shape =", gradients["dc_prev"].shape)
print("gradients[\"dWf\"][3][1] =", gradients["dWf"][3][1])
print("gradients[\"dWf\"].shape =", gradients["dWf"].shape)
print("gradients[\"dWi\"][1][2] =", gradients["dWi"][1][2])
print("gradients[\"dWi\"].shape =", gradients["dWi"].shape)
print("gradients[\"dWc\"][3][1] =", gradients["dWc"][3][1])
print("gradients[\"dWc\"].shape =", gradients["dWc"].shape)
print("gradients[\"dWo\"][1][2] =", gradients["dWo"][1][2])
print("gradients[\"dWo\"].shape =", gradients["dWo"].shape)
print("gradients[\"dbf\"][4] =", gradients["dbf"][4])
print("gradients[\"dbf\"].shape =", gradients["dbf"].shape)
print("gradients[\"dbi\"][4] =", gradients["dbi"][4])
print("gradients[\"dbi\"].shape =", gradients["dbi"].shape)
print("gradients[\"dbc\"][4] =", gradients["dbc"][4])
print("gradients[\"dbc\"].shape =", gradients["dbc"].shape)
print("gradients[\"dbo\"][4] =", gradients["dbo"][4])
print("gradients[\"dbo\"].shape =", gradients["dbo"].shape)

gradients["dxt"][1][2] = 3.230559115109188
gradients["dxt"].shape = (3, 10)
gradients["da_prev"][2][3] = -0.06396214197109241
gradients["da_prev"].shape = (5, 10)
gradients["dc_prev"][2][3] = 0.7975220387970015
gradients["dc_prev"].shape = (5, 10)
gradients["dWf"][3][1] = -0.14795483816449725
gradients["dWf"].shape = (5, 8)
gradients["dWi"][1][2] = 1.0574980552259903
gradients["dWi"].shape = (5, 8)
gradients["dWc"][3][1] = 2.3045621636876668
gradients["dWc"].shape = (5, 8)
gradients["dWo"][1][2] = 0.3313115952892108
gradients["dWo"].shape = (5, 8)
gradients["dbf"][4] = [0.18864637]
gradients["dbf"].shape = (5, 1)
gradients["dbi"][4] = [-0.40142491]
gradients["dbi"].shape = (5, 1)
gradients["dbc"][4] = [0.25587763]
gradients["dbc"].shape = (5, 1)
gradients["dbo"][4] = [0.13893342]
gradients["dbo"].shape = (5, 1)

Expected Output:

**gradients[“dxt”][1][2]** =	3.23055911511
**gradients[“dxt”].shape** =	(3, 10)
**gradients[“da_prev”][2][3]** =	-0.0639621419711
**gradients[“da_prev”].shape** =	(5, 10)
**gradients[“dc_prev”][2][3]** =	0.797522038797
**gradients[“dc_prev”].shape** =	(5, 10)
**gradients[“dWf”][3][1]** =	-0.147954838164
**gradients[“dWf”].shape** =	(5, 8)
**gradients[“dWi”][1][2]** =	1.05749805523
**gradients[“dWi”].shape** =	(5, 8)
**gradients[“dWc”][3][1]** =	2.30456216369
**gradients[“dWc”].shape** =	(5, 8)
**gradients[“dWo”][1][2]** =	0.331311595289
**gradients[“dWo”].shape** =	(5, 8)
**gradients[“dbf”][4]** =	[ 0.18864637]
**gradients[“dbf”].shape** =	(5, 1)
**gradients[“dbi”][4]** =	[-0.40142491]
**gradients[“dbi”].shape** =	(5, 1)
**gradients[“dbc”][4]** =	[ 0.25587763]
**gradients[“dbc”].shape** =	(5, 1)
**gradients[“dbo”][4]** =	[ 0.13893342]
**gradients[“dbo”].shape** =	(5, 1)

3.3 Backward pass through the LSTM RNN

This part is very similar to the rnn_backward function you implemented above. You will first create variables of the same dimension as your return variables. You will then iterate over all the time steps starting from the end and call the one step function you implemented for LSTM at each iteration. You will then update the parameters by summing them individually. Finally return a dictionary with the new gradients.

Instructions: Implement the lstm_backward function. Create a for loop starting from Tx$T_x$ and going backward. For each step call lstm_cell_backward and update the your old gradients by adding the new gradients to them. Note that dxt is not updated but is stored.

