TensorFlow-梯度下降

发表于 更新于 阅读次数： 阅读次数：

梯度下降法是一个一阶最优化算法，通常也称为最速下降法。要使用梯度下降法找到一个函数的局部极小值，必须向函数上当前点对于梯度（或者是近似梯度）的反方向的规定步长距离点进行迭代搜索。所以梯度下降法可以帮助我们求解某个函数的极小值或者最小值。对于n维问题就最优解，梯度下降法是最常用的方法之一。下面通过梯度下降法的前生今世来进行详细推导说明。

梯度下降法的前世

首先从简单的开始，看下面的一维函数：

1
f(x) = x^3 + 2 * x - 3

在数学中如果我们要求f(x) = 0处的解，我们可以通过如下误差等式来求得：

1
error = (f(x) - 0)^2

当error趋近于最小值时，也就是f(x) = 0处x的解，我们也可以通过图来观察：

通过这函数图，我们可以非常直观的发现，要想求得该函数的最小值，只要将x指定为函数图的最低谷。这在高中我们就已经掌握了该函数的最小值解法。我们可以通过对该函数进行求导（即斜率）：

1
derivative(x) = 6 * x^5 + 16 * x^3 - 18 * x^2 + 8 * x - 12

如果要得到最小值，只需令derivative(x) = 0，即x = 1。同时我们结合图与导函数可以知道：

当x < 1时，derivative < 0，斜率为负的；
当x > 1时，derivative > 0，斜率为正的；
当x 无限接近 1时，derivative也就无限=0，斜率为零。
通过上面的结论，我们可以使用如下表达式来代替x在函数中的移动

x = x - reate * derivative

当斜率为负的时候，x增大，当斜率为正的时候，x减小；因此x总是会向着低谷移动，使得error最小，从而求得 f(x) = 0处的解。其中的rate代表x逆着导数方向移动的距离，rate越大，x每次就移动的越多。反之移动的越少。
这是针对简单的函数，我们可以非常直观的求得它的导函数。为了应对复杂的函数，我们可以通过使用求导函数的定义来表达导函数:若函数f(x)在点x0处可导，那么有如下定义：

上面是都是公式推导，下面通过代码来实现，下面的代码都是使用python进行实现。

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
>>> def f(x):
...     return x**3 + 2 * x - 3
...
>>> def error(x):
...     return (f(x) - 0)**2
...
>>> def gradient_descent(x):
...     delta = 0.00000001
...     derivative = (error(x + delta) - error(x)) / delta
...     rate = 0.01
...     return x - rate * derivative
...
>>> x = 0.8
>>> for i in range(50):
...     x = gradient_descent(x)
...     print('x = {:6f}, f(x) = {:6f}'.format(x, f(x)))
...

执行上面程序，我们就能得到如下结果：

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
x = 0.869619, f(x) = -0.603123
x = 0.921110, f(x) = -0.376268
x = 0.955316, f(x) = -0.217521
x = 0.975927, f(x) = -0.118638
x = 0.987453, f(x) = -0.062266
x = 0.993586, f(x) = -0.031946
x = 0.996756, f(x) = -0.016187
x = 0.998369, f(x) = -0.008149
x = 0.999182, f(x) = -0.004088
x = 0.999590, f(x) = -0.002048
x = 0.999795, f(x) = -0.001025
x = 0.999897, f(x) = -0.000513
x = 0.999949, f(x) = -0.000256
x = 0.999974, f(x) = -0.000128
x = 0.999987, f(x) = -0.000064
x = 0.999994, f(x) = -0.000032
x = 0.999997, f(x) = -0.000016
x = 0.999998, f(x) = -0.000008
x = 0.999999, f(x) = -0.000004
x = 1.000000, f(x) = -0.000002
x = 1.000000, f(x) = -0.000001
x = 1.000000, f(x) = -0.000001
x = 1.000000, f(x) = -0.000000
x = 1.000000, f(x) = -0.000000
x = 1.000000, f(x) = -0.000000

通过上面的结果，也验证了我们最初的结论。x = 1时，f(x) = 0。
所以通过该方法，只要步数足够多，就能得到非常精确的值。

梯度下降法的今生

上面是对一维函数进行求解，那么对于多维函数又要如何求呢？我们接着看下面的函数，你会发现对于多维函数也是那么的简单。

1
f(x) = x[0] + 2 * x[1] + 4

同样的如果我们要求f(x) = 0处，x[0]与x[1]的值，也可以通过求error函数的最小值来间接求f(x)的解。跟一维函数唯一不同的是，要分别对x[0]与x[1]进行求导。在数学上叫做偏导数：

