线性回归
数学中的回归是指，现实中的变量之间存在一种函数关系，通过一批样本数据找出这个函数关系，即通过样本数据回归到真实的函数关系。
线性回归/Linear Regression是指，一些变量之间存在线性关系，通过一批样本数据找出这个关系，线性关系函数的图形是一条直线。
线性函数的方程如下：
y = wx + by=wx+b
线性回归就是根据一批样本数据，确定这个方程，即确定权重ww和偏置bb。
因此，要创建线性模型，需要:
- 应变量(y)
- 斜率或权重变量(w)
- 截距或偏置(b)
- 自变量(x)
让我们开始使用TensorFlow建立线性模型:
import tensorflow.compat.v1 as tf import numpy as np tf.compat.v1.disable_eager_execution() # 为参数斜率(W)创建变量，初始值为0.4 W = tf.Variable([.4], tf.float32) # 为参数截距(b)创建变量，初始值为-0.4 b = tf.Variable([-0.4], tf.float32) # 为自变量(用x表示)创建占位符 x = tf.placeholder(tf.float32) # 线性回归方程 linear_model = W * x + b # 初始化所有变量 sess = tf.compat.v1.Session() init = tf.compat.v1.global_variables_initializer() sess.run(init) # 运行回归模型，输出y值 print(sess.run(linear_model, feed_dict={x: [1, 2, 3, 4]}))
输出
C:\Anaconda3\python.exe "C:\Program Files\JetBrains\PyCharm 2019.1.1\helpers\pydev\pydevconsole.py" --mode=client --port=60639 import sys; print('Python %s on %s' % (sys.version, sys.platform)) sys.path.extend(['C:\\app\\PycharmProjects', 'C:/app/PycharmProjects']) Python 3.7.6 (default, Jan 8 2020, 20:23:39) [MSC v.1916 64 bit (AMD64)] Type 'copyright', 'credits' or 'license' for more information IPython 7.12.0 -- An enhanced Interactive Python. Type '?' for help. PyDev console: using IPython 7.12.0 Python 3.7.6 (default, Jan 8 2020, 20:23:39) [MSC v.1916 64 bit (AMD64)] on win32 runfile('C:/app/PycharmProjects/ArtificialIntelligence/test.py', wdir='C:/app/PycharmProjects/ArtificialIntelligence') WARNING:tensorflow:From C:\Anaconda3\lib\site-packages\tensorflow\python\ops\resource_variable_ops.py:1666: calling BaseResourceVariable.__init__ (from tensorflow.python.ops.resource_variable_ops) with constraint is deprecated and will be removed in a future version. Instructions for updating: If using Keras pass *_constraint arguments to layers. 2020-06-19 18:08:36.592548: I tensorflow/core/platform/cpu_feature_guard.cc:143] Your CPU supports instructions that this TensorFlow binary was not compiled to use: AVX2 2020-06-19 18:08:36.612575: I tensorflow/compiler/xla/service/service.cc:168] XLA service 0x1a941da5370 initialized for platform Host (this does not guarantee that XLA will be used). Devices: 2020-06-19 18:08:36.614292: I tensorflow/compiler/xla/service/service.cc:176] StreamExecutor device (0): Host, Default Version [0. 0.4 0.8000001 1.2 ]
上面的代码只是根据线性方程，输入x值，输出y值。
我们需要使用样本数据训练权重w和偏置b，根据输出的y值，计算误差(预测结果和已知结果之间的差异)，得到代价函数，利用梯度下降法求取代价函数的最小值，得到最终的权重w和偏置b。
代价函数
代价函数用于度量模型的实际输出和期望输出之间的差距。我们将使用常用的均方差作为代价函数：
E = \frac{1}{2}(t – y)^2E=21(t–y)2
- t – 目标输出
- y – 实际输出
- E – 均方差
# y占位符，接受样本中的y值 y = tf.placeholder(tf.float32) # 计算均方差 error = linear_model - y squared_errors = tf.square(error) loss = tf.reduce_sum(squared_errors) # 打印误差 print(sess.run(loss, feed_dict = {x:[1, 2, 3, 4], y:[2, 4, 6, 8]}))
完整代码
import tensorflow.compat.v1 as tf import numpy as np tf.compat.v1.disable_eager_execution() # 为参数斜率(W)创建变量，初始值为0.4 W = tf.Variable([.4], tf.float32) # 为参数截距(b)创建变量，初始值为-0.4 b = tf.Variable([-0.4], tf.float32) # 为自变量(用x表示)创建占位符 x = tf.placeholder(tf.float32) # 线性回归方程 linear_model = W * x + b # 初始化所有变量 sess = tf.compat.v1.Session() init = tf.compat.v1.global_variables_initializer() sess.run(init) # 运行回归模型，输出y值 print(sess.run(linear_model, feed_dict={x: [1, 2, 3, 4]})) # y占位符，接受样本中的y值 y = tf.placeholder(tf.float32) # 计算均方差 error = linear_model - y squared_errors = tf.square(error) loss = tf.reduce_sum(squared_errors) # 打印误差 print(sess.run(loss, feed_dict = {x:[1, 2, 3, 4], y:[2, 4, 6, 8]}))
输出
