cs231n：assignment1——Q4: Two-Layer Neural Network

自己写的cs231n的作业,希望给点意见,支出错误和不足.谢谢
参数调了好久，有一次调到57+%，当时没记参数,后来调不回来了

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two_layer_net.ipynb内容:

Implementing a Neural Network

In this exercise we will develop a neural network with fully-connected layers to perform classification, and test it out on the CIFAR-10 dataset.

# A bit of setup

import numpy as np
import matplotlib.pyplot as plt

from cs231n.classifiers.neural_net import TwoLayerNet

%matplotlib inline
plt.rcParams['figure.figsize'] = (10.0, 8.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

# for auto-reloading external modules
# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
%load_ext autoreload
%autoreload 2

def rel_error(x, y):
  """ returns relative error """
  return np.max(np.abs(x - y) / (np.maximum(1e-8, np.abs(x) + np.abs(y))))

We will use the class TwoLayerNet in the file cs231n/classifiers/neural_net.py to represent instances of our network. The network parameters are stored in the instance variable self.params where keys are string parameter names and values are numpy arrays. Below, we initialize toy data and a toy model that we will use to develop your implementation.

# Create a small net and some toy data to check your implementations.
# Note that we set the random seed for repeatable experiments.

input_size = 4
hidden_size = 10
num_classes = 3
num_inputs = 5

def init_toy_model():
  np.random.seed(0)
  return TwoLayerNet(input_size, hidden_size, num_classes, std=1e-1)

def init_toy_data():
  np.random.seed(1)
  X = 10 * np.random.randn(num_inputs, input_size)
  y = np.array([0, 1, 2, 2, 1])
  return X, y

net = init_toy_model()
X, y = init_toy_data()

Forward pass: compute scores

Open the file cs231n/classifiers/neural_net.py and look at the method TwoLayerNet.loss. This function is very similar to the loss functions you have written for the SVM and Softmax exercises: It takes the data and weights and computes the class scores, the loss, and the gradients on the parameters.

Implement the first part of the forward pass which uses the weights and biases to compute the scores for all inputs.

scores = net.loss(X)
print 'Your scores:'
print scores
print
print 'correct scores:'
correct_scores = np.asarray([
  [-0.81233741, -1.27654624, -0.70335995],
  [-0.17129677, -1.18803311, -0.47310444],
  [-0.51590475, -1.01354314, -0.8504215 ],
  [-0.15419291, -0.48629638, -0.52901952],
  [-0.00618733, -0.12435261, -0.15226949]])
print correct_scores
print

# The difference should be very small. We get < 1e-7
print 'Difference between your scores and correct scores:'
print np.sum(np.abs(scores - correct_scores))

Your scores:
[[-0.81233741 -1.27654624 -0.70335995]
 [-0.17129677 -1.18803311 -0.47310444]
 [-0.51590475 -1.01354314 -0.8504215 ]
 [-0.15419291 -0.48629638 -0.52901952]
 [-0.00618733 -0.12435261 -0.15226949]]

correct scores:
[[-0.81233741 -1.27654624 -0.70335995]
 [-0.17129677 -1.18803311 -0.47310444]
 [-0.51590475 -1.01354314 -0.8504215 ]
 [-0.15419291 -0.48629638 -0.52901952]
 [-0.00618733 -0.12435261 -0.15226949]]

Difference between your scores and correct scores:
3.68027209324e-08

Forward pass: compute loss

In the same function, implement the second part that computes the data and regularizaion loss.

loss, _ = net.loss(X, y, reg=0.1)
correct_loss = 1.30378789133

# should be very small, we get < 1e-12
print 'Difference between your loss and correct loss:'
print np.sum(np.abs(loss - correct_loss))

Difference between your loss and correct loss:
1.79412040779e-13

Backward pass

Implement the rest of the function. This will compute the gradient of the loss with respect to the variables W1, b1, W2, and b2. Now that you (hopefully!) have a correctly implemented forward pass, you can debug your backward pass using a numeric gradient check:

from cs231n.gradient_check import eval_numerical_gradient

# Use numeric gradient checking to check your implementation of the backward pass.
# If your implementation is correct, the difference between the numeric and
# analytic gradients should be less than 1e-8 for each of W1, W2, b1, and b2.