保持x[1]不变，对x[0]进行求导，即f(x)对x[0]的偏导数
保持x[0]不变，对x[1]进行求导，即f(x)对x[1]的偏导数
有了上面的理解基础，我们定义的gradient_descent如下：

1
2
3
4
5
6
7
8
9
>>> def gradient_descent(x):
...     delta = 0.00000001
...     derivative_x0 = (error([x[0] + delta, x[1]]) - error([x[0], x[1]])) / delta
...     derivative_x1 = (error([x[0], x[1] + delta]) - error([x[0], x[1]])) / delta
...     rate = 0.01
...     x[0] = x[0] - rate * derivative_x0
...     x[1] = x[1] - rate * derivative_x1
...     return [x[0], x[1]]
...

rate的作用不变，唯一的区别就是分别获取最新的x[0]与x[1]。下面是整个代码：

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
>>> def f(x):
...     return x[0] + 2 * x[1] + 4
...
>>> def error(x):
...     return (f(x) - 0)**2
...
>>> def gradient_descent(x):
...     delta = 0.00000001
...     derivative_x0 = (error([x[0] + delta, x[1]]) - error([x[0], x[1]])) / delta
...     derivative_x1 = (error([x[0], x[1] + delta]) - error([x[0], x[1]])) / delta
...     rate = 0.02
...     x[0] = x[0] - rate * derivative_x0
...     x[1] = x[1] - rate * derivative_x1
...     return [x[0], x[1]]
...
>>> x = [-0.5, -1.0]
>>> for i in range(100):
...     x = gradient_descent(x)
...     print('x = {:6f},{:6f}, f(x) = {:6f}'.format(x[0],x[1],f(x)))
...

输出结果为：

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
x = -0.560000,-1.120000, f(x) = 1.200000
x = -0.608000,-1.216000, f(x) = 0.960000
x = -0.646400,-1.292800, f(x) = 0.768000
x = -0.677120,-1.354240, f(x) = 0.614400
x = -0.701696,-1.403392, f(x) = 0.491520
x = -0.721357,-1.442714, f(x) = 0.393216
x = -0.737085,-1.474171, f(x) = 0.314573
x = -0.749668,-1.499337, f(x) = 0.251658
x = -0.759735,-1.519469, f(x) = 0.201327
x = -0.767788,-1.535575, f(x) = 0.161061
x = -0.774230,-1.548460, f(x) = 0.128849
x = -0.779384,-1.558768, f(x) = 0.103079
x = -0.783507,-1.567015, f(x) = 0.082463
x = -0.786806,-1.573612, f(x) = 0.065971
x = -0.789445,-1.578889, f(x) = 0.052777
x = -0.791556,-1.583112, f(x) = 0.042221
x = -0.793245,-1.586489, f(x) = 0.033777
x = -0.794596,-1.589191, f(x) = 0.027022
x = -0.795677,-1.591353, f(x) = 0.021617
x = -0.796541,-1.593082, f(x) = 0.017294
x = -0.797233,-1.594466, f(x) = 0.013835
x = -0.797786,-1.595573, f(x) = 0.011068
x = -0.798229,-1.596458, f(x) = 0.008854
x = -0.798583,-1.597167, f(x) = 0.007084
x = -0.798867,-1.597733, f(x) = 0.005667
x = -0.799093,-1.598187, f(x) = 0.004533
x = -0.799275,-1.598549, f(x) = 0.003627
x = -0.799420,-1.598839, f(x) = 0.002901
x = -0.799536,-1.599072, f(x) = 0.002321
x = -0.799629,-1.599257, f(x) = 0.001857
x = -0.799703,-1.599406, f(x) = 0.001486
x = -0.799762,-1.599525, f(x) = 0.001188
x = -0.799810,-1.599620, f(x) = 0.000951
x = -0.799848,-1.599696, f(x) = 0.000761
x = -0.799878,-1.599757, f(x) = 0.000608
x = -0.799903,-1.599805, f(x) = 0.000487
x = -0.799922,-1.599844, f(x) = 0.000389
x = -0.799938,-1.599875, f(x) = 0.000312
x = -0.799950,-1.599900, f(x) = 0.000249
x = -0.799960,-1.599920, f(x) = 0.000199
x = -0.799968,-1.599936, f(x) = 0.000159
x = -0.799974,-1.599949, f(x) = 0.000128
x = -0.799980,-1.599959, f(x) = 0.000102
x = -0.799984,-1.599967, f(x) = 0.000082
x = -0.799987,-1.599974, f(x) = 0.000065
x = -0.799990,-1.599979, f(x) = 0.000052
x = -0.799992,-1.599983, f(x) = 0.000042
x = -0.799993,-1.599987, f(x) = 0.000033
x = -0.799995,-1.599989, f(x) = 0.000027
x = -0.799996,-1.599991, f(x) = 0.000021
x = -0.799997,-1.599993, f(x) = 0.000017
x = -0.799997,-1.599995, f(x) = 0.000014
x = -0.799998,-1.599996, f(x) = 0.000011
x = -0.799998,-1.599997, f(x) = 0.000009
x = -0.799999,-1.599997, f(x) = 0.000007
x = -0.799999,-1.599998, f(x) = 0.000006
x = -0.799999,-1.599998, f(x) = 0.000004
x = -0.799999,-1.599999, f(x) = 0.000004
x = -0.799999,-1.599999, f(x) = 0.000003
x = -0.800000,-1.599999, f(x) = 0.000002
x = -0.800000,-1.599999, f(x) = 0.000002
x = -0.800000,-1.599999, f(x) = 0.000001
x = -0.800000,-1.600000, f(x) = 0.000001
x = -0.800000,-1.600000, f(x) = 0.000001
x = -0.800000,-1.600000, f(x) = 0.000001
x = -0.800000,-1.600000, f(x) = 0.000001
x = -0.800000,-1.600000, f(x) = 0.000000

