深度学习入门

Python Basic

$ python3

>>> 1 + 2
3

>>> 1 - 2
-1

>>> 4 * 5
20

>>> 7 / 5
1.4

>>> 3 ** 2
9

>>> type(10)
<class 'int'>

>>> type(2.718)
<class 'float'>

>>> type("hello")
<class 'str'>

>>> x = 10
>>> print(x)
10

>>> x = 100
>>> print(x)
100

>>> y = 3.14
>>> x * y
314.0

>>> type(x * y)
<class 'float'>

>>> a = [1, 2, 3, 4, 5]
>>> print(a)
[1, 2, 3, 4, 5]

>>> len(a)
5

>>> a[0]
1

>>> a[4]
5

>>> a[4] = 99
>>> print(a)
[1, 2, 3, 4, 99]

>>> a[0:2]
[1, 2]

>>> a[1:]
[2, 3, 4, 99]

>>> a[:3]
[1, 2, 3]

>>> a[:-1]
[1, 2, 3, 4]

>>> a[:-2]
[1, 2, 3]

>>> me = {'height' : 100}
>>> me['height']
100

>>> me['height'] = 70
>>> print(me)
{'height': 70}

>>> me['weight'] = 80
>>> print(me)
{'height': 70, 'weight': 80}

>>> hungry = True
>>> sleepy = False
>>> type(hungry)
<class 'bool'>

>>> not hungry
False

>>> hungry and sleepy

False

>>> hungry or sleepy
True

>>> hungry = True
>>> if hungry:
... print("I'm hungry")
...

"I'm hungry"

>>> hungry = False
>>> if hungry:
...     print("I'm hungry")
... else:
...     print("I'm not hungry")
... print("I'm sleepy")
...

"I'm not hungry"
"I'm sleepy"

>>> for i in [1, 2, 3]:
...     print(i)
...

1
2
3

>>> def hello():
...     print("Hello world")
...

>>> hello()
Hello world

>>> def hello(object):
...     print("Hello " + object + "!")
...

>>> hello("cat")
Hello cat!

>>>

Class

class Man:
	def __init__(self, name):
		self.name = name
		print("Initialized!")

	def hello(self):
		print("Hello " + self.name + "!")
	
	def goodbye(self):
		print("Good-bye " + self.name + "!")

m = Man("David")
m.hello()
m.goodbye()

Numpy

>>> import numpy as np
>>> x = np.array([1.0, 2.0, 3.0])
>>> print(x)
[1. 2. 3.]

>>> type(x)
<class 'numpy.ndarray'>

>>> x = np.array([1.0, 2.0, 3.0])
>>> y = np.array([2.0, 4.0, 6.0])
>>> x + y
array([3., 6., 9.])

>>> x - y
array([-1., -2., -3.])

>>> x * y
array([ 2., 8., 18.])

>>> x / y
array([0.5, 0.5, 0.5])

>>> x = np.array([1.0, 2.0, 3.0])
>>> x / 2.0
array([0.5, 1. , 1.5])

>>> A = np.array([[1,2], [3,4]])
>>> print(A)
[[1 2]
[3 4]]

>>> A.shape
(2, 2)

>>> A.dtype
dtype('int64')

>>> B = np.array([[3,0], [0,6]])
>>> A + B
array([[ 4, 2],
[ 3, 10]])

>>> A * B
array([[ 3, 0],
[ 0, 24]])

>>> print(A)
[[1 2]
[3 4]]

>>> A * 10
array([[10, 20],
[30, 40]])

>>> A = np.array([[1,2], [3,4]])
>>> B = np.array([10, 20])
>>> A * B
array([[10, 40],
[30, 80]])

>>> X = np.array([[51,55], [14, 19], [0, 4]])
>>> print(X)
[[51 55]
[14 19]
[ 0 4]]

>>> X[0] # get the first row of this array
array([51, 55])

>>> X[0][1] # get the number of (0, 1)
55

>>> for row in X:
... print(row)
...

