利用 tf.gradients 在 TensorFlow 中实现梯度下降

作者：chen_h
微信号 & QQ：862251340
微信公众号：coderpai
简书地址：http://www.jianshu.com/p/13e0...

---

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: 0.898

Tensorflow 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/13e0...

CoderPai 是一个专注于算法实战的平台，从基础的算法到人工智能算法都有设计。如果你对算法实战感兴趣，请快快关注我们吧。加入AI实战微信群，AI实战QQ群，ACM算法微信群，ACM算法QQ群。长按或者扫描如下二维码，关注 “CoderPai” 微信号（coderpai）