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lab_1/scripts/classifiers/__init__.py

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+from scripts.classifiers.k_nearest_neighbor import *
+from scripts.classifiers.linear_classifier import *

lab_1/scripts/classifiers/k_nearest_neighbor.py

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+from builtins import range
+from builtins import object
+import numpy as np
+from past.builtins import xrange
+
+
+class KNearestNeighbor(object):
+    """ a kNN classifier with L2 distance """
+
+    def __init__(self):
+        pass
+
+    def train(self, X, y):
+        """
+        Train the classifier. For k-nearest neighbors this is just
+        memorizing the training data.
+
+        Inputs:
+        - X: A numpy array of shape (num_train, D) containing the training data
+          consisting of num_train samples each of dimension D.
+        - y: A numpy array of shape (N,) containing the training labels, where
+             y[i] is the label for X[i].
+        """
+        self.X_train = X
+        self.y_train = y
+
+    def predict(self, X, k=1, num_loops=0):
+        """
+        Predict labels for test data using this classifier.
+
+        Inputs:
+        - X: A numpy array of shape (num_test, D) containing test data consisting
+             of num_test samples each of dimension D.
+        - k: The number of nearest neighbors that vote for the predicted labels.
+        - num_loops: Determines which implementation to use to compute distances
+          between training points and testing points.
+
+        Returns:
+        - y: A numpy array of shape (num_test,) containing predicted labels for the
+          test data, where y[i] is the predicted label for the test point X[i].
+        """
+        if num_loops == 0:
+            dists = self.compute_distances_no_loops(X)
+        elif num_loops == 1:
+            dists = self.compute_distances_one_loop(X)
+        elif num_loops == 2:
+            dists = self.compute_distances_two_loops(X)
+        else:
+            raise ValueError('Invalid value %d for num_loops' % num_loops)
+
+        return self.predict_labels(dists, k=k)
+
+    def compute_distances_two_loops(self, X):
+        """
+        Compute the distance between each test point in X and each training point
+        in self.X_train using a nested loop over both the training data and the
+        test data.
+
+        Inputs:
+        - X: A numpy array of shape (num_test, D) containing test data.
+
+        Returns:
+        - dists: A numpy array of shape (num_test, num_train) where dists[i, j]
+          is the Euclidean distance between the ith test point and the jth training
+          point.
+        """
+        num_test = X.shape[0]
+        num_train = self.X_train.shape[0]
+        dists = np.zeros((num_test, num_train))
+        for i in range(num_test):
+            for j in range(num_train):
+                #####################################################################
+                # TODO:                                                             #
+                # Compute the l2 distance between the ith test point and the jth    #
+                # training point, and store the result in dists[i, j]. You should   #
+                # not use a loop over dimension, nor use np.linalg.norm().          #
+                #####################################################################
+                # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+                pass
+
+                # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+        return dists
+
+    def compute_distances_one_loop(self, X):
+        """
+        Compute the distance between each test point in X and each training point
+        in self.X_train using a single loop over the test data.
+
+        Input / Output: Same as compute_distances_two_loops
+        """
+        num_test = X.shape[0]
+        num_train = self.X_train.shape[0]
+        dists = np.zeros((num_test, num_train))
+        for i in range(num_test):
+            #######################################################################
+            # TODO:                                                               #
+            # Compute the l2 distance between the ith test point and all training #
+            # points, and store the result in dists[i, :].                        #
+            # Do not use np.linalg.norm().                                        #
+            #######################################################################
+            # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            pass
+
+            # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+        return dists
+
+    def compute_distances_no_loops(self, X):
+        """
+        Compute the distance between each test point in X and each training point
+        in self.X_train using no explicit loops.
+
+        Input / Output: Same as compute_distances_two_loops
+        """
+        num_test = X.shape[0]
+        num_train = self.X_train.shape[0]
+        dists = np.zeros((num_test, num_train))
+        #########################################################################
+        # TODO:                                                                 #
+        # Compute the l2 distance between all test points and all training      #
+        # points without using any explicit loops, and store the result in      #
+        # dists.                                                                #
+        #                                                                       #
+        # You should implement this function using only basic array operations; #
+        # in particular you should not use functions from scipy,                #
+        # nor use np.linalg.norm().                                             #
+        #                                                                       #
+        # HINT: Try to formulate the l2 distance using matrix multiplication    #
+        #       and two broadcast sums.                                         #
+        #########################################################################
+        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        pass
+
+        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+        return dists
+
+    def predict_labels(self, dists, k=1):
+        """
+        Given a matrix of distances between test points and training points,
+        predict a label for each test point.
+
+        Inputs:
+        - dists: A numpy array of shape (num_test, num_train) where dists[i, j]
+          gives the distance betwen the ith test point and the jth training point.
+
+        Returns:
+        - y: A numpy array of shape (num_test,) containing predicted labels for the
+          test data, where y[i] is the predicted label for the test point X[i].
+        """
+        num_test = dists.shape[0]
+        y_pred = np.zeros(num_test)
+        for i in range(num_test):
+            # A list of length k storing the labels of the k nearest neighbors to
+            # the ith test point.
+            closest_y = []
+            #########################################################################
+            # TODO:                                                                 #
+            # Use the distance matrix to find the k nearest neighbors of the ith    #
+            # testing point, and use self.y_train to find the labels of these       #
+            # neighbors. Store these labels in closest_y.                           #
+            # Hint: Look up the function numpy.argsort.                             #
+            #########################################################################
+            # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            pass
+
+            # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+            #########################################################################
+            # TODO:                                                                 #
+            # Now that you have found the labels of the k nearest neighbors, you    #
+            # need to find the most common label in the list closest_y of labels.   #
+            # Store this label in y_pred[i]. Break ties by choosing the smaller     #
+            # label.                                                                #
+            #########################################################################
+            # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            pass
+
+            # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        return y_pred