loss, grads = net.loss(X, y, reg=0.1)

# these should all be less than 1e-8 or so
for param_name in grads:
  f = lambda W: net.loss(X, y, reg=0.1)[0]
  param_grad_num = eval_numerical_gradient(f, net.params[param_name], verbose=False)
  print '%s max relative error: %e' % (param_name, rel_error(param_grad_num, grads[param_name]))

W1 max relative error: 3.669857e-09
W2 max relative error: 3.440708e-09
b2 max relative error: 3.865028e-11
b1 max relative error: 1.125423e-09

Train the network

To train the network we will use stochastic gradient descent (SGD), similar to the SVM and Softmax classifiers. Look at the function TwoLayerNet.train and fill in the missing sections to implement the training procedure. This should be very similar to the training procedure you used for the SVM and Softmax classifiers. You will also have to implement TwoLayerNet.predict, as the training process periodically performs prediction to keep track of accuracy over time while the network trains.

Once you have implemented the method, run the code below to train a two-layer network on toy data. You should achieve a training loss less than 0.2.

net = init_toy_model()
stats = net.train(X, y, X, y,
            learning_rate=1e-1, reg=1e-5,
            num_iters=100, verbose=False)

print 'Final training loss: ', stats['loss_history'][-1]

# plot the loss history
plt.plot(stats['loss_history'])
plt.xlabel('iteration')
plt.ylabel('training loss')
plt.title('Training Loss history')
plt.show()

Final training loss:  0.0171496079387

Load the data

Now that you have implemented a two-layer network that passes gradient checks and works on toy data, it’s time to load up our favorite CIFAR-10 data so we can use it to train a classifier on a real dataset.

from cs231n.data_utils import load_CIFAR10

def get_CIFAR10_data(num_training=49000, num_validation=1000, num_test=1000):
    """
    Load the CIFAR-10 dataset from disk and perform preprocessing to prepare
    it for the two-layer neural net classifier. These are the same steps as
    we used for the SVM, but condensed to a single function.  
    """
    # Load the raw CIFAR-10 data
    cifar10_dir = 'cs231n/datasets/cifar-10-batches-py'
    X_train, y_train, X_test, y_test = load_CIFAR10(cifar10_dir)

    # Subsample the data
    mask = range(num_training, num_training + num_validation)
    X_val = X_train[mask]
    y_val = y_train[mask]
    mask = range(num_training)
    X_train = X_train[mask]
    y_train = y_train[mask]
    mask = range(num_test)
    X_test = X_test[mask]
    y_test = y_test[mask]

    # Normalize the data: subtract the mean image
    mean_image = np.mean(X_train, axis=0)
    X_train -= mean_image
    X_val -= mean_image
    X_test -= mean_image

    # Reshape data to rows
    X_train = X_train.reshape(num_training, -1)
    X_val = X_val.reshape(num_validation, -1)
    X_test = X_test.reshape(num_test, -1)

    return X_train, y_train, X_val, y_val, X_test, y_test


# Invoke the above function to get our data.
X_train, y_train, X_val, y_val, X_test, y_test = get_CIFAR10_data()
print 'Train data shape: ', X_train.shape
print 'Train labels shape: ', y_train.shape
print 'Validation data shape: ', X_val.shape
print 'Validation labels shape: ', y_val.shape
print 'Test data shape: ', X_test.shape
print 'Test labels shape: ', y_test.shape

Train data shape:  (49000, 3072)
Train labels shape:  (49000,)
Validation data shape:  (1000, 3072)
Validation labels shape:  (1000,)
Test data shape:  (1000, 3072)
Test labels shape:  (1000,)

Train a network

To train our network we will use SGD with momentum. In addition, we will adjust the learning rate with an exponential learning rate schedule as optimization proceeds; after each epoch, we will reduce the learning rate by multiplying it by a decay rate.

input_size = 32 * 32 * 3
hidden_size = 50
num_classes = 10
net = TwoLayerNet(input_size, hidden_size, num_classes)

