Welcome to Course 5's first assignment! In this assignment, you will implement key components of a 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 $x^{\langle t \rangle}$ (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 unidirectional RNN to take information from the past to process later inputs. A bidirectional RNN can take context from both the past and the future.
Notation:
Superscript $[l]$ denotes an object associated with the $l^{th}$ layer.
Superscript $(i)$ denotes an object associated with the $i^{th}$ example.
Superscript $\langle t \rangle$ denotes an object at the $t^{th}$ time-step.
Subscript $i$ denotes the $i^{th}$ entry of a vector.
Example:
numpy
.#### START CODE HERE
and
#### END CODE HERE
# GRADED FUNCTION: routine_name
rnn_cell_forward
:rnn_forward
xt
in a separate line of code to clarify what code is expected; added additional hints.lstm_forward
c
and c_next
to help students avoid copy-by-reference mistakes.xt
in a separate line to make this step explicit.Let's first import all the packages that you will need during this assignment.
import numpy as np
from rnn_utils import *
Later this week, you will generate music using an RNN. The basic RNN that you will implement has the structure below. In this example, $T_x = T_y$.
xt
.a_prev
or a_next
, depending on the function that's being implemented.y_pred
: $\hat{y}$yt_pred
: $\hat{y}^{\langle t \rangle}$Here's how you can implement an RNN:
Steps:
A recurrent neural network can be seen as the repeated use 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.
rnn_cell_forward
, also calculates the prediction $\hat{y}^{\langle t \rangle}$Exercise: Implement the RNN-cell described in Figure (2).
Instructions:
softmax
.cache
.cache
softmax
function that you can use. It is located in the file 'rnn_utils.py' and has been imported.# 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(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
np.random.seed(1)
xt_tmp = np.random.randn(3,10)
a_prev_tmp = np.random.randn(5,10)
parameters_tmp = {}
parameters_tmp['Waa'] = np.random.randn(5,5)
parameters_tmp['Wax'] = np.random.randn(5,3)
parameters_tmp['Wya'] = np.random.randn(2,5)
parameters_tmp['ba'] = np.random.randn(5,1)
parameters_tmp['by'] = np.random.randn(2,1)
a_next_tmp, yt_pred_tmp, cache_tmp = rnn_cell_forward(xt_tmp, a_prev_tmp, parameters_tmp)
print("a_next[4] = \n", a_next_tmp[4])
print("a_next.shape = \n", a_next_tmp.shape)
print("yt_pred[1] =\n", yt_pred_tmp[1])
print("yt_pred.shape = \n", yt_pred_tmp.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_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)
Exercise: Code the forward propagation of the RNN described in Figure (3).
Instructions:
a_next
by setting it equal to the initial hidden state, $a_{0}$.a_next
), the prediction $\hat{y}^{\langle t \rangle}$ and the cache by running rnn_cell_forward
.yt_pred
) in the 3D tensor $\hat{y}_{pred}$ at the $t^{th}$ position.var_name[:,:,i]
.# 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 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
np.random.seed(1)
x_tmp = np.random.randn(3,10,4)
a0_tmp = np.random.randn(5,10)
parameters_tmp = {}
parameters_tmp['Waa'] = np.random.randn(5,5)
parameters_tmp['Wax'] = np.random.randn(5,3)
parameters_tmp['Wya'] = np.random.randn(2,5)
parameters_tmp['ba'] = np.random.randn(5,1)
parameters_tmp['by'] = np.random.randn(2,1)
a_tmp, y_pred_tmp, caches_tmp = rnn_forward(x_tmp, a0_tmp, parameters_tmp)
print("a[4][1] = \n", a_tmp[4][1])
print("a.shape = \n", a_tmp.shape)
print("y_pred[1][3] =\n", y_pred_tmp[1][3])
print("y_pred.shape = \n", y_pred_tmp.shape)
print("caches[1][1][3] =\n", caches_tmp[1][1][3])
print("len(caches) = \n", len(caches_tmp))
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_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
Congratulations! You've successfully built the forward propagation of a recurrent neural network from scratch.
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.
The following figure shows the operations of an LSTM-cell.
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 $T_x$ time-steps.
The variable names in the code are similar to the equations, with slight differences.
Wf
: forget gate weight $\mathbf{W}_{f}$Wb
: forget gate bias $\mathbf{W}_{b}$ft
: forget gate $\Gamma_f^{\langle t \rangle}$cct
: candidate value $\mathbf{\tilde{c}}^{\langle t \rangle}$In the code, we'll use the variable names found in the academic literature. These variables don't use "u" to denote "update".
