TensorFlow(2)Softmax Regression

墨蓝 2022-05-16 10:46 154阅读 0赞

## Softmax Regression ##

#### Chapter ####

*  Basics
    
    1.  generate random Tensors
    2.  Three usual activation function in Neural Network
    3.  Softmax funcion
 *  Softmax Regression
    
    1.  Logistic Regression
    2.  Softmax Regression
    3.  Examples

### Basics ###

*  generate random Tensors
 *  Three usual activation function in Neural Network
 *  Softmax funcion

#### generate random Tensors ####

Two generally type of random distribute:

*  uniform
 *  normal

In TensorFlow, they come in tf.random\_normal() and tf.random\_uniform() funcion.

Example:

import tensorflow as tf # 正态分布 # tf.random_normal(shape,mean=0.0,stddev=1.0,dtype=tf.float32,seed=None,name=None) b = tf.random_normal([5, 5],seed = 1234) # 平均分布 # tf.random_uniform(shape,minval=0.0,maxval=1.0,dtype=tf.flaot32,seed=None,name=None) c = tf.random_uniform([5,5], seed=12) with tf.Session() as sees: mat1, mat2 = sees.run([b,c]) print(mat1) print(mat2)

output:

[[ 0.51340485 -0.25581399  0.65199131  1.39236379  0.37256798]
     [ 0.20336303  1.24701834 -0.98333126  0.50439858  0.98600131]
     [-1.70049322  1.51739979  0.06326418  1.07656693  0.03294745]
     [-2.32479048 -1.44697022  0.56703895  0.10577387 -0.90796399]
     [-1.05107033 -1.63305104  1.22501576  0.83072805  1.28771544]]
    [[ 0.63615251  0.92409146  0.67627728  0.50212514  0.96114957]
     [ 0.88567221  0.04360652  0.29879451  0.46695721  0.05440903]
     [ 0.82479727  0.28480017  0.98824406  0.67999697  0.66409671]
     [ 0.75018144  0.31693625  0.51552784  0.75187266  0.44445455]
     [ 0.07454526  0.04856801  0.35572827  0.2670846   0.4779768 ]]

#### Three usual activation function in Neural Network ####

*  logistic function: (sigmoid function)

f ( x ) = 1 1 + e − x f(x)=\\frac\{1\}\{1+e^\{-x\}\} f(x)=1\+e−x1

!\[logistic function\](http://images.cnblogs.com/cnblogs\_com/jclian91/1145646/o\_logistic.png)

*  tanh function

t a n h ( x ) = s i n h ( x ) c o s h ( x ) = e x − e − x e x + e − x tanh(x)=\\frac\{sinh(x)\}\{cosh(x)\}=\\frac\{e^\{x\}-e^\{-x\}\}\{e^\{x\}+e^\{-x\}\} tanh(x)=cosh(x)sinh(x)=ex\+e−xex−e−x

!\[tanh function\](http://images.cnblogs.com/cnblogs\_com/jclian91/1145646/o\_tanh.png)

*  ReLU (Rectified Linear Unit) function

f ( x ) = \{ 0 , if x  ≤  0 x , if  x   >  0 f(x) = \\begin\{cases\} 0, & \\text\{if x $\\le$ 0\} \\\\ x, & \\text\{if $x$ $\\gt$ 0\} \\end\{cases\} f(x)=\{ 0,x,if x ≤ 0if x > 0

!\[ReLU function\](http://images.cnblogs.com/cnblogs\_com/jclian91/1145646/o\_ReLU.png)

TensorFlow Code:

# 常用的神经网络的三种激活函数(sigmoid, tanh, ReLu) import tensorflow as tf import numpy as np import matplotlib.pyplot as plt g = tf.Graph() with g.as_default() as g: x = tf.placeholder(dtype=tf.float64) y = tf.nn.sigmoid(x) # sigmoid函数 # y = tf.nn.tanh(x) # tanh函数 # y = tf.nn.relu(x) # ReLU函数 with tf.Session(graph=g) as sess: # x的范围为-10到10，均匀地取1000个点 x_value = np.linspace(-10, 10, 1000) x, y = sess.run([x,y], feed_dict={ x: x_value}) # 绘制函数图像 plt.plot(x,y) plt.title('logistic Function') plt.xlabel('x') plt.ylabel('y') plt.show()

#### Softmax Function ####

The **softmax function** is a generalization of the logistic function that “squashes” a K-dimensional vector of arbitrary real values to a K-dimensional vector of real values, where each entry is in the range (0, 1\], and all the entries add up to 1.

The function is :

!\[\](https://wikimedia.org/api/rest\_v1/media/math/render/svg/86ead0d067436010ffe21c29fa4bf956eb023ff6) !\[\](https://wikimedia.org/api/rest\_v1/media/math/render/svg/e348290cf48ddbb6e9a6ef4e39363568b67c09d3)    In TensorFlow, we can use \*\*tf.nn.softmax()\*\* function.

