# Multiclass Support Vector Machine exercise¶

Complete and hand in this completed worksheet (including its outputs and any supporting code outside of the worksheet) with your assignment submission. For more details see the assignments page on the course website.

In this exercise you will:

• implement a fully-vectorized loss function for the SVM
• implement the fully-vectorized expression for its analytic gradient
• check your implementation using numerical gradient
• use a validation set to tune the learning rate and regularization strength
• optimize the loss function with SGD
• visualize the final learned weights
In [3]:
# Run some setup code for this notebook.

import random
import numpy as np
from cs231n.data_utils import load_CIFAR10
import matplotlib.pyplot as plt

# This is a bit of magic to make matplotlib figures appear inline in the
# notebook rather than in a new window.
%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'

# Some more magic so that the notebook will reload external python modules;
# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
%load_ext autoreload
%autoreload 2

The autoreload extension is already loaded. To reload it, use:
%reload_ext autoreload


## CIFAR-10 Data Loading and Preprocessing¶

In [4]:
# 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)

# As a sanity check, we print out the size of the training and test data.
print 'Training data shape: ', X_train.shape
print 'Training labels shape: ', y_train.shape
print 'Test data shape: ', X_test.shape
print 'Test labels shape: ', y_test.shape

Training data shape:  (50000, 32, 32, 3)
Training labels shape:  (50000,)
Test data shape:  (10000, 32, 32, 3)
Test labels shape:  (10000,)

In [5]:
# Visualize some examples from the dataset.
# We show a few examples of training images from each class.
classes = ['plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck']
num_classes = len(classes)
samples_per_class = 7
for y, cls in enumerate(classes):
idxs = np.flatnonzero(y_train == y)
idxs = np.random.choice(idxs, samples_per_class, replace=False)
for i, idx in enumerate(idxs):
plt_idx = i * num_classes + y + 1
plt.subplot(samples_per_class, num_classes, plt_idx)
plt.imshow(X_train[idx].astype('uint8'))
plt.axis('off')
if i == 0:
plt.title(cls)
plt.show()

In [6]:
# Split the data into train, val, and test sets. In addition we will
# create a small development set as a subset of the training data;
# we can use this for development so our code runs faster.
num_training = 49000
num_validation = 1000
num_test = 1000
num_dev = 500

# Our validation set will be num_validation points from the original
# training set.
mask = range(num_training, num_training + num_validation)
X_val = X_train[mask]
y_val = y_train[mask]

# Our training set will be the first num_train points from the original
# training set.
mask = range(num_training)
X_train = X_train[mask]
y_train = y_train[mask]

# We will also make a development set, which is a small subset of
# the training set.
mask = np.random.choice(num_training, num_dev, replace=False)
X_dev = X_train[mask]
y_dev = y_train[mask]

# We use the first num_test points of the original test set as our
# test set.
mask = range(num_test)
X_test = X_test[mask]
y_test = y_test[mask]

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, 32, 32, 3)
Train labels shape:  (49000,)
Validation data shape:  (1000, 32, 32, 3)
Validation labels shape:  (1000,)
Test data shape:  (1000, 32, 32, 3)
Test labels shape:  (1000,)

In [7]:
# Preprocessing: reshape the image data into rows
X_train = np.reshape(X_train, (X_train.shape[0], -1))
X_val = np.reshape(X_val, (X_val.shape[0], -1))
X_test = np.reshape(X_test, (X_test.shape[0], -1))
X_dev = np.reshape(X_dev, (X_dev.shape[0], -1))

# As a sanity check, print out the shapes of the data
print 'Training data shape: ', X_train.shape
print 'Validation data shape: ', X_val.shape
print 'Test data shape: ', X_test.shape
print 'dev data shape: ', X_dev.shape

Training data shape:  (49000, 3072)
Validation data shape:  (1000, 3072)
Test data shape:  (1000, 3072)
dev data shape:  (500, 3072)

In [8]:
# Preprocessing: subtract the mean image
# first: compute the image mean based on the training data
mean_image = np.mean(X_train, axis=0)
print mean_image[:10] # print a few of the elements
plt.figure(figsize=(4,4))
plt.imshow(mean_image.reshape((32,32,3)).astype('uint8')) # visualize the mean image
plt.show()

[ 130.64189796  135.98173469  132.47391837  130.05569388  135.34804082
131.75402041  130.96055102  136.14328571  132.47636735  131.48467347]

In [9]:
# second: subtract the mean image from train and test data
X_train -= mean_image
X_val -= mean_image
X_test -= mean_image
X_dev -= mean_image

In [10]:
# third: append the bias dimension of ones (i.e. bias trick) so that our SVM
# only has to worry about optimizing a single weight matrix W.
X_train = np.hstack([X_train, np.ones((X_train.shape[0], 1))])
X_val = np.hstack([X_val, np.ones((X_val.shape[0], 1))])
X_test = np.hstack([X_test, np.ones((X_test.shape[0], 1))])
X_dev = np.hstack([X_dev, np.ones((X_dev.shape[0], 1))])

print X_train.shape, X_val.shape, X_test.shape, X_dev.shape

(49000, 3073) (1000, 3073) (1000, 3073) (500, 3073)


## SVM Classifier¶

Your code for this section will all be written inside cs231n/classifiers/linear_svm.py.

