优达学城深度学习之五—

梯度下降算法推导与实现

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd#Some helper functions for plotting and drawing linesdef plot_points(X, y):admitted = X[np.argwhere(y==1)]rejected = X[np.argwhere(y==0)]plt.scatter([s[0][0] for s in rejected], [s[0][1] for s in rejected], s = 25, color = 'blue', edgecolor = 'k')plt.scatter([s[0][0] for s in admitted], [s[0][1] for s in admitted], s = 25, color = 'red', edgecolor = 'k')def display(m, b, color='g--'):plt.xlim(-0.05,1.05)plt.ylim(-0.05,1.05)x = np.arange(-10, 10, 0.1)plt.plot(x, m*x+b, color)
#读取与绘制数据
data = pd.read_csv('data.csv', header=None)
X = np.array(data[[0,1]])
y = np.array(data[2])
plot_points(X,y)
plt.show()# Implement the following functions# Activation (sigmoid) function
def sigmoid(x):return 1/(1+np.exp(-x))# Output (prediction) formula
def output_formula(features, weights, bias):sigmoid(np.dot(features, weights) + bias)# Error (log-loss) formula
def error_formula(y, output):return - y*np.log(output) - (1 - y) * np.log(1-output)# Gradient descent step
def update_weights(x, y, weights, bias, learnrate):output = output_formula(x, weights, bias)d_error = -(y - output)weights -= learnrate * d_error * xbias -= learnrate * d_errorreturn weights, bias
np.random.seed(44)epochs = 100
learnrate = 0.01def train(features, targets, epochs, learnrate, graph_lines=False):errors = []n_records, n_features = features.shapelast_loss = Noneweights = np.random.normal(scale=1 / n_features**.5, size=n_features)bias = 0for e in range(epochs):del_w = np.zeros(weights.shape)for x, y in zip(features, targets):output = output_formula(x, weights, bias)error = error_formula(y, output)weights, bias = update_weights(x, y, weights, bias, learnrate)# Printing out the log-loss error on the training setout = output_formula(features, weights, bias)loss = np.mean(error_formula(targets, out))errors.append(loss)if e % (epochs / 10) == 0:print("\n========== Epoch", e,"==========")if last_loss and last_loss < loss:print("Train loss: ", loss, "  WARNING - Loss Increasing")else:print("Train loss: ", loss)last_loss = losspredictions = out > 0.5accuracy = np.mean(predictions == targets)print("Accuracy: ", accuracy)if graph_lines and e % (epochs / 100) == 0:display(-weights[0]/weights[1], -bias/weights[1])
# Plotting the solution boundaryplt.title("Solution boundary")display(-weights[0]/weights[1], -bias/weights[1], 'black')# Plotting the dataplot_points(features, targets)plt.show()# Plotting the errorplt.title("Error Plot")plt.xlabel('Number of epochs')plt.ylabel('Error')plt.plot(errors)plt.show()
#训练算法
train(X, y, epochs, learnrate, True)

反向传播

反向传播流程如下：

进行前向反馈运算。
将模型的输出与期望的输出进行比较。
计算误差。
向后运行前向反馈运算（反向传播），将误差分散到每个权重上。
更新权重，并获得更好的模型。
继续此流程，直到获得很好的模型。

