CS231n-2017 Assignment1 k-近邻方法、SVM、Softmax、两层神经网络

一、k近邻方法

1. 使用两层循环计算距离矩阵

训练数据X_train和测试数据X中每一行是一个样本点。距离矩阵dists中每一行为X中的一点与X_train中各个点的距离。

k_nearest_neighbor文件中的compute_distances_two_loops()函数：

def compute_distances_two_loops(self, X):
    
    num_test = X.shape[0]
    num_train = self.X_train.shape[0]
    dists = np.zeros((num_test, num_train))
    for i in tqdm(xrange(num_test)):
      for j in xrange(num_train):
        # TODO
        dists[i, j] = np.sqrt(np.sum((X[i] - self.X_train[j])**2))

    return dists

2. 实现分类函数

使用numpy.argsort()获取k近邻，使用numpy.bincount()实现k近邻的投票统计。

k_nearest_neighbor文件中的predict_labels()函数：

def predict_labels(self, dists, k=1):
    num_test = dists.shape[0]
    y_pred = np.zeros(num_test)
    indxRangeList = range(k)
    for i in xrange(num_test):
        closest_y = []
        # TODO
        closest_y = self.y_train[np.argsort(dists[i])[:k]]

        # TODO
        y_pred[i] = np.argmax(np.bincount(closest_y))
    return y_pred

3. 使用一层循环计算距离矩阵

每次取出X中的一个点，计算其与X_train中各个点的距离。

这时就可看出使用矩阵中的一行代表一个样本点的好处：按行进行索引后，取出的向量与另一个矩阵自动满足numpy的广播条件。

k_nearest_neighbor文件中的compute_distances_one_loop()函数：

def compute_distances_one_loop(self, X):
    num_test = X.shape[0]
    num_train = self.X_train.shape[0]
    dists = np.zeros((num_test, num_train))
    for i in tqdm(xrange(num_test)):
        # TODO
        dists[i, :] = np.sqrt(np.sum((X[i] - self.X_train)**2, axis = 1))
    return dists 

4. 使用向量化方法计算距离矩阵(不使用循环)

假设点用行向量表示，先考虑计算两个点$\vec{x}$和$\vec{y}$的距离:

$$\Vert \vec{x}-\vec{y}{\Vert }^2=(\vec{x}-\vec{y})\cdot (\vec{x}-\vec{y}{)}^T=\vec{x}{\vec{x}}^T-2\vec{x}{\vec{y}}^T+\vec{y}{\vec{y}}^T=\Vert \vec{x}{\Vert }^2-2\vec{x}{\vec{y}}^T+\Vert \vec{y}{\Vert }^2$$
若设两个点集为:

$$X=\left[\begin{array}{c}-{\vec{x}}_1-\\ -{\vec{x}}_2-\\ ⋮\\ -{\vec{x}}_n-\end{array}\right]Y=\left[\begin{array}{c}-{\vec{y}}_1-\\ -{\vec{y}}_2-\\ ⋮\\ -{\vec{y}}_m-\end{array}\right]$$

则$\Vert \vec{x}{\Vert }^2$和$\Vert \vec{y}{\Vert }^2$分别可用$X$和$Y$的按行取模的值替代；$\vec{x}{\vec{y}}^T$可用$X{Y}^T$替代。然后在计算距离矩阵时，与$X$相关的沿列方向进行广播，与$Y$相关的沿行方向进行广播。

k_nearest_neighbor文件中的compute_distances_no_loops()函数：

def compute_distances_no_loops(self, X):
    num_test = X.shape[0]
    num_train = self.X_train.shape[0]
    dists = np.zeros((num_test, num_train)) 
    
    # TODO
    XY = np.dot(X,self.X_train.T)
    X_norm2 = np.sum(np.square(X),axis=1,keepdims=True)
    Y_norm2 = np.sum(np.square(self.X_train),axis=1,keepdims=True)
    dists = np.sqrt(X_norm2 - 2*XY + Y_norm2.T)

    return dists

其中keepdims参数表示：对于numpy中归并类的函数(即如np.sum()这类由矩阵中的某些值得到一个值的函数)，保持所得结果的维数与原矩阵的维数相同。

5. k-folds选取超参数

• 训练数据集与验证数据集划分。

使用np.array_split()函数，默认沿行划分。

X_train_folds = np.array_split(X_train, num_folds)
y_train_folds = np.array_split(y_train, num_folds)

• 针对超参数$k$进行参数筛选。

对每一个$k$值，使用num_folds-1折的数据作为训练集，使用剩余的一折数据作为验证集，得到模型的num_folds个准确率。

for k in k_choices:
    if k in k_to_accuracies:
        continue
    else:
        k_to_accuracies[k] = []
    for foldIndx in range(num_folds):
        X_train_cv = np.vstack(X_train_folds[0:foldIndx] + X_train_folds[foldIndx+1:])
        y_train_cv = np.hstack(y_train_folds[0:foldIndx] + y_train_folds[foldIndx+1:])
        classifier.train(X_train_cv, y_train_cv)
        dists = classifier.compute_distances_no_loops(X_train_folds[foldIndx])
        y_pred = classifier.predict_labels(dists, k)