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, 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))



    # 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])
        # Store or add the gradient to the parameters' previous step's gradient
        dx[:,:,t] = gradients["dxt"]
        dWf = gradients["dWf"]
        dWi = gradients["dWi"]
        dWc = gradients["dWc"]
        dWo = gradients["dWo"]
        dbf = gradients["dbf"]
        dbi = gradients["dbi"]
        dbc = gradients["dbc"]
        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

np.random.seed(1)
x = np.random.randn(3,10,7)
a0 = np.random.randn(5,10)
Wf = np.random.randn(5, 5+3)
bf = np.random.randn(5,1)
Wi = np.random.randn(5, 5+3)
bi = np.random.randn(5,1)
Wo = np.random.randn(5, 5+3)
bo = np.random.randn(5,1)
Wc = np.random.randn(5, 5+3)
bc = np.random.randn(5,1)

parameters = {"Wf": Wf, "Wi": Wi, "Wo": Wo, "Wc": Wc, "Wy": Wy, "bf": bf, "bi": bi, "bo": bo, "bc": bc, "by": by}

a, y, c, caches = lstm_forward(x, a0, parameters)

da = np.random.randn(5, 10, 4)
gradients = lstm_backward(da, caches)

print("gradients[\"dx\"][1][2] =", gradients["dx"][1][2])
print("gradients[\"dx\"].shape =", gradients["dx"].shape)
print("gradients[\"da0\"][2][3] =", gradients["da0"][2][3])
print("gradients[\"da0\"].shape =", gradients["da0"].shape)
print("gradients[\"dWf\"][3][1] =", gradients["dWf"][3][1])
print("gradients[\"dWf\"].shape =", gradients["dWf"].shape)
print("gradients[\"dWi\"][1][2] =", gradients["dWi"][1][2])
print("gradients[\"dWi\"].shape =", gradients["dWi"].shape)
print("gradients[\"dWc\"][3][1] =", gradients["dWc"][3][1])
print("gradients[\"dWc\"].shape =", gradients["dWc"].shape)
print("gradients[\"dWo\"][1][2] =", gradients["dWo"][1][2])
print("gradients[\"dWo\"].shape =", gradients["dWo"].shape)
print("gradients[\"dbf\"][4] =", gradients["dbf"][4])
print("gradients[\"dbf\"].shape =", gradients["dbf"].shape)
print("gradients[\"dbi\"][4] =", gradients["dbi"][4])
print("gradients[\"dbi\"].shape =", gradients["dbi"].shape)
print("gradients[\"dbc\"][4] =", gradients["dbc"][4])
print("gradients[\"dbc\"].shape =", gradients["dbc"].shape)
print("gradients[\"dbo\"][4] =", gradients["dbo"][4])
print("gradients[\"dbo\"].shape =", gradients["dbo"].shape)

gradients["dx"][1][2] = [-0.00173313  0.08287442 -0.30545663 -0.43281115]
gradients["dx"].shape = (3, 10, 4)
gradients["da0"][2][3] = -0.09591150195400469
gradients["da0"].shape = (5, 10)
gradients["dWf"][3][1] = 0.0
gradients["dWf"].shape = (5, 8)
gradients["dWi"][1][2] = -0.062451768586453626
gradients["dWi"].shape = (5, 8)
gradients["dWc"][3][1] = -0.0002859524353767697
gradients["dWc"].shape = (5, 8)
gradients["dWo"][1][2] = -0.0024024489673653724
gradients["dWo"].shape = (5, 8)
gradients["dbf"][4] = [0.]
gradients["dbf"].shape = (5, 1)
gradients["dbi"][4] = [-0.01339511]
gradients["dbi"].shape = (5, 1)
gradients["dbc"][4] = [-0.21783941]
gradients["dbc"].shape = (5, 1)
gradients["dbo"][4] = [0.22190701]
gradients["dbo"].shape = (5, 1)

Expected Output:

**gradients[“dx”][1][2]** =	[-0.00173313 0.08287442 -0.30545663 -0.43281115]
**gradients[“dx”].shape** =	(3, 10, 4)
**gradients[“da0”][2][3]** =	-0.095911501954
**gradients[“da0”].shape** =	(5, 10)
**gradients[“dWf”][3][1]** =	-0.0698198561274
**gradients[“dWf”].shape** =	(5, 8)
**gradients[“dWi”][1][2]** =	0.102371820249
**gradients[“dWi”].shape** =	(5, 8)
**gradients[“dWc”][3][1]** =	-0.0624983794927
**gradients[“dWc”].shape** =	(5, 8)
**gradients[“dWo”][1][2]** =	0.0484389131444
**gradients[“dWo”].shape** =	(5, 8)
**gradients[“dbf”][4]** =	[-0.0565788]
**gradients[“dbf”].shape** =	(5, 1)
**gradients[“dbi”][4]** =	[-0.06997391]
**gradients[“dbi”].shape** =	(5, 1)
**gradients[“dbc”][4]** =	[-0.27441821]
**gradients[“dbc”].shape** =	(5, 1)
**gradients[“dbo”][4]** =	[ 0.16532821]
**gradients[“dbo”].shape** =	(5, 1)

Congratulations !

Congratulations on completing this assignment. You now understand how recurrent neural networks work!

Lets go on to the next exercise, where you’ll use an RNN to build a character-level language model.

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