Python Code:

# -*- coding: utf-8 -*- import tensorflow as tf import numpy as np A = np.array([1.0,2.0,3.0,4.0,5.0,6.0]) exp_A = np.exp(A) softmax_A = exp_A/sum(exp_A) print(softmax_A) tf_a = tf.constant(A) with tf.Session() as sess: print(sess.run(tf.nn.softmax(tf_a)))

output:

[ 0.00426978  0.01160646  0.03154963  0.08576079  0.23312201  0.63369132]
    [ 0.00426978  0.01160646  0.03154963  0.08576079  0.23312201  0.63369132]

### Softmax Regression ###

*  Logistic Regression
 *  Softmax Regression
 *  Examples

#### Logistic Regression ####

Think logistic regression in a neural-network way.

![logistic regression][]

Suppose there are **m** samples, each sample in **n**\-dimension with label in \{0,1\}, so the **loss function** is:  
 ϕ ( x ) = ∑ j = 0 n w j x j h θ ( x ) = 1 1 + e − ϕ ( x ) \\phi(x) = \\sum\\limits\_\{j=0\}^\{n\}w\_\{j\}x\_\{j\}\\\\ h\_\{\\theta\}(x) =\\frac\{1\}\{1+e^\{-\\phi(x)\}\} ϕ(x)=j=0∑nwjxjhθ(x)=1\+e−ϕ(x)1

!\[\](http://deeplearning.stanford.edu/wiki/images/math/f/a/6/fa6565f1e7b91831e306ec404ccc1156.png)

We use **gradient descent method** to find the **optimal parameter**  θ \\theta θ (n+1 parameters:  w 0 , w 1 , . . . , w n w\_0,w\_1,...,w\_n w0,w1,...,wn) in order to **minimize the loss**.

#### Softmax Regression ####

Softmax Regression is a generalize type of Logistic Function which is useful for **multi-class classification**.

![softmax][]

Suppose there are **m** samples, each sample in **n**\-dimension with label in \{1,2,…,k\}, so the **loss function** is:

J ( θ ) = − 1 m \[ ∑ i = 1 m ∑ j = 1 k 1 \{ y ( i ) = j \} × log ⁡ ( p ( y ( i ) = j ∣ x i ; θ ) ) \] J(\\theta) = -\\frac\{1\}\{m\}\\Big\[\\sum\\limits\_\{i=1\}^\{m\}\\sum\\limits\_\{j=1\}^\{k\}1\{\\\{y^\{(i)\}=j\\\}\}\\times \\log\{(p(y^\{(i)\}=j|x^\{i\};\\theta))\}\\Big\] J(θ)=−m1\[i=1∑mj=1∑k1\{ y(i)=j\}×log(p(y(i)=j∣xi;θ))\]

We use **gradient descent method** to find the **optimal parameter**  θ \\theta θ ( k × ( n + 1 ) k\\times (n+1) k×(n\+1) parameters) in order to **minimize the loss**.

TensorFlow Code:

import tensorflow as tf import numpy as np import logging logging.basicConfig(level = logging.INFO, format='%(asctime)s - %(levelname)s: %(message)s') logger = logging.getLogger(__name__) class Softmax: def __init__(self, input_size, class_num, epoch = 1000, learning_rate = 0.1, save_model_path='./model.ckpt'): self.save_model_path = save_model_path # 模型保存目录 self.epoch = epoch # 循环次数 self.learning_rate = learning_rate # 学习率 x = tf.placeholder(dtype=tf.float32, shape=[None, input_size]) # 特征 y_true = tf.placeholder(dtype=tf.float32, shape=[None, class_num]) # 标签 # 定义softmax回归的网络结构 W = tf.Variable(tf.random_normal([input_size, class_num]), name='weight') b = tf.Variable(tf.random_normal([class_num]), name='bias') # tf.nn.softmax computes softmax activations # softmax = exp(logits) / reduce_sum(exp(logits), dim) self.hypothesis = tf.nn.softmax(tf.matmul(x, W) + b) self.x = x self.y_true = y_true # Cross entropy loss self.loss = tf.reduce_mean(-tf.reduce_sum(self.y_true * tf.log(self.hypothesis), axis=1)) # 选择optimizer使loss达到最小, 使用梯度下降法减少 loss，学习率是self.learning_rate self.train_model = tf.train.GradientDescentOptimizer(self.learning_rate).minimize(self.loss) self.saver = tf.train.Saver() logger.info('Initialize the model...') # 训练并保存模型 def train(self, x_data, y_data): logger.info('Training the model...') with tf.Session() as sess: # 对所有变量进行初始化 sess.run(tf.global_variables_initializer()) feed_dict = { self.x: x_data, self.y_true: y_data} # 进行迭代学习 for i in range(self.epoch+1): sess.run(self.train_model, feed_dict=feed_dict) if i % int(self.epoch/10) == 0: # to see the step improvement print('已训练%d次, loss: %s.' % (i, sess.run(self.loss, feed_dict=feed_dict))) # 保存ANN模型 logger.info('Saving the model...') self.saver.save(sess, self.save_model_path) # E # 预测数据 def predict(self, data): with tf.Session() as sess: logger.info('Restoring the model...') self.saver.restore(sess, self.save_model_path) # A predict = sess.run(self.hypothesis, feed_dict={ self.x: data}) # B predict_class = sess.run(tf.argmax(predict,1)) print("预测值为：%s." % predict_class) return predict # 使用Softmax类 # 样本数据 x_data = np.array([[1, 2, 1, 1], [2, 1, 3, 2], [3, 1, 3, 4], [4, 1, 5, 5], [1, 7, 5, 5], [1, 2, 5, 6], [1, 6, 6, 6], [1, 7, 7, 7]]) y_data = np.array([[0, 0, 1], [0, 0, 1], [0, 0, 1], [0, 1, 0], [0, 1, 0], [0, 1, 0], [1, 0, 0], [1, 0, 0]]) input_size = x_data.shape[1] class_num = y_data.shape[1] save_path = 'E://logs/model.ckpt' st = Softmax(input_size, class_num, 1000, 0.1, save_path) st.train(x_data, y_data) st.predict(np.array([[1, 3, 4, 3]]))

