cs231n作业1——Softmax

参考文章：cs231n assignment1——softmax

Softmax

softmax其实和SVM差别不大，两者损失函数不同，softmax就是把各个类的得分转化成了概率。
损失函数：

def softmax_loss_naive(W, X, y, reg):
    loss = 0.0
    dW = np.zeros_like(W)
    num_classes = W.shape[1]
    num_train = X.shape[0]
    for i in range(num_train):
        scores = X[i].dot(W)                # 矩阵点乘：第 i 张照片在各类别上的得分
        scores -= np.max(scores)            # 减去最大得分，减小计算量
        correct_class_score = scores[y[i]]  # 接下来三行是损失函数的计算
        exp_sum = np.sum(np.exp(scores))
        loss += -correct_class_score + np.log(exp_sum) # np.log()以e为底
        for j in range(num_classes):
            if j == y[i]:
                dW[:, y[i]] += (np.exp(scores[y[i]])/exp_sum-1)*X[i]
            else:
                dW[:, j] += np.exp(scores[j])/exp_sum*X[i]    
    
    loss /= num_train                      # 求平均损失
    loss += reg * np.sum(W * W)            # 损失加上正则化惩罚
    dW /= num_train                        # 求平均梯度
    dW += 2.0*reg*W

    return loss, dW

用向量法实现 Softmax

def softmax_loss_vectorized(W, X, y, reg):
    loss = 0.0
    dW = np.zeros_like(W)

    num_classes = W.shape[1]
    num_train = X.shape[0]
    scores = X.dot(W)                                                  # N*C 的矩阵
    scores -= np.max(scores, axis=1, keepdims=True)                    # 减去每行（每张图片对于每一类）的最大值
    correct_class_score = scores[range(num_train),y]
    exp_sum = np.sum(np.exp(scores), axis=1, keepdims=True)            # 按行求和，并保持为二维（列向量）
    loss = -np.sum(correct_class_score) + np.sum(np.log(exp_sum))      # 损失函数公式并求和
    loss = loss/num_train + reg * np.sum(W * W)
    
    med = np.exp(scores)/exp_sum         # 对于j!=yi的情况，dw=np.exp(scores[j])/exp_sum*X[i]
    med[range(num_train),y] -= 1         # 对于j=yi的情况，dw=(np.exp(scores[j])/exp_sum-1)*X[i]
    dW = X.T.dot(med)                    # 最后同时乘以 X[i]
    dW /= num_train
    dW += 2.0*reg*W

    return loss, dW

之后用随机梯度下降法优化损失函数，最后进行超参数的选择。