cs231n Assignment#1 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

# 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

CIFAR-10 Data Loading and Preprocessing

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

# 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()

?np.random.choice

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

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

# 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]

# 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

# third: append (at the last) 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

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

​

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.

# 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: 8.831645

​

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:

# 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: -13.865929 analytic: -13.865929, relative error: 1.283977e-12
numerical: 7.842142 analytic: 7.735021, relative error: 6.876784e-03
numerical: 3.464393 analytic: 3.464393, relative error: 9.040092e-11
numerical: -23.034911 analytic: -23.034911, relative error: 6.876266e-12
numerical: -0.185311 analytic: -0.185311, relative error: 2.538774e-10
numerical: 25.825504 analytic: 25.825504, relative error: 1.336035e-11
numerical: 4.457836 analytic: 4.457836, relative error: 1.015819e-10
numerical: 3.184691 analytic: 3.184691, relative error: 8.849109e-11
numerical: 10.428446 analytic: 10.374317, relative error: 2.601982e-03
numerical: 12.479957 analytic: 12.479957, relative error: 6.825191e-12
numerical: 12.237949 analytic: 12.326308, relative error: 3.597051e-03
numerical: 4.377103 analytic: 4.377103, relative error: 3.904758e-11
numerical: -1.951930 analytic: -1.951930, relative error: 1.432276e-10
numerical: 33.752503 analytic: 33.752503, relative error: 4.254520e-12
numerical: 11.367149 analytic: 11.367149, relative error: 1.682727e-11
numerical: 16.461879 analytic: 16.461879, relative error: 4.766805e-12
numerical: 3.814562 analytic: 3.814562, relative error: 1.087469e-10
numerical: 13.931226 analytic: 13.931226, relative error: 9.578349e-12
numerical: -27.291095 analytic: -27.395406, relative error: 1.907445e-03
numerical: -7.610407 analytic: -7.610407, relative error: 1.015282e-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: Maybe the SVM loss function is not differentiable on that dimension

?np.max

np.sum(np.maximum(0, X_dev.dot(W) - X_dev.dot(W)[np.arange(len(y_dev)), [y_dev]].T + 1))

4915.822409730994

# 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: 8.831645e+00 computed in 0.071000s
Vectorized loss: 8.831645e+00 computed in 0.000000s
difference: 0.000000

# 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 0.084000s
Vectorized loss and gradient: computed in 0.005000s
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 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 791.772037
iteration 100 / 1500: loss 286.021346
iteration 200 / 1500: loss 107.673095
iteration 300 / 1500: loss 41.812791
iteration 400 / 1500: loss 18.665578
iteration 500 / 1500: loss 10.614984
iteration 600 / 1500: loss 6.664814
iteration 700 / 1500: loss 6.509693
iteration 800 / 1500: loss 5.792204
iteration 900 / 1500: loss 4.986855
iteration 1000 / 1500: loss 5.914691
iteration 1100 / 1500: loss 5.058078
iteration 1200 / 1500: loss 5.491475
iteration 1300 / 1500: loss 5.609450
iteration 1400 / 1500: loss 5.376595
That took 5.454000s