# Train the network
stats = net.train(X_train, y_train, X_val, y_val,
            num_iters=1000, batch_size=200,
            learning_rate=1e-4, learning_rate_decay=0.95,
            reg=0.5, verbose=True)

# Predict on the validation set
val_acc = (net.predict(X_val) == y_val).mean()
print 'Validation accuracy: ', val_acc

iteration 0 / 1000: loss 2.302954
iteration 100 / 1000: loss 2.302550
iteration 200 / 1000: loss 2.297648
iteration 300 / 1000: loss 2.259602
iteration 400 / 1000: loss 2.204170
iteration 500 / 1000: loss 2.118565
iteration 600 / 1000: loss 2.051535
iteration 700 / 1000: loss 1.988466
iteration 800 / 1000: loss 2.006591
iteration 900 / 1000: loss 1.951473
Validation accuracy:  0.287

Debug the training

With the default parameters we provided above, you should get a validation accuracy of about 0.29 on the validation set. This isn’t very good.

One strategy for getting insight into what’s wrong is to plot the loss function and the accuracies on the training and validation sets during optimization.

Another strategy is to visualize the weights that were learned in the first layer of the network. In most neural networks trained on visual data, the first layer weights typically show some visible structure when visualized.

# Plot the loss function and train / validation accuracies
plt.subplot(2, 1, 1)
plt.plot(stats['loss_history'])
plt.title('Loss history')
plt.xlabel('Iteration')
plt.ylabel('Loss')

plt.subplot(2, 1, 2)
plt.plot(stats['train_acc_history'], label='train')
plt.plot(stats['val_acc_history'], label='val')
plt.title('Classification accuracy history')
plt.xlabel('Epoch')
plt.ylabel('Clasification accuracy')
plt.show()

from cs231n.vis_utils import visualize_grid

# Visualize the weights of the network

def show_net_weights(net):
  W1 = net.params['W1']
  W1 = W1.reshape(32, 32, 3, -1).transpose(3, 0, 1, 2)
  plt.imshow(visualize_grid(W1, padding=3).astype('uint8'))
  plt.gca().axis('off')
  plt.show()

show_net_weights(net)

Tune your hyperparameters

What’s wrong?. Looking at the visualizations above, we see that the loss is decreasing more or less linearly, which seems to suggest that the learning rate may be too low. Moreover, there is no gap between the training and validation accuracy, suggesting that the model we used has low capacity, and that we should increase its size. On the other hand, with a very large model we would expect to see more overfitting, which would manifest itself as a very large gap between the training and validation accuracy.

Tuning. Tuning the hyperparameters and developing intuition for how they affect the final performance is a large part of using Neural Networks, so we want you to get a lot of practice. Below, you should experiment with different values of the various hyperparameters, including hidden layer size, learning rate, numer of training epochs, and regularization strength. You might also consider tuning the learning rate decay, but you should be able to get good performance using the default value.

Approximate results. You should be aim to achieve a classification accuracy of greater than 48% on the validation set. Our best network gets over 52% on the validation set.

Experiment: You goal in this exercise is to get as good of a result on CIFAR-10 as you can, with a fully-connected Neural Network. For every 1% above 52% on the Test set we will award you with one extra bonus point. Feel free implement your own techniques (e.g. PCA to reduce dimensionality, or adding dropout, or adding features to the solver, etc.).

best_net = None # store the best model into this 

#################################################################################
# TODO: Tune hyperparameters using the validation set. Store your best trained  #
# model in best_net.                                                            #
#                                                                               #
# To help debug your network, it may help to use visualizations similar to the  #
# ones we used above; these visualizations will have significant qualitative    #
# differences from the ones we saw above for the poorly tuned network.          #
#                                                                               #
# Tweaking hyperparameters by hand can be fun, but you might find it useful to  #
# write code to sweep through possible combinations of hyperparameters          #
# automatically like we did on the previous exercises.                          #
#################################################################################
best_acc = -1
input_size = 32 * 32 * 3

best_stats = None

#hidden_size_choice = [x*100+50 for x in xrange(11)]
#reg_choice = [0.1, 0.5, 5, 15, 50, 100, 1000]
#learning_rate_choice = [1e-4, 5e-4, 1e-3, 5e-3, 1e-2, 1e-1, 1]
#batch_size_choice = [8, 40, 80, 160, 500, 1000]


