python计算函数梯度的解析表达式_python实现梯度下降算法的实例详解

这里选的python版本是2.7，因为我之前用python3试了几次，发现在画3d图的时候会报错，所以改用了2.7。

数据集选择

数据集我选了一个包含两个变量，三个参数的数据集，这样可以画出3d图形对结果进行验证。

部分函数总结

symbols()函数：首先要安装sympy库才可以使用。用法：

>>> x1 = symbols('x2')

>>> x1 + 1

x2 + 1

在这个例子中，x1和x2是不一样的，x2代表的是一个函数的变量，而x1代表的是python中的一个变量，它可以表示函数的变量，也可以表示其他的任何量，它替代x2进行函数的计算。实际使用的时候我们可以将x1,x2都命名为x，但是我们要知道他们俩的区别。

再看看这个例子：

>>> x = symbols('x')

>>> expr = x + 1

>>> x = 2

>>> print(expr)

x + 1

作为python变量的x被2这个数值覆盖了，所以它现在不再表示函数变量x，而expr依然是函数变量x+1的别名，所以结果依然是x+1。

subs()函数：既然普通的方法无法为函数变量赋值，那就肯定有函数来实现这个功能，用法：

>>> (1 + x*y).subs(x, pi)#一个参数时的用法

pi*y + 1

>>> (1 + x*y).subs({x:pi, y:2})#多个参数时的用法

1 + 2*pi

diff()函数：求偏导数，用法：result=diff(fun,x)，这个就是求fun函数对x变量的偏导数，结果result也是一个变量，需要赋值才能得到准确结果。

代码实现：

from __future__ import division

from sympy import symbols, diff, expand

import numpy as np

import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D

data = {'x1': [100, 50, 100, 100, 50, 80, 75, 65, 90, 90],

'x2': [4, 3, 4, 2, 2, 2, 3, 4, 3, 2],

'y': [9.3, 4.8, 8.9, 6.5, 4.2, 6.2, 7.4, 6.0, 7.6, 6.1]}#初始化数据集

theta0, theta1, theta2 = symbols('theta0 theta1 theta2', real=True)  # y=theta0+theta1*x1+theta2*x2,定义参数

costfuc = 0 * theta0

for i in range(10):

costfuc += (theta0 + theta1 * data['x1'][i] + theta2 * data['x2'][i] - data['y'][i]) ** 2

costfuc /= 20#初始化代价函数

dtheta0 = diff(costfuc, theta0)

dtheta1 = diff(costfuc, theta1)

dtheta2 = diff(costfuc, theta2)

rtheta0 = 1

rtheta1 = 1

rtheta2 = 1#为参数赋初始值

costvalue = costfuc.subs({theta0: rtheta0, theta1: rtheta1, theta2: rtheta2})

newcostvalue = 0#用cost的值的变化程度来判断是否已经到最小值了

count = 0

alpha = 0.0001#设置学习率，一定要设置的比较小，否则无法到达最小值

while (costvalue - newcostvalue > 0.00001 or newcostvalue - costvalue > 0.00001) and count < 1000:

count += 1

costvalue = newcostvalue

rtheta0 = rtheta0 - alpha * dtheta0.subs({theta0: rtheta0, theta1: rtheta1, theta2: rtheta2})

rtheta1 = rtheta1 - alpha * dtheta1.subs({theta0: rtheta0, theta1: rtheta1, theta2: rtheta2})

rtheta2 = rtheta2 - alpha * dtheta2.subs({theta0: rtheta0, theta1: rtheta1, theta2: rtheta2})

newcostvalue = costfuc.subs({theta0: rtheta0, theta1: rtheta1, theta2: rtheta2})

rtheta0 = round(rtheta0, 4)

rtheta1 = round(rtheta1, 4)

rtheta2 = round(rtheta2, 4)#给结果保留4位小数，防止数值溢出

print(rtheta0, rtheta1, rtheta2)

fig = plt.figure()

ax = Axes3D(fig)

ax.scatter(data['x1'], data['x2'], data['y'])  # 绘制散点图

xx = np.arange(20, 100, 1)

yy = np.arange(1, 5, 0.05)

X, Y = np.meshgrid(xx, yy)

Z = X * rtheta1 + Y * rtheta2 + rtheta0

ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=plt.get_cmap('rainbow'))

plt.show()#绘制3d图进行验证

结果：

实例扩展：

'''

梯度下降算法

Batch Gradient Descent

Stochastic Gradient Descent SGD

'''

