平时大多数时间做图片相关的事情，所以本来没计划学习RNN的；后来想了想，反正都学到了CNN，就再坚持一下，把RNN也看了把，看完之后感觉很神奇，和CNN不一样的算法，感觉像是人的不同思维方式，感想吴大大的深入浅出，讲课和作业都很好。
序列模型总共三周：
- 第一周 循环序列模型
- 第二周 自然语言处理与词嵌入
- 第三周 序列模型和注意力机制
RNN
关于RNN网络网上有很多教程，我就用课上的笔记做一个概要的总结吧。这里的RNN是循环神经网络（Recurrent Neural Network）。
“每次的输出Y，与前一轮a和本轮x有关，而输给下一轮神经元的a也和前一轮a和本轮x有关。”这个很像我们说话或者阅读的前后结合思维方式，所以RNN是这种前后 以时间顺序连接的一个个神经元。
时间序列模型有多种结构，如下图：
- 一对一的；
- 一对多：音乐生成模型；
- 多对多：1、语句词义词性检测；2、不同语言翻译。
RNN程序
像之前构建 DNN 网络一样，这个作业的重点在于前向、后向传播（主要是求导）的计算。
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, cache
# 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_x：每个样本每个时刻的向量长度； m：样本个数； T_x：时间维度
n_y, n_a = parameters["Wya"].shape # 参数是共享的，所以Wya只有两个维度
### 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
# 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 save gate, numpy array of shape (n_a, n_a + n_x)
bi -- Bias of the save 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 focus gate, numpy array of shape (n_a, n_a + n_x)
bo -- Bias of the focus 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 tilda),
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)
concatenate_ap_xt = np.concatenate((a_prev, xt), axis=0)
# concatenate_ap_xt = np.zeros([n_a + n_x, m])
# concatenate_ap_xt[: n_a, :] = a_prev
# concatenate_ap_xt[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, concatenate_ap_xt) + bf)
it = sigmoid(np.dot(Wi, concatenate_ap_xt) + bi)
ot = sigmoid(np.dot(Wo, concatenate_ap_xt) + bo)
cct = np.tanh(np.dot(Wc, concatenate_ap_xt) + bc)
c_next = ft * c_prev + it * cct
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
# 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 save gate, numpy array of shape (n_a, n_a + n_x)
bi -- Bias of the save 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 focus gate, numpy array of shape (n_a, n_a + n_x)
bo -- Bias of the focus 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 xt and Wy (≈2 lines)
n_x, m, T_x = x.shape
n_y, n_a = parameters["Wy"].shape # 参数是共享的，所以Wya只有两个维度
# 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)) # 输出是有m个的，即每个样本对应一个输出
# Initialize a_next and c_next (≈2 lines)
a_next = a0
c_next = np.zeros((n_a, m)) # 为每个样本定义一个c，需要在T维度上进行传播，所以只有两维，可以看做 C[:,：,0]
# 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_pred, cache = lstm_cell_forward(x[:,:,t], a_next, c_next, parameters=parameters) # 对每个t时刻，求这个时刻所有样本的lstmcell
# 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_pred
# Save the value of the next cell state (≈1 line)
c[:,:,t] = c_next
# Append the cache into caches (≈1 line)
caches.append(cache)
### END CODE HERE ###
# store values needed for backward propagation in cache
caches = (caches, x)
return a, y, c, caches
def rnn_cell_backward(da_next, cache):
"""
Implements the backward pass for the RNN-cell (single time-step).