# 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()

# 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: 0.364980
validation accuracy: 0.378000

# 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-8, 1e-7, 2e-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 learning_rate in learning_rates:
    for regularization_strength in regularization_strengths:
        svm = LinearSVM()
        loss_hist = svm.train(
            X_train, y_train, learning_rate, \
            regularization_strength, num_iters=1500, batch_size=200)
        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 validation_accuracy > best_val:
            best_val = validation_accuracy
            best_svm = svm
################################################################################
#                              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 1.000000e-08 reg 1.000000e+04 train accuracy: 0.221898 val accuracy: 0.247000
lr 1.000000e-08 reg 2.000000e+04 train accuracy: 0.233653 val accuracy: 0.258000
lr 1.000000e-08 reg 3.000000e+04 train accuracy: 0.234694 val accuracy: 0.225000
lr 1.000000e-08 reg 4.000000e+04 train accuracy: 0.255959 val accuracy: 0.249000
lr 1.000000e-08 reg 5.000000e+04 train accuracy: 0.259755 val accuracy: 0.273000
lr 1.000000e-08 reg 6.000000e+04 train accuracy: 0.267408 val accuracy: 0.269000
lr 1.000000e-08 reg 7.000000e+04 train accuracy: 0.269102 val accuracy: 0.287000
lr 1.000000e-08 reg 8.000000e+04 train accuracy: 0.277102 val accuracy: 0.285000
lr 1.000000e-08 reg 1.000000e+05 train accuracy: 0.295306 val accuracy: 0.301000
lr 1.000000e-07 reg 1.000000e+04 train accuracy: 0.369388 val accuracy: 0.374000
lr 1.000000e-07 reg 2.000000e+04 train accuracy: 0.380265 val accuracy: 0.390000
lr 1.000000e-07 reg 3.000000e+04 train accuracy: 0.375490 val accuracy: 0.378000
lr 1.000000e-07 reg 4.000000e+04 train accuracy: 0.375633 val accuracy: 0.385000
lr 1.000000e-07 reg 5.000000e+04 train accuracy: 0.369694 val accuracy: 0.375000
lr 1.000000e-07 reg 6.000000e+04 train accuracy: 0.372469 val accuracy: 0.383000
lr 1.000000e-07 reg 7.000000e+04 train accuracy: 0.356000 val accuracy: 0.370000
lr 1.000000e-07 reg 8.000000e+04 train accuracy: 0.352816 val accuracy: 0.355000
lr 1.000000e-07 reg 1.000000e+05 train accuracy: 0.356796 val accuracy: 0.377000
lr 2.000000e-07 reg 1.000000e+04 train accuracy: 0.393510 val accuracy: 0.395000
lr 2.000000e-07 reg 2.000000e+04 train accuracy: 0.377020 val accuracy: 0.382000
lr 2.000000e-07 reg 3.000000e+04 train accuracy: 0.363857 val accuracy: 0.373000
lr 2.000000e-07 reg 4.000000e+04 train accuracy: 0.368714 val accuracy: 0.372000
lr 2.000000e-07 reg 5.000000e+04 train accuracy: 0.361531 val accuracy: 0.364000
lr 2.000000e-07 reg 6.000000e+04 train accuracy: 0.354714 val accuracy: 0.368000
lr 2.000000e-07 reg 7.000000e+04 train accuracy: 0.348306 val accuracy: 0.365000
lr 2.000000e-07 reg 8.000000e+04 train accuracy: 0.358082 val accuracy: 0.378000
lr 2.000000e-07 reg 1.000000e+05 train accuracy: 0.347898 val accuracy: 0.358000
best validation accuracy achieved during cross-validation: 0.395000

# 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(3, 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(3, 1, 3)
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()

# 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: 0.383000

​

x = np.array([[[0], [1], [2]]])
np.squeeze(x)

array([0, 1, 2])

# 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

Codes

linear_svm.py

import numpy as np
from random import shuffle


def svm_loss_naive(W, X, y, reg):
    """
    Structured SVM loss function, naive implementation (with loops).

    Inputs have dimension D, there are C classes, and we operate on minibatches
    of N examples.

    Inputs:
    - W: A numpy array of shape (D, C) containing weights.
    - X: A numpy array of shape (N, D) containing a minibatch of data.
    - y: A numpy array of shape (N,) containing training labels; y[i] = c means
      that X[i] has label c, where 0 <= c < C.
    - reg: (float) regularization strength

    Returns a tuple of:
    - loss as single float
    - gradient with respect to weights W; an array of same shape as W
    """
    dW = np.zeros(W.shape)  # initialize the gradient as zero

    # compute the loss and the gradient
    num_classes = W.shape[1]
    num_train = X.shape[0]
    loss = 0.0

    for i in xrange(num_train):
        scores = X[i].dot(W)
        correct_class_score = scores[y[i]]

        for j in xrange(num_classes):
            if j == y[i]:
                continue
            margin = scores[j] - correct_class_score + 1  # note delta = 1
            if margin > 0:
                dW[:, y[i]] -= X[i, :]
                dW[:, j] += X[i, :]
                loss += margin

    # Right now the loss is a sum over all training examples, but we want it
    # to be an average instead so we divide by num_train.
    loss /= num_train

    dW /= num_train

    # Add regularization to the loss.
    loss += 0.5 * reg * np.sum(W * W)
    dW += reg * W

    #############################################################################
    # TODO:                                                                     #
    # Compute the gradient of the loss function and store it dW.                #
    # Rather that first computing the loss and then computing the derivative,     #
    # it may be simpler to compute the derivative at the same time that the     #
    # loss is being computed. As a result you may need to modify some of the    #
    # code above to compute the gradient.                                       #
    #############################################################################

    return loss, dW


def svm_loss_vectorized(W, X, y, reg):
    """
    Structured SVM loss function, vectorized implementation.

    Inputs and outputs are the same as svm_loss_naive.
    """
    loss = 0.0
    dW = np.zeros(W.shape)  # initialize the gradient as zero

    #############################################################################
    # TODO:                                                                     #
    # Implement a vectorized version of the structured SVM loss, storing the    #
    # result in loss.                                                           #
    #############################################################################
    num_train = X.shape[0]
    delta = 1.0

    scores = X.dot(W)
    correct_class_score = scores[np.arange(num_train), y]

    margins = np.maximum(
        0, scores - correct_class_score[:, np.newaxis] + delta)
    margins[np.arange(num_train), y] = 0

    loss = np.sum(margins)

    loss /= num_train

    loss += 0.5 * reg * np.sum(W.T.dot(W))

    #############################################################################
    #                             END OF YOUR CODE                              #
    #############################################################################

