以下代码来自Deep Learning for Computer Vision with Python第九章。
一、梯度下降法（Gradient Decent）
# import the necessary packages
from sklearn.model_selection import train_test_split
from sklearn.metrics import classification_report
from sklearn.datasets import make_blobs
import matplotlib.pyplot as plt
import numpy as np
import argparse
def sigmoid_activation(x):
# compute the sigmoid activation value for a given input
return 1.0 / (1 + np.exp(-x))
def predict(X, W):
# take the dot product between our features and weight matrix
preds = sigmoid_activation(X.dot(W))
# apply a step function to threshold the outputs to binary
# class labels
preds[preds <= 0.5] = 0
preds[preds > 0] = 1
# return the predictions
return preds
# construct the argument parse and parse the arguments
ap = argparse.ArgumentParser()
ap.add_argument("-e", "--epochs", type=float, default=100,
help="# of epochs")
ap.add_argument("-a", "--alpha", type=float, default=0.01,
help="learning rate")
args = vars(ap.parse_args())
# generate a 2-class classification problem with 1,000 data points,
# where each data point is a 2D feature vector
(X, y) = make_blobs(n_samples=1000, n_features=2, centers=2,
cluster_std=1.5, random_state=1)
y = y.reshape((y.shape[0], 1))
# insert a column of 1's as the last entry in the feature
# matrix -- this little trick allows us to treat the bias
# as a trainable parameter within the weight matrix
X = np.c_[X, np.ones((X.shape[0]))]
# partition the data into training and testing splits using 50% of
# the data for training and the remaining 50% for testing
(trainX, testX, trainY, testY) = train_test_split(X, y,
test_size=0.5, random_state=42)
# initialize our weight matrix and list of losses
print("[INFO] training...")
W = np.random.randn(X.shape[1], 1)
losses = []
# loop over the desired number of epochs
for epoch in np.arange(0, args["epochs"]):
# take the dot product between our features 'X' and the weight
# matrix 'W', then pass this value through our sigmoid activation
# function, thereby giving us our predictions on the dataset
preds = sigmoid_activation(trainX.dot(W))
# now that we have our predictions, we need to determine the
# 'error', which is the difference between our predictions and
# the true values
error = preds - trainY
loss = np.sum(error ** 2)
losses.append(loss)
# the gradient descent update is the dot product between our
# features and the error of the predictions
gradient = trainX.T.dot(error)
# in the update stage, all we need to do is "nudge" the weight
# matrix in the negative direction of the gradient (hence the
# term "gradient descent" by taking a small step towards a set
# of "more optimal" parameters
W += -args["alpha"] * gradient
# check to see if an update should be displayed
if epoch == 0 or (epoch + 1) % 5 == 0:
print("[INFO] epoch={}, loss={:.7f}".format(int(epoch + 1),
loss))
# evaluate our model
print("[INFO] evaluating...")
preds = predict(testX, W)
print(classification_report(testY, preds))
# plot the (testing) classification data
plt.style.use("ggplot")
plt.figure()
plt.title("Data")
plt.scatter(testX[:, 0], testX[:, 1], marker="o", c=testY, s=30)
# construct a figure that plots the loss over time
plt.style.use("ggplot")
plt.figure()
plt.plot(np.arange(0, args["epochs"]), losses)
plt.title("Training Loss")
plt.xlabel("Epoch #")
plt.ylabel("Loss")
plt.show()
本例子的神经网络是只有两层，输入3，输出1，（3-1）。且输入3个神经元中，最后一个是输入为1。是为了将偏移(bias)b值放到权重矩阵W中。
Python语言，使用了sklearn、matplotlib、numpy、imutils这几个库。
这个例程中，学习的内容如下：
1、细胞元激活函数
本例子采用sigmoid函数。
sigmoid函数曲线如下：
理论上，神经网络中每个神经元只有两种状态：有反应、无反应，即1和0。但这里允许神经元具有0-1V之间的任意电压。且输入输出符合Sigmoid曲线。
2、predict预测函数
预测函数中，把输入的变量X（3行1列矩阵）经过转置变成1行3列，乘以权值W（3行1列），得到输出。
3、网络初始化
使用make_blobs函数生成了1000个样品，每个样品两个参数。即输入矩阵是1000行2列。输出只有一个参数，是1行1000列。
X = np.c_[X, np.ones((X.shape[0]))]这语句可以在输入矩阵最低加上一行1，同时把权值W（weights）矩阵初始化为3行1列，最后1列是偏置b（bias）。
线性分类基本公式是：
可以把b放进权重W矩阵的最后一行，这样的好处是，可以在训练W矩阵时，也训练了b参数。
train_test_split函数可以将样品（X,y）按比例分配成一部分用于训练，一部分用于测试。
4、网络训练
这例子，更新权重矩阵W的频率是把全部训练样品处理一次，才更新一次的权重。因此学习速度十分缓慢。
error是全部训练样品的预测结果，和实际结果y想减。
损失函数是error中每个元素的平方和：loss = np.sum(error ** 2)
更新权值的公式是
error = preds - trainY
gradient = trainX.T.dot(error)
W += -args["alpha"] * gradient
5、网络测试
测试使用了classification_report.第一个参数是实际值，第二个参数是预测值。报告可自动生成精度、测试样品数量。
print(classification_report(testY, predict(testX, W)))
6、文件执行结果
========= RESTART: E:\FENG\workspace_python\ch9_gradient_descent.py =========
[INFO] training...
