In [2]:
import numpy as np
import matplotlib.pyplot as plt
In [10]:
def f(x):
return x * x - 2 * x + 1
def g(x):
return 2 * x - 2
x = np.linspace(-5,7,100)
y = f(x)
In [11]:
print x
[-5. -4.87878788 -4.75757576 -4.63636364 -4.51515152 -4.39393939
-4.27272727 -4.15151515 -4.03030303 -3.90909091 -3.78787879 -3.66666667
-3.54545455 -3.42424242 -3.3030303 -3.18181818 -3.06060606 -2.93939394
-2.81818182 -2.6969697 -2.57575758 -2.45454545 -2.33333333 -2.21212121
-2.09090909 -1.96969697 -1.84848485 -1.72727273 -1.60606061 -1.48484848
-1.36363636 -1.24242424 -1.12121212 -1. -0.87878788 -0.75757576
-0.63636364 -0.51515152 -0.39393939 -0.27272727 -0.15151515 -0.03030303
0.09090909 0.21212121 0.33333333 0.45454545 0.57575758 0.6969697
0.81818182 0.93939394 1.06060606 1.18181818 1.3030303 1.42424242
1.54545455 1.66666667 1.78787879 1.90909091 2.03030303 2.15151515
2.27272727 2.39393939 2.51515152 2.63636364 2.75757576 2.87878788
3. 3.12121212 3.24242424 3.36363636 3.48484848 3.60606061
3.72727273 3.84848485 3.96969697 4.09090909 4.21212121 4.33333333
4.45454545 4.57575758 4.6969697 4.81818182 4.93939394 5.06060606
5.18181818 5.3030303 5.42424242 5.54545455 5.66666667 5.78787879
5.90909091 6.03030303 6.15151515 6.27272727 6.39393939 6.51515152
6.63636364 6.75757576 6.87878788 7. ]
In [12]:
print y
[ 3.60000000e+01 3.45601469e+01 3.31496786e+01 3.17685950e+01
3.04168962e+01 2.90945822e+01 2.78016529e+01 2.65381084e+01
2.53039486e+01 2.40991736e+01 2.29237833e+01 2.17777778e+01
2.06611570e+01 1.95739210e+01 1.85160698e+01 1.74876033e+01
1.64885216e+01 1.55188246e+01 1.45785124e+01 1.36675849e+01
1.27860422e+01 1.19338843e+01 1.11111111e+01 1.03177227e+01
9.55371901e+00 8.81910009e+00 8.11386593e+00 7.43801653e+00
6.79155188e+00 6.17447199e+00 5.58677686e+00 5.02846648e+00
4.49954086e+00 4.00000000e+00 3.52984389e+00 3.08907254e+00
2.67768595e+00 2.29568411e+00 1.94306703e+00 1.61983471e+00
1.32598714e+00 1.06152433e+00 8.26446281e-01 6.20752984e-01
4.44444444e-01 2.97520661e-01 1.79981635e-01 9.18273646e-02
3.30578512e-02 3.67309458e-03 3.67309458e-03 3.30578512e-02
9.18273646e-02 1.79981635e-01 2.97520661e-01 4.44444444e-01
6.20752984e-01 8.26446281e-01 1.06152433e+00 1.32598714e+00
1.61983471e+00 1.94306703e+00 2.29568411e+00 2.67768595e+00
3.08907254e+00 3.52984389e+00 4.00000000e+00 4.49954086e+00
5.02846648e+00 5.58677686e+00 6.17447199e+00 6.79155188e+00
7.43801653e+00 8.11386593e+00 8.81910009e+00 9.55371901e+00
1.03177227e+01 1.11111111e+01 1.19338843e+01 1.27860422e+01
1.36675849e+01 1.45785124e+01 1.55188246e+01 1.64885216e+01
1.74876033e+01 1.85160698e+01 1.95739210e+01 2.06611570e+01
2.17777778e+01 2.29237833e+01 2.40991736e+01 2.53039486e+01
2.65381084e+01 2.78016529e+01 2.90945822e+01 3.04168962e+01
3.17685950e+01 3.31496786e+01 3.45601469e+01 3.60000000e+01]
In [13]:
%matplotlib inline
plt.plot(x, y)
Out[13]:
[<matplotlib.lines.Line2D at 0x10640a350>]
In [33]:
def gd(x_start, step, g): # gd代表了Gradient Descent
x = x_start
for i in range(20):
grad = g(x)
x -= grad * step
print '[ Epoch {0} ] grad = {1}, x = {2}'.format(i, grad, x)
if abs(grad) < 1e-6:
break;
return x
gd(5,0.1,g)
[ Epoch 0 ] grad = 8, x = 4.2
[ Epoch 1 ] grad = 6.4, x = 3.56
[ Epoch 2 ] grad = 5.12, x = 3.048
[ Epoch 3 ] grad = 4.096, x = 2.6384
[ Epoch 4 ] grad = 3.2768, x = 2.31072
[ Epoch 5 ] grad = 2.62144, x = 2.048576
[ Epoch 6 ] grad = 2.097152, x = 1.8388608
[ Epoch 7 ] grad = 1.6777216, x = 1.67108864
[ Epoch 8 ] grad = 1.34217728, x = 1.536870912
[ Epoch 9 ] grad = 1.073741824, x = 1.4294967296
[ Epoch 10 ] grad = 0.8589934592, x = 1.34359738368