hidden_size_choice = [400]
learning_rate_choice = [3e-3]
reg_choice = [0.02, 0.05, 0.1]
batch_size_choice =[500]
num_iters_choice = [1200]

for batch_size_curr in batch_size_choice:
    for reg_cur in reg_choice:
        for learning_rate_curr in learning_rate_choice:
            for hidden_size_curr in hidden_size_choice:
                for num_iters_curr in num_iters_choice:
                    print 
                    print "current training hidden_size:",hidden_size_curr
                    print "current training learning_rate:",learning_rate_curr
                    print "current training reg:",reg_cur
                    print "current training batch_size:",batch_size_curr
                    net = TwoLayerNet(input_size, hidden_size_curr, num_classes)
                    best_stats = net.train(X_train, y_train, X_val, y_val,
                            num_iters=num_iters_curr, batch_size=batch_size_curr,
                            learning_rate=learning_rate_curr, learning_rate_decay=0.95,
                            reg=reg_cur, verbose=True)
                    val_acc = (net.predict(X_val) == y_val).mean()
                    print "current val_acc:",val_acc
                    if val_acc>best_acc:
                        best_acc = val_acc
                        best_net = net
                        best_stats = stats
                        print 
                        print "best_acc:",best_acc
                        print "best hidden_size:",best_net.params['W1'].shape[1]
                        print "best learning_rate:",best_net.hyper_params['learning_rate']
                        print "best reg:",best_net.hyper_params['reg']
                        print "best batch_size:",best_net.hyper_params['batch_size']
                        print 
#################################################################################
#                               END OF YOUR CODE                                #
#################################################################################

current training hidden_size: 400
current training learning_rate: 0.003
current training reg: 0.02
current training batch_size: 500
iteration 0 / 1200: loss 2.302679
iteration 100 / 1200: loss 1.651489
iteration 200 / 1200: loss 1.500087
iteration 300 / 1200: loss 1.391165
iteration 400 / 1200: loss 1.515288
iteration 500 / 1200: loss 1.409726
iteration 600 / 1200: loss 1.450177
iteration 700 / 1200: loss 1.439996
iteration 800 / 1200: loss 1.286857
iteration 900 / 1200: loss 1.289027
iteration 1000 / 1200: loss 1.310876
iteration 1100 / 1200: loss 1.150956
current val_acc: 0.54

best_acc: 0.54
best hidden_size: 400
best learning_rate: 0.003
best reg: 0.02
best batch_size: 500


current training hidden_size: 400
current training learning_rate: 0.003
current training reg: 0.05
current training batch_size: 500
iteration 0 / 1200: loss 2.302859
iteration 100 / 1200: loss 1.761263
iteration 200 / 1200: loss 1.579761
iteration 300 / 1200: loss 1.472029
iteration 400 / 1200: loss 1.458600
iteration 500 / 1200: loss 1.414810
iteration 600 / 1200: loss 1.425350
iteration 700 / 1200: loss 1.366904
iteration 800 / 1200: loss 1.374242
iteration 900 / 1200: loss 1.415730
iteration 1000 / 1200: loss 1.152137
iteration 1100 / 1200: loss 1.198664
current val_acc: 0.514

current training hidden_size: 400
current training learning_rate: 0.003
current training reg: 0.1
current training batch_size: 500
iteration 0 / 1200: loss 2.303143
iteration 100 / 1200: loss 1.722455
iteration 200 / 1200: loss 1.530982
iteration 300 / 1200: loss 1.543712
iteration 400 / 1200: loss 1.400823
iteration 500 / 1200: loss 1.451125
iteration 600 / 1200: loss 1.402639
iteration 700 / 1200: loss 1.476569
iteration 800 / 1200: loss 1.349223
iteration 900 / 1200: loss 1.191459
iteration 1000 / 1200: loss 1.279797
iteration 1100 / 1200: loss 1.268143
current val_acc: 0.509

