svm


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 [1]:
# Run some setup code for this notebook.

import random
import numpy as np
from skynet.utils.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

CIFAR-10 Data Loading and Preprocessing


In [2]:
# Load the raw CIFAR-10 data.
cifar10_dir = '../skynet/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 [3]:
# 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 [4]:
# 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 = list(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 = list(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 = list(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 [5]:
# 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 [6]:
# 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 [7]:
# 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 [8]:
# 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 linear/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 [9]:
# Evaluate the naive implementation of the loss we provided for you:
from skynet.linear.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: 9.375891

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 [10]:
# 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 skynet.utils.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: 36.220637 analytic: 36.220637, relative error: 2.415254e-12
numerical: 12.393411 analytic: 12.393411, relative error: 3.009594e-11
numerical: 8.305584 analytic: 8.305584, relative error: 9.826594e-12
numerical: -13.496080 analytic: -13.496080, relative error: 4.277157e-11
numerical: -13.361409 analytic: -13.361409, relative error: 1.917824e-12
numerical: 22.509639 analytic: 22.509639, relative error: 4.390694e-12
numerical: 2.626211 analytic: 2.626211, relative error: 7.762939e-11
numerical: 15.661103 analytic: 15.661103, relative error: 1.162615e-11
numerical: 0.104081 analytic: 0.104081, relative error: 6.080488e-11
numerical: 41.829621 analytic: 41.829621, relative error: 1.110371e-11
numerical: -11.474561 analytic: -11.474561, relative error: 5.971728e-12
numerical: -21.493125 analytic: -21.493125, relative error: 1.543082e-11
numerical: -24.824344 analytic: -24.824344, relative error: 1.094673e-11
numerical: 30.581471 analytic: 30.581471, relative error: 8.052818e-12
numerical: 30.914171 analytic: 30.914171, relative error: 9.653320e-12
numerical: 1.876637 analytic: 1.876637, relative error: 4.272100e-10
numerical: 18.616917 analytic: 18.616917, relative error: 1.502463e-11
numerical: -6.888913 analytic: -6.888913, relative error: 2.255717e-11
numerical: -4.290447 analytic: -4.290447, relative error: 9.999123e-11
numerical: -42.901126 analytic: -42.901126, relative error: 2.482777e-12

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. The SVM loss function is not strictly speaking differentiable. The point is at the very hinge of the loss function. for [;f(x) = max(-x,0);] at x=0, there is no real gradient. For example we grad check at x=0.001, your computation will return 0 and at x=-0.001, your computation will return -1. However, the numerical computation does a finite approximation. They do not have gradients 0 or 1


In [11]:
# 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 skynet.linear.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: 9.375891e+00 computed in 0.140923s
Vectorized loss: 9.375891e+00 computed in 0.005879s
difference: 0.000000

In [12]:
# 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 0.167694s
Vectorized loss and gradient: computed in 0.004317s
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 [13]:
# In the file linear_classifier.py, implement SGD in the function
# LinearClassifier.train() and then run it with the code below.
from skynet.linear 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 790.821987
iteration 100 / 1500: loss 288.418142
iteration 200 / 1500: loss 108.657054
iteration 300 / 1500: loss 43.006753
iteration 400 / 1500: loss 19.097151
iteration 500 / 1500: loss 10.353906
iteration 600 / 1500: loss 7.145315
iteration 700 / 1500: loss 5.980948
iteration 800 / 1500: loss 5.501130
iteration 900 / 1500: loss 5.003601
iteration 1000 / 1500: loss 5.220317
iteration 1100 / 1500: loss 5.769452
iteration 1200 / 1500: loss 6.181881
iteration 1300 / 1500: loss 5.726748
iteration 1400 / 1500: loss 5.072970
That took 7.388011s

In [14]:
# 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 [15]:
# 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: 0.369735
validation accuracy: 0.380000

