tensorflow-梯度降低,有這一篇就足夠了

前言

最近機器學習愈來愈火了，前段時間斯丹福大學副教授吳恩達都親自錄製了關於Deep Learning Specialization的教程，在國內掀起了巨大的學習熱潮。本着不被時代拋棄的念頭，本身也開始研究有關機器學習的知識。都說機器學習的學習難度很是大，但不親自嘗試一下又怎麼會知道其中的奧妙與樂趣呢？只有不斷的嘗試才能找到最適合本身的道路。

請容忍我上述的自我煽情，下面進入主題。這篇文章主要對機器學習中所遇到的GradientDescent(梯度降低)進行全面分析，相信你看了這篇文章以後，對GradientDescent將完全弄明白其中的原理。

梯度降低的概念

梯度降低法是一個一階最優化算法，一般也稱爲最速降低法。要使用梯度降低法找到一個函數的局部極小值，必須向函數上當前點對於梯度（或者是近似梯度）的反方向的規定步長距離點進行迭代搜索。因此梯度降低法能夠幫助咱們求解某個函數的極小值或者最小值。對於n維問題就最優解，梯度降低法是最經常使用的方法之一。下面經過梯度降低法的前生今世來進行詳細推導說明。

梯度降低法的前世

首先從簡單的開始，看下面的一維函數：

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

在數學中若是咱們要求f(x) = 0處的解，咱們能夠經過以下偏差等式來求得：

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

當error趨近於最小值時，也就是f(x) = 0處x的解，咱們也能夠經過圖來觀察：

經過這函數圖，咱們能夠很是直觀的發現，要想求得該函數的最小值，只要將x指定爲函數圖的最低谷。這在高中咱們就已經掌握了該函數的最小值解法。咱們能夠經過對該函數進行求導（即斜率）：

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

若是要獲得最小值，只需令derivative(x) = 0，即x = 1。同時咱們結合圖與導函數能夠知道：

• 當x < 1時，derivative < 0，斜率爲負的；
• 當x > 1時，derivative > 0，斜率爲正的；
• 當x 無限接近 1時，derivative也就無限=0，斜率爲零。

經過上面的結論，咱們可使用以下表達式來代替x在函數中的移動

x = x - reate * derivative

當斜率爲負的時候，x增大，當斜率爲正的時候，x減少；所以x老是會向着低谷移動，使得error最小，從而求得 f(x) = 0處的解。其中的rate表明x逆着導數方向移動的距離，rate越大，x每次就移動的越多。反之移動的越少。

這是針對簡單的函數，咱們能夠很是直觀的求得它的導函數。爲了應對複雜的函數，咱們能夠經過使用求導函數的定義來表達導函數:若函數f(x)在點x0處可導，那麼有以下定義：

上面是都是公式推導，下面經過代碼來實現，下面的代碼都是使用python進行實現。

>>> 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)))
...

執行上面程序，咱們就能獲得以下結果：

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

經過上面的結果，也驗證了咱們最初的結論。x = 1時，f(x) = 0。
因此經過該方法，只要步數足夠多，就能獲得很是精確的值。

梯度降低法的此生

上面是對一維函數進行求解，那麼對於多維函數又要如何求呢？咱們接着看下面的函數，你會發現對於多維函數也是那麼的簡單。

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

一樣的若是咱們要求f(x) = 0處，x[0]與x[1]的值，也能夠經過求error函數的最小值來間接求f(x)的解。跟一維函數惟一不一樣的是，要分別對x[0]與x[1]進行求導。在數學上叫作偏導數：

• 保持x[1]不變，對x[0]進行求導，即f(x)對x[0]的偏導數
• 保持x[0]不變，對x[1]進行求導，即f(x)對x[1]的偏導數

有了上面的理解基礎，咱們定義的gradient_descent以下：

>>> 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的做用不變，惟一的區別就是分別獲取最新的x[0]與x[1]。下面是整個代碼：

>>> 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)))
...

輸出結果爲：

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

細心的你可能會發現， f(x) = 0不止這一個解還能夠是 x = -2, -1。這是由於梯度降低法只是對 當前所處的凹谷進行梯度降低求解，對於 error函數並不表明只有一個 f(x) = 0的凹谷。因此梯度降低法只能求得局部解，但不必定能求得所有的解。固然若是對於很是複雜的函數，可以求得局部解也是很是不錯的。

tensorflow中的運用

經過上面的示例，相信對梯度降低也有了一個基本的認識。如今咱們回到最開始的地方，在tensorflow中使用gradientDescent。

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的官網示例，上面代碼定義了函數linear_model = W * x + b,其中的error函數爲linear_model - y。目的是對一組x_train與y_train進行簡單的訓練求解W與b。爲了求得這一組數據的最優解，將每一組的error相加從而獲得loss，最後再對loss進行梯度降低求解最優值。

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

在這裏rate爲0.01,由於這個示例也是多維函數，因此也要用到偏導數來進行逐步向最優解靠近。

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

最後使用梯度降低進行循環推導，下面給出一些推導過程當中的相關結果

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

這裏就不推理驗證了，若是看了上面的梯度降低的前世此生，相信可以自主的推導出來。那麼咱們直接看最後的結果，能夠估算爲W = -1.0與b = 1.0，將他們帶入上面的loss獲得的結果爲0.0，即偏差損失值最小，因此W = -1.0與b = 1.0就是x_train與y_train這組數據的最優解。

好了，關於梯度降低的內容就到這了，但願可以幫助到你；若有不足之處歡迎來討論，若是感受這篇文章不錯的話，能夠關注個人博客，或者掃描下方二維碼關注：怪談時間到了 公衆號，查看個人其它文章。

博客地址

關注