adadelta算法_机器学习中的优化算法(3)-AdaGrad, Adadelta(附Python示例)

import math

import numpy as np

import matplotlib.pyplot as plt

RATIO = 3 # 椭圆的长宽比

LIMIT = 1.2 # 图像的坐标轴范围

class PlotComparaison(object):

"""多种优化器来优化函数 x1^2 + x2^2 * RATIO^2.每次参数改变为(d1, d2).梯度为(dx1, dx2)t+1次迭代,标准GD,d1_{t+1} = - eta * dx1d2_{t+1} = - eta * dx2带Momentum,d1_{t+1} = eta * (mu * d1_t - dx1_{t+1})d2_{t+1} = eta * (mu * d2_t - dx2_{t+1})带Nesterov Accent；,d1_{t+1} = eta * (mu * d1_t - dx1^{nag}_{t+1})d2_{t+1} = eta * (mu * d2_t - dx2^{nag}_{t+1})其中(dx1^{nag}, dx2^{nag})为(x1 + eta * mu * d1_t, x2 + eta * mu * d2_t)处的梯度RMSProp,w1_{t+1} = beta2 * w1_t + (1 - beta2) * dx1_t^2w2_{t+1} = beta2 * w2_t + (1 - beta2) * dx2_t^2d1_{t+1} = - eta * dx1_t / (sqrt(w1_{t+1}) + epsilon)d2_{t+1} = - eta * dx2_t / (sqrt(w2_{t+1}) + epsilon)Adam,每次参数改变为(d1, d2)v1_{t+1} = beta1 * v1_t + (1 - beta1) * dx1_tv2_{t+1} = beta1 * v2_t + (1 - beta1) * dx2_tw1_{t+1} = beta2 * w1_t + (1 - beta2) * dx1_t^2w2_{t+1} = beta2 * w2_t + (1 - beta2) * dx2_t^2v1_corrected = v1_{t+1} / (1 - beta1^{t+1})v2_corrected = v2_{t+1} / (1 - beta1^{t+1})w1_corrected = w1_{t+1} / (1 - beta2^{t+1})w2_corrected = w2_{t+1} / (1 - beta2^{t+1})d1_{t+1} = - eta * v1_corrected / (sqrt(w1_corrected) + epsilon)d2_{t+1} = - eta * v2_corrected / (sqrt(w2_corrected) + epsilon)AdaGrad,w1_{t+1} = w1_t + dx1_t^2w2_{t+1} = w2_t + dx2_t^2d1_{t+1} = - eta * dx1_t / sqrt(w1_{t+1} + epsilon)d2_{t+1} = - eta * dx2_t / sqrt(w2_{t+1} + epsilon)Adadeltaupdate1_{t+1} = rho * update1_t + (1 - rho) * d1_t^2update2_{t+1} = rho * update2_t + (1 - rho) * d2_t^2w1_{t+1} = rho * w1_t + (1 - rho) * dx1_t^2w2_{t+1} = rho * w2_t + (1 - rho) * dx2_t^2d1_{t+1} = - dx1 * rms(update1_{t+1}) / rms(w1_{t+1})d2_{t+1} = - dx2 * rms(update2_{t+1}) / rms(w2_{t+1})定义 rms(x) = sqrt(x + epsilon)"""

def __init__(self, eta=0.1, mu=0.9, beta1=0.9, beta2=0.99, rho=0.9, epsilon=1e-10, angles=None, contour_values=None,

stop_condition=1e-4):

# 全部算法的学习率

self.eta = eta

# 启发式学习的终止条件

self.stop_condition = stop_condition

# Nesterov Momentum超参数

self.mu = mu

# RMSProp超参数

self.beta1 = beta1

self.beta2 = beta2

self.epsilon = epsilon

# Adadelta的超参数

self.rho = rho

# 用正态分布随机生成初始点

self.x1_init, self.x2_init = np.random.uniform(LIMIT / 2, LIMIT), np.random.uniform(LIMIT / 2, LIMIT) / RATIO

self.x1, self.x2 = self.x1_init, self.x2_init

# 等高线相关

if angles == None:

angles = np.arange(0, 2 * math.pi, 0.01)

self.angles = angles

if contour_values == None:

contour_values = [0.25 * i for i in range(1, 5)]

self.contour_values = contour_values

setattr(self, "contour_colors", None)

def draw_common(self, title):

"""画等高线,最优点和设置图片各种属性"""

# 坐标轴尺度一致

plt.gca().set_aspect(1)