#自己加的(insert by myself)
#在上面调好的范围内微调
test_net = TwoLayerNet(input_size, 450, num_classes)
test_stats = test_net.train(X_train, y_train, X_val, y_val,
                       num_iters=1800, batch_size=500,
                       learning_rate=2e-3, learning_rate_decay=0.95,
                       reg=0.02, verbose=True)
test_val_acc = (test_net.predict(X_val) == y_val).mean()
print 
print "test_acc:",test_val_acc
print "test hidden_size:",test_net.hyper_params['hidden_size']
print "test learning_rate:",test_net.hyper_params['learning_rate']
print "test reg:",test_net.hyper_params['reg']
print "test batch_size:",test_net.hyper_params['batch_size']
print "test num_iter:",test_net.hyper_params['num_iter']

iteration 0 / 1800: loss 2.302743
iteration 100 / 1800: loss 1.635457
iteration 200 / 1800: loss 1.517586
iteration 300 / 1800: loss 1.529778
iteration 400 / 1800: loss 1.442434
iteration 500 / 1800: loss 1.374035
iteration 600 / 1800: loss 1.355994
iteration 700 / 1800: loss 1.322699
iteration 800 / 1800: loss 1.254596
iteration 900 / 1800: loss 1.260026
iteration 1000 / 1800: loss 1.164887
iteration 1100 / 1800: loss 1.162341
iteration 1200 / 1800: loss 1.170499
iteration 1300 / 1800: loss 1.165954
iteration 1400 / 1800: loss 1.129984
iteration 1500 / 1800: loss 1.118211
iteration 1600 / 1800: loss 1.088840
iteration 1700 / 1800: loss 1.041198

test_acc: 0.551
test hidden_size: 450
test learning_rate: 0.002
test reg: 0.02
test batch_size: 500
test num_iter: 1800

#自己加的(insert by myself)
# Plot the loss function and train / validation accuracies
plt.subplot(2, 1, 1)
plt.plot(test_stats['loss_history'])
plt.title('Loss history')
plt.xlabel('Iteration')
plt.ylabel('Loss')

plt.subplot(2, 1, 2)
plt.plot(test_stats['train_acc_history'], label='train')
plt.plot(test_stats['val_acc_history'], label='val')
plt.title('Classification accuracy history')
plt.xlabel('Epoch')
plt.ylabel('Clasification accuracy')
plt.show()

# visualize the weights of the best network
show_net_weights(best_net)
print "best hidden_size:",best_net.hyper_params['hidden_size']
print "learning_rate",best_net.hyper_params['learning_rate']
print "reg",best_net.hyper_params['reg']
print "batch_size",best_net.hyper_params['batch_size']

best hidden_size: 400
learning_rate 0.003
reg 0.02
batch_size 500

Run on the test set

When you are done experimenting, you should evaluate your final trained network on the test set; you should get above 48%.

We will give you extra bonus point for every 1% of accuracy above 52%.

final_test_acc = (best_net.predict(X_test) == y_test).mean()
print 'Test accuracy: ', final_test_acc
test_acc = (test_net.predict(X_test) == y_test).mean()
print 'Test accuracy: ', test_acc

Test accuracy:  0.533
Test accuracy:  0.546

neural_net.py内容:

import numpy as np
import matplotlib.pyplot as plt


class TwoLayerNet(object):
  """
  A two-layer fully-connected neural network. The net has an input dimension of
  N, a hidden layer dimension of H, and performs classification over C classes.
  We train the network with a softmax loss function and L2 regularization on the
  weight matrices. The network uses a ReLU nonlinearity after the first fully
  connected layer.

