吴恩达 Coursera Deep Learning 第五课 Sequence Models 第一周编程作业 1（部分选做）

Building your Recurrent Neural Network - Step by Step

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.

Notation:

Superscript [l][l] denotes
an object associated with the lthlth layer.

Example: a[4]a[4] is
the 4th4th layer
activation. W[5]W[5] and b[5]b[5] are
the 5th5th layer
parameters.

Superscript (i)(i) denotes
an object associated with the ithith example.

Example: x(i)x(i) is
the ithith training
example input.

Superscript ⟨t⟩⟨t⟩ denotes
an object at the tthtth time-step.

Example: x⟨t⟩x⟨t⟩ is
the input x at the tthtth time-step. x(i)⟨t⟩x(i)⟨t⟩ is
the input at the tthtth timestep
of example ii.

Lowerscript ii denotes
the ithith entry
of a vector.

Example: a[l]iai[l] denotes
the ithith entry
of the activations in layer ll.

We assume that you are already familiar with

numpy

and/or have completed the previous courses of the specialization.
Let's get started!

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

import numpy as np
from rnn_utils import *

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=TyTx=Ty.


Figure 1:
Basic RNN model

Here's how you can implement an RNN:

Steps:

Implement the calculations needed for one time-step of the RNN.
Implement a loop over TxTx 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 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.



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:

Compute the hidden state with tanh activation: a⟨t⟩=tanh(Waaa⟨t−1⟩+Waxx⟨t⟩+ba)a⟨t⟩=tanh⁡(Waaa⟨t−1⟩+Waxx⟨t⟩+ba).
Using your new hidden state a⟨t⟩a⟨t⟩,
compute the prediction ŷ ⟨t⟩=softmax(Wyaa⟨t⟩+by)y^⟨t⟩=softmax(Wyaa⟨t⟩+by).
We provided you a function:

softmax

.
Store (a⟨t⟩,a⟨t−1⟩,x⟨t⟩,parameters)(a⟨t⟩,a⟨t−1⟩,x⟨t⟩,parameters) in
cache
Return a⟨t⟩a⟨t⟩ , y⟨t⟩y⟨t⟩ and
cache

We will vectorize over mm examples.
Thus, x⟨t⟩x⟨t⟩ will
have dimension (nx,m)(nx,m),
and a⟨t⟩a⟨t⟩ will
have dimension (na,m)(na,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
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

np.random.seed(1)
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⟨Tx⟩) is carried over TxTx time steps. The network outputs y=(y⟨1⟩,y⟨2⟩,...,y⟨Tx⟩)y=(y⟨1⟩,y⟨2⟩,...,y⟨Tx⟩).Exercise: Code the forward propagation of the RNN described in Figure (3).Instructions:
Create a vector of zeros (aa) that will store all the hidden states computed by the RNN.
Initialize the "next" hidden state as a0a0 (initial hidden state).
Start looping over each time step, your incremental index is tt :Update the "next" hidden state and the cache by running [code]rnn_cell_forward

Store the "next" hidden state in aa (tthtth position)
Store the prediction in y
Add the cache to the list of caches

Return a, y and cachesyy a

# 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 = (cache)

### END CODE HERE ###

# store values needed for backward propagation in cache
caches = (caches, x)

return a, y_pred, caches

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[/code]
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 $y^{\langle t \rangle}$ can be estimated using mainly "local" context (meaning information from inputs $x^{\langle t' \rangle}$ where $t'$ is not too far from $t$).

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.
2 - Long Short-Term Memory (LSTM) network
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 TxTx time-steps.

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, such as whether the subject is singular or plural. 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:

Γ⟨t⟩f=σ(Wf[a⟨t−1⟩,x⟨t⟩]+bf)(1)

Here, WfWf are
weights that govern the forget gate's behavior. We concatenate [a⟨t−1⟩,x⟨t⟩][a⟨t−1⟩,x⟨t⟩] and
multiply by WfWf.
The equation above results in a vector Γ⟨t⟩fΓf⟨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⟩fΓf⟨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.

- 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:

Γ⟨t⟩u=σ(Wu[a⟨t−1⟩,x{t}]+bu)(2)

Similar to the forget gate, here Γ⟨t⟩uΓu⟨t⟩ is
again a vector of values between 0 and 1. This will be multiplied element-wise with c̃ ⟨t⟩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:

c̃ ⟨t⟩=tanh(Wc[a⟨t−1⟩,x⟨t⟩]+bc)(3)

Finally, the new cell state is:

c⟨t⟩=Γ⟨t⟩f∗c⟨t−1⟩+Γ⟨t⟩u∗c̃ ⟨t⟩(4)

- 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)

a⟨t⟩=Γ⟨t⟩o∗tanh(c⟨t⟩)(6)

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

2.1 - LSTM cell

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

Instructions:

Concatenate a⟨t−1⟩a⟨t−1⟩ and x⟨t⟩x⟨t⟩ in
a single matrix: concat=[a⟨t−1⟩x⟨t⟩]concat=[a⟨t−1⟩x⟨t⟩]
Compute all the formulas 1-6. You can use

sigmoid()

(provided) and

np.tanh()

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

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

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 TxTx inputs.

Figure 4: LSTM over multiple time-steps.

Exercise: Implement

lstm_forward()

to run an LSTM over TxTx 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 = []
Wy = parameters["Wy"]

### START CODE HERE ###
# Retrieve dimensions from shapes of x and Wy (≈2 lines)
n_x, m, T_x = x.shape
n_y, n_a = 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 = (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] =  0.172117767533
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

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.

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

JJ 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)(∂J∂Wax,∂J∂Waa,∂J∂b) to
update the parameters (Wax,Waa,ba)(Wax,Waa,ba).

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 tanhtanh is 1−tanh(x)21−tanh⁡(x)2.
You can find the complete proof here. Note that: sec(x)2=1−tanh(x)2sec⁡(x)2=1−tanh⁡(x)2

Similarly for ∂a⟨t⟩∂Wax,∂a⟨t⟩∂Waa,∂a⟨t⟩∂b,
the derivative of tanh(u)tanh⁡(u) is (1−tanh(u)2)du(1

The final two equations also follow same rule and are derived using the tanhtanh 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).

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:
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=1)

### 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] = -0.460564103059
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)