上传文件至 'assignment1/classifiers'

assignment1/classifiers/__init__.py

@@ -0,0 +1,2 @@
+from daseCV.classifiers.k_nearest_neighbor import *
+from daseCV.classifiers.linear_classifier import *

assignment1/classifiers/k_nearest_neighbor.py

@@ -0,0 +1,186 @@
+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: 
+                #计算第i个测试点与第j个训练点之间的l2距离，并将结果存储在dists[i，j]中。
+                #你不应使用循环和np.linalg.norm()函数。    
+                #####################################################################
+                # *****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:                                                               
+            #计算第i个测试点与所有训练点之间的l2距离，并将结果存储在dists[i，：]中。
+            #不要使用np.linalg.norm()。                                     
+            #######################################################################
+            # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            # 注意np.sum中要加上维度axis=1才能得出正确的结果
+            # 关于axis的介绍
+            # https://zhuanlan.zhihu.com/p/30960190
+            # 以及np.sum的介绍
+            # https://docs.scipy.org/doc/numpy/reference/generated/numpy.sum.html
+            
+            # self.X_train (5000,3072) X[i] (1,3072) (self.X_train - X[i]) (5000,3072)
+            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:                                                                 
+        #在不使用任何显式循环的情况下，计算所有测试点和所有训练点之间的l2距离，
+        #并将结果存储在dists中。                                                     
+        #您应该仅使用基本的数组操作来实现此功能。
+        #不可以使用scipy中的函数以及函数np.linalg.norm()。
+        #
+        #提示：尝试使用矩阵乘法和广播总和来计算l2距离。                           
+        #########################################################################
+        # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        # (x-y)^2 = x^2 + y^2 - 2xy
+        # reshape是为了让两个矩阵有个维度为1，这样子便可进行广播
+        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:                                                                   
+            #使用距离矩阵查找第i个测试点的k个最近邻居，
+            #并使用self.y_train查找这些邻居的标签。
+            #将这些标签存储在closest_y中。
+            #
+            #提示：查阅函数numpy.argsort。                            
+            #########################################################################
+            # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+            
+            # numpy.argsort 返回排序好的数列的索引
+            pass
+
+            # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+            #########################################################################
+            # TODO:                                                                 
+            #
+            #现在，你已经找到了k个最近邻的标签，接着需要在closest_y中找到最可能的标签。                                                     #将此标签存储在y_pred [i]中。如果有两个标签可能性一样的话选择索引更小的那个。
+            #########################################################################
+            # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+            y_pred[i] = np.bincount(closest_y).argmax()
+
+            # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+        return y_pred

assignment1/classifiers/linear_classifier.py

@@ -0,0 +1,136 @@
+from __future__ import print_function
+
+from builtins import range
+from builtins import object
+import numpy as np
+from daseCV.classifiers.linear_svm import *
+from daseCV.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:                               
+            # 从训练数据及其相应的标签中采样batch_size大小的样本，以用于本轮梯度下降。
+            # 将数据存储在X_batch中，并将其相应的标签存储在y_batch中:
+            # 采样后，X_batch的形状为(batch_size,dim)，y_batch的形状(batch_size,)                         
+            #                                                                       
+            # 提示：使用np.random.choice生成索引。 可重复的采样比不可重复的采样要快一点。
+            #########################################################################
+            # *****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:                                                                 
+            # 使用梯度和学习率更新权重。       
+            #########################################################################
+            # *****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:                                  
+        # 实现此方法。将预测的标签存储在y_pred中。           
+        ###########################################################################
+        # *****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)

assignment1/classifiers/linear_svm.py

@@ -0,0 +1,97 @@
+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
+                dW[:,j] += X[i] # dW计算
+                dW[:,y[i]] += -X[i] # dW计算
+
+    # 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：
+    # 计算损失函数的梯度并将其存储为dW。
+    # 与其先计算损失再计算梯度，还不如在计算损失的同时计算梯度更简单。
+    # 因此，您可能需要修改上面的一些代码来计算梯度。
+    #############################################################################
+    # *****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: 
+    # 实现一个向量化SVM损失计算方法,并将结果存储到loss中
+    #############################################################################
+    
+    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+    
+    pass
+
+    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    #############################################################################
+    # TODO:                                                              
+    # 实现一个向量化的梯度计算方法,并将结果存储到dW中                                
+    #                                                                           
+    # 提示:与其从头计算梯度,不如利用一些计算loss时的中间变量                                    
+    #############################################################################
+    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    pass
+
+    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    return loss, dW

assignment1/classifiers/neural_net.py

@@ -0,0 +1,220 @@
+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: 执行向前传播，计算输入数据的每个类的score。
+        # 将结果存储在scores变量中，该变量应该是一个(N, C)维的数组。
+        #############################################################################
+        # *****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: 完成向前传播，计算损失。
+        # 这应该包括数据损失和W1和W2的L2正则化项。
+        # 将结果存储在变量loss中，它应该是一个标量。
+        # 使用Softmax损失函数。
+        #############################################################################
+        # *****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: 计算反向传播，计算权重和偏置值的梯度, 将结果存储在grads字典中。
+        # 例如，grads['W1']存储W1的梯度，并且和W1是相同大小的矩阵。
+        #############################################################################
+        # *****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: 创建一个随机的数据和标签的mini-batch，存储在X_batch和y_batch中。
+            #########################################################################
+            # *****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: 使用grads字典中的梯度来更新网络参数(参数存储在字典self.params中)
+            # 使用随机梯度下降法。
+            #########################################################################
+            # *****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

assignment1/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: 使用显式循环计算softmax损失及其梯度。
+    # 将损失和梯度分别保存在loss和dW中。
+    # 如果你不小心，很容易遇到数值不稳定的情况。 
+    # 不要忘了正则化！                                                           
+    #############################################################################
+    # *****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: 不使用显式循环计算softmax损失及其梯度。
+    # 将损失和梯度分别保存在loss和dW中。
+    # 如果你不小心，很容易遇到数值不稳定的情况。 
+    # 不要忘了正则化！
+    #############################################################################
+    # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    pass
+
+    # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
+
+    return loss, dW