# 根据等高线的值生成坐标和颜色

# 海拔越高颜色越深

num_contour = len(self.contour_values)

if not self.contour_colors:

self.contour_colors = [(i / num_contour, i / num_contour, i / num_contour) for i in range(num_contour)]

self.contour_colors.reverse()

self.contours = [

[

list(map(lambda x: math.sin(x) * math.sqrt(val), self.angles)),

list(map(lambda x: math.cos(x) * math.sqrt(val) / RATIO, self.angles))

]

for val in self.contour_values

]

# 画等高线

for i in range(num_contour):

plt.plot(self.contours[i][0],

self.contours[i][1],

linewidth=1,

linestyle='-',

color=self.contour_colors[i],

label="y={}".format(round(self.contour_values[i], 2))

)

# 画最优点

plt.text(0, 0, 'x*')

# 图片标题

plt.title(title)

# 设置坐标轴名字和范围

plt.xlabel("x1")

plt.ylabel("x2")

plt.xlim((-LIMIT, LIMIT))

plt.ylim((-LIMIT, LIMIT))

# 显示图例

plt.legend(loc=1)

def forward_gd(self):

"""SGD一次迭代"""

self.d1 = -self.eta * self.dx1

self.d2 = -self.eta * self.dx2

self.ite += 1

def draw_gd(self, num_ite=5):

"""画基础SGD的迭代优化.包括每次迭代的点,以及表示每次迭代改变的箭头"""

# 初始化

setattr(self, "ite", 0)

setattr(self, "x1", self.x1_init)

setattr(self, "x2", self.x2_init)

# 画每次迭代

self.point_colors = [(i / num_ite, 0, 0) for i in range(num_ite)]

plt.scatter(self.x1, self.x2, color=self.point_colors[0])

for _ in range(num_ite):

self.forward_gd()

# 迭代的箭头

plt.arrow(self.x1, self.x2, self.d1, self.d2,

length_includes_head=True,

linestyle=':',

label='{}ite'.format(self.ite),

color='b',

head_width=0.08

)

self.x1 += self.d1

self.x2 += self.d2

print("第{}次迭代后,坐标为({},{})".format(self.ite, self.x1, self.x2))

plt.scatter(self.x1, self.x2) # 迭代的点

if self.loss < self.stop_condition:

break

def forward_momentum(self):

"""带Momentum的SGD一次迭代"""

self.d1 = self.eta * (self.mu * self.d1_pre - self.dx1)

self.d2 = self.eta * (self.mu * self.d2_pre - self.dx2)

self.ite += 1

self.d1_pre, self.d2_pre = self.d1, self.d2

def draw_momentum(self, num_ite=5):

"""画带Momentum的迭代优化."""

# 初始化

setattr(self, "ite", 0)

setattr(self, "x1", self.x1_init)

setattr(self, "x2", self.x2_init)

setattr(self, "d1_pre", 0)

setattr(self, "d2_pre", 0)

# 画每次迭代

self.point_colors = [(i / num_ite, 0, 0) for i in range(num_ite)]

plt.scatter(self.x1, self.x2, color=self.point_colors[0])

for _ in range(num_ite):

self.forward_momentum()

# 迭代的箭头

plt.arrow(self.x1, self.x2, self.d1, self.d2,

length_includes_head=True,

linestyle=':',

label='{}ite'.format(self.ite),

color='b',

head_width=0.08

)

self.x1 += self.d1

self.x2 += self.d2

print("第{}次迭代后,坐标为({},{})".format(self.ite, self.x1, self.x2))

plt.scatter(self.x1, self.x2) # 迭代的点

if self.loss < self.stop_condition:

break

def forward_nag(self):

"""Nesterov Accelerated的SGD一次迭代"""

self.d1 = self.eta * (self.mu * self.d1_pre - self.dx1_nag)

self.d2 = self.eta * (self.mu * self.d2_pre - self.dx2_nag)

self.ite += 1

self.d1_pre, self.d2_pre = self.d1, self.d2

def draw_nag(self, num_ite=5):

"""画Nesterov Accelerated的迭代优化."""

# 初始化

setattr(self, "ite", 0)

setattr(self, "x1", self.x1_init)

setattr(self, "x2", self.x2_init)

setattr(self, "d1_pre", 0)

setattr(self, "d2_pre", 0)

# 画每次迭代

self.point_colors = [(i / num_ite, 0, 0) for i in range(num_ite)]

plt.scatter(self.x1, self.x2, color=self.point_colors[0])

for _ in range(num_ite):

self.forward_nag()

