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深度学习6（多分类+交叉熵损失原理+手写数字识别案例TensorFlow）

news 2025/7/7 6:04:43

多分类

softmax回归

多分类

我们已经接触过二分类问题，神经网络输出层只有一个神经元，表示预测输出是正类的概率P(y =1m),g>0.5则判断为正类，反之判断为负类。那么对于多分类问题怎么办?

对于多分类问题，用 N表示种类个数，那么神经网络的输出层的神经元个数必须为L[output]=N,每个神经元的输出依次对应属于N个类别当中某个具体类别的概率，即 P(y=N1|),…,P(y= Nn|z)。

softmax回归

Softmax回归（也称为多项逻辑回归）是逻辑回归在多分类问题上的自然扩展，它是深度学习中最基础也是最重要的多分类方法之一。与二分类的逻辑回归不同，Softmax回归可以同时处理多个类别的分类问题，并且将输出转化为概率分布，满足 0 ≤ P ≤ 1

$z=Wx+b$

$P=\frac{e^{z} }{\sum_{i=1}^{K}e_{i} ^{z} }$

交叉熵损失

对于softmax回归(逻辑回归代价函数的推广，都可称之为交叉损失)，它的代价函数公式为:

$L(\hat{y} , y) = -\sum_{i=1}^{k} y_i \log g_i$

其实将逻辑回归看作是一个多分类（两个分类），它的表达形式也可以与之统一，

$J(\theta) = -\frac{1}{m} \left[ \sum_{i=1}^{m} (1 - y^{(i)}) \log(1 - h_\theta(x^{(i)})) + y^{(i)} \log h_\theta(x^{(i)}) \right]$

$= -\frac{1}{m} \left[ \sum_{i=1}^{m} \sum_{j=0}^{1} 1\left\{y^{(i)} = j\right\} \log p(y^{(i)} = j \mid x^{(i)}; \theta) \right]$

手写数字识别案例

本次代码没有手写实现正向传播、反向传播等细节，都使用tensorflow包调用其API来实现，

封装更简洁，大大增加了代码可读性。这里引入的tensorflow的版本可能落后，完全复制可能不一定复现成功。

本次使用的样本是通过下载网上的数据集，并通过mnist = tf.keras.datasets.mnist.load_data()引入数据。

import tensorflow.compat.v1 as tf
tf.disable_v2_behavior()
mnist = tf.keras.datasets.mnist.load_data()def fc_nn():# 获取数据mnist = tf.keras.datasets.mnist.load_data()with tf.variable_scope("resource"):# 特征Xx = tf.placeholder(tf.float32, [None, 784], name="X_data")# 目标值Y（修正：改为float32以匹配one-hot编码）y = tf.placeholder(tf.float32, [None, 10], name="label")with tf.variable_scope("hidden"):# [none,784] * [784 ,64] + [64] = [none , 64]# 修正：将mean和stddev参数移到random_normal函数内部weight_hidden = tf.Variable(tf.random_normal([784,64], mean=0.0, stddev=1.0), name="weight_hidden")bias_hidden = tf.Variable(tf.random_normal([64], mean=0.0, stddev=1.0), name="bias_hidden")A1 = tf.matmul(x, weight_hidden) + bias_hiddenwith tf.variable_scope("fc"):# [none , 64] * [64 , 10] + [10] = [none , 10]weight_fc = tf.Variable(tf.random_normal([64,10], mean=0.0, stddev=1.0), name="weight_fc")bias_fc = tf.Variable(tf.random_normal([10], mean=0.0, stddev=1.0), name="bias_fc")y_pred = tf.matmul(A1, weight_fc) + bias_fcwith tf.variable_scope("compute_loss"):# 计算损失all_loss = tf.nn.softmax_cross_entropy_with_logits(labels=y, logits=y_pred)loss = tf.reduce_mean(all_loss)with tf.variable_scope("optimize"):train_op = tf.train.GradientDescentOptimizer(0.1).minimize(loss)with tf.Session() as sess:# 运行初始化变量(必须做)sess.run(tf.global_variables_initializer())# 循环for i in range(2000):# 修正：添加批量数据获取batch_x, batch_y = mnist.train.next_batch(100)# 修正：添加feed_dict参数传递数据loss_run, _ = sess.run([loss, train_op], feed_dict={x: batch_x, y: batch_y})print("迭代第%d步，损失为：%f"%(i, loss_run))if __name__ == '__main__':fc_nn()