[51 55]
[14 19]
[0 4]

>>> X = X.flatten()
>>> print(X)
[51 55 14 19 0 4]

>>> X[np.array([0, 2, 4])] # get the number whose the index is 0, 2, 4
array([51, 14, 0])

>>> X > 15 # filter the number that greater than 15
array([ True, True, False, True, False, False])

>>> X[X > 15]
array([51, 55, 19])

>>>

Matplotlib

Plot the sin function

import numpy as np
import matplotlib.pyplot as plt

x = np.arange(0, 6, 0.1)
y = np.sin(x)

plt.plot(x, y)
plt.show()

Plot the image of function sin and cos

import numpy as np
import matplotlib.pyplot as plt

  

x = np.arange(0, 6, 0.1)
y1 = np.sin(x)
y2 = np.cos(x)

plt.plot(x, y1, label="sin")
plt.plot(x, y2, linestyle = "--", label="cos")

plt.xlabel("x")
plt.ylabel("y")
plt.title('sin & cos')

plt.legend()
plt.show()

Display the image

import matplotlib.pyplot as plt
from matplotlib.image import imread

img = imread('lena.png') # the path of the image
plt.imshow(img)
plt.show()

Perceptron

感知机的运行原理

The equation below can represent the behavior of the perceptron

$y=\{\begin{array}{c}0(w_1{x}_1+w_2{x}_2\le \theta )\\ 1(w_1{x}_1+w_2{x}_2>\theta )\end{array}$

Simple implementation

def AND(x1, x2):
	w1, w2, theta = 0.5, 0.5, 0.7
	tmp = w1 * x1 + w2 * x2
	
	if tmp <= theta:
		return 0
	elif tmp > theta:
		return 1

print(AND(0, 0))
print(AND(0, 1))
print(AND(1, 0))
print(AND(1, 1))

导入权重和偏置

$b$ is called bias, $w_1$ and $w_2$ are called weight

$y=\{\begin{array}{c}0(b+w_1{x}_1+w_2{x}_2\le 0)\\ 1(b+w_1{x}_1+w_2{x}_2>0)\end{array}$

use numpy to complete a simple neuron

import numpy as np
x = np.array([0, 1]) # input
w = np.array([0.5, 0.5]) # weight
b = -0.7 # bias

print(w * x)
print(np.sum(w * x))
print(np.sum(w * x) + b)

use bias and weight to complete an AND gate

def AND(x1, x2):
	x = np.array([x1, x2])
	w = np.array([0.5, 0.5])
	b = -0.7
	tmp = np.sum(w * x) + b
	if tmp <= 0:
		return 0
	else:
		return 1

NAND gate, OR gate

def NAND(x1, x2):
	x = np.array([x1, x2])
	w = np.array([-0.5, -0.5]) # just bias and weight different from AND
	b = 0.7
	tmp = np.sum(w * x) + b
	
	if tmp <= 0:
		return 0
	else:
		return 1


def OR(x1, x2):
	x = np.array([x1, x2])
	w = np.array([0.5, 0.5]) # just bias and weight different from AND
	b = -0.2
	tmp = np.sum(w * x) + b
	
	if tmp <= 0:
		return 0
	else:
		return 1

Use AND, NAND, OR gate to build NOR gate

def XOR(x1, x2):
	s1 = NAND(x1, x2)
	s2 = OR(x1, x2)
	y = AND(s1, s2)
	return y

Neuron Network

$y=\{\begin{array}{c}0(b+w_1{x}_1+w_2{x}_2\le 0)\\ 1(b+w_1{x}_1+w_2{x}_2>0)\end{array}$