  In other words, the network has the following architecture:

  input - fully connected layer - ReLU - fully connected layer - softmax

  The outputs of the second fully-connected layer are the scores for each class.
  """

  def __init__(self, input_size, hidden_size, output_size, std=1e-4):
    """
    Initialize the model. Weights are initialized to small random values and
    biases are initialized to zero. Weights and biases are stored in the
    variable self.params, which is a dictionary with the following keys:

    W1: First layer weights; has shape (D, H)
    b1: First layer biases; has shape (H,)
    W2: Second layer weights; has shape (H, C)
    b2: Second layer biases; has shape (C,)

    Inputs:
    - input_size: The dimension D of the input data.
    - hidden_size: The number of neurons H in the hidden layer.
    - output_size: The number of classes C.
    """
    self.params = {}
    self.params['W1'] = std * np.random.randn(input_size, hidden_size)
    self.params['b1'] = np.zeros(hidden_size)
    self.params['W2'] = std * np.random.randn(hidden_size, output_size)
    self.params['b2'] = np.zeros(output_size)

  def loss(self, X, y=None, reg=0.0):
    """
    Compute the loss and gradients for a two layer fully connected neural
    network.

    Inputs:
    - X: Input data of shape (N, D). Each X[i] is a training sample.
    - y: Vector of training labels. y[i] is the label for X[i], and each y[i] is
      an integer in the range 0 <= y[i] < C. This parameter is optional; if it
      is not passed then we only return scores, and if it is passed then we
      instead return the loss and gradients.
    - reg: Regularization strength.

    Returns:
    If y is None, return a matrix scores of shape (N, C) where scores[i, c] is
    the score for class c on input X[i].

    If y is not None, instead return a tuple of:
    - loss: Loss (data loss and regularization loss) for this batch of training
      samples.
    - grads: Dictionary mapping parameter names to gradients of those parameters
      with respect to the loss function; has the same keys as self.params.
    """
    # Unpack variables from the params dictionary
    W1, b1 = self.params['W1'], self.params['b1']
    W2, b2 = self.params['W2'], self.params['b2']
    N, D = X.shape

    # Compute the forward pass
    scores = None
    #############################################################################
    # TODO: Perform the forward pass, computing the class scores for the input. #
    # Store the result in the scores variable, which should be an array of      #
    # shape (N, C).                                                             #
    #############################################################################
    z2 = X.dot(W1) + b1
    a2 = np.zeros_like(z2)
    a2 = np.maximum(z2, 0)
    scores = a2.dot(W2) + b2
    #############################################################################
    #                              END OF YOUR CODE                             #
    #############################################################################

    # If the targets are not given then jump out, we're done
    if y is None:
      return scores

    # Compute the loss
    loss = None
    #############################################################################
    # TODO: Finish the forward pass, and compute the loss. This should include  #
    # both the data loss and L2 regularization for W1 and W2. Store the result  #
    # in the variable loss, which should be a scalar. Use the Softmax           #
    # classifier loss. So that your results match ours, multiply the            #
    # regularization loss by 0.5                                                #
    #############################################################################
    exp_scores = np.exp(scores)
    row_sum = exp_scores.sum(axis=1).reshape((N, 1))
    norm_scores = exp_scores / row_sum
    data_loss = -1.0/N * np.log(norm_scores[np.arange(N), y]).sum()
    reg_loss = 0.5 * reg * (np.sum(W1*W1) + np.sum(W2*W2))
    loss = data_loss + reg_loss
    #############################################################################
    #                              END OF YOUR CODE                             #
    #############################################################################

    # Backward pass: compute gradients
    grads = {}
    #############################################################################
    # TODO: Compute the backward pass, computing the derivatives of the weights #
    # and biases. Store the results in the grads dictionary. For example,       #
    # grads['W1'] should store the gradient on W1, and be a matrix of same size #
    #############################################################################
    delta3 = np.zeros_like(norm_scores)    #delta3 = dloss / dz3
    delta3[np.arange(N), y] -= 1
    delta3 += norm_scores
    grads['W2'] = a2.T.dot(delta3) / N + reg * W2
    #grads['b2'] = np.ones((1,N)).dot(delta3) / N
    grads['b2'] = np.ones(N).dot(delta3) / N

    da2_dz2 = np.zeros_like(z2)
    da2_dz2[z2>0] = 1
    delta2 = delta3.dot(W2.T) * da2_dz2
    grads['W1'] = X.T.dot(delta2) / N + reg * W1
    grads['b1'] = np.ones(N).dot(delta2) / N
    #############################################################################
    #                              END OF YOUR CODE                             #
    #############################################################################

    return loss, grads

  def train(self, X, y, X_val, y_val,
            learning_rate=1e-3, learning_rate_decay=0.95,
            reg=1e-5, num_iters=100,
            batch_size=200, verbose=False):
    """
    Train this neural network using stochastic gradient descent.

