import math
import numpy as np
from opt_utils import *
import matplotlib.pyplot as plt
# 标准梯度下降
def update_parameters_with_gd(parameters, grads, learning_rate):
L = len(parameters) // 2
for l in range(1, L + 1):
parameters[f"W{l}"] = parameters[f"W{l}"] - learning_rate * grads[f"dW{l}"]
parameters[f"b{l}"] = parameters[f"b{l}"] - learning_rate * grads[f"db{l}"]
return parameters
# mini-batch梯度下降
def random_mini_batches(X, Y, mini_batch_size=64, seed=0):
np.random.seed(seed)
m = X.shape[1]
mini_batches = []
permutation = list(np.random.permutation(m))
# 洗牌
shuffled_X = X[:, permutation]
shuffled_Y = Y[:, permutation].reshape((1, m))
# 分割
num_complete_minibatches = math.floor(m / mini_batch_size)
for k in range(0, num_complete_minibatches):
mini_batch_X = shuffled_X[:, k * mini_batch_size:(k + 1) * mini_batch_size]
mini_batch_Y = shuffled_Y[:, k * mini_batch_size:(k + 1) * mini_batch_size]
mini_batch = (mini_batch_X, mini_batch_Y)
mini_batches.append(mini_batch)
if m % mini_batch_size != 0:
mini_batch_X = shuffled_X[:, num_complete_minibatches * mini_batch_size:]
mini_batch_Y = shuffled_Y[:, num_complete_minibatches * mini_batch_size:]
mini_batch = (mini_batch_X, mini_batch_Y)
mini_batches.append(mini_batch)
return mini_batches
# 动量梯度下降
def initialize_velocity(parameters):
L = len(parameters) // 2
v = {}
for l in range(1, L + 1):
v[f"dW{l}"] = np.zeros_like(parameters[f"W{l}"])
v[f"db{l}"] = np.zeros_like(parameters[f"b{l}"])
return v
def update_parameters_with_momentum(parameters, grads, v, beta, learning_rate):
L = len(parameters) // 2
for l in range(L):
v["dW" + str(l + 1)] = beta * v["dW" + str(l + 1)] + (1 - beta) * grads['dW' + str(l + 1)]
v["db" + str(l + 1)] = beta * v["db" + str(l + 1)] + (1 - beta) * grads['db' + str(l + 1)]
parameters["W" + str(l + 1)] = parameters["W" + str(l + 1)] - learning_rate * v["dW" + str(l + 1)]
parameters["b" + str(l + 1)] = parameters["b" + str(l + 1)] - learning_rate * v["db" + str(l + 1)]
return parameters, v
# Adam梯度下降
def initialize_adam(parameters):
L = len(parameters) // 2
v = {}
s = {}
for l in range(L):
v["dW" + str(l + 1)] = np.zeros_like(parameters["W" + str(l + 1)])
v["db" + str(l + 1)] = np.zeros_like(parameters["b" + str(l + 1)])
s["dW" + str(l + 1)] = np.zeros_like(parameters["W" + str(l + 1)])
s["db" + str(l + 1)] = np.zeros_like(parameters["b" + str(l + 1)])
return v, s
def update_parameters_with_adam(parameters, grads, v, s, t, learning_rate=0.01,
beta1=0.9, beta2=0.999, epsilon=1e-8):
L = len(parameters) // 2
v_corrected = {}
s_corrected = {}
for l in range(L):
v["dW" + str(l + 1)] = beta1 * v["dW" + str(l + 1)] + (1 - beta1) * grads['dW' + str(l + 1)]
v["db" + str(l + 1)] = beta1 * v["db" + str(l + 1)] + (1 - beta1) * grads['db' + str(l + 1)]
v_corrected["dW" + str(l + 1)] = v["dW" + str(l + 1)] / (1 - np.power(beta1, t))
v_corrected["db" + str(l + 1)] = v["db" + str(l + 1)] / (1 - np.power(beta1, t))
s["dW" + str(l + 1)] = beta2 * s["dW" + str(l + 1)] + (1 - beta2) * np.power(grads['dW' + str(l + 1)], 2)
s["db" + str(l + 1)] = beta2 * s["db" + str(l + 1)] + (1 - beta2) * np.power(grads['db' + str(l + 1)], 2)
s_corrected["dW" + str(l + 1)] = s["dW" + str(l + 1)] / (1 - np.power(beta2, t))
s_corrected["db" + str(l + 1)] = s["db" + str(l + 1)] / (1 - np.power(beta2, t))
parameters["W" + str(l + 1)] = parameters["W" + str(l + 1)] - learning_rate * v_corrected[