[INFO] epoch=1, loss=155.6216601
[INFO] epoch=5, loss=0.1092728
[INFO] epoch=10, loss=0.1032095
[INFO] epoch=15, loss=0.0976591
[INFO] epoch=20, loss=0.0925605
[INFO] epoch=25, loss=0.0878624
[INFO] epoch=30, loss=0.0835212
[INFO] epoch=35, loss=0.0794996
[INFO] epoch=40, loss=0.0757656
[INFO] epoch=45, loss=0.0722911
[INFO] epoch=50, loss=0.0690518
[INFO] epoch=55, loss=0.0660262
[INFO] epoch=60, loss=0.0631954
[INFO] epoch=65, loss=0.0605427
[INFO] epoch=70, loss=0.0580530
[INFO] epoch=75, loss=0.0557131
[INFO] epoch=80, loss=0.0535110
[INFO] epoch=85, loss=0.0514360
[INFO] epoch=90, loss=0.0494784
[INFO] epoch=95, loss=0.0476294
[INFO] epoch=100, loss=0.0458811
[INFO] evaluating...
precision recall f1-score support
0 1.00 1.00 1.00 250
1 1.00 1.00 1.00 250
avg / total 1.00 1.00 1.00 500
Traceback (most recent call last):
File "E:\FENG\workspace_python\ch9_gradient_descent.py", line 92, in <module>
plt.scatter(testX[:, 0], testX[:, 1], marker="o", c=testY, s=30)
File "D:\ProgramFiles\Python27\lib\site-packages\matplotlib\pyplot.py", line 3470, in scatter
edgecolors=edgecolors, data=data, **kwargs)
File "D:\ProgramFiles\Python27\lib\site-packages\matplotlib\__init__.py", line 1855, in inner
return func(ax, *args, **kwargs)
File "D:\ProgramFiles\Python27\lib\site-packages\matplotlib\axes\_axes.py", line 4279, in scatter
.format(c.shape, x.size, y.size))
ValueError: c of shape (500, 1) not acceptable as a color sequence for x with size 500, y with size 500
二、随机梯度下降法（Stochastic Gradient Decent）
# import the necessary packages
from sklearn.model_selection import train_test_split
from sklearn.metrics import classification_report
from sklearn.datasets import make_blobs
import matplotlib.pyplot as plt
import numpy as np
import argparse
def sigmoid_activation(x):
# compute the sigmoid activation value for a given input
return 1.0 / (1 + np.exp(-x))
def predict(X, W):
# take the dot product between our features and weight matrix
preds = sigmoid_activation(X.dot(W))
# apply a step function to threshold the outputs to binary
# class labels
preds[preds <= 0.5] = 0
preds[preds > 0] = 1
# return the predictions
return preds
def next_batch(X, y, batchSize):
# loop over our dataset ‘X‘ in mini-batches, yielding a tuple of
# the current batched data and labels
for i in np.arange(0, X.shape[0], batchSize):
yield (X[i:i + batchSize], y[i:i + batchSize])
# construct the argument parse and parse the arguments
ap = argparse.ArgumentParser()
ap.add_argument("-e", "--epochs", type=float, default=100,
help="# of epochs")
ap.add_argument("-a", "--alpha", type=float, default=0.01,
help="learning rate")
ap.add_argument("-b", "--batch-size", type=int, default=32,
help="size of SGD mini-batches")
args = vars(ap.parse_args())
# generate a 2-class classification problem with 1,000 data points,
# where each data point is a 2D feature vector
(X, y) = make_blobs(n_samples=1000, n_features=2, centers=2,
cluster_std=1.5, random_state=1)
y = y.reshape((y.shape[0], 1))
# insert a column of 1’s as the last entry in the feature
# matrix -- this little trick allows us to treat the bias
# as a trainable parameter within the weight matrix
X = np.c_[X, np.ones((X.shape[0]))]
# partition the data into training and testing splits using 50% of
# the data for training and the remaining 50% for testing
(trainX, testX, trainY, testY) = train_test_split(X, y,
test_size=0.5, random_state=42)
# initialize our weight matrix and list of losses
print("[INFO] training...")