[ Epoch 11 ] grad = 0.68719476736, x = 1.27487790694
[ Epoch 12 ] grad = 0.549755813888, x = 1.21990232556
[ Epoch 13 ] grad = 0.43980465111, x = 1.17592186044
[ Epoch 14 ] grad = 0.351843720888, x = 1.14073748836
[ Epoch 15 ] grad = 0.281474976711, x = 1.11258999068
[ Epoch 16 ] grad = 0.225179981369, x = 1.09007199255
[ Epoch 17 ] grad = 0.180143985095, x = 1.07205759404
[ Epoch 18 ] grad = 0.144115188076, x = 1.05764607523
[ Epoch 19 ] grad = 0.115292150461, x = 1.04611686018
Out[33]:
1.0461168601842739
In [34]:
gd(5,100,g)
[ Epoch 0 ] grad = 8, x = -795
[ Epoch 1 ] grad = -1592, x = 158405
[ Epoch 2 ] grad = 316808, x = -31522395
[ Epoch 3 ] grad = -63044792, x = 6272956805
[ Epoch 4 ] grad = 12545913608, x = -1248318403995
[ Epoch 5 ] grad = -2496636807992, x = 248415362395205
[ Epoch 6 ] grad = 496830724790408, x = -49434657116645595
[ Epoch 7 ] grad = -98869314233291192, x = 9837496766212473605
[ Epoch 8 ] grad = 19674993532424947208, x = -1957661856476282247195
[ Epoch 9 ] grad = -3915323712952564494392, x = 389574709438780167192005
[ Epoch 10 ] grad = 779149418877560334384008, x = -77525367178317253271208795
[ Epoch 11 ] grad = -155050734356634506542417592, x = 15427548068485133400970550405
[ Epoch 12 ] grad = 30855096136970266801941100808, x = -3070082065628541546793139530395
[ Epoch 13 ] grad = -6140164131257083093586279060792, x = 610946331060079767811834766548805
[ Epoch 14 ] grad = 1221892662120159535623669533097608, x = -121578319880955873794555118543211995
[ Epoch 15 ] grad = -243156639761911747589110237086423992, x = 24194085656310218885116468590099187205
[ Epoch 16 ] grad = 48388171312620437770232937180198374408, x = -4814623045605733558138177249429738253595
[ Epoch 17 ] grad = -9629246091211467116276354498859476507192, x = 958109986075540978069497272636517912465605
[ Epoch 18 ] grad = 1916219972151081956138994545273035824931208, x = -190663887229032654635829957254667064580655195
[ Epoch 19 ] grad = -381327774458065309271659914509334129161310392, x = 37942113558577498272530161493678745851550384005
Out[34]:
37942113558577498272530161493678745851550384005L
In [35]:
gd(4,1,g)
[ Epoch 0 ] grad = 6, x = -2
[ Epoch 1 ] grad = -6, x = 4
[ Epoch 2 ] grad = 6, x = -2
[ Epoch 3 ] grad = -6, x = 4
[ Epoch 4 ] grad = 6, x = -2
[ Epoch 5 ] grad = -6, x = 4
[ Epoch 6 ] grad = 6, x = -2
[ Epoch 7 ] grad = -6, x = 4
[ Epoch 8 ] grad = 6, x = -2
[ Epoch 9 ] grad = -6, x = 4
[ Epoch 10 ] grad = 6, x = -2
[ Epoch 11 ] grad = -6, x = 4
[ Epoch 12 ] grad = 6, x = -2
[ Epoch 13 ] grad = -6, x = 4
[ Epoch 14 ] grad = 6, x = -2
[ Epoch 15 ] grad = -6, x = 4
[ Epoch 16 ] grad = 6, x = -2
[ Epoch 17 ] grad = -6, x = 4
[ Epoch 18 ] grad = 6, x = -2
[ Epoch 19 ] grad = -6, x = 4
Out[35]:
4
In [32]:
def f2(x):
return 4 * x * x - 4 * x + 1
def g2(x):
return 8 * x -4
gd(5,0.25,g2)
[ Epoch 0 ] grad = 36, x = -4.0
[ Epoch 1 ] grad = -36.0, x = 5.0
[ Epoch 2 ] grad = 36.0, x = -4.0
[ Epoch 3 ] grad = -36.0, x = 5.0
[ Epoch 4 ] grad = 36.0, x = -4.0
[ Epoch 5 ] grad = -36.0, x = 5.0
[ Epoch 6 ] grad = 36.0, x = -4.0
[ Epoch 7 ] grad = -36.0, x = 5.0
[ Epoch 8 ] grad = 36.0, x = -4.0
[ Epoch 9 ] grad = -36.0, x = 5.0
[ Epoch 10 ] grad = 36.0, x = -4.0
[ Epoch 11 ] grad = -36.0, x = 5.0
[ Epoch 12 ] grad = 36.0, x = -4.0
[ Epoch 13 ] grad = -36.0, x = 5.0
[ Epoch 14 ] grad = 36.0, x = -4.0
[ Epoch 15 ] grad = -36.0, x = 5.0
[ Epoch 16 ] grad = 36.0, x = -4.0
[ Epoch 17 ] grad = -36.0, x = 5.0
[ Epoch 18 ] grad = 36.0, x = -4.0
[ Epoch 19 ] grad = -36.0, x = 5.0
Out[32]:
5.0
Content source: hsmyy/zhihuzhuanlan
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