改善深层神经网络 - 第二课第一周作业 1.2 Regularization
来自吴恩达深度学习视频，如果直接看代码对你来说有困难，参见：https://blog.csdn.net/u013733326/article/details/79847918#正则化模型
请注意原博客与作业有一些差异而且计算L2 regularization时有一个错误，这篇博客完全正确且与原作业一致
这次作业对同一模型在不使用正则化，使用L2正则化，使用Dropout三种情况进行了对比。https://github.com/Hongze-Wang/Deep-Learning-Andrew-Ng/tree/master/homework 戳这里看完整版
# import packages
import numpy as np
import matplotlib.pyplot as plt
from reg_utils import sigmoid, relu, plot_decision_boundary, initialize_parameters, load_2D_dataset, predict_dec
from reg_utils import compute_cost, predict, forward_propagation, backward_propagation, update_parameters
import sklearn
import sklearn.datasets
import scipy.io
from testCases import *
%matplotlib inline
plt.rcParams['figure.figsize'] = (7.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'
会出现可忽略警告：
C:\Users\wangh\reg_utils.py:85: SyntaxWarning: assertion is always true, perhaps remove parentheses?
assert(parameters['W' + str(l)].shape == layer_dims[l], layer_dims[l-1])
C:\Users\wangh\reg_utils.py:86: SyntaxWarning: assertion is always true, perhaps remove parentheses?
assert(parameters['W' + str(l)].shape == layer_dims[l], 1)
train_X, train_Y, test_X, test_Y = load_2D_dataset()
def model(X, Y, learning_rate = 0.3, num_iterations = 30000, print_cost = True, lambd = 0, keep_prob = 1):
"""
Implements a three-layer neural network: LINEAR->RELU->LINEAR->RELU->LINEAR->SIGMOID.
Arguments:
X -- input data, of shape (input size, number of examples)
Y -- true "label" vector (1 for blue dot / 0 for red dot), of shape (output size, number of examples)
learning_rate -- learning rate of the optimization
num_iterations -- number of iterations of the optimization loop
print_cost -- If True, print the cost every 10000 iterations
lambd -- regularization hyperparameter, scalar
keep_prob - probability of keeping a neuron active during drop-out, scalar.
Returns:
parameters -- parameters learned by the model. They can then be used to predict.
"""
grads = {}
costs = [] # to keep track of the cost
m = X.shape[1] # number of examples
layers_dims = [X.shape[0], 20, 3, 1]
# Initialize parameters dictionary.
parameters = initialize_parameters(layers_dims)
# Loop (gradient descent)
for i in range(0, num_iterations):
# Forward propagation: LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID.
if keep_prob == 1:
a3, cache = forward_propagation(X, parameters)
elif keep_prob < 1:
a3, cache = forward_propagation_with_dropout(X, parameters, keep_prob)
# Cost function
if lambd == 0:
cost = compute_cost(a3, Y)
else:
cost = compute_cost_with_regularization(a3, Y, parameters, lambd)
# Backward propagation.
assert(lambd==0 or keep_prob==1) # it is possible to use both L2 regularization and dropout,
# but this assignment will only explore one at a time
if lambd == 0 and keep_prob == 1:
grads = backward_propagation(X, Y, cache)
elif lambd != 0:
grads = backward_propagation_with_regularization(X, Y, cache, lambd)
elif keep_prob < 1:
grads = backward_propagation_with_dropout(X, Y, cache, keep_prob)
# Update parameters.
parameters = update_parameters(parameters, grads, learning_rate)
# Print the loss every 10000 iterations
if print_cost and i % 10000 == 0:
print("Cost after iteration {}: {}".format(i, cost))
if print_cost and i % 1000 == 0:
costs.append(cost)
# plot the cost
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('iterations (x1,000)')
plt.title("Learning rate =" + str(learning_rate))
plt.show()
return parameters
parameters = model(train_X, train_Y)
print ("On the training set:")
predictions_train = predict(train_X, train_Y, parameters)
print ("On the test set:")
predictions_test = predict(test_X, test_Y, parameters)
Cost after iteration 0: 0.6557412523481002
Cost after iteration 10000: 0.1632998752572419
Cost after iteration 20000: 0.13851642423239133
On the training set:
Accuracy: 0.9478672985781991
On the test set:
Accuracy: 0.915
plt.title("Model without regularization")
axes = plt.gca()
axes.set_xlim([-0.75,0.40])
axes.set_ylim([-0.75,0.65])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)
# GRADED FUNCTION: compute_cost_with_regularization
def compute_cost_with_regularization(A3, Y, parameters, lambd):
"""
Implement the cost function with L2 regularization. See formula (2) above.