lab_1/scripts/classifiers/linear_classifier.py

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+from __future__ import print_function
+
+from builtins import range
+from builtins import object
+import numpy as np
+from scripts.classifiers.linear_svm import *
+from scripts.classifiers.softmax import *
+from past.builtins import xrange
+
+
+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 range(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 (batch_size, dim)   #
+            # 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.              #
+            #########################################################################
+            # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            pass
+
+            # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            # 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.          #
+            #########################################################################
+            # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            pass
+
+            # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            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.            #
+        ###########################################################################
+        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        pass
+
+        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+        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)

lab_1/scripts/classifiers/linear_svm.py

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+from builtins import range
+import numpy as np
+from random import shuffle
+from past.builtins import xrange
+
+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 range(num_train):
+        scores = X[i].dot(W)
+        correct_class_score = scores[y[i]]
+        for j in range(num_classes):
+            if j == y[i]:
+                continue
+            margin = scores[j] - correct_class_score + 1 # note delta = 1
+            if margin > 0:
+                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
+
+    # Add regularization to the loss.
+    loss += reg * np.sum(W * W)
+
+    #############################################################################
+    # TODO:                                                                     #
+    # Compute the gradient of the loss function and store it dW.                #
+    # Rather than 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.                                       #
+    #############################################################################
+    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    pass
+
+    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+    
+    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.                                                           #
+    #############################################################################
+    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    pass
+
+    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    #############################################################################
+    # 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.                                                                     #
+    #############################################################################
+    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    pass
+
+    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    return loss, dW