实战演练：利用神经网络来预测学生录取情况

数据集来源： http://www.ats.ucla.edu/

# Importing pandas and numpy
import pandas as pd
import numpy as np# Reading the csv file into a pandas DataFrame
data = pd.read_csv('student_data.csv')# Printing out the first 10 rows of our data
data[:10]
#绘制数据
# Importing matplotlib
import matplotlib.pyplot as plt
%matplotlib inline
# Function to help us plot
def plot_points(data):X = np.array(data[["gre","gpa"]])y = np.array(data["admit"])admitted = X[np.argwhere(y==1)]rejected = X[np.argwhere(y==0)]plt.scatter([s[0][0] for s in rejected], [s[0][1] for s in rejected], s = 25, color = 'red', edgecolor = 'k')plt.scatter([s[0][0] for s in admitted], [s[0][1] for s in admitted], s = 25, color = 'cyan', edgecolor = 'k')plt.xlabel('Test (GRE)')plt.ylabel('Grades (GPA)')# Plotting the points
plot_points(data)
plt.show()
# Separating the ranks
data_rank1 = data[data["rank"]==1]
data_rank2 = data[data["rank"]==2]
data_rank3 = data[data["rank"]==3]
data_rank4 = data[data["rank"]==4]# Plotting the graphs
plot_points(data_rank1)
plt.title("Rank 1")
plt.show()
plot_points(data_rank2)
plt.title("Rank 2")
plt.show()
plot_points(data_rank3)
plt.title("Rank 3")
plt.show()
plot_points(data_rank4)
plt.title("Rank 4")
plt.show()
#将评级进行one-shot编码
# TODO:  Make dummy variables for rank
one_hot_data = pd.concat([data, pd.get_dummies(data['rank'], prefix='rank')], axis=1)# TODO: Drop the previous rank column
one_hot_data = one_hot_data.drop('rank', axis=1)# Print the first 10 rows of our data
one_hot_data[:10]
#缩放数据
# Making a copy of our data
processed_data = one_hot_data[:]# TODO: Scale the columns
processed_data['gre']=processed_data['gre']/800
processed_data['gpa']=processed_data['gpa']/4.0# Printing the first 10 rows of our procesed data
processed_data[:10]
#将数据分成训练集和测试集
sample = np.random.choice(processed_data.index, size=int(len(processed_data)*0.9), replace=False)
train_data, test_data = processed_data.iloc[sample], processed_data.drop(sample)print("Number of training samples is", len(train_data))
print("Number of testing samples is", len(test_data))
print(train_data[:10])
print(test_data[:10])
#将数据分成特征和目标
features = train_data.drop('admit', axis=1)
targets = train_data['admit']
features_test = test_data.drop('admit', axis=1)
targets_test = test_data['admit']print(features[:10])
print(targets[:10])
#训练二层神经网络Activation (sigmoid) function
def sigmoid(x):return 1 / (1 + np.exp(-x))
def sigmoid_prime(x):return sigmoid(x) * (1-sigmoid(x))
def error_formula(y, output):return - y*np.log(output) - (1 - y) * np.log(1-output)
#误差反向传播
# TODO: Write the error term formula
def error_term_formula(y, output):return (y-output)*sigmoid_prime(x)
def error_term_formula(x, y, output):return (y-output) * output * (1 - output)
# Neural Network hyperparameters
epochs = 1000
learnrate = 0.5# Training function
def train_nn(features, targets, epochs, learnrate):# Use to same seed to make debugging easiernp.random.seed(42)n_records, n_features = features.shapelast_loss = None# Initialize weightsweights = np.random.normal(scale=1 / n_features**.5, size=n_features)for e in range(epochs):del_w = np.zeros(weights.shape)for x, y in zip(features.values, targets):# Loop through all records, x is the input, y is the target# Activation of the output unit#   Notice we multiply the inputs and the weights here #   rather than storing h as a separate variable output = sigmoid(np.dot(x, weights))# The error, the target minus the network outputerror = error_formula(y, output)# The error term#   Notice we calulate f'(h) here instead of defining a separate#   sigmoid_prime function. This just makes it faster because we#   can re-use the result of the sigmoid function stored in#   the output variableerror_term = error_term_formula(x,y, output)# The gradient descent step, the error times the gradient times the inputsdel_w += error_term * x# Update the weights here. The learning rate times the # change in weights, divided by the number of records to averageweights += learnrate * del_w / n_records# Printing out the error on the training setif e % (epochs / 10) == 0:out = sigmoid(np.dot(features, weights))loss = np.mean((out - targets) ** 2)print("Epoch:", e)if last_loss and last_loss < loss:print("Train loss: ", loss, "  WARNING - Loss Increasing")else:print("Train loss: ", loss)last_loss = lossprint("=========")print("Finished training!")return weightsweights = train_nn(features, targets, epochs, learnrate)
#计算测试数据的准确度
# Calculate accuracy on test data
tes_out = sigmoid(np.dot(features_test, weights))
predictions = tes_out > 0.5
accuracy = np.mean(predictions == targets_test)
print("Prediction accuracy: {:.3f}".format(accuracy))

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