引入$h(x)$

$y=h(b+w_1{x}_1+w_2{x}_2)$

$h(x)=\{\begin{array}{c}0(x\le 0)\\ 1(x>0)\end{array}$

激活函数

activation function

$a=b+w_1{x}_1+w_2{x}_2$
$y=h(a)$

sigmoid 函数

Sigmoid function

$h(x)=\frac{1}{1+exp(-x)}$ , $exp(-x)$ represent $e^{-x}$

def sigmoid(x):
	return 1 / (1 + np.exp(-x))

Function of step

import numpy as np
import matplotlib.pylab as plt

# def step_function(x):
# if x > 0:
#     return 1
# else:
#     return 0


# def step_function(x):
# y = x > 0
# return y.astype(np.int)

def step_function(x):
	return np.array(x > 0, dtype=np.int64)


x = np.arange(-5.0, 5.0, 0.1)
y = step_function(x)
plt.plot(x, y)
plt.ylim(-0.1, 1.1) # the arange of axis y plt.show()
plt.show()

Sigmoid function and step function:

import numpy as np
import matplotlib.pylab as plt


# def step_function(x):
#     if x > 0:
#         return 1
#     else:
#         return 0


# def step_function(x):
#     y = x > 0
#     return y.astype(np.int)


def step_function(x):
	return np.array(x > 0, dtype=np.int64)


def sigmoid(x):
	return 1 / (1 + np.exp(-x))


x = np.arange(-5.0, 5.0, 0.1)
y1 = step_function(x)
y2 = sigmoid(x)
plt.plot(x, y1, linestyle = "--")
plt.plot(x, y2)
plt.ylim(-0.1, 1.1) # the arange of axis y plt.show()
plt.show()

ReLU函数

ReLU function

$h(x)=\{\begin{array}{c}x(x>0)\\ 0(x\le 0)\end{array}$

def relu(x):
	return np.maximum(0, x)

多维数组

>>> import numpy as np
>>> A = np.array([1, 2, 3, 4])
>>> print(A)
[1 2 3 4]
>>> np.ndim(A)
1
>>> A.shape
(4,)
>>> A.shape[0]
4

>>> B = np.array([[1,2], [3,4], [5,6]])
>>> print(B)
[[1 2]
 [3 4]
 [5 6]]
>>> np.ndim(B)
2
>>> B.shape
(3, 2)

矩阵乘法

>>> A = np.array([[1,2], [3,4]])
>>> A.shape
(2, 2)
>>> B = np.array([[5,6], [7,8]])
>>> B.shape
(2, 2)
>>> np.dot(A, B)
array([[19, 22],
       [43, 50]])

>>> A = np.array([[1,2,3], [4,5,6]])
>>> A.shape
(2, 3)

>>> B = np.array([[1,2], [3,4], [5,6]])
>>> B.shape
(3, 2)

>>> np.dot(A, B)
array([[22, 28],
       [49, 64]])

>>> C = np.array([[1,2], [3,4]])  
>>> C.shape
(2, 2)  
>>> A.shape  
(2, 3)  
>>> np.dot(A, C)  
Traceback (most recent call last):  
	File "", line 1, in <module>  
ValueError: shapes (2,3) and (2,2) not aligned: 3 (dim 1) != 2 (dim 0)

>>> A = np.array([[1,2], [3, 4], [5,6]])  
>>> A.shape  
(3, 2)  
  
>>> B = np.array([7,8])    
>>> B.shape  
(2,)  
  
>>> np.dot(A, B)  
array([23, 53, 83])

>>> A = np.array([[1,2], [3, 4], [5,6]])  
>>> A.shape  
(3, 2)  
  
>>> B = np.array([7,8])    
>>> B.shape  
(2,)  

>>> np.dot(A, B)  
array([23, 53, 83])

神经网络内积

>>> X = np.array([1, 2])  
>>> X.shape  
(2,)  
  
>>> W = np.array([[1, 3, 5], [2, 4, 6]])  
>>> print(W)  
[[1 3 5]  
[2 4 6]]  
  