lab_1/scripts/classifiers/neural_net.py

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+from __future__ import print_function
+
+from builtins import range
+from builtins import object
+import numpy as np
+import matplotlib.pyplot as plt
+from past.builtins import xrange
+
+class TwoLayerNet(object):
+    """
+    A two-layer fully-connected neural network. The net has an input dimension of
+    N, a hidden layer dimension of H, and performs classification over C classes.
+    We train the network with a softmax loss function and L2 regularization on the
+    weight matrices. The network uses a ReLU nonlinearity after the first fully
+    connected layer.
+
+    In other words, the network has the following architecture:
+
+    input - fully connected layer - ReLU - fully connected layer - softmax
+
+    The outputs of the second fully-connected layer are the scores for each class.
+    """
+
+    def __init__(self, input_size, hidden_size, output_size, std=1e-4):
+        """
+        Initialize the model. Weights are initialized to small random values and
+        biases are initialized to zero. Weights and biases are stored in the
+        variable self.params, which is a dictionary with the following keys:
+
+        W1: First layer weights; has shape (D, H)
+        b1: First layer biases; has shape (H,)
+        W2: Second layer weights; has shape (H, C)
+        b2: Second layer biases; has shape (C,)
+
+        Inputs:
+        - input_size: The dimension D of the input data.
+        - hidden_size: The number of neurons H in the hidden layer.
+        - output_size: The number of classes C.
+        """
+        self.params = {}
+        self.params['W1'] = std * np.random.randn(input_size, hidden_size)
+        self.params['b1'] = np.zeros(hidden_size)
+        self.params['W2'] = std * np.random.randn(hidden_size, output_size)
+        self.params['b2'] = np.zeros(output_size)
+
+    def loss(self, X, y=None, reg=0.0):
+        """
+        Compute the loss and gradients for a two layer fully connected neural
+        network.
+
+        Inputs:
+        - X: Input data of shape (N, D). Each X[i] is a training sample.
+        - y: Vector of training labels. y[i] is the label for X[i], and each y[i] is
+          an integer in the range 0 <= y[i] < C. This parameter is optional; if it
+          is not passed then we only return scores, and if it is passed then we
+          instead return the loss and gradients.
+        - reg: Regularization strength.
+
+        Returns:
+        If y is None, return a matrix scores of shape (N, C) where scores[i, c] is
+        the score for class c on input X[i].
+
+        If y is not None, instead return a tuple of:
+        - loss: Loss (data loss and regularization loss) for this batch of training
+          samples.
+        - grads: Dictionary mapping parameter names to gradients of those parameters
+          with respect to the loss function; has the same keys as self.params.
+        """
+        # Unpack variables from the params dictionary
+        W1, b1 = self.params['W1'], self.params['b1']
+        W2, b2 = self.params['W2'], self.params['b2']
+        N, D = X.shape
+
+        # Compute the forward pass
+        scores = None
+        #############################################################################
+        # TODO: Perform the forward pass, computing the class scores for the input. #
+        # Store the result in the scores variable, which should be an array of      #
+        # shape (N, C).                                                             #
+        #############################################################################
+        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        pass
+
+        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        # If the targets are not given then jump out, we're done
+        if y is None:
+            return scores
+
+        # Compute the loss
+        loss = None
+        #############################################################################
+        # TODO: Finish the forward pass, and compute the loss. This should include  #
+        # both the data loss and L2 regularization for W1 and W2. Store the result  #
+        # in the variable loss, which should be a scalar. Use the Softmax           #
+        # classifier loss.                                                          #
+        #############################################################################
+        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        pass
+
+        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        # Backward pass: compute gradients
+        grads = {}
+        #############################################################################
+        # TODO: Compute the backward pass, computing the derivatives of the weights #
+        # and biases. Store the results in the grads dictionary. For example,       #
+        # grads['W1'] should store the gradient on W1, and be a matrix of same size #
+        #############################################################################
+        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        pass
+
+        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        return loss, grads
+
+    def train(self, X, y, X_val, y_val,
+              learning_rate=1e-3, learning_rate_decay=0.95,
+              reg=5e-6, num_iters=100,
+              batch_size=200, verbose=False):
+        """
+        Train this neural network using stochastic gradient descent.
+
+        Inputs:
+        - X: A numpy array of shape (N, D) giving training data.
+        - y: A numpy array f shape (N,) giving training labels; y[i] = c means that
+          X[i] has label c, where 0 <= c < C.
+        - X_val: A numpy array of shape (N_val, D) giving validation data.
+        - y_val: A numpy array of shape (N_val,) giving validation labels.
+        - learning_rate: Scalar giving learning rate for optimization.
+        - learning_rate_decay: Scalar giving factor used to decay the learning rate
+          after each epoch.
+        - reg: Scalar giving regularization strength.
+        - num_iters: Number of steps to take when optimizing.
+        - batch_size: Number of training examples to use per step.
+        - verbose: boolean; if true print progress during optimization.
+        """
+        num_train = X.shape[0]
+        iterations_per_epoch = max(num_train / batch_size, 1)
+
+        # Use SGD to optimize the parameters in self.model
+        loss_history = []
+        train_acc_history = []
+        val_acc_history = []
+
+        for it in range(num_iters):
+            X_batch = None
+            y_batch = None
+
+            #########################################################################
+            # TODO: Create a random minibatch of training data and labels, storing  #
+            # them in X_batch and y_batch respectively.                             #
+            #########################################################################
+            # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            pass
+
+            # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            # Compute loss and gradients using the current minibatch
+            loss, grads = self.loss(X_batch, y=y_batch, reg=reg)
+            loss_history.append(loss)
+
+            #########################################################################
+            # TODO: Use the gradients in the grads dictionary to update the         #
+            # parameters of the network (stored in the dictionary self.params)      #
+            # using stochastic gradient descent. You'll need to use the gradients   #
+            # stored in the grads dictionary defined above.                         #
+            #########################################################################
+            # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            pass
+
+            # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            if verbose and it % 100 == 0:
+                print('iteration %d / %d: loss %f' % (it, num_iters, loss))
+
+            # Every epoch, check train and val accuracy and decay learning rate.
+            if it % iterations_per_epoch == 0:
+                # Check accuracy
+                train_acc = (self.predict(X_batch) == y_batch).mean()
+                val_acc = (self.predict(X_val) == y_val).mean()
+                train_acc_history.append(train_acc)
+                val_acc_history.append(val_acc)
+
+                # Decay learning rate
+                learning_rate *= learning_rate_decay
+
+        return {
+          'loss_history': loss_history,
+          'train_acc_history': train_acc_history,
+          'val_acc_history': val_acc_history,
+        }
+
+    def predict(self, X):
+        """
+        Use the trained weights of this two-layer network to predict labels for
+        data points. For each data point we predict scores for each of the C
+        classes, and assign each data point to the class with the highest score.
+
+        Inputs:
+        - X: A numpy array of shape (N, D) giving N D-dimensional data points to
+          classify.
+
+        Returns:
+        - y_pred: A numpy array of shape (N,) giving predicted labels for each of
+          the elements of X. For all i, y_pred[i] = c means that X[i] is predicted
+          to have class c, where 0 <= c < C.
+        """
+        y_pred = None
+
+        ###########################################################################
+        # TODO: Implement this function; it should be VERY simple!                #
+        ###########################################################################
+        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        pass
+
+        # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        return y_pred

