Deep Learning day 2

ニューラルネットワーク学習

from IPython.display import Image
import numpy as np
import matplotlib.pyplot as plt

Image("e 4.2.png", width=200)

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

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

cross_entropy_error(np.array(y), np.array(t))

0.510825457099338

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

cross_entropy_error(np.array(y), np.array(t))

2.302584092994546

Image("e 4.3.png", width=300)

def cross_entropy_error(y ,t ):
    delta = 1e-7
    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 + delta))/ batch_size

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

cross_entropy_error(np.array(y), np.array(t))

0.510825457099338

Image("e 4.5.png", width=200)

def function_1(x):
    return 0.01*x**2 + 0.1*x

x = np.arange(0.0, 20.0, 0.1)
y = function_1(x)
plt.xlabel("x")
plt.ylabel("f(x)")
plt.plot(x, y)

[<matplotlib.lines.Line2D at 0x2ad385e8be0>]

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

x = np.arange(0.0, 20.0, 0.1)
y = function_1(x)

a = numerical_diff(function_1, 5)
b = function_1(5) - (a*5)
y2 = (a * x) + b

plt.xlabel("x")
plt.ylabel("f(x)")
plt.plot(x, y)
plt.plot(x, y2)

[<matplotlib.lines.Line2D at 0x2ad38af1940>]

def numerical_graient(f, x):
    h = 1e-4
    grad = np.zeros_like(x)
    
    for idx in range(x.size):
        tmp_val = x[idx]
        x[idx] = tmp_val + h
        fxh1 = f(x)  #f(x+h)        
        
        x[idx] = tmp_val - h
        fxh2 = f(x)  #f(x-h)
        
        grad[idx] = (fxh1 - fxh2)/(2*h)
        x[idx] = tmp_val # 값 복원
    
    return grad

Image("e 4.6.png", width=200)

def function_2(x):
    return x[0]**2 + x[1]**2

numerical_graient(function_2, np.array([3.0, 4.0]))

array([6., 8.])

Image("e 4.7.png", width=150)

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

init_x = np.array([-3.0, 4.0])
gradient_descent(function_2, init_x = init_x, lr=0.1, step_num =100)

array([-6.11110793e-10,  8.14814391e-10])

def numerical_gradient(f, x):
    h = 1e-4 # 0.0001
    grad = np.zeros_like(x)
    
    it = np.nditer(x, flags=['multi_index'], op_flags=['readwrite'])
    while not it.finished:
        idx = it.multi_index
        tmp_val = x[idx]
        x[idx] = float(tmp_val) + h
        fxh1 = f(x) # f(x+h)
        
        x[idx] = tmp_val - h 
        fxh2 = f(x) # f(x-h)
        grad[idx] = (fxh1 - fxh2) / (2*h)
        
        x[idx] = tmp_val # 값 복원
        it.iternext()   
        
    return grad

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

class simpleNet:
    def __init__(self):
        self.W = np.random.randn(2,3) #정규분포로 초기화
    
    def predict(self, x):
        return np.dot(x, self.W)
    
    def loss(self, x, t):
        z = self.predict(x)
        y = softmax(z)
        loss = cross_entropy_error(y, t)
        return loss

x = np.array([0.6, 0.9])
t = np.array([0, 0, 1])

net = simpleNet()
f = lambda w: net.loss(x, t)
dw = numerical_gradient(f, net.W)

net.W # 랜덤하게 임의로 설정한 가중치

array([[-0.48115741,  0.54270006, -0.80327648],
       [-0.41143265, -0.67968633, -0.94115196]])

dw # 기울기 벡터

array([[ 0.20245512,  0.29394845, -0.49640358],
       [ 0.30368269,  0.44092268, -0.74460537]])