Wi
is the update gate weight $\mathbf{W}_i$ (not "Wu")bi
is the update gate bias $\mathbf{b}_i$ (not "bu")it
is the forget gate $\mathbf{\Gamma}_i^{\langle t \rangle}$ (not "ut")c
: cell state, including all time steps, $\mathbf{c}$ shape $(n_{a}, m, T)$c_next
: new (next) cell state, $\mathbf{c}^{\langle t \rangle}$ shape $(n_{a}, m)$c_prev
: previous cell state, $\mathbf{c}^{\langle t-1 \rangle}$, shape $(n_{a}, m)$Wo
: output gate weight, $\mathbf{W_o}$bo
: output gate bias, $\mathbf{b_o}$ot
: output gate, $\mathbf{\Gamma}_{o}^{\langle t \rangle}$a
: hidden state, including time steps. $\mathbf{a}$ has shape $(n_{a}, m, T_{x})$a_next
: hidden state for next time step. $\mathbf{a}^{\langle t \rangle}$ has shape $(n_{a}, m)$The equation is: $$\mathbf{y}^{\langle t \rangle}_{pred} = \textrm{softmax}(\mathbf{W}_{y} \mathbf{a}^{\langle t \rangle} + \mathbf{b}_{y})$$
y_pred
: prediction, including all time steps. $\mathbf{y}_{pred}$ has shape $(n_{y}, m, T_{x})$. Note that $(T_{y} = T_{x})$ for this example.yt_pred
: prediction for the current time step $t$. $\mathbf{y}^{\langle t \rangle}_{pred}$ has shape $(n_{y}, m)$Exercise: Implement the LSTM cell described in the Figure (4).
Instructions:
axis
parameter.sigmoid()
and softmax
are imported from rnn_utils.py
.Wi
, bi
refer to the weights and biases of the update gate. There are no variables named "Wu" or "bu" in this function.# 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 cell state (memory)
"""
# Retrieve parameters from "parameters"
Wf = parameters["Wf"] # forget gate weight
bf = parameters["bf"]
Wi = parameters["Wi"] # update gate weight (notice the variable name)
bi = parameters["bi"] # (notice the variable name)
Wc = parameters["Wc"] # candidate value weight
bc = parameters["bc"]
Wo = parameters["Wo"] # output gate weight
bo = parameters["bo"]
Wy = parameters["Wy"] # prediction weight
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 = 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_tmp = np.random.randn(3,10)
a_prev_tmp = np.random.randn(5,10)
c_prev_tmp = np.random.randn(5,10)
parameters_tmp = {}
parameters_tmp['Wf'] = np.random.randn(5, 5+3)
parameters_tmp['bf'] = np.random.randn(5,1)
parameters_tmp['Wi'] = np.random.randn(5, 5+3)
parameters_tmp['bi'] = np.random.randn(5,1)
parameters_tmp['Wo'] = np.random.randn(5, 5+3)
parameters_tmp['bo'] = np.random.randn(5,1)
parameters_tmp['Wc'] = np.random.randn(5, 5+3)
parameters_tmp['bc'] = np.random.randn(5,1)
parameters_tmp['Wy'] = np.random.randn(2,5)
parameters_tmp['by'] = np.random.randn(2,1)
a_next_tmp, c_next_tmp, yt_tmp, cache_tmp = lstm_cell_forward(xt_tmp, a_prev_tmp, c_prev_tmp, parameters_tmp)
print("a_next[4] = \n", a_next_tmp[4])
print("a_next.shape = ", c_next_tmp.shape)
print("c_next[2] = \n", c_next_tmp[2])
print("c_next.shape = ", c_next_tmp.shape)
print("yt[1] =", yt_tmp[1])
print("yt.shape = ", yt_tmp.shape)
print("cache[1][3] =\n", cache_tmp[1][3])
print("len(cache) = ", len(cache_tmp))
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
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 $T_x$ inputs.
Exercise: Implement lstm_forward()
to run an LSTM over $T_x$ time-steps.
Instructions
x
and parameters
.a_next
.a0
.c_next
.c_next
as its own variable with its own location in memory. Do not initialize it as a slice of the 3D tensor $c$. In other words, don't do c_next = c[:,:,0]
.lstm_cell_forward
function that you defined previously, to get the hidden state, cell state, prediction, and cache.# 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 (4).