Arguments:
da_next -- Gradient of loss with respect to next hidden state
cache -- python dictionary containing useful values (output of rnn_step_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)
dWax = np.dot(dtanh, xt.T)
dxt = np.dot(Wax.T, dtanh)
# compute the gradient with respect to Waa (≈2 lines)
dWaa = np.dot(dtanh, a_prev.T)
da_prev = np.dot(Waa.T, dtanh)
# 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
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 gradients
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 input 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 information from "cache"
(a_next, c_next, a_prev, c_prev, ft, it, cct, ot, xt, parameters) = cache
### START CODE HERE ###
# Retrieve dimensions from xt's and a_next's shape (≈2 lines)
n_x, m = xt.shape
n_a, m = a_next.shape
# Compute gates related derivatives, you can find their values can be found by looking carefully at equations (7) to (10) (≈4 lines)
dot = da_next * np.tanh(c_next) * ot * (1 - ot)
dcct = (dc_next * it + ot * (1 - np.square(np.tanh(c_next))) * it * da_next) * (1 - np.square(cct))
dit = (dc_next * cct + ot * (1 - np.square(np.tanh(c_next))) * cct * da_next) * it * (1 - it)
dft = (dc_next * c_prev + ot * (1 - np.square(np.tanh(c_next))) * c_prev * da_next) * ft * (1 - ft)
## Code equations (7) to (10) (≈4 lines)
##dit = None
##dft = None
##dot = None
##dcct = None
##
# Compute parameters related derivatives. Use equations (11)-(14) (≈8 lines)
concat = np.concatenate((a_prev, xt), axis=0).T
dWf = np.dot(dft, concat)
dWi = np.dot(dit, concat)
dWc = np.dot(dcct, concat)
dWo = np.dot(dot, concat)
dbf = np.sum(dft, axis=1, keepdims=True)
dbi = np.sum(dit, axis=1, keepdims=True)
dbc = np.sum(dcct, axis=1, keepdims=True)
dbo = np.sum(dot, axis=1, keepdims=True)
# Compute derivatives w.r.t previous hidden state, previous memory state and input. Use equations (15)-(17). (≈3 lines)
da_prev = np.dot(parameters["Wf"][:, :n_a].T, dft) + np.dot(parameters["Wc"][:, :n_a].T, dcct) + np.dot(
parameters["Wi"][:, :n_a].T, dit) + np.dot(parameters["Wo"][:, :n_a].T, dot)
dc_prev = dc_next * ft + ot * (1 - np.square(np.tanh(c_next))) * ft * da_next
dxt = np.dot(parameters["Wf"][:, n_a:].T, dft) + np.dot(parameters["Wc"][:, n_a:].T, dcct) + np.dot(
parameters["Wi"][:, n_a:].T, dit) + np.dot(parameters["Wo"][:, n_a:].T, dot)
### END CODE HERE ###
# 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
def lstm_backward(da, caches):
"""
Implement the backward pass for the RNN with LSTM-cell (over a whole sequence).
Arguments:
da -- Gradients w.r.t the hidden states, numpy-array of shape (n_a, m, T_x)
dc -- Gradients w.r.t the memory states, numpy-array of shape (n_a, m, T_x)
caches -- cache storing information from the forward pass (lstm_forward)
Returns:
gradients -- python dictionary containing:
dx -- Gradient of inputs, of shape (n_x, m, T_x)
da0 -- Gradient w.r.t. the previous hidden state, numpy array of shape (n_a, m)
dWf -- Gradient w.r.t. the weight matrix of the forget gate, numpy array of shape (n_a, n_a + n_x)
dWi -- Gradient w.r.t. the weight matrix of the update gate, numpy array of shape (n_a, n_a + n_x)
dWc -- Gradient w.r.t. the weight matrix of the memory gate, numpy array of shape (n_a, n_a + n_x)
dWo -- Gradient w.r.t. the weight matrix of the save gate, numpy array of shape (n_a, n_a + n_x)
dbf -- Gradient w.r.t. biases of the forget gate, of shape (n_a, 1)
dbi -- Gradient w.r.t. biases of the update gate, of shape (n_a, 1)
dbc -- Gradient w.r.t. biases of the memory gate, of shape (n_a, 1)
dbo -- Gradient w.r.t. biases of the save gate, of shape (n_a, 1)
"""
# Retrieve values from the first cache (t=1) of caches.
(caches, x) = caches
(a1, c1, a0, c0, f1, i1, cc1, o1, x1, parameters) = caches[0]
### START CODE HERE ###
# Retrieve dimensions from da's and x1's shapes (≈2 lines)
n_a, m, T_x = da.shape
n_x, m = x1.shape
# initialize the gradients with the right sizes (≈12 lines)
dx = np.zeros([n_x, m, T_x])
da0 = np.zeros([n_a, m])
da_prevt = np.zeros([n_a, m])
dc_prevt = np.zeros([n_a, m])
dWf = np.zeros([n_a, n_a + n_x])
dWi = np.zeros([n_a, n_a + n_x])
dWc = np.zeros([n_a, n_a + n_x])
dWo = np.zeros([n_a, n_a + n_x])
dbf = np.zeros([n_a, 1])
dbi = np.zeros([n_a, 1])
dbc = np.zeros([n_a, 1])
dbo = np.zeros([n_a, 1])
# loop back over the whole sequence
for t in reversed(range(T_x)):
# Compute all gradients using lstm_cell_backward
gradients = lstm_cell_backward(da[:,:,t],dc_prevt,caches[t])
# da_prevt, dc_prevt = gradients['da_prev'], gradients["dc_prev"]
# Store or add the gradient to the parameters' previous step's gradient
dx[:,:,t] = gradients['dxt']
dWf = dWf+gradients['dWf']
dWi = dWi+gradients['dWi']
dWc = dWc+gradients['dWc']
dWo = dWo+gradients['dWo']
dbf = dbf+gradients['dbf']
dbi = dbi+gradients['dbi']
dbc = dbc+gradients['dbc']
dbo = 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
if __name__ == '__main__':
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)
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)