C:\Anaconda3\python.exe "C:\Program Files\JetBrains\PyCharm 2019.1.1\helpers\pydev\pydevconsole.py" --mode=client --port=64343 import sys; print('Python %s on %s' % (sys.version, sys.platform)) sys.path.extend(['C:\\app\\PycharmProjects', 'C:/app/PycharmProjects']) Python 3.7.6 (default, Jan 8 2020, 20:23:39) [MSC v.1916 64 bit (AMD64)] Type 'copyright', 'credits' or 'license' for more information IPython 7.12.0 -- An enhanced Interactive Python. Type '?' for help. PyDev console: using IPython 7.12.0 Python 3.7.6 (default, Jan 8 2020, 20:23:39) [MSC v.1916 64 bit (AMD64)] on win32 runfile('C:/app/PycharmProjects/ArtificialIntelligence/test.py', wdir='C:/app/PycharmProjects/ArtificialIntelligence') WARNING:tensorflow:From C:\Anaconda3\lib\site-packages\tensorflow\python\ops\resource_variable_ops.py:1666: calling BaseResourceVariable.__init__ (from tensorflow.python.ops.resource_variable_ops) with constraint is deprecated and will be removed in a future version. Instructions for updating: If using Keras pass *_constraint arguments to layers. 2020-06-19 18:15:29.396415: I tensorflow/core/platform/cpu_feature_guard.cc:143] Your CPU supports instructions that this TensorFlow binary was not compiled to use: AVX2 2020-06-19 18:15:29.415583: I tensorflow/compiler/xla/service/service.cc:168] XLA service 0x17166c35f50 initialized for platform Host (this does not guarantee that XLA will be used). Devices: 2020-06-19 18:15:29.417842: I tensorflow/compiler/xla/service/service.cc:176] StreamExecutor device (0): Host, Default Version [0. 0.4 0.8000001 1.2 ] 90.24
可以看到输出的误差值很大。因此，我们需要调整权重(W)和偏差(b)，以减少误差。
模型训练
TensorFlow提供了优化器，可以缓慢地更改每个变量(权重w，偏置b)，最小化代价函数。
最简单的优化器是梯度下降优化器，它根据代价函数对变量的变化率(导数)来修改对应变量，进行迭代得到代价函数的最小值。
# 创建梯度下降优化器实例，学习率为0.01 optimizer = tf.train.GradientDescentOptimizer(0.01) # 使用优化器最小化代价函数 train = optimizer.minimize(loss) # 在1000次迭代中最小化误差，这样在迭代时，将使用优化器根据误差修改模型参数w & b以最小化误差 for i in range(1000): sess.run(train, {x:[1, 2, 3, 4], y:[2, 4, 6, 8]}) # 打印权重和偏差 print(sess.run([W, b]))
完整代码：
import tensorflow.compat.v1 as tf import numpy as np tf.compat.v1.disable_eager_execution() # 为参数斜率(W)创建变量，初始值为0.4 W = tf.Variable([.4], tf.float32) # 为参数截距(b)创建变量，初始值为-0.4 b = tf.Variable([-0.4], tf.float32) # 为自变量(用x表示)创建占位符 x = tf.placeholder(tf.float32) # 线性回归方程 linear_model = W * x + b # 初始化所有变量 sess = tf.compat.v1.Session() init = tf.compat.v1.global_variables_initializer() sess.run(init) # 运行回归模型，输出y值 print(sess.run(linear_model, feed_dict={x: [1, 2, 3, 4]})) # y占位符，接受样本中的y值 y = tf.placeholder(tf.float32) # 计算均方差 error = linear_model - y squared_errors = tf.square(error) loss = tf.reduce_sum(squared_errors) # 打印误差 print(sess.run(loss, feed_dict = {x:[1, 2, 3, 4], y:[2, 4, 6, 8]})) # 创建梯度下降优化器实例，学习率为0.01 optimizer = tf.train.GradientDescentOptimizer(0.01) # 使用优化器最小化代价函数 train = optimizer.minimize(loss) # 在1000次迭代中最小化误差，这样在迭代时，将使用优化器根据误差修改模型参数w & b以最小化误差 for i in range(1000): sess.run(train, {x:[1, 2, 3, 4], y:[2, 4, 6, 8]}) # 打印权重和偏差 print(sess.run([W, b]))
输出
C:\Anaconda3\python.exe "C:\Program Files\JetBrains\PyCharm 2019.1.1\helpers\pydev\pydevconsole.py" --mode=client --port=64387 import sys; print('Python %s on %s' % (sys.version, sys.platform)) sys.path.extend(['C:\\app\\PycharmProjects', 'C:/app/PycharmProjects']) Python 3.7.6 (default, Jan 8 2020, 20:23:39) [MSC v.1916 64 bit (AMD64)] Type 'copyright', 'credits' or 'license' for more information IPython 7.12.0 -- An enhanced Interactive Python. Type '?' for help. PyDev console: using IPython 7.12.0 Python 3.7.6 (default, Jan 8 2020, 20:23:39) [MSC v.1916 64 bit (AMD64)] on win32 runfile('C:/app/PycharmProjects/ArtificialIntelligence/test.py', wdir='C:/app/PycharmProjects/ArtificialIntelligence') WARNING:tensorflow:From C:\Anaconda3\lib\site-packages\tensorflow\python\ops\resource_variable_ops.py:1666: calling BaseResourceVariable.__init__ (from tensorflow.python.ops.resource_variable_ops) with constraint is deprecated and will be removed in a future version. Instructions for updating: If using Keras pass *_constraint arguments to layers. 2020-06-19 18:16:36.829150: I tensorflow/core/platform/cpu_feature_guard.cc:143] Your CPU supports instructions that this TensorFlow binary was not compiled to use: AVX2 2020-06-19 18:16:36.848335: I tensorflow/compiler/xla/service/service.cc:168] XLA service 0x193806eb120 initialized for platform Host (this does not guarantee that XLA will be used). Devices: 2020-06-19 18:16:36.850094: I tensorflow/compiler/xla/service/service.cc:176] StreamExecutor device (0): Host, Default Version [0. 0.4 0.8000001 1.2 ] 90.24 [array([1.9999996], dtype=float32), array([9.863052e-07], dtype=float32)]