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神经网络入门教程

分类: 文章 • 2024-03-02 18:11:28

神经元

考虑一个2入 xT=[x1 x2]x^T=[x_1\ x_2] 1出 yy 的神经元
y=f(wTx+b) y=f(w^Tx+b)
wT=[w1 w2]w^T=[w_1\ w_2]为权重，bb 为偏置，ff 为**函数

import numpy as np
def sigmoid(x): # Our activation function: f(x) = 1 / (1 + e^(-x))
return 1 / (1 + np.exp(-x))
class Neuron:
  def __init__(self, weights, bias):
    self.weights = weights
    self.bias = bias
  def feedforward(self, inputs):
    # Weight inputs, add bias, then use the activation function
    total = np.dot(self.weights, inputs) + self.bias
    return sigmoid(total)
weights = np.array([0, 1]) # w1 = 0, w2 = 1
bias = 4                   # b = 4
n = Neuron(weights, bias)
x = np.array([2, 3])       # x1 = 2, x2 = 3
print(n.feedforward(x))    # 0.9990889488055994

神经网络

神经网络就是把一堆神经元连接在一起，有2个输入、一个包含2个神经元的隐藏层（h1h_1和h2h_2）、包含1个神经元的输出层o1o_1。

class OurNeuralNetwork:
  '''
  A neural network with:
    - 2 inputs
    - a hidden layer with 2 neurons (h1, h2)
    - an output layer with 1 neuron (o1)
  Each neuron has the same weights and bias:
    - w = [0, 1]
    - b = 0
  '''
  def __init__(self):
    weights = np.array([0, 1])
    bias = 0
    # The Neuron class here is from the previous section
    self.h1 = Neuron(weights, bias)
    self.h2 = Neuron(weights, bias)
    self.o1 = Neuron(weights, bias)
  def feedforward(self, x):
    out_h1 = self.h1.feedforward(x)
    out_h2 = self.h2.feedforward(x)
    # The inputs for o1 are the outputs from h1 and h2
    out_o1 = self.o1.feedforward(np.array([out_h1, out_h2]))
    return out_o1
network = OurNeuralNetwork()
x = np.array([2, 3])
print(network.feedforward(x)) # 0.7216325609518421

训练神经网络

损失定义
MSE=1n∑i=1n(ytrue−ypred)2 MSE=\frac{1}{n}\sum_{i=1}^n(y_{true}-y_{pred})^2
训练神经网络就是将损失最小化

import numpy as np
def mse_loss(y_true, y_pred):
  # y_true and y_pred are numpy arrays of the same length.
  return ((y_true - y_pred) ** 2).mean()
y_true = np.array([1, 0, 0, 1])
y_pred = np.array([0, 0, 0, 0])
print(mse_loss(y_true, y_pred)) # 0.5

预测值是由一系列网络权重和偏置计算出来的,所以损失函数实际上是包含多个权重、偏置的多元函数
神经网络入门教程
L(w1,w2,w3,w4,w5,w6,b1,b2,b3) L(w_1,w_2,w_3,w_4,w_5,w_6,b_1,b_2,b_3)
若要判断调整权重 w1w_1 对损失函数的影响，需要考虑偏导数的正负
∂L∂w1=∂L∂ypred∗∂ypred∂h1∗∂h1∂w1 \frac{\partial L}{\partial w_{1}}=\frac{\partial L}{\partial y_{p r e d}} * \frac{\partial y_{p r e d}}{\partial h_{1}} * \frac{\partial h_{1}}{\partial w_{1}}
这种向后计算偏导数的系统称为反向传播（backpropagation）

随机梯度下降

w1←w1−η∂L∂w1 w_{1} \leftarrow w_{1}-\eta \frac{\partial L}{\partial w_{1}}
用这种方法去逐步改变网络的权重ww和偏置bb，损失函数会缓慢地降低，从而改进神经网络

训练流程如下：
1、从数据集中选择一个样本；
2、计算损失函数对所有权重和偏置的偏导数；
3、使用更新公式更新每个权重和偏置；
4、回到第1步。

import numpy as np

def sigmoid(x):
  # Sigmoid activation function: f(x) = 1 / (1 + e^(-x))
  return 1 / (1 + np.exp(-x))

def deriv_sigmoid(x):
  # Derivative of sigmoid: f'(x) = f(x) * (1 - f(x))
  fx = sigmoid(x)
  return fx * (1 - fx)

def mse_loss(y_true, y_pred):
  # y_true and y_pred are numpy arrays of the same length.
  return ((y_true - y_pred) ** 2).mean()

class OurNeuralNetwork:
  '''
  A neural network with:
    - 2 inputs
    - a hidden layer with 2 neurons (h1, h2)
    - an output layer with 1 neuron (o1)