细心的你可能会发现，f(x) = 0不止这一个解还可以是x = -2, -1。这是因为梯度下降法只是对当前所处的凹谷进行梯度下降求解，对于error函数并不代表只有一个f(x) = 0的凹谷。所以梯度下降法只能求得局部解，但不一定能求得全部的解。当然如果对于非常复杂的函数，能够求得局部解也是非常不错的。

tensorflow中的运用

通过上面的示例，相信对梯度下降也有了一个基本的认识。现在我们回到最开始的地方，在tensorflow中使用gradientDescent。

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
import tensorflow as tf
 
# Model parameters
W = tf.Variable([.3], dtype=tf.float32)
b = tf.Variable([-.3], dtype=tf.float32)
# Model input and output
x = tf.placeholder(tf.float32)
linear_model = W*x + b
y = tf.placeholder(tf.float32)
 
# loss
loss = tf.reduce_sum(tf.square(linear_model - y)) # sum of the squares
# optimizer
optimizer = tf.train.GradientDescentOptimizer(0.01)
train = optimizer.minimize(loss)
 
# training data
x_train = [1, 2, 3, 4]
y_train = [0, -1, -2, -3]
# training loop
init = tf.global_variables_initializer()
sess = tf.Session()
sess.run(init) # reset values to wrong
for i in range(1000):
  sess.run(train, {x: x_train, y: y_train})
 
# evaluate training accuracy
curr_W, curr_b, curr_loss = sess.run([W, b, loss], {x: x_train, y: y_train})
print("W: %s b: %s loss: %s"%(curr_W, curr_b, curr_loss))

上面的是tensorflow的官网示例，上面代码定义了函数linear_model = W * x + b,其中的error函数为linear_model - y。目的是对一组x_train与y_train进行简单的训练求解W与b。为了求得这一组数据的最优解，将每一组的error相加从而得到loss，最后再对loss进行梯度下降求解最优值。

1
2
optimizer = tf.train.GradientDescentOptimizer(0.01)
train = optimizer.minimize(loss)

在这里rate为0.01,因为这个示例也是多维函数，所以也要用到偏导数来进行逐步向最优解靠近。

1
2
for i in range(1000):
  sess.run(train, {x: x_train, y: y_train})

最后使用梯度下降进行循环推导，下面给出一些推导过程中的相关结果

1
2
3
4
5
6
7
8
9
10
11
12
13
W: [-0.21999997] b: [-0.456] loss: 4.01814
W: [-0.39679998] b: [-0.49552] loss: 1.81987
W: [-0.45961601] b: [-0.4965184] loss: 1.54482
W: [-0.48454273] b: [-0.48487374] loss: 1.48251
W: [-0.49684232] b: [-0.46917531] loss: 1.4444
W: [-0.50490189] b: [-0.45227283] loss: 1.4097
W: [-0.5115062] b: [-0.43511063] loss: 1.3761
....
....
....
W: [-0.99999678] b: [ 0.99999058] loss: 5.84635e-11
W: [-0.99999684] b: [ 0.9999907] loss: 5.77707e-11
W: [-0.9999969] b: [ 0.99999082] loss: 5.69997e-11

这里就不推理验证了，如果看了上面的梯度下降的前世今生，相信能够自主的推导出来。那么我们直接看最后的结果，可以估算为W = -1.0与b = 1.0，将他们带入上面的loss得到的结果为0.0，即误差损失值最小，所以W = -1.0与b = 1.0就是x_train与y_train这组数据的最优解。