SVM, Softmax损失函数[通俗易懂]

def svm_loss_naive(W, X, y, reg):
    """ # SVM 损失函数 native版本 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
    # 对于每一个样本，累加loss
    for i in xrange(num_train):
        scores = X[i].dot(W)     # (1, C)
        correct_class_score = scores[y[i]]
        for j in xrange(num_classes):
            if j == y[i]:
                continue
            # 根据 SVM 损失函数计算
            margin = scores[j] - correct_class_score + 1    # note delta = 1
            # 当 margin>0 时，才会有损失，此时也会有梯度的累加
            if margin > 0:      # max(0, yi - yc + 1)
                loss += margin
                 # 根据公式：∇Wyi Li = - xiT(∑j≠yi1(xiWj - xiWyi +1>0)) + 2λWyi
                dW[:, y[i]] += -X[i, :]   # y[i] 是正确的类
                # 根据公式： ∇Wj Li = xiT 1(xiWj - xiWyi +1>0) + 2λWj ,
                dW[:, j] += X[i, :]

    # 训练数据平均损失
    loss /= num_train
    dW /= num_train

    # 正则损失
    loss += 0.5 * reg * np.sum(W * W)
    dW += reg * W

    #
    return loss, dW


#
def svm_loss_vectorized(W, X, y, reg):
    """ SVM 损失函数 向量化版本 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
    scores = X.dot(W)        # N by C 样本数*类别数
    num_train = X.shape[0]
    num_classes = W.shape[1]

    scores_correct = scores[np.arange(num_train), y]
    scores_correct = np.reshape(scores_correct, (num_train, 1))  # N*1 每个样本的正确类别

    margins = scores - scores_correct + 1.0     # N by C 计算scores矩阵中每一处的损失
    margins[np.arange(num_train), y] = 0.0      # 每个样本的正确类别损失置0
    margins[margins <= 0] = 0.0                 # max(0, x)
    loss += np.sum(margins) / num_train         # 累加所有损失，取平均
    loss += 0.5 * reg * np.sum(W * W)           # 正则

    # compute the gradient
    margins[margins > 0] = 1.0                  # max(0, x) 大于0的梯度计为1
    row_sum = np.sum(margins, axis=1)           # N*1 每个样本累加
    margins[np.arange(num_train), y] = -row_sum  # 类正确的位置 = -梯度累加
    dW += np.dot(X.T, margins)/num_train + reg * W     # D by C
    return loss, dW

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)
    dW_each = np.zeros_like(W)
    #
    num_train, dim = X.shape
    num_class = W.shape[1]
    f = X.dot(W)        # 样本数*类别数 分值
    #
    f_max = np.reshape(np.max(f, axis=1), (num_train, 1))
    # 计算对数概率 prob.shape=N*10 每一行与一个样本相对应 每一行的概率和为1
    # 其中 f_max 是每行的最大值，exp(x)中x的值过大而出现数值不稳定问题
    prob = np.exp(f - f_max) / np.sum(np.exp(f - f_max), axis=1, keepdims=True)
    #
    y_trueClass = np.zeros_like(prob)
    y_trueClass[np.arange(num_train), y] = 1.0     # 每行只有正确的类别处为1，其余为0
    #
    for i in range(num_train):
        for j in range(num_class):
            loss += -(y_trueClass[i, j] * np.log(prob[i, j]))
            dW_each[:, j] = -(y_trueClass[i, j] - prob[i, j]) * X[i, :]
        dW += dW_each
    loss /= num_train
    loss += 0.5 * reg * np.sum(W * W)
    dW /= num_train
    dW += reg * W

    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. """
    loss = 0.0
    dW = np.zeros_like(W)    # D by C
    num_train, dim = X.shape

    f = X.dot(W)    # N by C
    # Considering the Numeric Stability
    f_max = np.reshape(np.max(f, axis=1), (num_train, 1))   # N by 1
    prob = np.exp(f - f_max) / np.sum(np.exp(f - f_max), axis=1, keepdims=True)
    y_trueClass = np.zeros_like(prob)
    y_trueClass[range(num_train), y] = 1.0    # N by C

    # 计算损失 y_trueClass是N*C维度 np.log(prob)也是N*C的维度
    loss += -np.sum(y_trueClass * np.log(prob)) / num_train + 0.5 * reg * np.sum(W * W)

    # 计算损失 X.T = (D*N) y_truclass-prob = (N*C)
    dW += -np.dot(X.T, y_trueClass - prob) / num_train + reg * W

    return loss, dW