(DeepLearning MOOC) Lesson 2: Deep Neural Networks

[TOC]

Linear models

matrix multiplication: fast with GPU
numerically stable
cannot cocatenate linear units → equivalent to one big matrix...

⇒ add non-linear units in between

rectified linear units (RELU)

chain rule: efficient computationally

back propagation

easy to compute the gradient as long as the function Y(X) is made of simple blocks with simple deritivates.
most deep-learning framework can do it automatically for you.

N.B. The backprop block takes 2x memory/compute wrt the forward prop blocks.

first neural network: RELU units between linear classifiers:

Tensor flow

tensors define computations, and they are nodes in a computation graph.
To actually run the optimization, use sessions...

define a computation graph:

batch_size = 128  
num_hidden = 1024

graph = tf.Graph()  
with graph.as_default():

  # Input data. For the training data, we use a placeholder that will be fed  
  # at run time with a training minibatch.  
  tf_train_dataset = tf.placeholder(tf.float32,  
                                    shape=(batch_size, image_size * image_size))  
  tf_train_labels = tf.placeholder(tf.float32, shape=(batch_size, num_labels))  
  tf_valid_dataset = tf.constant(valid_dataset)  
  tf_test_dataset = tf.constant(test_dataset)

  # Variables for linear layer 1  
  W1 = tf.Variable(  
    tf.truncated_normal([image_size * image_size, num_hidden]))  
  b1 = tf.Variable(tf.zeros([num_hidden]))

  # Hidden RELU input computation  
  y1 = tf.matmul(tf_train_dataset, W1) + b1  
  # Hidden RELU output computation  
  X1 = tf.nn.relu(y1)

  # Variables for linear layer 2  
  W2 = tf.Variable(  
    tf.truncated_normal([num_hidden, num_labels]))#W2  
  b2 = tf.Variable(tf.zeros([num_labels])) #b2  
  # logit (y2) output  
  logits = tf.matmul(X1, W2) + b2  
  loss = tf.reduce_mean(  
    tf.nn.softmax_cross_entropy_with_logits(logits, tf_train_labels))

  def getlogits(X):  
    y1 = tf.matmul(X, W1) + b1  
    X1 = tf.nn.relu(y1)  
    return tf.matmul(X1, W2) + b2

  # Optimizer.  
  optimizer = tf.train.GradientDescentOptimizer(0.5).minimize(loss)

  # Predictions for the training, validation, and test data.  
  train_prediction = tf.nn.softmax(logits)  
  valid_prediction = tf.nn.softmax( getlogits(tf_valid_dataset) )  
  test_prediction = tf.nn.softmax( getlogits(tf_test_dataset))

run sgd optimization:

num_steps = 3001

with tf.Session(graph=graph) as session:  
  tf.initialize_all_variables().run()  
  print("Initialized")  
  for step in range(num_steps):  
    # Pick an offset within the training data, which has been randomized.  
    # Note: we could use better randomization across epochs.  
    offset = (step * batch_size) % (train_labels.shape[0] - batch_size)  
    # Generate a minibatch.  
    batch_data = train_dataset[offset:(offset + batch_size), :]  
    batch_labels = train_labels[offset:(offset + batch_size), :]  
    # Prepare a dictionary telling the session where to feed the minibatch.  
    # The key of the dictionary is the placeholder node of the graph to be fed,  
    # and the value is the numpy array to feed to it.  
    feed_dict = {tf_train_dataset : batch_data, tf_train_labels : batch_labels}  
    _, l, predictions = session.run(  
      [optimizer, loss, train_prediction], feed_dict=feed_dict)  
    if (step % 500 == 0):  
      print("Minibatch loss at step %d: %f" % (step, l))  
      print("Minibatch accuracy: %.1f%%" % accuracy(predictions, batch_labels))  
      print("Validation accuracy: %.1f%%" % accuracy(  
        valid_prediction.eval(), valid_labels))  
  print("Test accuracy: %.1f%%" % accuracy(test_prediction.eval(), test_labels))

Regularization

  • early termination: stop when cannot improve in validation performance.
  • L2 regularization: adding L2 norm of

Dropout

def. activation is the output of last layer that flows into the next layer.
dropout: randomly set half of activations to 0.

rational: forcing your model to learn reduadant representations (consus over an ensemble of nns...)...

N.B.
for evaluation no longer dropout, ye = average of activations, trick to let ye=E(yt), in training, multiply the remaining activations by 2.


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