output:

2018-08-17 17:01:37,854 - INFO: Initialize the model...
    2018-08-17 17:01:37,854 - INFO: Training the model...
    已训练0次, loss: 5.31073.
    已训练100次, loss: 0.624064.
    已训练200次, loss: 0.559649.
    已训练300次, loss: 0.511522.
    已训练400次, loss: 0.470231.
    已训练500次, loss: 0.432752.
    已训练600次, loss: 0.397538.
    已训练700次, loss: 0.363444.
    已训练800次, loss: 0.329436.
    已训练900次, loss: 0.294607.
    已训练1000次, loss: 0.259079.
    2018-08-17 17:01:38,289 - INFO: Saving the model...
    2018-08-17 17:01:39,062 - INFO: Restoring the model...
    INFO:tensorflow:Restoring parameters from E://logs/model.ckpt
    2018-08-17 17:01:39,062 - INFO: Restoring parameters from E://logs/model.ckpt
    预测值为：[0].

#### Examples ####

**iris.csv**: [https://github.com/percent4/Storing\_pictures/blob/master/iris.csv][https_github.com_percent4_Storing_pictures_blob_master_iris.csv]

TensorFlow Code:

Softmax 类：TF\_Softmax.py

import tensorflow as tf import logging logging.basicConfig(level = logging.INFO, format='%(asctime)s - %(levelname)s: %(message)s') logger = logging.getLogger(__name__) class Softmax: def __init__(self, input_size, class_num, epoch = 1000, learning_rate = 0.1, save_model_path='./model.ckpt'): self.save_model_path = save_model_path # 模型保存目录 self.epoch = epoch # 循环次数 self.learning_rate = learning_rate # 学习率 x = tf.placeholder(dtype=tf.float32, shape=[None, input_size]) # 特征 y_true = tf.placeholder(dtype=tf.float32, shape=[None, class_num]) # 标签 # 定义softmax回归的网络结构 W = tf.Variable(tf.random_normal([input_size, class_num], name='weight')) b = tf.Variable(tf.random_normal([class_num], name='bias')) # tf.nn.softmax computes softmax activations # softmax = exp(logits) / reduce_sum(exp(logits), dim) self.hypothesis = tf.nn.softmax(tf.matmul(x, W) + b) self.x = x self.y_true = y_true # Cross entropy loss self.loss = tf.reduce_mean(-tf.reduce_sum(self.y_true * tf.log(self.hypothesis), axis=1)) # 选择optimizer使loss达到最小, 使用梯度下降法减少 loss，学习率是self.learning_rate self.train_model = tf.train.GradientDescentOptimizer(self.learning_rate).minimize(self.loss) self.saver = tf.train.Saver() logger.info('Initialize the model...') # 训练并保存模型 def train(self, x_data, y_data): logger.info('Training the model...') with tf.Session() as sess: # 对所有变量进行初始化 sess.run(tf.global_variables_initializer()) feed_dict = { self.x: x_data, self.y_true: y_data} # 进行迭代学习 for i in range(self.epoch+1): sess.run(self.train_model, feed_dict=feed_dict) if i % int(self.epoch/10) == 0: # to see the step improvement print('已训练%d次, loss: %s.' % (i, sess.run(self.loss, feed_dict=feed_dict))) # 保存ANN模型 logger.info('Saving the model...') self.saver.save(sess, self.save_model_path) # E # 预测数据 def predict(self, data): with tf.Session() as sess: logger.info('Restoring the model...') self.saver.restore(sess, self.save_model_path) # A predict = sess.run(self.hypothesis, feed_dict={ self.x: data}) # B # print("预测值为：%s." % predict) return predict