Arguments:
A3 -- post-activation, output of forward propagation, of shape (output size, number of examples)
Y -- "true" labels vector, of shape (output size, number of examples)
parameters -- python dictionary containing parameters of the model
Returns:
cost - value of the regularized loss function (formula (2))
"""
m = Y.shape[1]
W1 = parameters["W1"]
W2 = parameters["W2"]
W3 = parameters["W3"]
cross_entropy_cost = compute_cost(A3, Y) # This gives you the cross-entropy part of the cost
### START CODE HERE ### (approx. 1 line)
L2_regularization_cost = lambd * (np.sum(np.square(W1)) + np.sum(np.square(W2)) + np.sum(np.square(W3))) / (m*2)
### END CODER HERE ###
cost = cross_entropy_cost + L2_regularization_cost
return cost
A3, Y_assess, parameters = compute_cost_with_regularization_test_case()
print("cost = " + str(compute_cost_with_regularization(A3, Y_assess, parameters, lambd = 0.1)))
cost = 1.7864859451590758
# GRADED FUNCTION: backward_propagation_with_regularization
def backward_propagation_with_regularization(X, Y, cache, lambd):
"""
Implements the backward propagation of our baseline model to which we added an L2 regularization.
Arguments:
X -- input dataset, of shape (input size, number of examples)
Y -- "true" labels vector, of shape (output size, number of examples)
cache -- cache output from forward_propagation()
lambd -- regularization hyperparameter, scalar
Returns:
gradients -- A dictionary with the gradients with respect to each parameter, activation and pre-activation variables
"""
m = X.shape[1]
(Z1, A1, W1, b1, Z2, A2, W2, b2, Z3, A3, W3, b3) = cache
dZ3 = A3 - Y
### START CODE HERE ### (approx. 1 line)
dW3 = 1./m * np.dot(dZ3, A2.T) + lambd/m*W3
### END CODE HERE ###
db3 = 1./m * np.sum(dZ3, axis=1, keepdims = True)
dA2 = np.dot(W3.T, dZ3)
dZ2 = np.multiply(dA2, np.int64(A2 > 0))
### START CODE HERE ### (approx. 1 line)
dW2 = 1./m * np.dot(dZ2, A1.T) + lambd/m*W2
### END CODE HERE ###
db2 = 1./m * np.sum(dZ2, axis=1, keepdims = True)
dA1 = np.dot(W2.T, dZ2)
dZ1 = np.multiply(dA1, np.int64(A1 > 0))
### START CODE HERE ### (approx. 1 line)
dW1 = 1./m * np.dot(dZ1, X.T) + lambd/m*W1
### END CODE HERE ###
db1 = 1./m * np.sum(dZ1, axis=1, keepdims = True)
gradients = {"dZ3": dZ3, "dW3": dW3, "db3": db3,"dA2": dA2,
"dZ2": dZ2, "dW2": dW2, "db2": db2, "dA1": dA1,
"dZ1": dZ1, "dW1": dW1, "db1": db1}
return gradients
X_assess, Y_assess, cache = backward_propagation_with_regularization_test_case()
grads = backward_propagation_with_regularization(X_assess, Y_assess, cache, lambd = 0.7)
print ("dW1 = "+ str(grads["dW1"]))
print ("dW2 = "+ str(grads["dW2"]))
print ("dW3 = "+ str(grads["dW3"]))
dW1 = [[-0.25604646 0.12298827 -0.28297129]
[-0.17706303 0.34536094 -0.4410571 ]]
dW2 = [[ 0.79276486 0.85133918]
[-0.0957219 -0.01720463]
[-0.13100772 -0.03750433]]
dW3 = [[-1.77691347 -0.11832879 -0.09397446]]
parameters = model(train_X, train_Y, lambd = 0.60) # (我把keep_prob从0.7改成了0.6 取得了测试集9535%的准确度)
print ("On the train set:")
predictions_train = predict(train_X, train_Y, parameters)