>>> W.shape
(2, 3)  
>>> Y = np.dot(X, W)  
>>> print(Y)
[ 5 11 17]

多层神经网络

$w_{12}^{(1)}$ 中 $(1)$ 表示第1层的权重，1 表示后一层的第一个神经元，2 表示前一层的第2个神经元
权重右下角按照“后一层的索引号、前一层的索引号”的顺序排列

各层间信号传递的实现

例如：

$a_1^{(1)}=w_{11}^{(1)}{x}_1+w_{12}^{(1)}{x}_2+b_1^{(1)}$

如果使用矩阵的乘法运算，则可以将第1层的加权和表示成下面的式
$A^{(1)}=X{W}^{(1)}+B^{(1)}$

其中， $A^{(1)},X,W^{(1)},B^{(1)}$ 如下所示：

$A^{(1)}=(a_1^{(1)}{a}_2^{(1)}{a}_3^{(1)})$, $X=(x_1{x}_2)$, $B^{(1)}=(b_1^{(1)}{b}_2^{(1)}{b}_3^{(1)})$, $W^{(1)}=\left(\begin{array}{ccc}{w}_{11}^{(1)}& w_{21}^{(1)}& w_{31}^{(1)}\\ w_{12}^{(1)}& w_{22}^{(1)}& w_{32}^{(1)}\end{array}\right)$

X = np.array([1.0, 0.5])
W1 = np.array([[0.1, 0.3, 0.5], [0.2, 0.4, 0.6]])
B1 = np.array([0.1, 0.2, 0.3])

print(W1.shape) # (2, 3)
print(X.shape) # (2,)
print(B1.shape) # (3,)  

A1 = np.dot(X, W1) + B1 # (2,) * (2, 3) + (3,) = (3,)

$W1$ 为 $2\times 3$ 的数组，$X$ 为元素个数为 2 的一维数组，是用 $sigmoid$ 作为激活函数

Z1 = sigmoid(A1)  
  
print(A1) # [0.3, 0.7, 1.1]  
print(Z1) # [0.57444252, 0.66818777, 0.75026011]

第一层的输出变成第二层的输入

W2 = np.array([[0.1, 0.4], [0.2, 0.5], [0.3, 0.6]])
B2 = np.array([0.1, 0.2])

print(Z1.shape) # (3,)
print(W2.shape) # (3, 2)
print(B2.shape) # (2,)

A2 = np.dot(Z1, W2) + B2 # (3,) * (3, 2) + (2,) = (2,)
Z2 = sigmoid(A2) # (2,)

激活函数

def identity_function(x):  
	return x  

W3 = np.array([[0.1, 0.3], [0.2, 0.4]])  
B3 = np.array([0.1, 0.2])  

A3 = np.dot(Z2, W3) + B3  
Y = identity_function(A3) # 或者Y = A3

代码实现

import numpy as np

def sigmoid(x):
	return 1 / (1 + np.exp(-x))


def identity_function(x):
	return x


def init_network():
	network = {}
	network['W1'] = np.array([[0.1, 0.3, 0.5], [0.2, 0.4, 0.6]])
	network['b1'] = np.array([0.1, 0.2, 0.3])
	network['W2'] = np.array([[0.1, 0.4], [0.2, 0.5], [0.3, 0.6]])
	network['b2'] = np.array([0.1, 0.2])
	network['W3'] = np.array([[0.1, 0.3], [0.2, 0.4]])
	network['b3'] = np.array([0.1, 0.2])
	return network


def forward(network, x):
	W1, W2, W3 = network['W1'], network['W2'], network['W3']
	b1, b2, b3 = network['b1'], network['b2'], network['b3']
	
	a1 = np.dot(x, W1) + b1
	z1 = sigmoid(a1)
	a2 = np.dot(z1, W2) + b2
	
	z2 = sigmoid(a2)
	a3 = np.dot(z2, W3) + b3
	y = identity_function(a3)
	
