## [深度学习] 吴恩达深度学习公开课第五课 序列模型第一周 一步一步实现循环神经网络 作业一答案

### 表示这个东西有点头疼，下面是代码

``````import numpy as np
from rnn_utils import *

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
a_next = np.tanh(np.dot(Waa,a_prev) + np.dot(Wax,xt) + 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

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 parameters["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

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)
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.dot(Wf,concat) + bf)
it = sigmoid(np.dot(Wi,concat) + bi)
cct = np.tanh(np.dot(Wc,concat) + bc)
c_next = it * cct + ft * c_prev
ot = sigmoid(np.dot(Wo,concat) + bo)
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

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 parameters['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

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_cell_forward())

Returns:
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 - np.square(a_next)) * 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}

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:
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)
(cache, x) = caches
(a1, a0, x1, parameters) = cache[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)
# Retrieve derivatives from gradients (≈ 1 line)
# 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}

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:
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

# Compute parameters related derivatives. Use equations (11)-(14) (≈8 lines)
concat = np.concatenate((a_prev, xt), axis=0).T
dWf = np.dot(dft, concat)
dWi = np.dot(dit, concat)
dWc = np.dot(dcct, concat)
dWo = np.dot(dot, concat)
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['Wc'][:, :n_a].T,dcct) +np.dot(parameters['Wi'][:, :n_a].T,dit)+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['Wc'][:, n_a:].T,dcct) + np.dot(parameters['Wi'][:, n_a:].T,dit) + \
np.dot(parameters['Wo'][:, n_a:].T,dot)
### END CODE HERE ###

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}

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:
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

### 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}