        # Compute and print the fraction of correctly predicted examples
        num_correct = np.sum(y_pred == y_train_folds[foldIndx])
        accuracy = float(num_correct) / y_train_folds[foldIndx].shape[0]
        k_to_accuracies[k].append(accuracy)

所得准确率统计如下图所示。

图 1. 不同k值下的k近邻分类准确率

《

可见在k_choices = [1, 3, 5, 8, 10, 12, 15, 20, 50, 100]范围内，当k=10时，所得准确率最高。此时k近邻模型在测试集上的准确率约为28.2%。

二、支持向量机方法

对单一样本$(\vec{x},y)$，假设模型给出的在各个类别上的得分为 s ⃗ = { s i } = x ⃗ ⋅ W \vec{s}=\{s_i\}=\vec{x}\cdot W s ={si​}=x ⋅W（行向量），则损失函数为：$L={\sum }_{i̸=y}\max (0,s_i-s_y+1)$，梯度为：

$$\begin{array}{lcr}{\mathrm{c}\mathrm{o}\mathrm{l}}_i{\frac{dL}{dW}|}_{i̸=y}& =& {\vec{x}}^T\cdot (s_i-s_y+1>0)\\ & & \\ {\mathrm{c}\mathrm{o}\mathrm{l}}_y\frac{dL}{dW}& =& -{\vec{x}}^T\cdot {\sum }_{i̸=y}(s_i-s_y+1>0)\end{array}$$

1. 计算损失函数与梯度的朴素方法(以循环方式)

linear_svm文件中svm_loss_naive()函数：

def svm_loss_naive(W, X, y, reg):

    dW = np.zeros(W.shape) # initialize the gradient as zero

    # TODO
    # 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:
            loss += margin
            dW[:, j] += X[i,:].T
            dW[:, y[i]] -= 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 += reg * np.sum(W * W)
    dW += reg * 2 * W
    
    return loss, dW

2. 使用向量化方法计算损失与梯度

事实上，实现朴素的计算方法，对整理出向量化实现方法是很有帮助的。

linear_svm文件中svm_loss_vectorized()函数：

def svm_loss_vectorized(W, X, y, reg):
    loss = 0.0
    dW = np.zeros(W.shape) # initialize the gradient as zero

    # TODO
    num_train = X.shape[0]
    scores = X.dot(W)
    correct_scores = scores[np.arange(num_train), y][:, newaxis]
    scores = scores - correct_scores + 1
    scores[range(num_train), y] = 0
    scores = np.maximum(0, scores)
    loss = np.sum(scores) / num_train
    loss += reg * np.sum(W * W)
    # TODO
    scores[scores > 0] = 1
    scores[range(num_train), y] -= np.sum(scores, axis=1)
    dW = dW + X.T.dot(scores)
    dW /= num_train
    dW += reg * 2 * W

    return loss, dW

3. 随机梯度算法

linear_classifier文件中LinearClassifier.train()函数：

def train(self, X, y, learning_rate=1e-3, reg=1e-5, num_iters=100,
            batch_size=200, verbose=False):
    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
      idx = np.random.choice(num_train, batch_size)
      X_batch = X[idx,:]
      y_batch = y[idx]

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

      # perform parameter update
      # TODO
      self.W -= learning_rate * grad

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

    return loss_history

4. 预测函数

linear_classfier文件中LinearClassifier.predict()函数：使用numpy.argmax()函数获取得分最高的位置索引。

def predict(self, X):
    y_pred = np.argmax(X.dot(self.W), axis=1)
    return y_pred

5. 超参数选取

for lr in learning_rates:
    for rs in regularization_strengths:
        svm = LinearSVM()
        loss_hist = svm.train(X_train, y_train, learning_rate=lr, reg=rs,
                      num_iters=1000, verbose=False)
        
        y_val_pred = svm.predict(X_val)
        y_train_pred = svm.predict(X_train)
        train_acc = np.mean(y_train == y_train_pred)
        val_acc = np.mean(y_val == y_val_pred)
        results[(lr, rs)] = (train_acc, val_acc)
        if val_acc > best_val:
            best_val = val_acc
            best_svm = svm

