作者：chen_h
微信号 & QQ：862251340
微信公众号：coderpai
简书地址：http://www.jianshu.com/p/13e024c8ea44
我喜欢 TensorFlow 的其中一个原因是它可以自动的计算函数的梯度。我们只需要设计我们的函数，然后去调用 tf.gradients 函数就可以了。是不是非常简单。
接下来让我们来举个例子，具体说明一下。
使用 TensorFlow 内置的优化器对 MNIST 数据集进行 softmax 回归
在使用 tf.gradients 实现梯度下降之前，我们先尝试使用 TensorFlow 的内置优化器（比如 GradientDescentOptimizer）来解决MNIST数据集分类问题。
import tensorflow as tf
# Import MNIST data
from tensorflow.examples.tutorials.mnist import input_data
mnist = input_data.read_data_sets("/tmp/data/", one_hot=True)
# Parameters
learning_rate = 0.01
training_epochs = 10
batch_size = 100
display_step = 1
# tf Graph Input
x = tf.placeholder(tf.float32, [None, 784]) # mnist data image of shape 28*28=784
y = tf.placeholder(tf.float32, [None, 10]) # 0-9 digits recognition => 10 classes
# Set model weights
W = tf.Variable(tf.zeros([784, 10]))
b = tf.Variable(tf.zeros([10]))
# Construct model
pred = tf.nn.softmax(tf.matmul(x, W) + b) # Softmax
# Minimize error using cross entropy
cost = tf.reduce_mean(-tf.reduce_sum(y*tf.log(pred), reduction_indices=1))
optimizer = tf.train.GradientDescentOptimizer(learning_rate).minimize(cost)
# Start training
with tf.Session() as sess:
sess.run(tf.global_variables_initializer())
# Training cycle
for epoch in range(training_epochs):
avg_cost = 0.
total_batch = int(mnist.train.num_examples/batch_size)
# Loop over all batches
for i in range(total_batch):
batch_xs, batch_ys = mnist.train.next_batch(batch_size)
# Fit training using batch data
_, c = sess.run([optimizer, cost], feed_dict={x: batch_xs,
y: batch_ys})
# print(__w)
# Compute average loss
avg_cost += c / total_batch
# Display logs per epoch step
if (epoch+1) % display_step == 0:
# print(sess.run(W))
print ("Epoch:", '%04d' % (epoch+1), "cost=", "{:.9f}".format(avg_cost))
print ("Optimization Finished!")
# Test model
correct_prediction = tf.equal(tf.argmax(pred, 1), tf.argmax(y, 1))
# Calculate accuracy for 3000 examples
accuracy = tf.reduce_mean(tf.cast(correct_prediction, tf.float32))
print ("Accuracy:", accuracy.eval({x: mnist.test.images[:3000], y: mnist.test.labels[:3000]}))