    Inputs:
    - X: A numpy array of shape (N, D) giving training data.
    - y: A numpy array f shape (N,) giving training labels; y[i] = c means that
      X[i] has label c, where 0 <= c < C.
    - X_val: A numpy array of shape (N_val, D) giving validation data.
    - y_val: A numpy array of shape (N_val,) giving validation labels.
    - learning_rate: Scalar giving learning rate for optimization.
    - learning_rate_decay: Scalar giving factor used to decay the learning rate
      after each epoch.
    - reg: Scalar giving regularization strength.
    - num_iters: Number of steps to take when optimizing.
    - batch_size: Number of training examples to use per step.
    - verbose: boolean; if true print progress during optimization.
    """
    self.hyper_params = {}
    self.hyper_params['learning_rate'] = learning_rate
    self.hyper_params['reg'] = reg
    self.hyper_params['batch_size'] = batch_size
    self.hyper_params['hidden_size'] = self.params['W1'].shape[1]
    self.hyper_params['num_iter'] = num_iters

    num_train = X.shape[0]
    iterations_per_epoch = max(num_train / batch_size, 1)

    # Use SGD to optimize the parameters in self.model
    loss_history = []
    train_acc_history = []
    val_acc_history = []

    for it in xrange(num_iters):
      X_batch = None
      y_batch = None

      #########################################################################
      # TODO: Create a random minibatch of training data and labels, storing  #
      # them in X_batch and y_batch respectively.                             #
      #########################################################################
      batch_inx = np.random.choice(num_train, batch_size)
      X_batch = X[batch_inx,:]
      y_batch = y[batch_inx]
      #########################################################################
      #                             END OF YOUR CODE                          #
      #########################################################################

      # Compute loss and gradients using the current minibatch
      loss, grads = self.loss(X_batch, y=y_batch, reg=reg)
      loss_history.append(loss)

      #########################################################################
      # TODO: Use the gradients in the grads dictionary to update the         #
      # parameters of the network (stored in the dictionary self.params)      #
      # using stochastic gradient descent. You'll need to use the gradients   #
      # stored in the grads dictionary defined above.                         #
      #########################################################################
      self.params['W1'] -= learning_rate * grads['W1']
      self.params['b1'] -= learning_rate * grads['b1']
      self.params['W2'] -= learning_rate * grads['W2']
      self.params['b2'] -= learning_rate * grads['b2']
      #########################################################################
      #                             END OF YOUR CODE                          #
      #########################################################################

      if verbose and it % 100 == 0:
        print 'iteration %d / %d: loss %f' % (it, num_iters, loss)

      # Every epoch, check train and val accuracy and decay learning rate.
      if it % iterations_per_epoch == 0:
        # Check accuracy
        train_acc = (self.predict(X_batch) == y_batch).mean()
        val_acc = (self.predict(X_val) == y_val).mean()
        train_acc_history.append(train_acc)
        val_acc_history.append(val_acc)
        # Decay learning rate
        learning_rate *= learning_rate_decay

    return {
      'loss_history': loss_history,
      'train_acc_history': train_acc_history,
      'val_acc_history': val_acc_history,
    }

  def predict(self, X):
    """
    Use the trained weights of this two-layer network to predict labels for
    data points. For each data point we predict scores for each of the C
    classes, and assign each data point to the class with the highest score.

    Inputs:
    - X: A numpy array of shape (N, D) giving N D-dimensional data points to
      classify.

    Returns:
    - y_pred: A numpy array of shape (N,) giving predicted labels for each of
      the elements of X. For all i, y_pred[i] = c means that X[i] is predicted
      to have class c, where 0 <= c < C.
    """
    y_pred = None

    ###########################################################################
    # TODO: Implement this function; it should be VERY simple!                #
    ###########################################################################
    scores = self.loss(X)
    y_pred = np.argmax(scores, axis=1)
    ###########################################################################
    #                              END OF YOUR CODE                           #
    ###########################################################################

    return y_pred