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)
c -- The value of the cell state, numpy array of shape (n_a, 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 ###
Wy = parameters['Wy'] # saving parameters['Wy'] in a local variable in case students use Wy instead of parameters['Wy']
# Retrieve dimensions from shapes of x and parameters['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.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_tmp = np.random.randn(3,10,7)
a0_tmp = np.random.randn(5,10)
parameters_tmp = {}
parameters_tmp['Wf'] = np.random.randn(5, 5+3)
parameters_tmp['bf'] = np.random.randn(5,1)
parameters_tmp['Wi'] = np.random.randn(5, 5+3)
parameters_tmp['bi']= np.random.randn(5,1)
parameters_tmp['Wo'] = np.random.randn(5, 5+3)
parameters_tmp['bo'] = np.random.randn(5,1)
parameters_tmp['Wc'] = np.random.randn(5, 5+3)
parameters_tmp['bc'] = np.random.randn(5,1)
parameters_tmp['Wy'] = np.random.randn(2,5)
parameters_tmp['by'] = np.random.randn(2,1)
a_tmp, y_tmp, c_tmp, caches_tmp = lstm_forward(x_tmp, a0_tmp, parameters_tmp)
print("a[4][3][6] = ", a_tmp[4][3][6])
print("a.shape = ", a_tmp.shape)
print("y[1][4][3] =", y_tmp[1][4][3])
print("y.shape = ", y_tmp.shape)
print("caches[1][1][1] =\n", caches_tmp[1][1][1])
print("c[1][2][1]", c_tmp[1][2][1])
print("len(caches) = ", len(caches_tmp))
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
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.
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 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.
We will start by computing the backward pass for the basic RNN-cell.
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 $\tanh$ is $1-\tanh(x)^2$. You can find the complete proof here. Note that: $ \text{sech}(x)^2 = 1 - \tanh(x)^2$
Similarly for $\frac{ \partial a^{\langle t \rangle} } {\partial W_{ax}}, \frac{ \partial a^{\langle t \rangle} } {\partial W_{aa}}, \frac{ \partial a^{\langle t \rangle} } {\partial b}$, the derivative of $\tanh(u)$ is $(1-\tanh(u)^2)du$.
The final two equations also follow the same rule and are derived using the $\tanh$ 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).
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-a_next*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,keepdims=True,axis=-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_tmp = np.random.randn(3,10)
a_prev_tmp = np.random.randn(5,10)
parameters_tmp = {}
parameters_tmp['Wax'] = np.random.randn(5,3)
parameters_tmp['Waa'] = np.random.randn(5,5)
parameters_tmp['Wya'] = np.random.randn(2,5)
parameters_tmp['ba'] = np.random.randn(5,1)
parameters_tmp['by'] = np.random.randn(2,1)
a_next_tmp, yt_tmp, cache_tmp = rnn_cell_forward(xt_tmp, a_prev_tmp, parameters_tmp)
da_next_tmp = np.random.randn(5,10)
gradients_tmp = rnn_cell_backward(da_next_tmp, cache_tmp)
print("gradients[\"dxt\"][1][2] =", gradients_tmp["dxt"][1][2])
print("gradients[\"dxt\"].shape =", gradients_tmp["dxt"].shape)
print("gradients[\"da_prev\"][2][3] =", gradients_tmp["da_prev"][2][3])
print("gradients[\"da_prev\"].shape =", gradients_tmp["da_prev"].shape)
print("gradients[\"dWax\"][3][1] =", gradients_tmp["dWax"][3][1])
print("gradients[\"dWax\"].shape =", gradients_tmp["dWax"].shape)
print("gradients[\"dWaa\"][1][2] =", gradients_tmp["dWaa"][1][2])
print("gradients[\"dWaa\"].shape =", gradients_tmp["dWaa"].shape)
print("gradients[\"dba\"][4] =", gradients_tmp["dba"][4])
print("gradients[\"dba\"].shape =", gradients_tmp["dba"].shape)
gradients["dxt"][1][2] = -1.3872130506 gradients["dxt"].shape = (3, 10) gradients["da_prev"][2][3] = -0.152399493774 gradients["da_prev"].shape = (5, 10) gradients["dWax"][3][1] = 0.410772824935 gradients["dWax"].shape = (5, 3) gradients["dWaa"][1][2] = 1.15034506685 gradients["dWaa"].shape = (5, 5) gradients["dba"][4] = [ 0.20023491] 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) |
Computing the gradients of the cost with respect to $a^{\langle t \rangle}$ at every time-step $t$ 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 $db_a$, $dW_{aa}$, $dW_{ax}$ and you store $dx$.