"dW" + str(l + 1)] / np.sqrt(s_corrected["dW" + str(l + 1)] + epsilon)
parameters["b" + str(l + 1)] = parameters["b" + str(l + 1)] - learning_rate * v_corrected[
"db" + str(l + 1)] / np.sqrt(s_corrected["db" + str(l + 1)] + epsilon)
return parameters, v, s
def model(X, Y, layers_dims, optimizer, learning_rate=0.0007, mini_batch_size=64, beta=0.9, beta1=0.9, beta2=0.999,
epsilon=1e-8, num_epochs=10000, print_cost=True):
L = len(layers_dims)
costs = []
t = 0
seed = 10
parameters = initialize_parameters(layers_dims)
if optimizer == "gd":
pass
elif optimizer == "momentum":
v = initialize_velocity(parameters)
elif optimizer == "adam":
v, s = initialize_adam(parameters)
# 一个epoch 就是遍历整个数据集一遍，一个epoch有多个mini-batch
for i in range(num_epochs):
seed = seed + 1
minibatches = random_mini_batches(X, Y, mini_batch_size, seed)
for minibatch in minibatches:
(minibatch_X, minibatch_Y) = minibatch
a3, caches = forward_propagation(minibatch_X, parameters)
cost = compute_cost(a3, minibatch_Y)
grads = backward_propagation(minibatch_X, minibatch_Y, caches)
if optimizer == "gd":
parameters = update_parameters_with_gd(parameters, grads, learning_rate)
elif optimizer == "momentum":
parameters, v = update_parameters_with_momentum(parameters, grads, v, beta, learning_rate)
elif optimizer == "adam":
t = t + 1
parameters, v, s = update_parameters_with_adam(parameters, grads, v, s, t, learning_rate, beta1, beta2,
epsilon)
if print_cost and i % 1000 == 0:
print("Cost after epoch %i: %f" % (i, cost))
if print_cost and i % 100 == 0:
costs.append(cost)
return parameters, costs
def training1(train_X, train_Y, layers_dims):
# 标准mini-batch梯度下降
parameters, costs = model(train_X, train_Y, layers_dims, optimizer="gd")
p = predict_dec(parameters, train_X)
print("Accuracy: " + str(np.mean((p[0, :] == train_Y[0, :]))))
plt.subplot(131)
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('epochs (per 100)')
plt.title("Learning rate = " + str(0.0007))
# plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y.ravel())
def training2(train_X, train_Y, layers_dims):
# mini-batch动量梯度下降
parameters, costs = model(train_X, train_Y, layers_dims, optimizer="momentum")
p = predict_dec(parameters, train_X)
print("Accuracy: " + str(np.mean((p[0, :] == train_Y[0, :]))))
plt.subplot(132)
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('epochs (per 100)')
# plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y.ravel())
def training3(train_X, train_Y, layers_dims):
# mini-batch Adam梯度下降
parameters, costs = model(train_X, train_Y, layers_dims, optimizer="adam")
p = predict_dec(parameters, train_X)
print("Accuracy: " + str(np.mean((p[0, :] == train_Y[0, :]))))
plt.subplot(133)
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('epochs (per 100)')
# plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y.ravel())
if __name__ == "__main__":
train_X, train_Y = load_dataset()
layers_dims = [train_X.shape[0], 5, 2, 1]
print("mini batch with gd:")
training1(train_X, train_Y, layers_dims)
print("mini batch with momentum:")
training2(train_X, train_Y, layers_dims)
print("mini batch with adam:")
training3(train_X, train_Y, layers_dims)
plt.show()
util:
import numpy as np
import matplotlib.pyplot as plt
import h5py
import scipy.io
import sklearn
import sklearn.datasets
def sigmoid(x):
"""
Compute the sigmoid of x
Arguments:
x -- A scalar or numpy array of any size.