As you can see, we have prefilled the function compute_loss_naive which uses for loops to evaluate the multiclass SVM loss function.

In [11]:
# Evaluate the naive implementation of the loss we provided for you:
from cs231n.classifiers.linear_svm import svm_loss_naive
import time

# generate a random SVM weight matrix of small numbers
W = np.random.randn(3073, 10) * 0.0001

loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.00001)
print 'loss: %f' % (loss, )

loss: 9.091887


The grad returned from the function above is right now all zero. Derive and implement the gradient for the SVM cost function and implement it inline inside the function svm_loss_naive. You will find it helpful to interleave your new code inside the existing function.

To check that you have correctly implemented the gradient correctly, you can numerically estimate the gradient of the loss function and compare the numeric estimate to the gradient that you computed. We have provided code that does this for you:

In [12]:
# Once you've implemented the gradient, recompute it with the code below
# and gradient check it with the function we provided for you

# Compute the loss and its gradient at W.
loss, grad = svm_loss_naive(W, X_dev, y_dev, 0.0)

# Numerically compute the gradient along several randomly chosen dimensions, and
# compare them with your analytically computed gradient. The numbers should match
# almost exactly along all dimensions.
from cs231n.gradient_check import grad_check_sparse
f = lambda w: svm_loss_naive(w, X_dev, y_dev, 0.0)[0]
grad_numerical = grad_check_sparse(f, W, grad)

# do the gradient check once again with regularization turned on
# you didn't forget the regularization gradient did you?
loss, grad = svm_loss_naive(W, X_dev, y_dev, 1e2)
f = lambda w: svm_loss_naive(w, X_dev, y_dev, 1e2)[0]
grad_numerical = grad_check_sparse(f, W, grad)

numerical: -12.507484 analytic: -12.552117, relative error: 1.781078e-03
numerical: 15.528538 analytic: 15.517544, relative error: 3.541001e-04
numerical: 1.229498 analytic: 1.151596, relative error: 3.271679e-02
numerical: -11.124613 analytic: -10.993818, relative error: 5.913415e-03
numerical: -2.115227 analytic: -2.115227, relative error: 3.942145e-11
numerical: -2.843209 analytic: -2.843209, relative error: 7.242355e-12
numerical: -10.525466 analytic: -10.525466, relative error: 4.763067e-11
numerical: 16.099935 analytic: 16.099935, relative error: 1.005312e-11
numerical: 4.847796 analytic: 4.847796, relative error: 5.423503e-11
numerical: -20.459558 analytic: -20.540553, relative error: 1.975482e-03
numerical: -2.737533 analytic: -2.737533, relative error: 1.368054e-11
numerical: -3.517704 analytic: -3.578824, relative error: 8.612594e-03
numerical: 11.933287 analytic: 11.933287, relative error: 3.901512e-11
numerical: -25.301047 analytic: -25.301047, relative error: 6.251885e-12
numerical: 11.532505 analytic: 11.532505, relative error: 3.159781e-12
numerical: -6.066317 analytic: -6.066317, relative error: 1.438522e-11
numerical: 23.579876 analytic: 23.579876, relative error: 8.338224e-12
numerical: 0.293745 analytic: 0.293745, relative error: 1.520530e-09
numerical: 29.850090 analytic: 29.850090, relative error: 3.450347e-13
numerical: 4.410050 analytic: 4.324150, relative error: 9.834895e-03


### Inline Question 1:¶

It is possible that once in a while a dimension in the gradcheck will not match exactly. What could such a discrepancy be caused by? Is it a reason for concern? What is a simple example in one dimension where a gradient check could fail? Hint: the SVM loss function is not strictly speaking differentiable

Your Answer: fill this in.