# 迭代的箭头

plt.arrow(self.x1, self.x2, self.d1, self.d2,

length_includes_head=True,

linestyle=':',

label='{}ite'.format(self.ite),

color='b',

head_width=0.08

)

self.x1 += self.d1

self.x2 += self.d2

print("第{}次迭代后,坐标为({},{})".format(self.ite, self.x1, self.x2))

plt.scatter(self.x1, self.x2) # 迭代的点

if self.loss < self.stop_condition:

break

def forward_rmsprop(self):

"""RMSProp一次迭代"""

w1 = self.beta2 * self.w1_pre + (1 - self.beta2) * (self.dx1 ** 2)

w2 = self.beta2 * self.w2_pre + (1 - self.beta2) * (self.dx2 ** 2)

self.ite += 1

self.w1_pre, self.w2_pre = w1, w2

self.d1 = -self.eta * self.dx1 / (math.sqrt(w1) + self.epsilon)

self.d2 = -self.eta * self.dx2 / (math.sqrt(w2) + self.epsilon)

def draw_rmsprop(self, num_ite=5):

"""画RMSProp的迭代优化."""

# 初始化

setattr(self, "ite", 0)

setattr(self, "x1", self.x1_init)

setattr(self, "x2", self.x2_init)

setattr(self, "w1_pre", 0)

setattr(self, "w2_pre", 0)

# 画每次迭代

self.point_colors = [(i / num_ite, 0, 0) for i in range(num_ite)]

plt.scatter(self.x1, self.x2, color=self.point_colors[0])

for _ in range(num_ite):

self.forward_rmsprop()

# 迭代的箭头

plt.arrow(self.x1, self.x2, self.d1, self.d2,

length_includes_head=True,

linestyle=':',

label='{}ite'.format(self.ite),

color='b',

head_width=0.08

)

self.x1 += self.d1

self.x2 += self.d2

print("第{}次迭代后,坐标为({},{})".format(self.ite, self.x1, self.x2))

plt.scatter(self.x1, self.x2) # 迭代的点

if self.loss < self.stop_condition:

break

def forward_adam(self):

"""AdaM一次迭代"""

w1 = self.beta2 * self.w1_pre + (1 - self.beta2) * (self.dx1 ** 2)

w2 = self.beta2 * self.w2_pre + (1 - self.beta2) * (self.dx2 ** 2)

v1 = self.beta1 * self.v1_pre + (1 - self.beta1) * self.dx1

v2 = self.beta1 * self.v2_pre + (1 - self.beta1) * self.dx2

self.ite += 1

self.v1_pre, self.v2_pre = v1, v2

self.w1_pre, self.w2_pre = w1, w2

v1_corr = v1 / (1 - math.pow(self.beta1, self.ite))

v2_corr = v2 / (1 - math.pow(self.beta1, self.ite))

w1_corr = w1 / (1 - math.pow(self.beta2, self.ite))

w2_corr = w2 / (1 - math.pow(self.beta2, self.ite))

self.d1 = -self.eta * v1_corr / (math.sqrt(w1_corr) + self.epsilon)

self.d2 = -self.eta * v2_corr / (math.sqrt(w2_corr) + self.epsilon)

def draw_adam(self, num_ite=5):

"""画AdaM的迭代优化."""

# 初始化

setattr(self, "ite", 0)

setattr(self, "x1", self.x1_init)

setattr(self, "x2", self.x2_init)

setattr(self, "w1_pre", 0)

setattr(self, "w2_pre", 0)

setattr(self, "v1_pre", 0)

setattr(self, "v2_pre", 0)

# 画每次迭代

self.point_colors = [(i / num_ite, 0, 0) for i in range(num_ite)]

plt.scatter(self.x1, self.x2, color=self.point_colors[0])

for _ in range(num_ite):

self.forward_adam()

# 迭代的箭头

plt.arrow(self.x1, self.x2, self.d1, self.d2,

length_includes_head=True,

linestyle=':',

label='{}ite'.format(self.ite),

color='b',

head_width=0.08

)

self.x1 += self.d1

self.x2 += self.d2

print("第{}次迭代后,坐标为({},{})".format(self.ite, self.x1, self.x2))

plt.scatter(self.x1, self.x2) # 迭代的点

if self.loss < self.stop_condition:

break

def forward_adagrad(self):

"""AdaGrad一次迭代"""

w1 = self.w1_pre + self.dx1 ** 2

w2 = self.w2_pre + self.dx2 ** 2

self.ite += 1

self.w1_pre, self.w2_pre = w1, w2

self.d1 = -self.eta * self.dx1 / math.sqrt(w1 + self.epsilon)

self.d2 = -self.eta * self.dx2 / math.sqrt(w2 + self.epsilon)

def draw_adagrad(self, num_ite=5):

"""画AdaGrad的迭代优化."""