Return:
s -- sigmoid(x)
"""
s = 1/(1+np.exp(-x))
return s
def relu(x):
"""
Compute the relu of x
Arguments:
x -- A scalar or numpy array of any size.
Return:
s -- relu(x)
"""
s = np.maximum(0,x)
return s
def load_params_and_grads(seed=1):
np.random.seed(seed)
W1 = np.random.randn(2,3)
b1 = np.random.randn(2,1)
W2 = np.random.randn(3,3)
b2 = np.random.randn(3,1)
dW1 = np.random.randn(2,3)
db1 = np.random.randn(2,1)
dW2 = np.random.randn(3,3)
db2 = np.random.randn(3,1)
return W1, b1, W2, b2, dW1, db1, dW2, db2
def initialize_parameters(layer_dims):
"""
Arguments:
layer_dims -- python array (list) containing the dimensions of each layer in our network
Returns:
parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
W1 -- weight matrix of shape (layer_dims[l], layer_dims[l-1])
b1 -- bias vector of shape (layer_dims[l], 1)
Wl -- weight matrix of shape (layer_dims[l-1], layer_dims[l])
bl -- bias vector of shape (1, layer_dims[l])
Tips:
- For example: the layer_dims for the "Planar Data classification model" would have been [2,2,1].
This means W1's shape was (2,2), b1 was (1,2), W2 was (2,1) and b2 was (1,1). Now you have to generalize it!
- In the for loop, use parameters['W' + str(l)] to access Wl, where l is the iterative integer.
"""
np.random.seed(3)
parameters = {}
L = len(layer_dims) # number of layers in the network
for l in range(1, L):
parameters['W' + str(l)] = np.random.randn(layer_dims[l], layer_dims[l-1])* np.sqrt(2 / layer_dims[l-1])
parameters['b' + str(l)] = np.zeros((layer_dims[l], 1))
assert(parameters['W' + str(l)].shape == layer_dims[l], layer_dims[l-1])
assert(parameters['W' + str(l)].shape == layer_dims[l], 1)
return parameters
def compute_cost(a3, Y):
"""
Implement the cost function
Arguments:
a3 -- post-activation, output of forward propagation
Y -- "true" labels vector, same shape as a3
Returns:
cost - value of the cost function
"""
m = Y.shape[1]
logprobs = np.multiply(-np.log(a3),Y) + np.multiply(-np.log(1 - a3), 1 - Y)
cost = 1./m * np.sum(logprobs)
return cost
def forward_propagation(X, parameters):
"""
Implements the forward propagation (and computes the loss) presented in Figure 2.
Arguments:
X -- input dataset, of shape (input size, number of examples)
parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
W1 -- weight matrix of shape ()
b1 -- bias vector of shape ()
W2 -- weight matrix of shape ()
b2 -- bias vector of shape ()
W3 -- weight matrix of shape ()
b3 -- bias vector of shape ()
Returns:
loss -- the loss function (vanilla logistic loss)
"""
# 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)
z2 = np.dot(W2, a1) + b2
a2 = relu(z2)
z3 = np.dot(W3, a2) + b3
a3 = sigmoid(z3)
cache = (z1, a1, W1, b1, z2, a2, W2, b2, z3, a3, W3, b3)
return a3, cache
def backward_propagation(X, Y, cache):
"""
Implement the backward propagation presented in figure 2.
Arguments:
X -- input dataset, of shape (input size, number of examples)
Y -- true "label" vector (containing 0 if cat, 1 if non-cat)
cache -- cache output from forward_propagation()
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 = 1./m * (a3 - Y)
dW3 = np.dot(dz3, a2.T)
db3 = np.sum(dz3, axis=1, keepdims = True)
da2 = np.dot(W3.T, dz3)
dz2 = np.multiply(da2, np.int64(a2 > 0))
dW2 = np.dot(dz2, a1.T)
db2 = np.sum(dz2, axis=1, keepdims = True)
da1 = np.dot(W2.T, dz2)
dz1 = np.multiply(da1, np.int64(a1 > 0))
dW1 = np.dot(dz1, X.T)
db1 = 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
def predict(X, y, parameters):
"""
This function is used to predict the results of a n-layer neural network.