这里是通过AI修改后，符合版本的代码，谨慎尝试

import tensorflow.compat.v1 as tf
tf.disable_v2_behavior()def fc_nn():# 获取数据(x_train, y_train), (x_test, y_test) = tf.keras.datasets.mnist.load_data()# 预处理数据x_train = x_train.reshape(-1, 784).astype('float32') / 255.0y_train = tf.keras.utils.to_categorical(y_train, 10)with tf.variable_scope("resource"):# 特征Xx = tf.placeholder(tf.float32, [None, 784], name="X_data")# 目标值Yy = tf.placeholder(tf.float32, [None, 10], name="label")with tf.variable_scope("hidden"):# [none,784] * [784 ,64] + [64] = [none , 64]weight_hidden = tf.Variable(tf.random_normal([784,64], mean=0.0, stddev=1.0), name="weight_hidden")bias_hidden = tf.Variable(tf.random_normal([64], mean=0.0, stddev=1.0), name="bias_hidden")A1 = tf.matmul(x, weight_hidden) + bias_hiddenwith tf.variable_scope("fc"):# [none , 64] * [64 , 10] + [10] = [none , 10]weight_fc = tf.Variable(tf.random_normal([64,10], mean=0.0, stddev=1.0), name="weight_fc")bias_fc = tf.Variable(tf.random_normal([10], mean=0.0, stddev=1.0), name="bias_fc")y_pred = tf.matmul(A1, weight_fc) + bias_fcwith tf.variable_scope("compute_loss"):# 计算损失all_loss = tf.nn.softmax_cross_entropy_with_logits(labels=y, logits=y_pred)loss = tf.reduce_mean(all_loss)with tf.variable_scope("optimize"):train_op = tf.train.GradientDescentOptimizer(0.1).minimize(loss)with tf.Session() as sess:# 运行初始化变量(必须做)sess.run(tf.global_variables_initializer())# 循环for i in range(2000):# 获取批量数据start = (i * 100) % (len(x_train) - 100)end = start + 100batch_x = x_train[start:end]batch_y = y_train[start:end]loss_run, _ = sess.run([loss, train_op], feed_dict={x: batch_x, y: batch_y})print("迭代第%d步，损失为：%f"%(i, loss_run))if __name__ == '__main__':fc_nn()

改进代码（数据保存+TensorBoard显示图像）

import tensorflow.compat.v1 as tf
import ostf.disable_v2_behavior()def fc_nn():# 获取数据(x_train, y_train), (x_test, y_test) = tf.keras.datasets.mnist.load_data()# 预处理数据x_train = x_train.reshape(-1, 784).astype('float32') / 255.0y_train = tf.keras.utils.to_categorical(y_train, 10)# 创建日志目录log_dir = "./logs/fc_nn"if not os.path.exists(log_dir):os.makedirs(log_dir)with tf.variable_scope("resource"):# 特征Xx = tf.placeholder(tf.float32, [None, 784], name="X_data")# 目标值Yy = tf.placeholder(tf.float32, [None, 10], name="label")with tf.variable_scope("hidden"):# [none,784] * [784 ,64] + [64] = [none , 64]weight_hidden = tf.Variable(tf.random_normal([784,64], mean=0.0, stddev=1.0), name="weight_hidden")bias_hidden = tf.Variable(tf.random_normal([64], mean=0.0, stddev=1.0), name="bias_hidden")A1 = tf.matmul(x, weight_hidden) + bias_hidden# 添加激活函数和直方图记录A1 = tf.nn.relu(A1)tf.summary.histogram("hidden_activations", A1)with tf.variable_scope("fc"):# [none , 64] * [64 , 10] + [10] = [none , 10]weight_fc = tf.Variable(tf.random_normal([64,10], mean=0.0, stddev=1.0), name="weight_fc")bias_fc = tf.Variable(tf.random_normal([10], mean=0.0, stddev=1.0), name="bias_fc")y_pred = tf.matmul(A1, weight_fc) + bias_fc# 记录权重和偏置tf.summary.histogram("fc_weights", weight_fc)tf.summary.histogram("fc_biases", bias_fc)with tf.variable_scope("compute_loss"):# 计算损失all_loss = tf.nn.softmax_cross_entropy_with_logits(labels=y, logits=y_pred)loss = tf.reduce_mean(all_loss)# 记录损失tf.summary.scalar("loss", loss)with tf.variable_scope("accuracy"):# 计算准确率correct_pred = tf.equal(tf.argmax(y_pred, 1), tf.argmax(y, 1))accuracy = tf.reduce_mean(tf.cast(correct_pred, tf.float32))tf.summary.scalar("accuracy", accuracy)with tf.variable_scope("optimize"):train_op = tf.train.GradientDescentOptimizer(0.1).minimize(loss)# 合并所有summarymerged = tf.summary.merge_all()with tf.Session() as sess:# 创建FileWritertrain_writer = tf.summary.FileWriter(log_dir, sess.graph)# 运行初始化变量sess.run(tf.global_variables_initializer())# 训练循环for i in range(2000):# 获取批量数据start = (i * 100) % (len(x_train) - 100)end = start + 100batch_x = x_train[start:end]batch_y = y_train[start:end]# 运行训练并记录summary_, loss_run, summary = sess.run([train_op, loss, merged],feed_dict={x: batch_x, y: batch_y})# 写入summarytrain_writer.add_summary(summary, i)# 每100步打印一次信息if i % 100 == 0:acc = sess.run(accuracy, feed_dict={x: batch_x, y: batch_y})print(f"迭代第{i}步，损失: {loss_run:.4f}, 准确率: {acc:.4f}")# 关闭FileWritertrain_writer.close()if __name__ == '__main__':fc_nn()

该代码运行一次后会创建log文件保存数据，然后再python终端里输入