前馈神经网络优化器
引用的知乎上的文章内容，现在有些地方还不太明白，留待以后查看。
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 * dx1
d2_{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 Momentum,
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^2
w2_{t+1} = beta2 * w2_t + (1 - beta2) * dx2_t^2
d1_{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_t
v2_{t+1} = beta1 * v2_t + (1 - beta1) * dx2_t
w1_{t+1} = beta2 * w1_t + (1 - beta2) * dx1_t^2
w2_{t+1} = beta2 * w2_t + (1 - beta2) * dx2_t^2
v1_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)
"""
def __init__(self, eta=0.1, mu=0.9, beta1=0.9, beta2=0.99, 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
# 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
@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()
def main(num_ite=15):
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()
xixi.draw_adam(num_ite)
xixi.draw_common("Optimize x1^2+x2^2*{} Using AdaM".format(RATIO ** 2))
xixi.show()
main()