Arguments:
X -- data set of examples you would like to label
parameters -- parameters of the trained model
Returns:
p -- predictions for the given dataset X
"""
m = X.shape[1]
p = np.zeros((1,m), dtype = np.int)
# Forward propagation
a3, caches = forward_propagation(X, parameters)
# convert probas to 0/1 predictions
for i in range(0, a3.shape[1]):
if a3[0,i] > 0.5:
p[0,i] = 1
else:
p[0,i] = 0
# print results
#print ("predictions: " + str(p[0,:]))
#print ("true labels: " + str(y[0,:]))
print("Accuracy: " + str(np.mean((p[0,:] == y[0,:]))))
return p
def load_2D_dataset():
data = scipy.io.loadmat('datasets/data.mat')
train_X = data['X'].T
train_Y = data['y'].T
test_X = data['Xval'].T
test_Y = data['yval'].T
plt.scatter(train_X[0, :], train_X[1, :], c=train_Y, s=40, cmap=plt.cm.Spectral);
return train_X, train_Y, test_X, test_Y
def plot_decision_boundary(model, X, y):
# Set min and max values and give it some padding
x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1
y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1
h = 0.01
# Generate a grid of points with distance h between them
xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
# Predict the function value for the whole grid
Z = model(np.c_[xx.ravel(), yy.ravel()])
Z = Z.reshape(xx.shape)
# Plot the contour and training examples
plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
plt.ylabel('x2')
plt.xlabel('x1')
plt.scatter(X[0, :], X[1, :], c=y, cmap=plt.cm.Spectral)
plt.show()
def predict_dec(parameters, X):
"""
Used for plotting decision boundary.
Arguments:
parameters -- python dictionary containing your parameters
X -- input data of size (m, K)
Returns
predictions -- vector of predictions of our model (red: 0 / blue: 1)
"""
# Predict using forward propagation and a classification threshold of 0.5
a3, cache = forward_propagation(X, parameters)
predictions = (a3 > 0.5)
return predictions
def load_dataset():
np.random.seed(3)
train_X, train_Y = sklearn.datasets.make_moons(n_samples=300, noise=.2) #300 #0.2
# Visualize the data
# plt.scatter(train_X[:, 0], train_X[:, 1], c=train_Y, s=40, cmap=plt.cm.Spectral);
train_X = train_X.T
train_Y = train_Y.reshape((1, train_Y.shape[0]))
return train_X, train_Y
mini batch with gd:
Cost after epoch 0: 0.690736
Cost after epoch 1000: 0.685273
Cost after epoch 2000: 0.647072
Cost after epoch 3000: 0.619525
Cost after epoch 4000: 0.576584
Cost after epoch 5000: 0.607243
Cost after epoch 6000: 0.529403
Cost after epoch 7000: 0.460768
Cost after epoch 8000: 0.465586
Cost after epoch 9000: 0.464518
Accuracy: 0.7966666666666666
mini batch with momentum:
Cost after epoch 0: 0.690741
Cost after epoch 1000: 0.685341
Cost after epoch 2000: 0.647145
Cost after epoch 3000: 0.619594
Cost after epoch 4000: 0.576665
Cost after epoch 5000: 0.607324
Cost after epoch 6000: 0.529476
Cost after epoch 7000: 0.460936
Cost after epoch 8000: 0.465780
Cost after epoch 9000: 0.464740
Accuracy: 0.7966666666666666
mini batch with adam:
Cost after epoch 0: 0.690552
Cost after epoch 1000: 0.185501
Cost after epoch 2000: 0.150830
Cost after epoch 3000: 0.074454
Cost after epoch 4000: 0.125959
Cost after epoch 5000: 0.104344
Cost after epoch 6000: 0.100676
Cost after epoch 7000: 0.031652
Cost after epoch 8000: 0.111973
Cost after epoch 9000: 0.197940
Accuracy: 0.94
