深度学习-序列模型 吴恩达第五课第一周作业1答案(Building your Recurrent Neural Network - Step by Step)
import numpy as np
from rnn_utils import *# 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, cachenp.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)
# GRADED FUNCTION: rnn_forward
def rnn_forward(x, a0, parameters):
"""
Implement the forward propagation of the recurrent neural network described in Figure (3).
Arguments:
x -- Input data for every time-step, of shape (n_x, m, T_x).
a0 -- Initial hidden state, of shape (n_a, m)
parameters -- python dictionary containing:
Waa -- Weight matrix multiplying the hidden state, numpy array of shape (n_a, n_a)
Wax -- Weight matrix multiplying the input, numpy array of shape (n_a, n_x)
Wya -- Weight matrix relating the hidden-state to the output, numpy array of shape (n_y, n_a)
ba -- Bias numpy array of shape (n_a, 1)
by -- Bias relating the hidden-state to the output, numpy array of shape (n_y, 1)
Returns:
a -- Hidden states for every time-step, numpy array of shape (n_a, m, T_x)
y_pred -- Predictions for every time-step, numpy array of shape (n_y, m, T_x)
caches -- tuple of values needed for the backward pass, contains (list of caches, x)
"""
# Initialize "caches" which will contain the list of all caches
caches = []
# Retrieve dimensions from shapes of x and Wy
n_x, m, T_x = x.shape
n_y, n_a = parameters["Wya"].shape
### START CODE HERE ###
# initialize "a" and "y" with zeros (≈2 lines)
a = np.zeros((n_a, m, T_x))
y_pred = np.zeros((n_y, m, T_x))
# Initialize a_next (≈1 line)
a_next = a0
# loop over all time-steps
for t in range(T_x):
# Update next hidden state, compute the prediction, get the cache (≈1 line)
a_next, yt_pred, cache = rnn_cell_forward(x[:,:,t], a_next, parameters)
# Save the value of the new "next" hidden state in a (≈1 line)
a[:,:,t] = a_next
# Save the value of the prediction in y (≈1 line)
y_pred[:,:,t] = yt_pred
# Append "cache" to "caches" (≈1 line)
caches.append(cache)
### END CODE HERE ###
# store values needed for backward propagation in cache
caches = (caches, x)
return a, y_pred, cachesnp.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
# 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.zeros((n_a+n_x, 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 = np.multiply(it, cct) + np.multiply(ft, c_prev)
ot = sigmoid( np.dot( Wo, concat) + bo)
a_next = np.multiply( 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, cachenp.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
# 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((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, cachesnp.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
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.power(a_next, 2)) * da_next
# compute the gradient of the loss with respect to Wax (≈2 lines)
dxt = np.dot( Wax.T, dtanh)
dWax = np.dot( dtanh, xt.T)
# compute the gradient with respect to Waa (≈2 lines)
da_prev = np.dot( Waa.T, dtanh)
dWaa = np.dot( dtanh, a_prev.T)
# compute the gradient with respect to b (≈1 line)
dba = np.sum(dtanh, axis=1, keepdims=True)
### END CODE HERE ###
# Store the gradients in a python dictionary
gradients = {"dxt": dxt, "da_prev": da_prev, "dWax": dWax, "dWaa": dWaa, "dba": dba}
return gradientsnp.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] = 0.08429686538067724
gradients["da_prev"].shape = (5, 10)
gradients["dWax"][3][1] = 0.39308187392193034
gradients["dWax"].shape = (5, 3)
gradients["dWaa"][1][2] = -0.28483955786960663
gradients["dWaa"].shape = (5, 5)
gradients["dba"][4] = [0.80517166]
gradients["dba"].shape = (5, 1)
def rnn_backward(da, caches):
"""
Implement the backward pass for a RNN over an entire sequence of input data.
Arguments:
da -- Upstream gradients of all hidden states, of shape (n_a, m, T_x)
caches -- tuple containing information from the forward pass (rnn_forward)
Returns:
gradients -- python dictionary containing:
dx -- Gradient w.r.t. the input data, numpy-array of shape (n_x, m, T_x)
da0 -- Gradient w.r.t the initial hidden state, numpy-array of shape (n_a, m)
dWax -- Gradient w.r.t the input's weight matrix, numpy-array of shape (n_a, n_x)
dWaa -- Gradient w.r.t the hidden state's weight matrix, numpy-arrayof shape (n_a, n_a)
dba -- Gradient w.r.t the bias, of shape (n_a, 1)
"""
### START CODE HERE ###
# Retrieve values from the first cache (t=1) of caches (≈2 lines)
(caches, x) = caches
(a1, a0, x1, parameters) = caches[0]
# Retrieve dimensions from da's and x1's shapes (≈2 lines)
n_a, m, T_x = da.shape
n_x, m = x1.shape
# initialize the gradients with the right sizes (≈6 lines)
dx = np.zeros((n_x, m, T_x))
dWax = np.zeros((n_a, n_x))
dWaa = np.zeros((n_a, n_a))
dba = np.zeros((n_a, 1))
da0 = np.zeros((n_a, m))
da_prevt = np.zeros((n_a, m))
# Loop through all the time steps
for t in reversed(range(T_x)):
# Compute gradients at time step t. Choose wisely the "da_next" and the "cache" to use in the backward propagation step. (≈1 line)
gradients = rnn_cell_backward(da[:,:,t]+da_prevt, caches[t]) # !!!!
