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.


In [1]:
# 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.


In [2]:
# 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.


In [3]:
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:
3.68027209255e-08

Forward pass: compute loss

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


In [4]:
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:


In [5]:
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]))


b2 max relative error: 4.447677e-11
W2 max relative error: 3.440708e-09
W1 max relative error: 3.669858e-09
b1 max relative error: 2.738422e-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.


In [6]:
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.


In [7]:
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.


In [8]:
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.


In [9]:
# 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()



In [10]:
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.).

Hidden Dim Lr RS Val Acc Num Iter Test Acc
500 1e-4 0.5 0.302 1000
500 2e-4 0.5 0.374 1000
500 3e-4 0.5 0.409 1000
500 4e-4 0.5 0.443 1000
500 5e-4 0.5 0.463 1000
500 6e-4 0.5 0.467 1000
500 1e-3 0.5 0.498 1000
500 1e-4 0.5 0.383 5000
500 2e-4 0.5 0.461 5000
500 3e-4 0.5 0.469 5000
500 4e-4 0.5 0.497 5000
500 5e-4 0.5 0.506 5000
500 6e-4 0.5 0.516 5000
500 1e-3 0.5 0.541 5000
500 2e-3 0.5 0.567 5000
500 3e-3 0.5 0.578 5000 0.568
500 3e-3 0.8 0.569 5000 0.565
500 3e-3 0.5 0.574 10000 0.569

In [11]:
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.                          #
#################################################################################
input_size = 32 * 32 * 3
hidden_sizes = [500]
# learning_rates = [1e-4, 2e-4, 3e-4, 4e-4, 5e-4, 6e-4, 1e-3]
learning_rates = [3e-3]
regularization_strengths = [0.8]
num_classes = 10
best_val_acc = 0.0
best_hidden_size = None
best_learning_rate = None
best_regularization_strength = None
for hidden_size in hidden_sizes:
    for learning_rate in learning_rates:
        for regularization_strength in regularization_strengths:
            net = TwoLayerNet(input_size, hidden_size, num_classes)

            # Train the network
            stats = net.train(X_train, y_train, X_val, y_val,
                        num_iters=5000, batch_size=500,
                        learning_rate=learning_rate, learning_rate_decay=0.95,
                        reg=regularization_strength, verbose=True)

            # Predict on the validation set
            val_acc = (net.predict(X_val) == y_val).mean()
            if best_val_acc < val_acc:
                best_val_acc = val_acc
                best_net = net
                best_hidden_size = hidden_size
                best_learning_rate = learning_rate
                best_regularization_strength = regularization_strength
            print 'Validation accuracy: ', val_acc
#################################################################################
#                               END OF YOUR CODE                                #
#################################################################################


iteration 0 / 5000: loss 2.308811
iteration 100 / 5000: loss 1.748190
iteration 200 / 5000: loss 1.624208
iteration 300 / 5000: loss 1.576230
iteration 400 / 5000: loss 1.590158
iteration 500 / 5000: loss 1.540740
iteration 600 / 5000: loss 1.460016
iteration 700 / 5000: loss 1.512373
iteration 800 / 5000: loss 1.484931
iteration 900 / 5000: loss 1.519888
iteration 1000 / 5000: loss 1.434063
iteration 1100 / 5000: loss 1.447981
iteration 1200 / 5000: loss 1.438646
iteration 1300 / 5000: loss 1.498109
iteration 1400 / 5000: loss 1.430086
iteration 1500 / 5000: loss 1.396886
iteration 1600 / 5000: loss 1.467241
iteration 1700 / 5000: loss 1.379039
iteration 1800 / 5000: loss 1.436430
iteration 1900 / 5000: loss 1.375030
iteration 2000 / 5000: loss 1.417252
iteration 2100 / 5000: loss 1.411219
iteration 2200 / 5000: loss 1.312807
iteration 2300 / 5000: loss 1.367249
iteration 2400 / 5000: loss 1.354348
iteration 2500 / 5000: loss 1.287970
iteration 2600 / 5000: loss 1.369612
iteration 2700 / 5000: loss 1.285167
iteration 2800 / 5000: loss 1.400310
iteration 2900 / 5000: loss 1.340548
iteration 3000 / 5000: loss 1.247520
iteration 3100 / 5000: loss 1.307151
iteration 3200 / 5000: loss 1.364933
iteration 3300 / 5000: loss 1.319338
iteration 3400 / 5000: loss 1.344405
iteration 3500 / 5000: loss 1.372994
iteration 3600 / 5000: loss 1.319584
iteration 3700 / 5000: loss 1.258468
iteration 3800 / 5000: loss 1.288981
iteration 3900 / 5000: loss 1.243964
iteration 4000 / 5000: loss 1.339315
iteration 4100 / 5000: loss 1.301341
iteration 4200 / 5000: loss 1.282817
iteration 4300 / 5000: loss 1.223743
iteration 4400 / 5000: loss 1.337186
iteration 4500 / 5000: loss 1.297403
iteration 4600 / 5000: loss 1.208556
iteration 4700 / 5000: loss 1.311911
iteration 4800 / 5000: loss 1.335119
iteration 4900 / 5000: loss 1.226947
Validation accuracy:  0.55

In [12]:
# visualize the weights of the best network
show_net_weights(best_net)


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%.


In [13]:
test_acc = (best_net.predict(X_test) == y_test).mean()
print 'Test accuracy: ', test_acc


Test accuracy:  0.571