	return y


network = init_network()
x = np.array([1.0, 0.5])
y = forward(network, x)
print(y)

softmax 函数

$y_k=\frac{exp(a_k)}{{\sum }_{i=1}^{n}exp(a_i)}$

>>> import numpy as np
>>> a = np.array([0.3, 2.9, 4.0])
>>> exp_a = np.exp(a)
>>> print(exp_a)
[ 1.34985881 18.17414537 54.59815003]
>>> sum_exp_a = np.sum(exp_a)
>>> print(sum_exp_a)
74.1221542101633
>>> y = exp_a / sum_exp_a
>>> print(y)
[0.01821127 0.24519181 0.73659691]

Softmax function

def softmax(a):
	exp_a = np.exp(a)
	sum_exp_a = np.sum(exp_a)
	y = exp_a / sum_exp_a
	return y

$\begin{array}{ccc}{y}_k& =\frac{exp(a_k)}{{\sum }_{i=1}^{n}exp(a_i)}& =\frac{Cexp(a_k)}{C{\sum }_{i=1}^{n}exp(a_i)}\\ & & =\frac{exp(a_k+logC)}{{\sum }_{i=1}^{n}exp(a_i+logC)}\\ & & =\frac{exp(a_k+C^{\prime })}{{\sum }_{i=1}^{n}exp(a_i+C^{\prime })}\end{array}$

>>> a = np.array([1010, 1000, 990])
>>> np.exp(a) / np.sum(np.exp(a))
<stdin>:1: RuntimeWarning: invalid value encountered in divide
array([nan, nan, nan])
>>> c = np.max(a)
>>> a - c
array([  0, -10, -20])
>>> np.exp(a - c) / np.sum(np.exp(a - c))
array([9.99954600e-01, 4.53978686e-05, 2.06106005e-09])

def softmax(a):
	c = np.max(a)
	exp_a = np.exp(a - c) # 溢出对策 sum_exp_a = np.sum(exp_a)
	y = exp_a / sum_exp_a
	return y

Deep Learning

均方误差

mean squared error

$E=\frac{1}{2}{\sum }_k(y_k-t_k{)}^2$

import numpy as np

y = [0.1, 0.05, 0.6, 0.0, 0.05, 0.1, 0.0, 0.1, 0.0, 0.0]
t = [0, 0, 1, 0, 0, 0, 0, 0, 0, 0]

def mean_squared_error(y, t):
	return 0.5 * np.sum((y - t)**2)

print(mean_squared_error(np.array(y), np.array(t)))

交叉熵误差

cross entropy error

$E=-{\sum }_k{t}_{k}log{y}_k$

def cross_entropy_error(y, t):
	delta = 1e-7
	return -np.sum(t * np.log(y + delta))

mini-batch 学习

$E=-\frac{1}{N}{\sum }_n{\sum }_k{t}_{nk}log{y}_{nk}$

mini-batch 版交叉误差

def cross_entropy_error(y, t):
	if y.ndim == 1:
		t = t.reshape(1, t.size)
		y = y.reshape(1, y.size)
	
	batch_size = y.shape[0]
	return -np.sum(t * np.log(y + 1e-7)) / batch_size
#	return -np.sum(np.log(y[np.arange(batch_size), t] + 1e-7)) / batch_size

导数

numerical differentiation

$\frac{df(x)}{dx}=li{m}_{h\to 0}\frac{f(x+h)-f(x)}{h}$

def numerical_diff(f, x):
	h = 1e-4
	return (f(x + h) - f(x - h)) / (2 * h)

偏导数

$\frac{\partial f}{\partial x_0},\frac{\partial f}{\partial x_1}$

梯度

$(\frac{\partial f}{\partial x_0},\frac{\partial f}{\partial x_1})$

def numerical_gradient(f, x):
	h = 1e-4 # 0.0001
	grad = np.zeros_like(x) # 生成和x形状相同的数组
	
	for idx in range(x.size):
		tmp_val = x[idx]
		
