# Python:梯度下降 Lab (六十三)

## 工作区

• sigmoid: sigmoid激活函数。
• output_formula: 输出（预测）公式
• error_formula: 误差函数。
• update_weights: 更新权重的函数。
• 当你执行它们时，运行 train 函数，这将绘制连续梯度下降步骤中的几条直线。 它还会绘制误差函数，随着 epoch 数量的增加，你可以看到它正在降低。

## 实现梯度下降算法

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

#Some helper functions for plotting and drawing lines

def plot_points(X, y):
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()

## 待办： 实现基本函数

• Sigmoid 激活函数

$$\sigma(x) = \frac{1}{1+e^{-x}}$$

• 输出（预测）公式

$$\hat{y} = \sigma(w_1 x_1 + w_2 x_2 + b)$$

• 误差函数

$$Error(y, \hat{y}) = - y \log(\hat{y}) - (1-y) \log(1-\hat{y})$$

• 更新权重的函数

$$w_i^{'} \longleftarrow w_i + \alpha (y - \hat{y}) x_i$$

$$b^{'} \longleftarrow b + \alpha (y - \hat{y})$$

# 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):
return 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 * x
bias += learnrate * d_error
return weights, bias


## 训练函数

np.random.seed(44)

epochs = 100
learnrate = 0.01

def train(features, targets, epochs, learnrate, graph_lines=False):

errors = []
n_records, n_features = features.shape
last_loss = None
weights = np.random.normal(scale=1 / n_features**.5, size=n_features)
bias = 0
for 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 set
out = 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 = loss
predictions = out > 0.5
accuracy = 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 boundary
plt.title("Solution boundary")
display(-weights[0]/weights[1], -bias/weights[1], 'black')

# Plotting the data
plot_points(features, targets)
plt.show()

# Plotting the error
plt.title("Error Plot")
plt.xlabel('Number of epochs')
plt.ylabel('Error')
plt.plot(errors)
plt.show()

## 是时候来训练算法啦！

• 目前的训练损失与准确性的 10 次更新
• 获取的数据图和一些边界线的图。 最后一个是黑色的。请注意，随着我们遍历更多的 epoch ，线会越来越接近最佳状态。
• 误差函数的图。 请留意，随着我们遍历更多的 epoch，它会如何降低。
train(X, y, epochs, learnrate, True)
========== Epoch 0 ==========
Train loss:  0.713584519538
Accuracy:  0.4

========== Epoch 10 ==========
Train loss:  0.622583521045
Accuracy:  0.59

========== Epoch 20 ==========
Train loss:  0.554874408367
Accuracy:  0.74

========== Epoch 30 ==========
Train loss:  0.501606141872
Accuracy:  0.84

========== Epoch 40 ==========
Train loss:  0.459333464186
Accuracy:  0.86

========== Epoch 50 ==========
Train loss:  0.425255434335
Accuracy:  0.93

========== Epoch 60 ==========
Train loss:  0.397346157167
Accuracy:  0.93

========== Epoch 70 ==========
Train loss:  0.374146976524
Accuracy:  0.93

========== Epoch 80 ==========
Train loss:  0.354599733682
Accuracy:  0.94

========== Epoch 90 ==========
Train loss:  0.337927365888
Accuracy:  0.94