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.randint(num_train, size = batch_size)
      X_batch = X[mask]
      y_batch = y[mask]
      pass
      #########################################################################
      #                       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 -= grad * learning_rate
      #########################################################################
      #                       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: A numpy array of shape (N, D) containing training data; there are N
      training samples each of dimension D.

    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.
    """
    y_pred = np.zeros(X.shape[0])
    ###########################################################################
    # TODO:                                                                   #
    # Implement this method. Store the predicted labels in y_pred.            #
    ###########################################################################
    y_pred = np.argmax(np.dot(X, self.W), axis = 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
    """
    svm_loss_vectorized(self.W, X_batch, y_batch, reg)



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
  special = 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:
        loss += margin
        dW[:, y[i]] -= X[i,:].T
        dW[:, j] += X[i, :].T
  # 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
  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
  num_classes = W.shape[1]
  num_train = X.shape[0]
  np.set_printoptions(threshold=np.inf)
  #############################################################################
  # TODO:                                                                     #
  # Implement a vectorized version of the structured SVM loss, storing the    #
  # result in loss.                                                           #
  #############################################################################
  scores = X.dot(W)
  correct_class_score = scores[np.arange(num_train), y]
  tmpMat = scores.T - correct_class_score + 1
  tmpMat = tmpMat.T
  tmpMat[np.arange(num_train), y] = 0
  margin = np.maximum(tmpMat, np.zeros((num_train, num_classes)))
  loss = np.sum(margin)
  loss /= num_train
  loss += 0.5*reg*np.sum(W*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.                                                                     #
  #############################################################################
  binary = margin
  #print(binary.shape)
  binary[margin > 0] = 1
  col_sum = np.sum(binary, axis=1)
  #print(col_sum.shape)
  binary[np.arange(num_train), y] = -col_sum[range(num_train)]
  #print(binary)
  dW = np.dot(X.T, binary)
  dW /= num_train
  dW += reg*W
  #############################################################################
  #                             END OF YOUR CODE                              #
  #############################################################################
  return loss, dW