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 array of shape (n_a, n_a)
dba -- Gradient w.r.t the bias, of shape (n_a, 1)
"""
### START CODE HERE ###
# Retrieve values from the first cache (t=1) of caches (≈2 lines)
(caches, x) = caches
(a1, a0, x1, parameters) = caches[0]
# Retrieve dimensions from da's and x1's shapes (≈2 lines)
n_a, m, T_x = da.shape
n_x, m = x1.shape
# initialize the gradients with the right sizes (≈6 lines)
dx = np.zeros((n_x, m, T_x))
dWax = np.zeros((n_a, n_x))
dWaa = np.zeros((n_a, n_a))
dba = np.zeros((n_a, 1))
da0 = np.zeros((n_a, m))
da_prevt = np.zeros((n_a, m))
# Loop through all the time steps
for t in reversed(range(T_x)):
# Compute gradients at time step t. Choose wisely the "da_next" and the "cache" to use in the backward propagation step. (≈1 line)
gradients = rnn_cell_backward(da[:, :, t] + da_prevt, caches[t])
# Retrieve derivatives from gradients (≈ 1 line)
dxt, da_prevt, dWaxt, dWaat, dbat = gradients["dxt"], gradients["da_prev"], gradients["dWax"], gradients["dWaa"], gradients["dba"]
# Increment global derivatives w.r.t parameters by adding their derivative at time-step t (≈4 lines)
dx[:, :, t] = dxt
dWax += 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
np.random.seed(1)
x_tmp = np.random.randn(3,10,4)
a0_tmp = np.random.randn(5,10)
parameters_tmp = {}
parameters_tmp['Wax'] = np.random.randn(5,3)
parameters_tmp['Waa'] = np.random.randn(5,5)
parameters_tmp['Wya'] = np.random.randn(2,5)
parameters_tmp['ba'] = np.random.randn(5,1)
parameters_tmp['by'] = np.random.randn(2,1)
a_tmp, y_tmp, caches_tmp = rnn_forward(x_tmp, a0_tmp, parameters_tmp)
da_tmp = np.random.randn(5, 10, 4)
gradients_tmp = rnn_backward(da_tmp, caches_tmp)
print("gradients[\"dx\"][1][2] =", gradients_tmp["dx"][1][2])
print("gradients[\"dx\"].shape =", gradients_tmp["dx"].shape)
print("gradients[\"da0\"][2][3] =", gradients_tmp["da0"][2][3])
print("gradients[\"da0\"].shape =", gradients_tmp["da0"].shape)
print("gradients[\"dWax\"][3][1] =", gradients_tmp["dWax"][3][1])
print("gradients[\"dWax\"].shape =", gradients_tmp["dWax"].shape)
print("gradients[\"dWaa\"][1][2] =", gradients_tmp["dWaa"][1][2])
print("gradients[\"dWaa\"].shape =", gradients_tmp["dWaa"].shape)
print("gradients[\"dba\"][4] =", gradients_tmp["dba"][4])
print("gradients[\"dba\"].shape =", gradients_tmp["dba"].shape)
gradients["dx"][1][2] = [-2.07101689 -0.59255627 0.02466855 0.01483317] gradients["dx"].shape = (3, 10, 4) gradients["da0"][2][3] = -0.314942375127 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)
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) |
The LSTM backward pass is slightly 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.)
To calculate $db_f, db_u, db_c, db_o$ you just need to sum across the horizontal (axis= 1) axis on $d\Gamma_f^{\langle t \rangle}, d\Gamma_u^{\langle t \rangle}, d\tilde c^{\langle t \rangle}, d\Gamma_o^{\langle t \rangle}$ 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.
$$ da_{prev} = W_f^T*d\Gamma_f^{\langle t \rangle} + W_u^T * d\Gamma_u^{\langle t \rangle}+ W_c^T * d\tilde c^{\langle t \rangle} + W_o^T * d\Gamma_o^{\langle t \rangle} \tag{15}$$Here, the weights for equations 13 are the first n_a, (i.e. $W_f = W_f[:n_a,:]$ etc...)
$$ dc_{prev} = dc_{next}\Gamma_f^{\langle t \rangle} + \Gamma_o^{\langle t \rangle} * (1- \tanh(c_{next})^2)*\Gamma_f^{\langle t \rangle}*da_{next} \tag{16}$$$$ dx^{\langle t \rangle} = W_f^T*d\Gamma_f^{\langle t \rangle} + W_u^T * d\Gamma_u^{\langle t \rangle}+ W_c^T * d\tilde c_t + W_o^T * d\Gamma_o^{\langle t \rangle}\tag{17} $$where the weights for equation 15 are from n_a to the end, (i.e. $W_f = W_f[n_a:,:]$ etc...)