In [13]:
# Next implement the function svm_loss_vectorized; for now only compute the loss;
# we will implement the gradient in a moment.
tic = time.time()
loss_naive, grad_naive = svm_loss_naive(W, X_dev, y_dev, 0.00001)
toc = time.time()
print 'Naive loss: %e computed in %fs' % (loss_naive, toc - tic)

from cs231n.classifiers.linear_svm import svm_loss_vectorized
tic = time.time()
loss_vectorized, _ = svm_loss_vectorized(W, X_dev, y_dev, 0.00001)
toc = time.time()
print 'Vectorized loss: %e computed in %fs' % (loss_vectorized, toc - tic)

# The losses should match but your vectorized implementation should be much faster.
print 'difference: %f' % (loss_naive - loss_vectorized)

Naive loss: 9.091887e+00 computed in 0.119111s
Vectorized loss: 9.091887e+00 computed in 0.003597s
difference: -0.000000

In [14]:
# Complete the implementation of svm_loss_vectorized, and compute the gradient
# of the loss function in a vectorized way.

# The naive implementation and the vectorized implementation should match, but
# the vectorized version should still be much faster.
tic = time.time()
_, grad_naive = svm_loss_naive(W, X_dev, y_dev, 0.00001)
toc = time.time()
print 'Naive loss and gradient: computed in %fs' % (toc - tic)

tic = time.time()
_, grad_vectorized = svm_loss_vectorized(W, X_dev, y_dev, 0.00001)
toc = time.time()
print 'Vectorized loss and gradient: computed in %fs' % (toc - tic)

# The loss is a single number, so it is easy to compare the values computed
# by the two implementations. The gradient on the other hand is a matrix, so
# we use the Frobenius norm to compare them.
difference = np.linalg.norm(grad_naive - grad_vectorized, ord='fro')
print 'difference: %f' % difference

Naive loss and gradient: computed in 0.125091s
Vectorized loss and gradient: computed in 0.004289s
difference: 0.000000


### Stochastic Gradient Descent¶

We now have vectorized and efficient expressions for the loss, the gradient and our gradient matches the numerical gradient. We are therefore ready to do SGD to minimize the loss.

In [15]:
# In the file linear_classifier.py, implement SGD in the function
# LinearClassifier.train() and then run it with the code below.
from cs231n.classifiers import LinearSVM
svm = LinearSVM()
tic = time.time()
loss_hist = svm.train(X_train, y_train, learning_rate=1e-7, reg=5e4,
num_iters=1500, verbose=True)
toc = time.time()
print 'That took %fs' % (toc - tic)

iteration 0 / 1500: loss 791.117973
iteration 100 / 1500: loss 287.402696
iteration 200 / 1500: loss 108.607363
iteration 300 / 1500: loss 42.511326
iteration 400 / 1500: loss 19.098572
iteration 500 / 1500: loss 10.066096
iteration 600 / 1500: loss 7.267049
iteration 700 / 1500: loss 6.057420
iteration 800 / 1500: loss 5.770418
iteration 900 / 1500: loss 5.312655
iteration 1000 / 1500: loss 5.188154
iteration 1100 / 1500: loss 5.635616
iteration 1200 / 1500: loss 5.315875
iteration 1300 / 1500: loss 5.108222
iteration 1400 / 1500: loss 6.043795
That took 4.874408s

In [16]:
# A useful debugging strategy is to plot the loss as a function of
# iteration number:
plt.plot(loss_hist)
plt.xlabel('Iteration number')
plt.ylabel('Loss value')
plt.show()

In [17]:
# Write the LinearSVM.predict function and evaluate the performance on both the
# training and validation set
y_train_pred = svm.predict(X_train)
print 'training accuracy: %f' % (np.mean(y_train == y_train_pred), )
y_val_pred = svm.predict(X_val)
print 'validation accuracy: %f' % (np.mean(y_val == y_val_pred), )

training accuracy: 0.370429
validation accuracy: 0.373000

In [18]:
# Use the validation set to tune hyperparameters (regularization strength and
# learning rate). You should experiment with different ranges for the learning
# rates and regularization strengths; if you are careful you should be able to
# get a classification accuracy of about 0.4 on the validation set.
learning_rates = [1e-7, 5e-5]
regularization_strengths = [5e4, 1e5]
# learning_rates = [1e-7, 2e-7, 3e-7, 5e-5, 8e-7]
# regularization_strengths = [1e4, 2e4, 3e4, 4e4, 5e4, 6e4, 7e4, 8e4, 1e5]

# results is dictionary mapping tuples of the form
# (learning_rate, regularization_strength) to tuples of the form
# (training_accuracy, validation_accuracy). The accuracy is simply the fraction
# of data points that are correctly classified.
results = {}
best_val = -1   # The highest validation accuracy that we have seen so far.
best_svm = None # The LinearSVM object that achieved the highest validation rate.