#### Output
# Extracting /tmp/data/train-images-idx3-ubyte.gz
# Extracting /tmp/data/train-labels-idx1-ubyte.gz
# Extracting /tmp/data/t10k-images-idx3-ubyte.gz
# Extracting /tmp/data/t10k-labels-idx1-ubyte.gz
# Epoch: 0001 cost= 1.184285608
# Epoch: 0002 cost= 0.665428013
# Epoch: 0003 cost= 0.552858426
# Epoch: 0004 cost= 0.498728328
# Epoch: 0005 cost= 0.465593693
# Epoch: 0006 cost= 0.442609185
# Epoch: 0007 cost= 0.425552949
# Epoch: 0008 cost= 0.412188290
# Epoch: 0009 cost= 0.401390140
# Epoch: 0010 cost= 0.392354651
# Optimization Finished!
# Accuracy: 0.873333所以，我们在这里做的是利用内置的优化器来计算损失值。如果我们想自己计算渐变过程和更新权重，那应该怎么办？这就是 tf.gradients 的作用了。
使用 tf.gradients 对MNIST数据集进行 softmax 回归
通过梯度下降公式，权重的更新方式如下：
为了实现梯度下降，我将不使用优化器的代码，而是采用自己写的权重更新。
因为这里有权重矩阵 w 和偏差项矩阵 b，所以我们需要去计算这些矩阵的梯度。所以实现的代码如下：
# Computing the gradient of cost with respect to W and b
grad_W, grad_b = tf.gradients(xs=[W, b], ys=cost)
# Gradient Step
new_W = W.assign(W - learning_rate * grad_W)
new_b = b.assign(b - learning_rate * grad_b)这三行代码只是替代前面的一行代码，干嘛给自己造成这么大的麻烦呢？因为如果你需要自己的损失函数的梯度，并且你不想编写严格的数学函数，那么 TensorFlow 就可以帮助你了。
我们已经构建好了计算图，所以接下来我们只需要在会话中运行这个计算图就行了。让我来试试吧。
# Fit training using batch data
_, _, c = sess.run([new_W, new_b ,cost], feed_dict={x: batch_xs, y: batch_ys})我们不需要 new_W 和 new_b 的输出，所以我忽略了这些变量。
完整代码如下：
import tensorflow as tf
# Import MNIST data
from tensorflow.examples.tutorials.mnist import input_data
mnist = input_data.read_data_sets("/tmp/data/", one_hot=True)
# Parameters
learning_rate = 0.01
training_epochs = 10
batch_size = 100
display_step = 1
# Parameters
learning_rate = 0.01
training_epochs = 10
batch_size = 100
display_step = 1
# tf Graph Input
x = tf.placeholder(tf.float32, [None, 784]) # mnist data image of shape 28*28=784
y = tf.placeholder(tf.float32, [None, 10]) # 0-9 digits recognition => 10 classes
# Set model weights
W = tf.Variable(tf.zeros([784, 10]))
b = tf.Variable(tf.zeros([10]))
# Construct model
pred = tf.nn.softmax(tf.matmul(x, W) + b) # Softmax
# Minimize error using cross entropy
cost = tf.reduce_mean(-tf.reduce_sum(y*tf.log(pred), reduction_indices=1))
grad_W, grad_b = tf.gradients(xs=[W, b], ys=cost)
new_W = W.assign(W - learning_rate * grad_W)
new_b = b.assign(b - learning_rate * grad_b)
# Initialize the variables (i.e. assign their default value)
init = tf.global_variables_initializer()
# Start training
with tf.Session() as sess:
sess.run(init)
# Training cycle
for epoch in range(training_epochs):
avg_cost = 0.
total_batch = int(mnist.train.num_examples/batch_size)
# Loop over all batches
for i in range(total_batch):
batch_xs, batch_ys = mnist.train.next_batch(batch_size)
# Fit training using batch data
_, _, c = sess.run([new_W, new_b ,cost], feed_dict={x: batch_xs,
y: batch_ys})
# Compute average loss
avg_cost += c / total_batch
# Display logs per epoch step
if (epoch+1) % display_step == 0:
# print(sess.run(W))
print ("Epoch:", '%04d' % (epoch+1), "cost=", "{:.9f}".format(avg_cost))
print ("Optimization Finished!")
# Test model
correct_prediction = tf.equal(tf.argmax(pred, 1), tf.argmax(y, 1))
# Calculate accuracy for 3000 examples
accuracy = tf.reduce_mean(tf.cast(correct_prediction, tf.float32))
print ("Accuracy:", accuracy.eval({x: mnist.test.images[:3000], y: mnist.test.labels[:3000]}))