Exercise: Implement lstm_cell_backward
by implementing equations $7-17$ 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)
# Code equations (7) to (10) (≈4 lines)
dit = (da_next * ot * (1 - np.tanh(c_next) ** 2) + dc_next) * cct * (1 - it) * it
dft = (da_next * ot * (1 - np.tanh(c_next) ** 2) + dc_next) * c_prev * ft * (1 - ft)
dot = da_next * np.tanh(c_next) * ot * (1 - ot)
dcct = (da_next * ot * (1 - np.tanh(c_next) ** 2) + dc_next) * it * (1 - cct ** 2)
# Compute parameters related derivatives. Use equations (11)-(14) (≈8 lines)
dWf = np.dot(dft,np.concatenate((a_prev, xt), axis=0).T) # or use np.dot(dft, np.hstack([a_prev.T, xt.T]))
dWi = np.dot(dit,np.concatenate((a_prev, xt), axis=0).T)
dWc = np.dot(dcct,np.concatenate((a_prev, xt), axis=0).T)
dWo = np.dot(dot,np.concatenate((a_prev, xt), axis=0).T)
dbf = np.sum(dft,axis=1,keepdims=True)
dbi = np.sum(dit,axis=1,keepdims=True)
dbc = np.sum(dcct,axis=1,keepdims=True)
dbo = np.sum(dot,axis=1,keepdims=True)
# 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
np.random.seed(1)
xt_tmp = np.random.randn(3,10)
a_prev_tmp = np.random.randn(5,10)
c_prev_tmp = np.random.randn(5,10)
parameters_tmp = {}
parameters_tmp['Wf'] = np.random.randn(5, 5+3)
parameters_tmp['bf'] = np.random.randn(5,1)
parameters_tmp['Wi'] = np.random.randn(5, 5+3)
parameters_tmp['bi'] = np.random.randn(5,1)
parameters_tmp['Wo'] = np.random.randn(5, 5+3)
parameters_tmp['bo'] = np.random.randn(5,1)
parameters_tmp['Wc'] = np.random.randn(5, 5+3)
parameters_tmp['bc'] = np.random.randn(5,1)
parameters_tmp['Wy'] = np.random.randn(2,5)
parameters_tmp['by'] = np.random.randn(2,1)
a_next_tmp, c_next_tmp, yt_tmp, cache_tmp = lstm_cell_forward(xt_tmp, a_prev_tmp, c_prev_tmp, parameters_tmp)
da_next_tmp = np.random.randn(5,10)
dc_next_tmp = np.random.randn(5,10)
gradients_tmp = lstm_cell_backward(da_next_tmp, dc_next_tmp, cache_tmp)
print("gradients[\"dxt\"][1][2] =", gradients_tmp["dxt"][1][2])
print("gradients[\"dxt\"].shape =", gradients_tmp["dxt"].shape)
print("gradients[\"da_prev\"][2][3] =", gradients_tmp["da_prev"][2][3])
print("gradients[\"da_prev\"].shape =", gradients_tmp["da_prev"].shape)
print("gradients[\"dc_prev\"][2][3] =", gradients_tmp["dc_prev"][2][3])
print("gradients[\"dc_prev\"].shape =", gradients_tmp["dc_prev"].shape)
print("gradients[\"dWf\"][3][1] =", gradients_tmp["dWf"][3][1])
print("gradients[\"dWf\"].shape =", gradients_tmp["dWf"].shape)
print("gradients[\"dWi\"][1][2] =", gradients_tmp["dWi"][1][2])
print("gradients[\"dWi\"].shape =", gradients_tmp["dWi"].shape)
print("gradients[\"dWc\"][3][1] =", gradients_tmp["dWc"][3][1])
print("gradients[\"dWc\"].shape =", gradients_tmp["dWc"].shape)
print("gradients[\"dWo\"][1][2] =", gradients_tmp["dWo"][1][2])
print("gradients[\"dWo\"].shape =", gradients_tmp["dWo"].shape)
print("gradients[\"dbf\"][4] =", gradients_tmp["dbf"][4])
print("gradients[\"dbf\"].shape =", gradients_tmp["dbf"].shape)
print("gradients[\"dbi\"][4] =", gradients_tmp["dbi"][4])
print("gradients[\"dbi\"].shape =", gradients_tmp["dbi"].shape)
print("gradients[\"dbc\"][4] =", gradients_tmp["dbc"][4])
print("gradients[\"dbc\"].shape =", gradients_tmp["dbc"].shape)
print("gradients[\"dbo\"][4] =", gradients_tmp["dbo"][4])
print("gradients[\"dbo\"].shape =", gradients_tmp["dbo"].shape)