		# f(x+h)的计算
		x[idx] = tmp_val + h
		fxh1 = f(x)
		
		# f(x-h)的计算
		x[idx] = tmp_val - h
		fxh2 = f(x)
		
		grad[idx] = (fxh1 - fxh2) / (2 * h)
		x[idx] = tmp_val # 还原值
	
	return grad

梯度法

$x_0=x_0-\eta \frac{\partial f}{\partial x_0}$

$x_1=x_1-\eta \frac{\partial f}{\partial x_1}$

梯度下降法：

def gradient_descent(f, init_x, lr=0.01, step_num=100):
	x = init_x
	for i in range(step_num):
		grad = numerical_gradient(f, x)
		x -= lr * grad

	return x

神经网络的梯度

$W$ 为 $2\times 3$ 权重的神经网络，$L$ 表示损失函数

$W=\left(\begin{array}{ccc}{w}_{11}& w_{12}& w_{13}\\ w_{21}& w_{22}& w_{23}\end{array}\right)$

梯度用 $\frac{\partial L}{\partial W}$ 表示

$\frac{\partial L}{\partial W}=\left(\begin{array}{ccc}\frac{\partial L}{\partial w_{11}}& \frac{\partial L}{\partial w_{12}}& \frac{\partial L}{\partial w_{13}}\\ \frac{\partial L}{\partial w_{21}}& \frac{\partial L}{\partial w_{22}}& \frac{\partial L}{\partial w_{23}}\end{array}\right)$

简单层的实现

乘法层

class MulLayer:
	def __init__(self):
		self.x = None
		self.y = None
	
	def forward(self, x, y):
		self.x = x
		self.y = y
		out = x * y
		return out
	
	def backward(self, dout):
		dx = dout * self.y
		dy = dout * self.x
		return dx, dy

加法层

class AddLayer:  
	def __init__(self):    
		pass
	
	def forward(self, x, y):  
		out = x + y
		return out  
	
	def backward(self, dout):  
		dx = dout * 1  
		dy = dout * 1  
		return dx, dy

ReLU 层

class Relu:
	def __init__(self):
		self.mask = None
	
	def forward(self, x):
		self.mask = (x <= 0)
		out = x.copy()
		out[self.mask] = 0
		
		return out
	
	def backward(self, dout):
		dout[self.mask] = 0
		dx = dout
		
		return dx

误差反向传播法

$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial t}\frac{\partial t}{\partial x}$ ,

$y=\{\begin{array}{cc}x& (x>0)\\ 0& (x\le 0)\end{array}$ , $\frac{\partial y}{\partial x}=\{\begin{array}{cc}1& (x>0)\\ 0& (x\le 0)\end{array}$ ,

$y=\frac{1}{1+exp(-x)}$ , $\frac{\partial y}{\partial x}=-\frac{1}{x^2}=-y^2$

$\begin{array}{ccccccccc}x& \stackrel{\times (-1)}{\to }& -x& \stackrel{exp}{\to }& exp(-x)& \stackrel{+1}{\to }& 1+exp(-x)& \stackrel{/}{\to }& y\\ \frac{\partial L}{\partial y}{y}^{2}exp(-x)& \stackrel{\times (-1)}{\leftarrow }& \frac{\partial L}{\partial y}{y}^{2}exp(-x)& \stackrel{exp}{\leftarrow }& \frac{\partial L}{\partial y}{y}^2& \stackrel{+1}{\leftarrow }& \frac{\partial L}{\partial y}{y}^2& \stackrel{/}{\leftarrow }& \frac{\partial L}{\partial y}\end{array}$