print ("On the test set:")
predictions_test = predict(test_X, test_Y, parameters)
Cost after iteration 0: 0.6914902783181227
Cost after iteration 10000: 0.2616965878421321
Cost after iteration 20000: 0.2614637394190471
On the train set:
Accuracy: 0.9289099526066351
On the test set:
Accuracy: 0.955
plt.title("Model with L2-regularization")
axes = plt.gca()
axes.set_xlim([-0.75,0.40])
axes.set_ylim([-0.75,0.65])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)
# GRADED FUNCTION: forward_propagation_with_dropout
def forward_propagation_with_dropout(X, parameters, keep_prob = 0.5):
"""
Implements the forward propagation: LINEAR -> RELU + DROPOUT -> LINEAR -> RELU + DROPOUT -> LINEAR -> SIGMOID.
Arguments:
X -- input dataset, of shape (2, number of examples)
parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
W1 -- weight matrix of shape (20, 2)
b1 -- bias vector of shape (20, 1)
W2 -- weight matrix of shape (3, 20)
b2 -- bias vector of shape (3, 1)
W3 -- weight matrix of shape (1, 3)
b3 -- bias vector of shape (1, 1)
keep_prob - probability of keeping a neuron active during drop-out, scalar
Returns:
A3 -- last activation value, output of the forward propagation, of shape (1,1)
cache -- tuple, information stored for computing the backward propagation
"""
np.random.seed(1)
# retrieve parameters
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
W3 = parameters["W3"]
b3 = parameters["b3"]
# LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID
Z1 = np.dot(W1, X) + b1
A1 = relu(Z1)
### START CODE HERE ### (approx. 4 lines) # Steps 1-4 below correspond to the Steps 1-4 described above.
D1 = np.random.rand(A1.shape[0], A1.shape[1]) # Step 1: initialize matrix D1 = np.random.rand(..., ...)
D1 = D1 < keep_prob # Step 2: convert entries of D1 to 0 or 1 (using keep_prob as the threshold)
A1 = A1 * D1 # Step 3: shut down some neurons of A1
A1 = A1 / keep_prob # Step 4: scale the value of neurons that haven't been shut down
### END CODE HERE ###
Z2 = np.dot(W2, A1) + b2
A2 = relu(Z2)
### START CODE HERE ### (approx. 4 lines)
D2 = np.random.rand(A2.shape[0], A2.shape[1]) # Step 1: initialize matrix D2 = np.random.rand(..., ...)
D2 = D2 < keep_prob # Step 2: convert entries of D2 to 0 or 1 (using keep_prob as the threshold)
A2 = A2 * D2 # Step 3: shut down some neurons of A2
A2 = A2 / keep_prob # Step 4: scale the value of neurons that haven't been shut down
### END CODE HERE ###
Z3 = np.dot(W3, A2) + b3
A3 = sigmoid(Z3)
cache = (Z1, D1, A1, W1, b1, Z2, D2, A2, W2, b2, Z3, A3, W3, b3)
return A3, cache
X_assess, parameters = forward_propagation_with_dropout_test_case()
A3, cache = forward_propagation_with_dropout(X_assess, parameters, keep_prob = 0.7)
print ("A3 = " + str(A3))
A3 = [[0.36974721 0.00305176 0.04565099 0.49683389 0.36974721]]
# GRADED FUNCTION: backward_propagation_with_dropout
def backward_propagation_with_dropout(X, Y, cache, keep_prob):
"""
Implements the backward propagation of our baseline model to which we added dropout.