W = np.random.randn(X.shape[1], 1)
losses = []
# loop over the desired number of epochs
for epoch in np.arange(0, args["epochs"]):
# initialize the total loss for the epoch
epochLoss = []
# loop over our data in batches
for (batchX, batchY) in next_batch(X, y, args["batch_size"]):
# take the dot product between our current batch of features
# and the weight matrix, then pass this value through our
# activation function
preds = sigmoid_activation(batchX.dot(W))
# now that we have our predictions, we need to determine the
# ‘error‘, which is the difference between our predictions
# and the true values
error = preds - batchY
epochLoss.append(np.sum(error ** 2))
# the gradient descent update is the dot product between our
# current batch and the error on the batch
gradient = batchX.T.dot(error)
# in the update stage, all we need to do is "nudge" the
# weight matrix in the negative direction of the gradient
# (hence the term "gradient descent") by taking a small step
# towards a set of "more optimal" parameters
W += -args["alpha"] * gradient
# update our loss history by taking the average loss across all
# batches
loss = np.average(epochLoss)
losses.append(loss)
# check to see if an update should be displayed
if epoch == 0 or (epoch + 1) % 5 == 0:
print("[INFO] epoch={}, loss={:.7f}".format(int(epoch + 1),
loss))
# evaluate our model
print("[INFO] evaluating...")
preds = predict(testX, W)
print(classification_report(testY, preds))
# plot the (testing) classification data
plt.style.use("ggplot")
plt.figure()
plt.title("Data")
plt.scatter(testX[:, 0], testX[:, 1], marker="o", c=testY, s=30)
# construct a figure that plots the loss over time
plt.style.use("ggplot")
plt.figure()
plt.plot(np.arange(0, args["epochs"]), losses)
plt.title("Training Loss")
plt.xlabel("Epoch #")
plt.ylabel("Loss")
plt.show()与第一种方法不同之处在于，每处理一小量数据，即按照该段数据更新权值矩阵。
执行结果如下：
================ RESTART: E:\FENG\workspace_python\ch9_sgd.py ================
[INFO] training...
[INFO] epoch=1, loss=0.5633928
[INFO] epoch=5, loss=0.0116136
[INFO] epoch=10, loss=0.0063118
[INFO] epoch=15, loss=0.0058116
[INFO] epoch=20, loss=0.0054206
[INFO] epoch=25, loss=0.0050830
[INFO] epoch=30, loss=0.0047875
[INFO] epoch=35, loss=0.0045260
[INFO] epoch=40, loss=0.0042924
[INFO] epoch=45, loss=0.0040821
[INFO] epoch=50, loss=0.0038914
[INFO] epoch=55, loss=0.0037176
[INFO] epoch=60, loss=0.0035583
[INFO] epoch=65, loss=0.0034118
[INFO] epoch=70, loss=0.0032764
[INFO] epoch=75, loss=0.0031509
[INFO] epoch=80, loss=0.0030342
[INFO] epoch=85, loss=0.0029253
[INFO] epoch=90, loss=0.0028235
[INFO] epoch=95, loss=0.0027281
[INFO] epoch=100, loss=0.0026385
[INFO] evaluating...
precision recall f1-score support
0 1.00 1.00 1.00 250
1 1.00 1.00 1.00 250
avg / total 1.00 1.00 1.00 500
Traceback (most recent call last):
File "E:\FENG\workspace_python\ch9_sgd.py", line 109, in <module>
plt.scatter(testX[:, 0], testX[:, 1], marker="o", c=testY, s=30)
File "D:\ProgramFiles\Python27\lib\site-packages\matplotlib\pyplot.py", line 3470, in scatter
edgecolors=edgecolors, data=data, **kwargs)
File "D:\ProgramFiles\Python27\lib\site-packages\matplotlib\__init__.py", line 1855, in inner
return func(ax, *args, **kwargs)
File "D:\ProgramFiles\Python27\lib\site-packages\matplotlib\axes\_axes.py", line 4279, in scatter
.format(c.shape, x.size, y.size))
ValueError: c of shape (500, 1) not acceptable as a color sequence for x with size 500, y with size 500对比可见，SGD的损失下降比较快。