# Output
# Epoch: 0001 cost= 1.183741399
# Epoch: 0002 cost= 0.665312284
# Epoch: 0003 cost= 0.552796521
# Epoch: 0004 cost= 0.498697014
# Epoch: 0005 cost= 0.465521633
# Epoch: 0006 cost= 0.442611256
# Epoch: 0007 cost= 0.425528946
# Epoch: 0008 cost= 0.412203073
# Epoch: 0009 cost= 0.401364554
# Epoch: 0010 cost= 0.392398663
# Optimization Finished!
# Accuracy: 0.874使用梯度公式的 softmax 回归
我们对于权重 w 的梯度处理如下：
如前所示，不使用 tf.gradients 或使用 TensorFlow 的内置优化器，这样可以实现梯度方程。完整代码如下：
import tensorflow as tf
# Import MNIST data
from tensorflow.examples.tutorials.mnist import input_data
mnist = input_data.read_data_sets("/tmp/data/", one_hot=True)
# Parameters
learning_rate = 0.01
training_epochs = 10
batch_size = 100
display_step = 1
# Parameters
learning_rate = 0.01
training_epochs = 10
batch_size = 100
display_step = 1
# tf Graph Input
x = tf.placeholder(tf.float32, [None, 784]) # mnist data image of shape 28*28=784
y = tf.placeholder(tf.float32, [None, 10]) # 0-9 digits recognition => 10 classes
# Set model weights
W = tf.Variable(tf.zeros([784, 10]))
b = tf.Variable(tf.zeros([10]))
# Construct model
pred = tf.nn.softmax(tf.matmul(x, W)) # Softmax
# Minimize error using cross entropy
cost = tf.reduce_mean(-tf.reduce_sum(y*tf.log(pred), reduction_indices=1))
W_grad = - tf.matmul ( tf.transpose(x) , y - pred)
b_grad = - tf.reduce_mean( tf.matmul(tf.transpose(x), y - pred), reduction_indices=0)
new_W = W.assign(W - learning_rate * W_grad)
new_b = b.assign(b - learning_rate * b_grad)
init = tf.global_variables_initializer()
with tf.Session() as sess:
sess.run(init)
# Training cycle
for epoch in range(training_epochs):
avg_cost = 0.
total_batch = int(mnist.train.num_examples/batch_size)
# Loop over all batches
for i in range(total_batch):
batch_xs, batch_ys = mnist.train.next_batch(batch_size)
# Fit training using batch data
_, _, c = sess.run([new_W, new_b, cost], feed_dict={x: batch_xs, y: batch_ys})
# Compute average loss
avg_cost += c / total_batch
# Display logs per epoch step
if (epoch+1) % display_step == 0:
print ("Epoch:", '%04d' % (epoch+1), "cost=", "{:.9f}".format(avg_cost))
print ("Optimization Finished!")
# Test model
correct_prediction = tf.equal(tf.argmax(pred, 1), tf.argmax(y, 1))
# Calculate accuracy for 3000 examples
accuracy = tf.reduce_mean(tf.cast(correct_prediction, tf.float32))
print ("Accuracy:", accuracy.eval({x: mnist.test.images[:3000], y: mnist.test.labels[:3000]}))
# Output
# Extracting /tmp/data/train-images-idx3-ubyte.gz
# Extracting /tmp/data/train-labels-idx1-ubyte.gz
# Extracting /tmp/data/t10k-images-idx3-ubyte.gz
# Extracting /tmp/data/t10k-labels-idx1-ubyte.gz
# Epoch: 0001 cost= 0.432943137
# Epoch: 0002 cost= 0.330031527
# Epoch: 0003 cost= 0.313661941
# Epoch: 0004 cost= 0.306443773
# Epoch: 0005 cost= 0.300219418
# Epoch: 0006 cost= 0.298976618
# Epoch: 0007 cost= 0.293222957
# Epoch: 0008 cost= 0.291407861
# Epoch: 0009 cost= 0.288372261
# Epoch: 0010 cost= 0.286749691
# Optimization Finished!
# Accuracy: 0.898Tensorflow 是如何计算梯度的？
你可以在思考，TensorFlow是如何计算函数的梯度？
TensorFlow 使用的是一种称为 Automatic Differentiation 的方法，具体你可以查看 Wikipedia。
我希望这篇文章对你有帮会帮助。
算法直播课：请点击这里
作者：chen_h
微信号 & QQ：862251340
简书地址：http://www.jianshu.com/p/13e024c8ea44
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