三、Softmax分类器

分类器输出为被判判断是各类别的概率：
对单一样本$(\vec{x},y)$，假设模型给出的在各个类别上的概率为 p ⃗ = { p i } = e x ⃗ ⋅ W / ∥ e x ⃗ ⋅ W ∥ l = 1 \vec{p}=\{p_i\}=e^{\vec{x}\cdot W}/\|e^{\vec{x}\cdot W}\|_{l=1} p ​={pi​}=ex ⋅W/∥ex ⋅W∥l=1​（行向量），损失函数为：$L=-\log p_y=\log \Vert e^{\vec{x}\cdot W}{\Vert }_{l=1}-\vec{x}\cdot {\mathrm{c}\mathrm{o}\mathrm{l}}_{y}W$，则梯度为：

$${\mathrm{c}\mathrm{o}\mathrm{l}}_i{\frac{dL}{dW}|}_{i̸=y}=\frac{e^{\vec{x}\cdot W}}{\Vert e^{\vec{x}\cdot W}{\Vert }_{l=1}}{\vec{x}}^T-{\vec{x}}^T\cdot (i=y)=\left[p_i-(i=y)\right]\cdot {\vec{x}}^T$$

1. 计算损失函数与梯度的朴素方法(以循环方式)

softmax文件中softmax_loss_naive()函数：

def softmax_loss_naive(W, X, y, reg):
    # Initialize the loss and gradient to zero.
    loss = 0.0
    dW = np.zeros_like(W)

    # TODO
    num_train = X.shape[0]
    num_class = W.shape[1]
    for i in xrange(num_train):
    probs = np.exp(X[i,:].dot(W))
    probs = probs / np.sum(probs)
    loss -= np.log(probs[y[i]])
    for j in xrange(num_class):
        if j == y[i]:
            dW[:,j] += (probs[j] - 1)*X[i,:].T
        else:
            dW[:,j] += probs[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 += reg * np.sum(W * W)
    dW += reg * 2 * W

    return loss, dW

2. 使用向量化方法计算损失与梯度

softmax文件中的softmax_loss_vectorized()函数：

def softmax_loss_vectorized(W, X, y, reg):
  # Initialize the loss and gradient to zero.
  loss = 0.0
  dW = np.zeros_like(W)
  num_train = X.shape[0]
  num_class = W.shape[1]

  # TODO
  probs = np.exp(X.dot(W))
  probs = probs / np.sum(probs, axis = 1)[:, newaxis]
  loss = loss - np.sum(np.log(probs[range(num_train), y]))

  loss /= num_train
  loss += reg * np.sum(W * W)

  probs[range(num_train), y] = probs[range(num_train), y] - 1
  dW = dW + X.T.dot(probs)
  dW /= num_train
  dW += reg * 2 * W

  return loss, dW

3. 超参数选取

for lr in learning_rates:
    for rs in regularization_strengths:
        softmaxClassfier = Softmax()
        loss_hist = softmaxClassfier.train(X_train, y_train, learning_rate=lr, reg=rs,
                      num_iters=1000, verbose=False)
        
        y_val_pred = softmaxClassfier.predict(X_val)
        y_train_pred = softmaxClassfier.predict(X_train)
        train_acc = np.mean(y_train == y_train_pred)
        val_acc = np.mean(y_val == y_val_pred)
        results[(lr, rs)] = (train_acc, val_acc)
        if val_acc > best_val:
            best_val = val_acc
            best_softmax = softmaxClassfier

四、两层神经网络

1. 神经网络的前向传播与反向传播

这里采用和Softmax分类器同样的损失函数。则前向传播的过程很直白。而由BP算法的推导，反向传播的过程也很直白。
neural_net文件的loss()函数：

def loss(self, X, y=None, reg=0.0):
    # 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. 
    h1 = np.maximum(0, X.dot(W1) + b1)
    scores = h1.dot(W2) + b2
    
    # 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. 
    probs = np.exp(scores)
    probs = probs / np.sum(probs, axis = 1, keepdims=True)
    loss = 0 - np.sum(np.log(probs[range(N), y]+eps))

    loss /= N
    loss += reg * (np.sum(W1*W1) + np.sum(W2*W2))
    
    # Backward pass: compute gradients
    grads = {}

    # TODO: backward propagation
    probs[range(N), y] = probs[range(N), y] - 1
    gradW2 = h1.T.dot(probs)
    gradW2 /= N
    gradW2 += reg * 2 * W2

    gradB2 = np.sum(probs, axis=0)
    gradB2 /= N

    gradH1 = probs.dot(W2.T)
    gradW1 = X.T.dot((h1>0)*gradH1)
    gradW1 /= N 
    gradW1 += reg * 2 * W1

    gradB1 = np.sum((h1>0)*gradH1, axis=0)
    gradB1 /= N
    
    grads["W1"] = gradW1
    grads["W2"] = gradW2
    grads["b1"] = gradB1
    grads["b2"] = gradB2

    return loss, grads

2. 训练、预测、超参数选取

与其余分类器类似。

五、特征工程

采用两层神经网络，使用如下参数可在测试数据集上获得60%的准确率：

learning_rate = 1.5
regularization_strength = 0.001
num_iters = 5000
batch_size = 500