In [16]:
# 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 = [5e-8, 1e-7, 3e-7, 5e-7, 7e-7]
regularization_strengths = [1e4, 3e4, 5e4, 7e4, 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 learning_rate in learning_rates:
    for regularization_strength in regularization_strengths:
        svm = LinearSVM()
        loss_hist = svm.train(X_train, y_train, learning_rate, reg=regularization_strength,
                             num_iters=1500, verbose=False)
        y_train_pred = svm.predict(X_train)
        y_val_pred = svm.predict(X_val)
        training_accuracy = np.mean(y_train == y_train_pred)
        validation_accuracy = np.mean(y_val == y_val_pred)
        results[(learning_rate, regularization_strength)] = (training_accuracy, 
                                                             validation_accuracy)
        if best_val < validation_accuracy:
            best_val = validation_accuracy
            best_svm = svm
pass
################################################################################
#                              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)


lr 5.000000e-08 reg 1.000000e+04 train accuracy: 0.320224 val accuracy: 0.326000
lr 5.000000e-08 reg 3.000000e+04 train accuracy: 0.372592 val accuracy: 0.382000
lr 5.000000e-08 reg 5.000000e+04 train accuracy: 0.376306 val accuracy: 0.380000
lr 5.000000e-08 reg 7.000000e+04 train accuracy: 0.368653 val accuracy: 0.386000
lr 5.000000e-08 reg 1.000000e+05 train accuracy: 0.352388 val accuracy: 0.377000
lr 1.000000e-07 reg 1.000000e+04 train accuracy: 0.374082 val accuracy: 0.372000
lr 1.000000e-07 reg 3.000000e+04 train accuracy: 0.378714 val accuracy: 0.387000
lr 1.000000e-07 reg 5.000000e+04 train accuracy: 0.358776 val accuracy: 0.376000
lr 1.000000e-07 reg 7.000000e+04 train accuracy: 0.363000 val accuracy: 0.385000
lr 1.000000e-07 reg 1.000000e+05 train accuracy: 0.350020 val accuracy: 0.357000
lr 3.000000e-07 reg 1.000000e+04 train accuracy: 0.378837 val accuracy: 0.374000
lr 3.000000e-07 reg 3.000000e+04 train accuracy: 0.362939 val accuracy: 0.365000
lr 3.000000e-07 reg 5.000000e+04 train accuracy: 0.348122 val accuracy: 0.351000
lr 3.000000e-07 reg 7.000000e+04 train accuracy: 0.353592 val accuracy: 0.361000
lr 3.000000e-07 reg 1.000000e+05 train accuracy: 0.335714 val accuracy: 0.345000
lr 5.000000e-07 reg 1.000000e+04 train accuracy: 0.362796 val accuracy: 0.363000
lr 5.000000e-07 reg 3.000000e+04 train accuracy: 0.345714 val accuracy: 0.353000
lr 5.000000e-07 reg 5.000000e+04 train accuracy: 0.329306 val accuracy: 0.350000
lr 5.000000e-07 reg 7.000000e+04 train accuracy: 0.308286 val accuracy: 0.329000
lr 5.000000e-07 reg 1.000000e+05 train accuracy: 0.331286 val accuracy: 0.351000
lr 7.000000e-07 reg 1.000000e+04 train accuracy: 0.346633 val accuracy: 0.334000
lr 7.000000e-07 reg 3.000000e+04 train accuracy: 0.316735 val accuracy: 0.308000
lr 7.000000e-07 reg 5.000000e+04 train accuracy: 0.308633 val accuracy: 0.322000
lr 7.000000e-07 reg 7.000000e+04 train accuracy: 0.335571 val accuracy: 0.333000
lr 7.000000e-07 reg 1.000000e+05 train accuracy: 0.313776 val accuracy: 0.324000
best validation accuracy achieved during cross-validation: 0.387000

In [17]:
plt.figure(figsize=(10, 20))

# 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(4, 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(4, 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')

# plot training accuracy
sz = [results[x][0]*1000 for x in results] # default size of markers is 20
plt.subplot(4,2,1)
plt.scatter(x_scatter, y_scatter, sz)
plt.xlabel('log learning rate')
plt.ylabel('log regularization strength')
plt.title('CIFAR-10 training accuracy')

# plot validation accuracy
sz = [results[x][1]*1000 for x in results] # default size of markers is 20
plt.subplot(4,2,2)
plt.scatter(x_scatter, y_scatter, sz)
plt.xlabel('log learning rate')
plt.ylabel('log regularization strength')
plt.title('CIFAR-10 validation accuracy')

plt.show()



In [18]:
# 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: 0.369000

In [19]:
# 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 range(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

Then look like crude low-resolution templates of the object classes they describe along with the backgrounds they typically occur with. Due to sampling, the templates are very noisy.

And the horse weights look like two heads in both diretions. Checking the training set, we can find that there are indeed some horse images in which head is to the right. This may show the weakness of this model: it doesn't generalize well because it depends on raw pixel layout and colors.