__author__ = 'epleone'

import numpy as np

import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D

import sys

# 使用随机数种子， 让每次的随机数生成相同，方便调试

# np.random.seed(111111111)

class GradientDescent(object):

eps = 1.0e-8

max_iter = 1000000 # 暂时不需要

dim = 1

func_args = [2.1, 2.7] # [w_0, .., w_dim, b]

def __init__(self, func_arg=None, N=1000):

self.data_num = N

if func_arg is not None:

self.FuncArgs = func_arg

self._getData()

def _getData(self):

x = 20 * (np.random.rand(self.data_num, self.dim) - 0.5)

b_1 = np.ones((self.data_num, 1), dtype=np.float)

# x = np.concatenate((x, b_1), axis=1)

self.x = np.concatenate((x, b_1), axis=1)

def func(self, x):

# noise太大的话， 梯度下降法失去作用

noise = 0.01 * np.random.randn(self.data_num) + 0

w = np.array(self.func_args)

# y1 = w * self.x[0, ] # 直接相乘

y = np.dot(self.x, w) # 矩阵乘法

y += noise

return y

@property

def FuncArgs(self):

return self.func_args

@FuncArgs.setter

def FuncArgs(self, args):

if not isinstance(args, list):

raise Exception(

'args is not list, it should be like [w_0, ..., w_dim, b]')

if len(args) == 0:

raise Exception('args is empty list!!')

if len(args) == 1:

args.append(0.0)

self.func_args = args

self.dim = len(args) - 1

self._getData()

@property

def EPS(self):

return self.eps

@EPS.setter

def EPS(self, value):

if not isinstance(value, float) and not isinstance(value, int):

raise Exception("The type of eps should be an float number")

self.eps = value

def plotFunc(self):

# 一维画图

if self.dim == 1:

# x = np.sort(self.x, axis=0)

x = self.x

y = self.func(x)

fig, ax = plt.subplots()

ax.plot(x, y, 'o')

ax.set(xlabel='x ', ylabel='y', title='Loss Curve')

ax.grid()

plt.show()

# 二维画图

if self.dim == 2:

# x = np.sort(self.x, axis=0)

x = self.x

y = self.func(x)

xs = x[:, 0]

ys = x[:, 1]

zs = y

fig = plt.figure()

ax = fig.add_subplot(111, projection='3d')

ax.scatter(xs, ys, zs, c='r', marker='o')

ax.set_xlabel('X Label')

ax.set_ylabel('Y Label')

ax.set_zlabel('Z Label')

plt.show()

else:

# plt.axis('off')

plt.text(

0.5,

0.5,

"The dimension(x.dim > 2) \n is too high to draw",

size=17,

rotation=0.,

ha="center",

va="center",

bbox=dict(

boxstyle="round",

ec=(1., 0.5, 0.5),

fc=(1., 0.8, 0.8), ))

plt.draw()

plt.show()

# print('The dimension(x.dim > 2) is too high to draw')

# 梯度下降法只能求解凸函数

def _gradient_descent(self, bs, lr, epoch):

x = self.x

# shuffle数据集没有必要

# np.random.shuffle(x)

y = self.func(x)

w = np.ones((self.dim + 1, 1), dtype=float)

for e in range(epoch):

print('epoch:' + str(e), end=',')

# 批量梯度下降，bs为1时 等价单样本梯度下降

for i in range(0, self.data_num, bs):

y_ = np.dot(x[i:i + bs], w)

loss = y_ - y[i:i + bs].reshape(-1, 1)

d = loss * x[i:i + bs]

d = d.sum(axis=0) / bs

d = lr * d

d.shape = (-1, 1)

w = w - d

y_ = np.dot(self.x, w)

loss_ = abs((y_ - y).sum())

print('\tLoss = ' + str(loss_))

print('拟合的结果为:', end=',')

print(sum(w.tolist(), []))

print()

if loss_ < self.eps:

print('The Gradient Descent algorithm has converged!!\n')

break

pass

def __call__(self, bs=1, lr=0.1, epoch=10):

if sys.version_info < (3, 4):

raise RuntimeError('At least Python 3.4 is required')

if not isinstance(bs, int) or not isinstance(epoch, int):

raise Exception(

"The type of BatchSize/Epoch should be an integer number")

self._gradient_descent(bs, lr, epoch)

pass

pass

if __name__ == "__main__":

if sys.version_info < (3, 4):

raise RuntimeError('At least Python 3.4 is required')

gd = GradientDescent([1.2, 1.4, 2.1, 4.5, 2.1])

# gd = GradientDescent([1.2, 1.4, 2.1])

print("要拟合的参数结果是: ")

print(gd.FuncArgs)

print("===================\n\n")

# gd.EPS = 0.0

gd.plotFunc()

gd(10, 0.01)

print("Finished!")

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