快教程

10分钟入门神经网络 PyTorch 手写数字识别

慢教程

【深度学习Pytorch入门】

简单回归问题-1

梯度下降算法

梯度下降算法

$$loss=x^2\ast sin(x)$$

求导得：

$$f^{‘}(x)=2xsinx+x^{2}cosx$$

迭代式：
$$x^{‘}=x-\delta x$$

$\delta x$ 前乘上学习速度 $lr$ , 使得梯度慢慢往下降无限趋近合适解，在最优解附近波动 ,得到一个近似解

求解器

• sgd
• rmsprop
• adam

求解一个简单的二元一次方程

噪声

实际数据是含有高斯噪声的，我们拿来做观测值，通过观察数据分布为线型分布时，不断优化loss，即求loss极小值

• $y=w\ast x+b+ϵ$

• $ϵ\backsim N(0.01,1)$

求解loss极小值，求得 $y$ 近似于$WX+b$ 的取值：
$$loss=(WX+b-y{)}^2$$
最后
$$loss=\sum _i(w\ast x_i+b-y_i{)}^2$$
从而
$$w^{‘}\ast x+b^{‘}\to y^{‘}$$

简单回归问题-2

凸优化(感兴趣可以查阅资料)

• linear Regression

取值范围是连续的

$$\begin{gathered} wx+b \\ {w}_1+b \\ ....... \\ {w}_n+b \end{gathered}$$

用以上实际数据(8)预测 $WX+B$

# 梯度下降的应用
def compute_error_for_line_given_points(b,w,points):totalError = 0for i in range(0,len(points)):x = points[i,0]y = points[i,1]#  totalError += (y - (w * x + b ) ) ** 2b_gradient += -(2/N) * (y - ((w_current * x) + b_current))w_gradient += -(2/N) * x * (y - ((w_current * x) + b_current))new_b = b_current - (learningRate * b_gradient)new_w = w_current - (learningRate * w_gradient)return [new_b, new_w]#return totalError / float(len(points))def gradient_descent_runner(points, starting_b , starting_m, learning_rate, num_iterations):b = starting_bm = starting_mfor i in range(num_iterations):b, m = step_gradient(b,m,np.array(points),learning_rate)return [b,m]

• Logistic Regression

值域压缩到 [0-1] 的范围

• Classification

在上一种regression基础上，每个点的概率加起来为1

简单回归实战案例

import numpy as np# y = wx + b
def compute_error_for_line_given_points(b, w, points):totalError = 0for i in range(0, len(points)):x = points[i, 0]y = points[i, 1]totalError += (y - (w * x + b)) ** 2return totalError / float(len(points))def step_gradient(b_current, w_current, points, learningRate):b_gradient = 0w_gradient = 0N = float(len(points))for i in range(0, len(points)):x = points[i, 0]y = points[i, 1]b_gradient += -(2/N) * (y - ((w_current * x) + b_current))w_gradient += -(2/N) * x * (y - ((w_current * x) + b_current))new_b = b_current - (learningRate * b_gradient)new_m = w_current - (learningRate * w_gradient)return [new_b, new_m]def gradient_descent_runner(points, starting_b, starting_m, learning_rate, num_iterations):b = starting_bm = starting_mfor i in range(num_iterations):b, m = step_gradient(b, m, np.array(points), learning_rate)return [b, m]def run():points = np.genfromtxt("data.csv", delimiter=",")learning_rate = 0.0001initial_b = 0 # initial y-intercept guessinitial_m = 0 # initial slope guessnum_iterations = 1000print("Starting gradient descent at b = {0}, m = {1}, error = {2}".format(initial_b, initial_m,compute_error_for_line_given_points(initial_b, initial_m, points)))print("Running...")[b, m] = gradient_descent_runner(points, initial_b, initial_m, learning_rate, num_iterations)print("After {0} iterations b = {1}, m = {2}, error = {3}".format(num_iterations, b, m,compute_error_for_line_given_points(b, m, points)))if __name__ == '__main__':run()

# 跑完结果
Starting gradient descent at b = 0, m = 0, error = 5565.107834483211
Running...
After 1000 iterations b = 0.08893651993741346, m = 1.4777440851894448, error = 112.61481011613473

分类问题引入-1

MNIST数据集

• 每个数字有7000张图像
• 训练数据和测试数据划分为：60k 和 10k

H3:[1,d3]Y:[0/1/.../9]

(1) Nutshell

在最简单的二元一次线性方程基础上进行三次线性模型嵌套，使线性输出更稳定，每一次嵌套后的结果作为后一个的输入
p r e d = W 3 ∗ { W 2 [ W 1 X + b 1 ] + b 2 } + b 3 pred = W_3 *\{W_2[W_1X+b_1]+b_2\}+b_3\nonumber pred=W3​∗{W2​[W1​X+b1​]+b2​}+b3​

(2) Non-linear Factor

• segmoid

• ReLU

  • 梯度离散

    三层嵌套整流函数

    $H1=relu(XW1+b1)$

    $H2=relu(H1W2+b2)$

    $H3=relu(H2W3+b3)$

    增加了非线性变化的容错

(3) Gradient Descent

$objective=\sum (red-Y{)}^2$

• $[W1,W2,W3]$
• $[b1,b2,b3]$

说人话就是让模型愈来愈贴近真实的变化（从正常的字体，到倾斜，模糊，笔画奇特等字体），以便更好的预测