# 初始化

setattr(self, "ite", 0)

setattr(self, "x1", self.x1_init)

setattr(self, "x2", self.x2_init)

setattr(self, "w1_pre", 0)

setattr(self, "w2_pre", 0)

# 画每次迭代

self.point_colors = [(i / num_ite, 0, 0) for i in range(num_ite)]

plt.scatter(self.x1, self.x2, color=self.point_colors[0])

for _ in range(num_ite):

self.forward_adagrad()

# 迭代的箭头

plt.arrow(self.x1, self.x2, self.d1, self.d2,

length_includes_head=True,

linestyle=':',

label='{}ite'.format(self.ite),

color='b',

head_width=0.08

)

self.x1 += self.d1

self.x2 += self.d2

print("第{}次迭代后,坐标为({},{})".format(self.ite, self.x1, self.x2))

plt.scatter(self.x1, self.x2) # 迭代的点

if self.loss < self.stop_condition:

break

def forward_adadelta(self):

"""Adadelta一次迭代"""

w1 = self.rho * self.w1_pre + (1 - self.rho) * (self.dx1 ** 2)

w2 = self.rho * self.w2_pre + (1 - self.rho) * (self.dx2 ** 2)

update1 = self.rho * self.update1_pre + (1 - self.rho) * (self.d1 ** 2)

update2 = self.rho * self.update2_pre + (1 - self.rho) * (self.d2 ** 2)

self.ite += 1

self.update1_pre, self.update2_pre = update1, update2

self.w1_pre, self.w2_pre = w1, w2

self.d1 = - self.rms(update1) / self.rms(w1) * self.dx1

self.d2 = - self.rms(update2) / self.rms(w2) * self.dx2

def draw_adadelta(self, num_ite=5):

"""画Adadelta的迭代优化."""

# 初始化

for attr in ["w{}_pre", "update{}_pre", "d{}"]:

for dim in [1, 2]:

setattr(self, attr.format(dim), 0)

setattr(self, "ite", 0)

setattr(self, "x1", self.x1_init)

setattr(self, "x2", self.x2_init)

# 画每次迭代

self.point_colors = [(i / num_ite, 0, 0) for i in range(num_ite)]

plt.scatter(self.x1, self.x2, color=self.point_colors[0])

for _ in range(num_ite):

self.forward_adadelta()

# 迭代的箭头

plt.arrow(self.x1, self.x2, self.d1, self.d2,

length_includes_head=True,

linestyle=':',

label='{}ite'.format(self.ite),

color='b',

head_width=0.08

)

self.x1 += self.d1

self.x2 += self.d2

print("第{}次迭代后,坐标为({},{})".format(self.ite, self.x1, self.x2))

plt.scatter(self.x1, self.x2) # 迭代的点

if self.loss < self.stop_condition:

break

@property

def dx1(self, x1=None):

return self.x1 * 2

@property

def dx2(self):

return self.x2 * 2 * (RATIO ** 2)

@property

def dx1_nag(self, x1=None):

return (self.x1 + self.eta * self.mu * self.d1_pre) * 2

@property

def dx2_nag(self):

return (self.x2 + self.eta * self.mu * self.d2_pre) * 2 * (RATIO ** 2)

@property

def loss(self):

return self.x1 ** 2 + (RATIO * self.x2) ** 2

def rms(self, x):

return math.sqrt(x + self.epsilon)

def show(self):

# 设置图片大小

plt.figure(figsize=(20, 20))

# 展示

plt.show()

num_ite = 10

xixi = PlotComparaison()

print("起始点为({},{})".format(xixi.x1_init, xixi.x2_init))

xixi.draw_momentum(num_ite)

xixi.draw_common("Optimize x1^2+x2^2*{}Using SGD With Momentum".format(RATIO ** 2))

xixi.show()

xixi.draw_rmsprop(num_ite)

xixi.draw_common("Optimize x1^2+x2^2*{}Using RMSProp".format(RATIO ** 2))

xixi.show()

def adagrad(eta):

xixi.eta = eta

xixi.draw_adagrad(num_ite)

xixi.draw_common("Optimize x1^2+x2^2*{}Using AdaGrad with eta={}".format(RATIO ** 2, eta))

xixi.show()

adagrad(1e-1)

adagrad(1)

adagrad(10)

adagrad(100)

def adadelta(epsilon):

xixi.epsilon = epsilon

xixi.draw_adadelta(num_ite)

xixi.draw_common("Optimize x1^2+x2^2*{}Using Adadelta with epsilon={}".format(RATIO ** 2, epsilon))

xixi.show()

adadelta(1e-2)

adadelta(1e-1)

adadelta(1e-2)

adadelta(1e-3)