lab_1/scripts/classifiers/softmax.py

@@ -0,0 +1,65 @@
+from builtins import range
+import numpy as np
+from random import shuffle
+from past.builtins import xrange
+
+def softmax_loss_naive(W, X, y, reg):
+    """
+    Softmax 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
+    """
+    # Initialize the loss and gradient to zero.
+    loss = 0.0
+    dW = np.zeros_like(W)
+
+    #############################################################################
+    # TODO: Compute the softmax loss and its gradient using explicit loops.     #
+    # Store the loss in loss and the gradient in dW. If you are not careful     #
+    # here, it is easy to run into numeric instability. Don't forget the        #
+    # regularization!                                                           #
+    #############################################################################
+    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    pass
+
+    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    return loss, dW
+
+
+def softmax_loss_vectorized(W, X, y, reg):
+    """
+    Softmax loss function, vectorized version.
+
+    Inputs and outputs are the same as softmax_loss_naive.
+    """
+    # Initialize the loss and gradient to zero.
+    loss = 0.0
+    dW = np.zeros_like(W)
+
+    #############################################################################
+    # TODO: Compute the softmax loss and its gradient using no explicit loops.  #
+    # Store the loss in loss and the gradient in dW. If you are not careful     #
+    # here, it is easy to run into numeric instability. Don't forget the        #
+    # regularization!                                                           #
+    #############################################################################
+    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    pass
+
+    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    return loss, dW