预测数据：test\_softmax.py

""" 利用Softmax对IRIS数据集进行多分类 """ from TF_Softmax import Softmax import pandas as pd from sklearn.model_selection import train_test_split from sklearn.metrics import accuracy_score from sklearn.preprocessing import LabelBinarizer CSV_FILE_PATH = 'E://iris.csv' # CSV 文件路径 IRIS = pd.read_csv(CSV_FILE_PATH) # 读取CSV文件 target_variable = 'class' # 目标变量 # 数据集的特征 features = list(IRIS.columns) features.remove(target_variable) # 目标变量的类别 Class = IRIS[target_variable].unique() # 对目标变量进行重新编码 # 目标变量的类别字典 Class_dict = { } for i, clf in enumerate(Class): Class_dict[clf] = i+1 # 增加一列target, 将目标变量进行编码 IRIS['target'] = IRIS[target_variable].apply(lambda x: Class_dict[x]) # 对目标变量进行0-1编码 lb = LabelBinarizer() lb.fit(list(Class_dict.values())) transformed_labels = lb.transform(IRIS['target']) y_bin_labels = [] # 对多分类进行0-1编码的变量 for i in range(transformed_labels.shape[1]): y_bin_labels.append('y'+str(i)) IRIS['y'+str(i)] = transformed_labels[:, i] # print(IRIS.head(100)) # 数据是否标准化 # x_bar = (x-mean)/std IS_STANDARD = 'no' if IS_STANDARD == 'yes': for feature in features: mean = IRIS[feature].mean() std = IRIS[feature].std() IRIS[feature] = (IRIS[feature]-mean)/std # print(IRIS.head()) # 将数据集分为训练集和测试集，训练集80%, 测试集20% x_train, x_test, y_train, y_test = train_test_split(IRIS[features], IRIS[y_bin_labels], \ train_size = 0.8, test_size=0.2, random_state=123) # 使用Softmax进行预测 # 构建Softmax网络 input_size = x_train.shape[1] class_num = y_train.shape[1] # 模型保存地址 MODEL_SAVE_PATH = 'E://logs/softmax.ckpt' # Softmax初始化 ann = Softmax(input_size, class_num, 10000, 0.5, MODEL_SAVE_PATH) ann.train(x_train, y_train) # 训练ANN y_pred = ann.predict(x_test) # 预测数据 # 预测分类 prediction = [] for pred in y_pred: prediction.append(list(pred).index(max(pred))+1) # 计算预测的准确率 x_test['prediction'] = prediction x_test['label'] = IRIS['target'][y_test.index] print(x_test.head()) accuracy = accuracy_score(x_test['prediction'], x_test['label']) print('Softmax预测准确率为%.2f%%.'%(accuracy*100))

output:

2018-08-17 21:36:33,373 - INFO: Initialize the model...
    2018-08-17 21:36:33,373 - INFO: Training the model...
    已训练0次, loss: 8.691852.
    已训练1000次, loss: 0.060822684.
    已训练2000次, loss: 0.053384975.
    已训练3000次, loss: 0.04848685.
    已训练4000次, loss: 0.044897027.
    已训练5000次, loss: 0.042198572.
    已训练6000次, loss: 0.040111676.
    已训练7000次, loss: 0.038444195.
    已训练8000次, loss: 0.037071057.
    已训练9000次, loss: 0.035911743.
    已训练10000次, loss: 0.034913346.
    2018-08-17 21:36:37,012 - INFO: Saving the model...
    2018-08-17 21:36:37,215 - INFO: Restoring the model...
    INFO:tensorflow:Restoring parameters from E://logs/softmax.ckpt
    2018-08-17 21:36:37,230 - INFO: Restoring parameters from E://logs/softmax.ckpt
         sepal_length  sepal_width  petal_length  petal_width  prediction  label
    72            6.3          2.5           4.9          1.5           2      2
    112           6.8          3.0           5.5          2.1           3      3
    132           6.4          2.8           5.6          2.2           3      3
    88            5.6          3.0           4.1          1.3           2      2
    37            4.9          3.1           1.5          0.1           1      1
    Softmax预测准确率为96.67%.

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[logistic regression]: /images/20220516/38618c8cf1f642e7b57cc060f06d8b88.png
[softmax]: /images/20220516/72201411c73a4d37a5aab311a6b39e4b.png
[https_github.com_percent4_Storing_pictures_blob_master_iris.csv]: https://github.com/percent4/Storing_pictures/blob/master/iris.csv