################################################################################
# TODO:                                                                        #
# Write code that chooses the best hyperparameters by tuning on the validation #
# set. For each combination of hyperparameters, train a linear SVM on the      #
# training set, compute its accuracy on the training and validation sets, and  #
# store these numbers in the results dictionary. In addition, store the best   #
# validation accuracy in best_val and the LinearSVM object that achieves this  #
# accuracy in best_svm.                                                        #
#                                                                              #
# Hint: You should use a small value for num_iters as you develop your         #
# validation code so that the SVMs don't take much time to train; once you are #
# confident that your validation code works, you should rerun the validation   #
# code with a larger value for num_iters.                                      #
################################################################################
for rate in learning_rates:
for strength in regularization_strengths:
svm = LinearSVM()
svm.train(X_train, y_train, learning_rate=rate, reg=strength,
num_iters=1500, verbose=False)
learning_accuracy = np.mean(svm.predict(X_train) == y_train)
validation_accuracy = np.mean(svm.predict(X_val) == y_val)
if validation_accuracy > best_val:
best_val = validation_accuracy
best_svm = svm
results[(rate, strength)] = (learning_accuracy, validation_accuracy)
################################################################################
#                              END OF YOUR CODE                                #
################################################################################

# Print out results.
for lr, reg in sorted(results):
train_accuracy, val_accuracy = results[(lr, reg)]
print 'lr %e reg %e train accuracy: %f val accuracy: %f' % (
lr, reg, train_accuracy, val_accuracy)

print 'best validation accuracy achieved during cross-validation: %f' % best_val

cs231n/classifiers/linear_svm.py:88: RuntimeWarning: overflow encountered in double_scalars
loss = np.sum(margins) / num_train + 0.5 * reg * np.sum(W * W)
cs231n/classifiers/linear_svm.py:88: RuntimeWarning: overflow encountered in multiply
loss = np.sum(margins) / num_train + 0.5 * reg * np.sum(W * W)
cs231n/classifiers/linear_svm.py:109: RuntimeWarning: overflow encountered in multiply
dW += X.T.dot(margins) / num_train + reg * W
cs231n/classifiers/linear_svm.py:103: RuntimeWarning: invalid value encountered in greater
margins[margins > 0] = 1.0

lr 1.000000e-07 reg 5.000000e+04 train accuracy: 0.372245 val accuracy: 0.372000
lr 1.000000e-07 reg 1.000000e+05 train accuracy: 0.359306 val accuracy: 0.371000
lr 5.000000e-05 reg 5.000000e+04 train accuracy: 0.067245 val accuracy: 0.065000
lr 5.000000e-05 reg 1.000000e+05 train accuracy: 0.100265 val accuracy: 0.087000
best validation accuracy achieved during cross-validation: 0.372000

In [117]:
# Visualize the cross-validation results
import math
x_scatter = [math.log10(x[0]) for x in results]
y_scatter = [math.log10(x[1]) for x in results]

# plot training accuracy
marker_size = 100
colors = [results[x][0] for x in results]
plt.subplot(2, 1, 1)
plt.scatter(x_scatter, y_scatter, marker_size, c=colors)
plt.colorbar()
plt.xlabel('log learning rate')
plt.ylabel('log regularization strength')
plt.title('CIFAR-10 training accuracy')

# plot validation accuracy
colors = [results[x][1] for x in results] # default size of markers is 20
plt.subplot(2, 1, 2)
plt.scatter(x_scatter, y_scatter, marker_size, c=colors)
plt.colorbar()
plt.xlabel('log learning rate')
plt.ylabel('log regularization strength')
plt.title('CIFAR-10 validation accuracy')
plt.show()

In [118]:
# Evaluate the best svm on test set
y_test_pred = best_svm.predict(X_test)
test_accuracy = np.mean(y_test == y_test_pred)
print 'linear SVM on raw pixels final test set accuracy: %f' % test_accuracy

linear SVM on raw pixels final test set accuracy: 0.381000

In [119]:
# Visualize the learned weights for each class.
# Depending on your choice of learning rate and regularization strength, these may
# or may not be nice to look at.
w = best_svm.W[:-1,:] # strip out the bias
w = w.reshape(32, 32, 3, 10)
w_min, w_max = np.min(w), np.max(w)
classes = ['plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck']
for i in xrange(10):
plt.subplot(2, 5, i + 1)

# Rescale the weights to be between 0 and 255
wimg = 255.0 * (w[:, :, :, i].squeeze() - w_min) / (w_max - w_min)
plt.imshow(wimg.astype('uint8'))
plt.axis('off')
plt.title(classes[i])


### Inline question 2:¶

Describe what your visualized SVM weights look like, and offer a brief explanation for why they look they way that they do.

Your answer: fill this in