# Retrieve derivatives from gradients (≈ 1 line)
dxt, da_prevt, dWaxt, dWaat, dbat = gradients["dxt"], gradients["da_prev"], gradients["dWax"], gradients["dWaa"], gradients["dba"]
# Increment global derivatives w.r.t parameters by adding their derivative at time-step t (≈4 lines)
dx[:, :, t] = dxt
dWax += 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 gradientsnp.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] = -0.31494237512664996
gradients["da0"].shape = (5, 10)
gradients["dWax"][3][1] = 11.264104496527777
gradients["dWax"].shape = (5, 3)
gradients["dWaa"][1][2] = 2.303333126579893
gradients["dWaa"].shape = (5, 5)
gradients["dba"][4] = [-0.74747722]
gradients["dba"].shape = (5, 1)
原教程的反向推导有些错误，本人修正在以下连接（欢迎指正）
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+da_next*ot*( 1- np.square(np.tanh(c_next))))*it*( 1-np.square(cct))
dit = ( dc_next+da_next*ot*( 1- np.square(np.tanh(c_next))))*cct*it*(1-it)
dft = ( dc_next+da_next*ot*( 1- np.square(np.tanh(c_next))))*c_prev*ft*(1-ft)
# Compute parameters related derivatives. Use equations (11)-(14) (≈8 lines)
concat = np.concatenate((a_prev, xt), axis=0)
dWf = np.dot( dft, concat.T)
dWi = np.dot( dit, concat.T)
dWc = np.dot( dcct, concat.T)
dWo = np.dot( dot, concat.T)
dbf = np.sum( dft, axis=1, keepdims=True)
dbi = np.sum( dit, axis=1, keepdims=True)
dbc = np.sum( dcct, axis=1, keepdims=True)
dbo = np.sum( dot, axis=1, keepdims=True)
#
Wf = parameters["Wf"]
Wi = parameters["Wi"]
Wc = parameters["Wc"]
Wo = parameters["Wo"]
Wy = parameters["Wy"]
# Compute derivatives w.r.t previous hidden state, previous memory state and input. Use equations (15)-(17). (≈3 lines)
da_prev = np.dot( Wc[:,:n_a].T, dcct) + np.dot( Wi[:,:n_a].T, dit) + \
np.dot( Wf[:,:n_a].T, dft) + np.dot( Wo[:,:n_a].T, dot)
dc_prev = ( dc_next+da_next*ot*( 1- np.square(np.tanh(c_next)))) * ft
dxt = np.dot( Wc[:,n_a:].T, dcct) + np.dot( Wi[:,n_a:].T, dit) + \
np.dot( Wf[:,n_a:].T, dft) + np.dot( 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 gradientsnp.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] = -0.06396214197109233
gradients["da_prev"].shape = (5, 10)
gradients["dc_prev"][2][3] = 0.7975220387970015
gradients["dc_prev"].shape = (5, 10)
gradients["dWf"][3][1] = -0.14795483816449675
gradients["dWf"].shape = (5, 8)
gradients["dWi"][1][2] = 1.0574980552259903
gradients["dWi"].shape = (5, 8)
gradients["dWc"][3][1] = 2.304562163687667
gradients["dWc"].shape = (5, 8)
gradients["dWo"][1][2] = 0.3313115952892109
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)
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]+da_prevt, dc_prevt, caches[t])
# Store or add the gradient to the parameters' previous step's gradient
dxt, da_prevt, dc_prevt = gradients['dxt'], gradients['da_prev'], gradients['dc_prev']
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 = da_prevt
### 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 gradientsnp.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] = [ 0.00218254 0.28205375 -0.48292508 -0.43281115]
gradients["dx"].shape = (3, 10, 4)
gradients["da0"][2][3] = 0.31277031025726026
gradients["da0"].shape = (5, 10)
gradients["dWf"][3][1] = -0.08098023109383468
gradients["dWf"].shape = (5, 8)
gradients["dWi"][1][2] = 0.4051243309298186
gradients["dWi"].shape = (5, 8)
gradients["dWc"][3][1] = -0.07937467355121491
gradients["dWc"].shape = (5, 8)
gradients["dWo"][1][2] = 0.038948775762986956
gradients["dWo"].shape = (5, 8)
gradients["dbf"][4] = [-0.15745657]
gradients["dbf"].shape = (5, 1)
gradients["dbi"][4] = [-0.50848333]
gradients["dbi"].shape = (5, 1)
gradients["dbc"][4] = [-0.42510818]
gradients["dbc"].shape = (5, 1)
gradients["dbo"][4] = [-0.17958196]
gradients["dbo"].shape = (5, 1)