$\begin{array}{ccc}\frac{\partial L}{\partial y}{y}^{2}exp(-x)& =& \frac{\partial L}{\partial y}\frac{1}{(1+exp(-x){)}^2}exp(-x)\\ & =& \frac{\partial L}{\partial y}\frac{1}{1+exp(-x)}\frac{exp(-x)}{1+exp(-x)}\\ & =& \frac{\partial L}{\partial y}y(1-y)\end{array}$

class Sigmoid:
	def __init__(self):
		self.out = None
	
	def forward(self, x):
		out = 1 / (1 + np.exp(-x))
		self.out = out
		
		return out
	
	def backward(self, dout):
		dx = dout * (1.0 - self.out) * self.out
		
		return dx

$\frac{\partial L}{\partial X}=\frac{\partial L}{\partial Y}\cdot W^T$, $\frac{\partial L}{\partial W}=X^T\cdot \frac{\partial L}{\partial Y}$

$W=\left(\begin{array}{ccc}{w}_{11}& w_{12}& w_{13}\\ w_{21}& w_{22}& w_{23}\end{array}\right)$， W T = ( w 11 w 21 w 12 w 22 w 13 w 23 ) W^T = \begin{pmatrix} w_{11} & w_{21} \\ w_{12} & w_{22} \\ w_{13} & w_{23} \end{pmatrix} WT= ​w11​w12​w13​​w21​w22​w23​​ ​

$X=(x_0,x_1,...,x_n)$ , $\frac{\partial L}{\partial X}=(\frac{\partial L}{\partial x_0},\frac{\partial L}{\partial x_1},...,\frac{\partial L}{\partial x_n})$

Affine 层

class Affine:  
	def __init__(self, W, b):  
		self.W = W  
		self.b = b  
		self.x = None  
		self.dW = None  
		self.db = None  
	
	def forward(self, x):  
		self.x = x  
		out = np.dot(x, self.W) + self.b  
		return out  
	  
	def backward(self, dout):  
		dx = np.dot(dout, self.W.T)  
		self.dW = np.dot(self.x.T, dout)  
		self.db = np.sum(dout, axis=0)  
		return dx

Softmax-with-Loss 层

class SoftmaxWithLoss:  
	def __init__(self):  
		self.loss = None # 损失  
		self.y = None # softmax的输出  
		self.t = None # 监督数据（one-hot vector）  
	
	def forward(self, x, t):  
		self.t = t  
		self.y = softmax(x)  
		self.loss = cross_entropy_error(self.y, self.t)  
		return self.loss  
	  
	def backward(self, dout=1):  
		batch_size = self.t.shape[0]  
		dx = (self.y - self.t) / batch_size  
		return dx

Trick

SGD

$W\leftarrow W\leftarrow \eta \frac{\partial L}{\partial W}$

class SGD:
	def __init__(self, lr=0.01):
		self.lr = lr
	
	def update(self, params, grads):
		for key in params.keys():
			params[key] -= self.lr * grads[key]

Momentum

$v\leftarrow \alpha v-\eta \frac{\partial L}{\partial W}$
$W\leftarrow W+v$

class Momentum:
	def __init__(self, lr=0.01, momentum=0.9):
		self.lr = lr
		self.momentum = momentum
		self.v = None

	def update(self, params, grads):
		if self.v is None:
			self.v = {}
			for key, val in params.items():
				self.v[key] = np.zeros_like(val)

		for key in params.keys():
			self.v[key] = self.momentum * self.v[key] - self.lr * grads[key]
			params[key] += self.v[key]

AdaGrad

$h\leftarrow h+\frac{\partial L}{\partial W}\odot \frac{\partial L}{\partial W}$

$W\leftarrow W-\eta \frac{1}{\sqrt{h}}\frac{\partial L}{\partial W}$

class AdaGrad:
	def __init__(self, lr=0.01):
		self.lr = lr
		self.h = None

	def update(self, params, grads):
		if self.h is None:
			self.h = {}
			for key, val in params.items():
				self.h[key] = np.zeros_like(val)
		for key in params.items():
			self.h[key] += grads[key] * grads[key]
			params[key] -= self.lr * grads[key] / (np.sqrt(self.h[key]) + 1e-7)