  *** DISCLAIMER ***:
  The code below is intended to be simple and educational, NOT optimal.
  Real neural net code looks nothing like this. DO NOT use this code.
  Instead, read/run it to understand how this specific network works.
  '''
  def __init__(self):
    # Weights
    self.w1 = np.random.normal()
    self.w2 = np.random.normal()
    self.w3 = np.random.normal()
    self.w4 = np.random.normal()
    self.w5 = np.random.normal()
    self.w6 = np.random.normal()

    # Biases
    self.b1 = np.random.normal()
    self.b2 = np.random.normal()
    self.b3 = np.random.normal()

  def feedforward(self, x):
    # x is a numpy array with 2 elements.
    h1 = sigmoid(self.w1 * x[0] + self.w2 * x[1] + self.b1)
    h2 = sigmoid(self.w3 * x[0] + self.w4 * x[1] + self.b2)
    o1 = sigmoid(self.w5 * h1 + self.w6 * h2 + self.b3)
    return o1

  def train(self, data, all_y_trues):
    '''
    - data is a (n x 2) numpy array, n = # of samples in the dataset.
    - all_y_trues is a numpy array with n elements.
      Elements in all_y_trues correspond to those in data.
    '''
    learn_rate = 0.1
    epochs = 1000 # number of times to loop through the entire dataset

    for epoch in range(epochs):
      for x, y_true in zip(data, all_y_trues):
        # --- Do a feedforward (we'll need these values later)
        sum_h1 = self.w1 * x[0] + self.w2 * x[1] + self.b1
        h1 = sigmoid(sum_h1)

        sum_h2 = self.w3 * x[0] + self.w4 * x[1] + self.b2
        h2 = sigmoid(sum_h2)

        sum_o1 = self.w5 * h1 + self.w6 * h2 + self.b3
        o1 = sigmoid(sum_o1)
        y_pred = o1

        # --- Calculate partial derivatives.
        # --- Naming: d_L_d_w1 represents "partial L / partial w1"
        d_L_d_ypred = -2 * (y_true - y_pred)

        # Neuron o1
        d_ypred_d_w5 = h1 * deriv_sigmoid(sum_o1)
        d_ypred_d_w6 = h2 * deriv_sigmoid(sum_o1)
        d_ypred_d_b3 = deriv_sigmoid(sum_o1)

        d_ypred_d_h1 = self.w5 * deriv_sigmoid(sum_o1)
        d_ypred_d_h2 = self.w6 * deriv_sigmoid(sum_o1)

        # Neuron h1
        d_h1_d_w1 = x[0] * deriv_sigmoid(sum_h1)
        d_h1_d_w2 = x[1] * deriv_sigmoid(sum_h1)
        d_h1_d_b1 = deriv_sigmoid(sum_h1)

        # Neuron h2
        d_h2_d_w3 = x[0] * deriv_sigmoid(sum_h2)
        d_h2_d_w4 = x[1] * deriv_sigmoid(sum_h2)
        d_h2_d_b2 = deriv_sigmoid(sum_h2)

        # --- Update weights and biases
        # Neuron h1
        self.w1 -= learn_rate * d_L_d_ypred * d_ypred_d_h1 * d_h1_d_w1
        self.w2 -= learn_rate * d_L_d_ypred * d_ypred_d_h1 * d_h1_d_w2
        self.b1 -= learn_rate * d_L_d_ypred * d_ypred_d_h1 * d_h1_d_b1

        # Neuron h2
        self.w3 -= learn_rate * d_L_d_ypred * d_ypred_d_h2 * d_h2_d_w3
        self.w4 -= learn_rate * d_L_d_ypred * d_ypred_d_h2 * d_h2_d_w4
        self.b2 -= learn_rate * d_L_d_ypred * d_ypred_d_h2 * d_h2_d_b2

        # Neuron o1
        self.w5 -= learn_rate * d_L_d_ypred * d_ypred_d_w5
        self.w6 -= learn_rate * d_L_d_ypred * d_ypred_d_w6
        self.b3 -= learn_rate * d_L_d_ypred * d_ypred_d_b3

      # --- Calculate total loss at the end of each epoch
      if epoch % 10 == 0:
        y_preds = np.apply_along_axis(self.feedforward, 1, data)
        loss = mse_loss(all_y_trues, y_preds)
        print("Epoch %d loss: %.3f" % (epoch, loss))

# Define dataset
data = np.array([
  [-2, -1],  # Alice
  [25, 6],   # Bob
  [17, 4],   # Charlie
  [-15, -6], # Diana
])
all_y_trues = np.array([
  1, # Alice
  0, # Bob
  0, # Charlie
  1, # Diana
])

# Train our neural network!
network = OurNeuralNetwork()
network.train(data, all_y_trues)

做预测

# Make some predictions
emily = np.array([-7, -3]) # 128 pounds, 63 inches
frank = np.array([20, 2])  # 155 pounds, 68 inches
print("Emily: %.3f" % network.feedforward(emily)) # 0.951 - F
print("Frank: %.3f" % network.feedforward(frank)) # 0.039 - M

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