Arguments:
X -- input dataset, of shape (2, number of examples)
Y -- "true" labels vector, of shape (output size, number of examples)
cache -- cache output from forward_propagation_with_dropout()
keep_prob - probability of keeping a neuron active during drop-out, scalar
Returns:
gradients -- A dictionary with the gradients with respect to each parameter, activation and pre-activation variables
"""
m = X.shape[1]
(Z1, D1, A1, W1, b1, Z2, D2, A2, W2, b2, Z3, A3, W3, b3) = cache
dZ3 = A3 - Y
dW3 = 1./m * np.dot(dZ3, A2.T)
db3 = 1./m * np.sum(dZ3, axis=1, keepdims = True)
dA2 = np.dot(W3.T, dZ3)
### START CODE HERE ### (≈ 2 lines of code)
dA2 = dA2 * D2 # Step 1: Apply mask D2 to shut down the same neurons as during the forward propagation
dA2 = dA2 / keep_prob # Step 2: Scale the value of neurons that haven't been shut down
### END CODE HERE ###
dZ2 = np.multiply(dA2, np.int64(A2 > 0))
dW2 = 1./m * np.dot(dZ2, A1.T)
db2 = 1./m * np.sum(dZ2, axis=1, keepdims = True)
dA1 = np.dot(W2.T, dZ2)
### START CODE HERE ### (≈ 2 lines of code)
dA1 = dA1 * D1 # Step 1: Apply mask D1 to shut down the same neurons as during the forward propagation
dA1 = dA1 / keep_prob # Step 2: Scale the value of neurons that haven't been shut down
### END CODE HERE ###
dZ1 = np.multiply(dA1, np.int64(A1 > 0))
dW1 = 1./m * np.dot(dZ1, X.T)
db1 = 1./m * np.sum(dZ1, axis=1, keepdims = True)
gradients = {"dZ3": dZ3, "dW3": dW3, "db3": db3,"dA2": dA2,
"dZ2": dZ2, "dW2": dW2, "db2": db2, "dA1": dA1,
"dZ1": dZ1, "dW1": dW1, "db1": db1}
return gradients
X_assess, Y_assess, cache = backward_propagation_with_dropout_test_case()
gradients = backward_propagation_with_dropout(X_assess, Y_assess, cache, keep_prob = 0.8)
print ("dA1 = " + str(gradients["dA1"]))
print ("dA2 = " + str(gradients["dA2"]))
dA1 = [[ 0.36544439 0. -0.00188233 0. -0.17408748]
[ 0.65515713 0. -0.00337459 0. -0. ]]
dA2 = [[ 0.58180856 0. -0.00299679 0. -0.27715731]
[ 0. 0.53159854 -0. 0.53159854 -0.34089673]
[ 0. 0. -0.00292733 0. -0. ]]
parameters = model(train_X, train_Y, keep_prob = 0.86, learning_rate = 0.3)
print ("On the train set:")
predictions_train = predict(train_X, train_Y, parameters)
print ("On the test set:")
predictions_test = predict(test_X, test_Y, parameters)
Cost after iteration 0: 0.6543912405149825
C:\Users\wangh\reg_utils.py:236: RuntimeWarning: divide by zero encountered in log
logprobs = np.multiply(-np.log(a3),Y) + np.multiply(-np.log(1 - a3), 1 - Y)
C:\Users\wangh\reg_utils.py:236: RuntimeWarning: invalid value encountered in multiply
logprobs = np.multiply(-np.log(a3),Y) + np.multiply(-np.log(1 - a3), 1 - Y)
Cost after iteration 10000: 0.061016986574905605
Cost after iteration 20000: 0.060582435798513114
On the train set:
Accuracy: 0.9289099526066351
On the test set:
Accuracy: 0.95
plt.title("Model with dropout")
axes = plt.gca()
axes.set_xlim([-0.75,0.40])
axes.set_ylim([-0.75,0.65])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)
- 3-layer NN without regularization
训练集准确度 95% 测试集准确度 91.5% - 3-layer NN with L2-regularization
训练集准确度 94% 测试集准确度 93%
(我把keep_prob从0.7改成了0.6 取得了测试集95%的准确度) - 3-layer NN with dropout
训练集准确度 93% 测试集准确度 95%