Batch Normalization

${\mu }_B\leftarrow \frac{1}{m}{\sum }_{i=1}^m{x}_i$

${\sigma }_B^2\leftarrow \frac{1}{m}{\sum }_{i=1}^m(x_i-{\mu }_B{)}^2$

$\widehat{x_i}\leftarrow \frac{x_i-{\mu }_B}{\sqrt{{\sigma }_B^2+\epsilon }}$

$y_i\leftarrow \gamma \widehat{x_i}+\beta$

Dropout

class Dropout:  
	def __init__(self, dropout_ratio=0.5):  
		self.dropout_ratio = dropout_ratio
		self.mask = None
		
	def forward(self, x, train_flg=True):  
		if train_flg:
			self.mask = np.random.rand(*x.shape) > self.dropout_ratio
			return x * self.mask
		else:
			return x * (1.0 - self.dropout_ratio)
	
	def backward(self, dout):  
		return dout * self.mask

卷积神经网络

假设输入大小为 $(H,W)$ ，滤波器大小为 $(FH,FW)$ ，输出大小为 $(OH,OW)$ ，填充为 $P$，步幅为$S$ 。此时，输出大小可通过下式进行计算

$OH=\frac{H+2P-FH}{S}+1$ , $OW=\frac{W+2P-FW}{S}+1$

通道数为 $C$、高度为 $H$、长度为W的数据的形状可以写成 $(C,H,W)$。滤波器也一样，要按(channel, height,width) 的顺序书写。比如，通道数为 $C$、滤波器高度为 $FH$ (Filter Height)、长度为 $FW$ (Filter Width) 时，可以写成 $(C,FH,FW)$

$(C,H,W)\ast (C,FH,FW)\to (1,OH,OW)$

基于多个滤波器的卷积运算
$(C,H,W)\ast (FN,C,FH,FW)\to (FN,OH,OW)$

卷积运算的处理流
$(C,H,W)\ast (FN,C,FH,FW)\to (FN,OH,OW)+(FN,1,1)\to (FN,OH,OW)$

卷积运算的处理流（批处理）
$(N,C,H,W)\ast (FN,C,FH,FW)\to (N,FN,OH,OW)+(FN,1,1)\to (N,FN,OH,OW)$

class Convolution:
	def __init__(self, W, b, stride=1, pad=0):
		self.W = W
		self.b = b
		self.stride = stride
		self.pad = pad
	
	def forward(self, x):
		FN, C, FH, FW = self.W.shape
		N, C, H, W = x.shape
		out_h = int(1 + (H + 2*self.pad - FH) / self.stride)  
		out_w = int(1 + (W + 2*self.pad - FW) / self.stride)  
		
		col = im2col(x, FH, FW, self.stride, self.pad)  
		col_W = self.W.reshape(FN, -1).T # 滤波器的展开  
		out = np.dot(col, col_W) + self.b  
		out = out.reshape(N, out_h, out_w, -1).transpose(0, 3, 1, 2)  
		  
		return out

池化层

class Pooling:  
	def __init__(self, pool_h, pool_w, stride=1, pad=0):  
		self.pool_h = pool_h  
		self.pool_w = pool_w  
		self.stride = stride  
		self.pad = pad  
	
	def forward(self, x):  
		N, C, H, W = x.shape    
		out_h = int(1 + (H - self.pool_h) / self.stride)  
		out_w = int(1 + (W - self.pool_w) / self.stride)  
		
		# 展开(1) 
		col = im2col(x, self.pool_h, self.pool_w, self.stride, self.pad)  
		col = col.reshape(-1, self.pool_h*self.pool_w)  
		
		# 最大值(2)  
		out = np.max(col, axis=1)  
		
		# 转换(3)  
		out = out.reshape(N, out_h, out_w, C).transpose(0, 3, 1, 2)  
		return out