    #############################################################################
    # TODO:                                                                     #
    # Implement a vectorized version of the gradient for the structured SVM     #
    # loss, storing the result in dW.                                           #
    #                                                                           #
    # Hint: Instead of computing the gradient from scratch, it may be easier    #
    # to reuse some of the intermediate values that you used to compute the     #
    # loss.                                                                     #
    #############################################################################
    X_mask = np.zeros(margins.shape)
    X_mask[margins > 0] = 1

    count = np.sum(X_mask, axis=1)
    X_mask[np.arange(num_train), y] = -count

    dW = X.T.dot(X_mask)

    dW /= num_train

    dW += np.multiply(W, reg)
    #############################################################################
    #                             END OF YOUR CODE                              #
    #############################################################################

    return loss, dW

linear_classifier

import numpy as np
from cs231n.classifiers.linear_svm import *
from cs231n.classifiers.softmax import *


class LinearClassifier(object):

    def __init__(self):
        self.W = None

    def train(self, X, y, learning_rate=1e-3, reg=1e-5, num_iters=100,
              batch_size=200, verbose=False):
        """
        Train this linear classifier using stochastic gradient descent.

        Inputs:
        - X: A numpy array of shape (N, D) containing training data; there are N
          training samples each of dimension D.
        - y: A numpy array of shape (N,) containing training labels; y[i] = c
          means that X[i] has label 0 <= c < C for C classes.
        - learning_rate: (float) learning rate for optimization.
        - reg: (float) regularization strength.
        - num_iters: (integer) number of steps to take when optimizing
        - batch_size: (integer) number of training examples to use at each step.
        - verbose: (boolean) If true, print progress during optimization.

        Outputs:
        A list containing the value of the loss function at each training iteration.
        """
        num_train, dim = X.shape
        num_classes = np.max(y) + 1  # assume y takes values 0...K-1 where K is number of classes
        if self.W is None:
            # lazily initialize W
            self.W = 0.001 * np.random.randn(dim, num_classes)

        # Run stochastic gradient descent to optimize W
        loss_history = []
        for it in xrange(num_iters):
            X_batch = None
            y_batch = None

            #########################################################################
            # TODO:                                                                 #
            # Sample batch_size elements from the training data and their           #
            # corresponding labels to use in this round of gradient descent.        #
            # Store the data in X_batch and their corresponding labels in           #
            # y_batch; after sampling X_batch should have shape (dim, batch_size)   #
            # and y_batch should have shape (batch_size,)                           #
            #                                                                       #
            # Hint: Use np.random.choice to generate indices. Sampling with         #
            # replacement is faster than sampling without replacement.              #
            #########################################################################
            mask = np.random.choice(num_train, batch_size, replace=True)
            X_batch = X[mask]
            y_batch = y[mask]
            #########################################################################
            #                       END OF YOUR CODE                                #
            #########################################################################

            # evaluate loss and gradient
            loss, grad = self.loss(X_batch, y_batch, reg)
            loss_history.append(loss)

            # perform parameter update
            #########################################################################
            # TODO:                                                                 #
            # Update the weights using the gradient and the learning rate.          #
            #########################################################################
            self.W = self.W - learning_rate * grad
            #########################################################################
            #                       END OF YOUR CODE                                #
            #########################################################################

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

        return loss_history

    def predict(self, X):
        """
        Use the trained weights of this linear classifier to predict labels for
        data points.

        Inputs:
        - X: D x N array of training data. Each column is a D-dimensional point.

        Returns:
        - y_pred: Predicted labels for the data in X. y_pred is a 1-dimensional
          array of length N, and each element is an integer giving the predicted
          class.
        """
        X = X.T
        y_pred = np.zeros(X.shape[1])
        ###########################################################################
        # TODO:                                                                   #
        # Implement this method. Store the predicted labels in y_pred.            #
        ###########################################################################
        scores = X.T.dot(self.W)
        y_pred = np.argsort(scores, axis=1)[:, -1]
        ###########################################################################
        #                           END OF YOUR CODE                              #
        ###########################################################################
        return y_pred

    def loss(self, X_batch, y_batch, reg):
        """
        Compute the loss function and its derivative.
        Subclasses will override this.

        Inputs:
        - X_batch: A numpy array of shape (N, D) containing a minibatch of N
          data points; each point has dimension D.
        - y_batch: A numpy array of shape (N,) containing labels for the minibatch.
        - reg: (float) regularization strength.

        Returns: A tuple containing:
        - loss as a single float
        - gradient with respect to self.W; an array of the same shape as W
        """
        pass


class LinearSVM(LinearClassifier):
    """ A subclass that uses the Multiclass SVM loss function """

    def loss(self, X_batch, y_batch, reg):
        return svm_loss_vectorized(self.W, X_batch, y_batch, reg)


class Softmax(LinearClassifier):
    """ A subclass that uses the Softmax + Cross-entropy loss function """

    def loss(self, X_batch, y_batch, reg):
        return softmax_loss_vectorized(self.W, X_batch, y_batch, reg)