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)
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) |
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 $T_x$ 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)
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] + da_prevt, 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_tmp = np.random.randn(3,10,7)
a0_tmp = np.random.randn(5,10)
parameters_tmp = {}
parameters_tmp['Wf'] = np.random.randn(5, 5+3)
parameters_tmp['bf'] = np.random.randn(5,1)
parameters_tmp['Wi'] = np.random.randn(5, 5+3)
parameters_tmp['bi'] = np.random.randn(5,1)
parameters_tmp['Wo'] = np.random.randn(5, 5+3)
parameters_tmp['bo'] = np.random.randn(5,1)
parameters_tmp['Wc'] = np.random.randn(5, 5+3)
parameters_tmp['bc'] = np.random.randn(5,1)
parameters_tmp['Wy'] = np.random.randn(2,5)
parameters_tmp['by'] = np.random.randn(2,1)
a_tmp, y_tmp, c_tmp, caches_tmp = lstm_forward(x_tmp, a0_tmp, parameters_tmp)
da_tmp = np.random.randn(5, 10, 4)
gradients_tmp = lstm_backward(da_tmp, caches_tmp)
print("gradients[\"dx\"][1][2] =", gradients_tmp["dx"][1][2])
print("gradients[\"dx\"].shape =", gradients_tmp["dx"].shape)
print("gradients[\"da0\"][2][3] =", gradients_tmp["da0"][2][3])
print("gradients[\"da0\"].shape =", gradients_tmp["da0"].shape)
print("gradients[\"dWf\"][3][1] =", gradients_tmp["dWf"][3][1])
print("gradients[\"dWf\"].shape =", gradients_tmp["dWf"].shape)
print("gradients[\"dWi\"][1][2] =", gradients_tmp["dWi"][1][2])
print("gradients[\"dWi\"].shape =", gradients_tmp["dWi"].shape)
print("gradients[\"dWc\"][3][1] =", gradients_tmp["dWc"][3][1])
print("gradients[\"dWc\"].shape =", gradients_tmp["dWc"].shape)
print("gradients[\"dWo\"][1][2] =", gradients_tmp["dWo"][1][2])
print("gradients[\"dWo\"].shape =", gradients_tmp["dWo"].shape)
print("gradients[\"dbf\"][4] =", gradients_tmp["dbf"][4])
print("gradients[\"dbf\"].shape =", gradients_tmp["dbf"].shape)
print("gradients[\"dbi\"][4] =", gradients_tmp["dbi"][4])
print("gradients[\"dbi\"].shape =", gradients_tmp["dbi"].shape)
print("gradients[\"dbc\"][4] =", gradients_tmp["dbc"][4])
print("gradients[\"dbc\"].shape =", gradients_tmp["dbc"].shape)
print("gradients[\"dbo\"][4] =", gradients_tmp["dbo"][4])
print("gradients[\"dbo\"].shape =", gradients_tmp["dbo"].shape)
gradients["dx"][1][2] = [ 0.01980463 -0.02745056 -0.31327706 0.53886581] gradients["dx"].shape = (3, 10, 4) gradients["da0"][2][3] = -0.000284495289749 gradients["da0"].shape = (5, 10) gradients["dWf"][3][1] = -0.0153890043327 gradients["dWf"].shape = (5, 8) gradients["dWi"][1][2] = -0.10924217935 gradients["dWi"].shape = (5, 8) gradients["dWc"][3][1] = 0.0793905844933 gradients["dWc"].shape = (5, 8) gradients["dWo"][1][2] = -0.0810144543621 gradients["dWo"].shape = (5, 8) gradients["dbf"][4] = [-0.24148921] gradients["dbf"].shape = (5, 1) gradients["dbi"][4] = [-0.08824333] gradients["dbi"].shape = (5, 1) gradients["dbc"][4] = [ 0.14411048] gradients["dbc"].shape = (5, 1) gradients["dbo"][4] = [-0.45977321] 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 on completing this assignment. You now understand how recurrent neural networks work!
Let's go on to the next exercise, where you'll use an RNN to build a character-level language model.