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对话系统（二）-普通神经网络

發布時間：2024/10/8 windows 35 豆豆

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原理

流程

生成數據

生成權重

layer1：

layer2：

${x}$ ：輸入數據 ${(20, 5)}$
${w_{1}}$ ：第一層權重 ${(5, 3)}$
${w_{2}}$ ：第二層權重 ${(3, 2)}$
${a_{1}}$ ：乘積 ${(20, 3)}$
${h_{1}}$ ：過激活函數 ${(20, 3)}$
${a_{2}}$ ：乘積 ${(20, 2)}$
${h_{2}}$ ：過激活函數 ${(20, 2)}$

正向傳播

${x}$
${a_{1}}={x}*{w_{1}}$
${h_{1}}=sigmoid({a_{1}})$
${a_{2}}={h_{1}}*{w_{2}}$
${h_{2}}=sigmoid({a_{2}})$

推導

損失函數logloss： $J=?1m∑(ylog?y^+(1?y)log?(1?y^))\displaystyle J=-\frac{1}{m}\sum(y\log{\hat{y}}+(1-y)\log(1-\hat{y}))$

$?J?w2=?J?h2??h2?a2??a2?w2\displaystyle\frac{\partial{J}}{\partial{w_2}}=\frac{\partial{J}}{\partial{h_2}}*\frac{\partial{h_{2}}}{\partial{a_2}}*\frac{\partial{a_{2}}}{\partial{w_{2}}}$

$?J?w1=?J?h2??h2?a2??a2?h1??h1?a1??a1?w1\displaystyle\frac{\partial{J}}{\partial{w_1}}=\frac{\partial{J}}{\partial{h_2}}*\frac{\partial{h_{2}}}{\partial{a_2}}*\frac{\partial{a_{2}}}{\partial{h_{1}}}*\frac{\partial{h_{1}}}{\partial{a_{1}}}*\frac{\partial{a_{1}}}{\partial{w_{1}}}$

其中公共部分（前兩個偏導）為： $?J?h2??h2?a2\displaystyle\frac{\partial{J}}{\partial{h_2}}*\frac{\partial{h_{2}}}{\partial{a_2}}$

$?J?h2=?1m?y?h2h2(1?h2)\displaystyle\frac{\partial{J}}{\partial{h_2}}=-\frac{1}{m}*\frac{y-h_{2}}{h_{2}(1-h_{2})}$

$?h2?a2=h2(1?h2)\displaystyle\frac{\partial{h_{2}}}{\partial{a_2}}=h_{2}(1-h_{2})$

$?a2?w2=h1\displaystyle\frac{\partial{a_{2}}}{\partial{w_{2}}}=h_{1}$

$?a2?h1=w2\displaystyle\frac{\partial{a_{2}}}{\partial{h_{1}}}=w_{2}$

$?h1?a1=h1?(1?h1)\displaystyle\frac{\partial{h_{1}}}{\partial{a_{1}}}=h_{1}*(1-h_{1})$

$?a1?w1=x\displaystyle\frac{\partial{a_{1}}}{\partial{w_{1}}}=x$

求

{x_{1}}

代碼

用numpy實現

import numpy as nptrain_x_dim = 5 sample_1_num = 10 sample_0_num = 10 weight1_dim = 3 weight2_dim = 2train_x_1 = np.random.rand(sample_1_num, train_x_dim) train_x_0 = np.random.rand(sample_0_num, train_x_dim)*10train_y_1 = np.ones(sample_1_num) train_y_0 = np.zeros(sample_0_num)weight1 = np.random.rand(train_x_dim, weight1_dim)def sigmoid(x):return 1/(1+np.exp(-x))a1 = np.dot(train_x_1, weight1) h1 = sigmoid(a1)weight2 = np.random.rand(weight1_dim, weight2_dim) a2 = np.dot(h1, weight2) h2 = sigmoid(a2)def sigmoid_derv(x):return sigmoid(x)*(1-sigmoid(x))

用tf實現

from tensorflow import keras # load data fashion_mnist = keras.datasets.fashion_mnist (train_images, train_labels), (test_images, test_labels) = fashion_mnist.load_data() # build model model = keras.Sequential([keras.layers.Flatten(input_shape=(28, 28)),keras.layers.Dense(128, activation=tf.nn.relu),keras.layers.Dense(10, activation=tf.nn.softmax) ]) # compile model model.compile(optimizer=tf.train.AdamOptimizer(),loss='sparse_categorical_crossentropy',metrics=['accuracy']) # train model model.fit(train_images, train_labels, epochs=5) # evaluate test_loss, test_acc = model.evaluate(test_images, test_labels)

總結

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