集成学习之Boosting —— Gradient Boosting实现

Gradient Boosting的一般算法流程

1. 初始化： \(f_0(x) = \mathop{\arg\min}\limits_\gamma \sum\limits_{i=1}^N L(y_i, \gamma)\)

2. for m=1 to M:
   
   (a) 计算负梯度： \(\tilde{y}_i = -\frac{\partial L(y_i,f_{m-1}(x_i))}{\partial f_{m-1}(x_i)}, \qquad i = 1,2 \cdots N\)
   
   (b) 通过最小化平方误差，用基学习器\(h_m(x)\)拟合\(\tilde{y_i}\)，\(w_m = \mathop{\arg\min}\limits_w \sum\limits_{i=1}^{N} \left[\tilde{y}_i - h_m(x_i\,;\,w) \right]^2\)
   
   (c) 使用line search确定步长\(\rho_m\)，以使L最小，\(\rho_m = \mathop{\arg\min}\limits_{\rho} \sum\limits_{i=1}^{N} L(y_i,f_{m-1}(x_i) + \rho h_m(x_i\,;\,w_m))\)
   
   (d) \(f_m(x) = f_{m-1}(x) + \rho_m h_m(x\,;\,w_m)\)

3. 输出\(f_M(x)\)

• 另外具体实现了early_stopping，回归，分类和分步预测 (stage_predict，见完整代码)。

• Gradient Boostig一般有一个初始值存在，即上面第一步中的\(f_0(x)\)，在实现的时候这个初始值是不能乘学习率的，因为乘的话等于变相改变了初始值，会产生一些意想不到的结果 (很不幸我就犯过这个错误 ~) 。

  ​

# 先定义各类损失函数，回归有squared loss、huber loss；分类有logistic loss，modified huber loss
def SquaredLoss_NegGradient(y_pred, y):
    return y - y_pred

def Huberloss_NegGradient(y_pred, y, alpha):
    diff = y - y_pred
    delta = stats.scoreatpercentile(np.abs(diff), alpha * 100)
    g = np.where(np.abs(diff) > delta, delta * np.sign(diff), diff)
    return g

def logistic(p):
    return 1 / (1 + np.exp(-2 * p))

def LogisticLoss_NegGradient(y_pred, y):
    g = 2 * y / (1 + np.exp(1 + 2 * y * y_pred))  # logistic_loss = log(1+exp(-2*y*y_pred))
    return g

def modified_huber(p):
    return (np.clip(p, -1, 1) + 1) / 2

def Modified_Huber_NegGradient(y_pred, y):
    margin = y * y_pred
    g = np.where(margin >= 1, 0, np.where(margin >= -1, y * 2 * (1-margin), 4 * y))
    # modified_huber_loss = np.where(margin >= -1, max(0, (1-margin)^2), -4 * margin)
    return g

class GradientBoosting(object):
    def __init__(self, M, base_learner, learning_rate=1.0, method="regression", tol=None, subsample=None,
                 loss="square", alpha=0.9):
        self.M = M
        self.base_learner = base_learner
        self.learning_rate = learning_rate
        self.method = method
        self.tol = tol
        self.subsample = subsample
        self.loss = loss
        self.alpha = alpha

    def fit(self, X, y):
        # tol为early_stopping的阈值，如果使用early_stopping，则从训练集中分出验证集
        if self.tol is not None:
            X, X_val, y, y_val = train_test_split(X, y, random_state=2)
            former_loss = float("inf")
            count = 0
            tol_init = self.tol

        init_learner = self.base_learner
        y_pred = init_learner.fit(X, y).predict(X)   # 初始值
        self.base_learner_total = [init_learner]
        for m in range(self.M):

            if self.subsample is not None:  # subsample
                sample = [np.random.choice(len(X), int(self.subsample * len(X)), replace=False)]
                X_s, y_s, y_pred_s = X[sample], y[sample], y_pred[sample]
            else:
                X_s, y_s, y_pred_s = X, y, y_pred

            # 计算负梯度
            if self.method == "regression":
                if self.loss == "square":
                    response = SquaredLoss_NegGradient(y_pred_s, y_s)
                elif self.loss == "huber":
                    response = Huberloss_NegGradient(y_pred_s, y_s, self.alpha)
            elif self.method == "classification":
                if self.loss == "logistic":
                    response = LogisticLoss_NegGradient(y_pred_s, y_s)
                elif self.loss == "modified_huber":
                    response = Modified_Huber_NegGradient(y_pred_s, y_s)

            base_learner = clone(self.base_learner)
            y_pred += base_learner.fit(X_s, response).predict(X) * self.learning_rate
            self.base_learner_total.append(base_learner)

            '''early stopping'''
            if m % 10 == 0 and m > 300 and self.tol is not None:
                p = np.array([self.base_learner_total[m].predict(X_val) for m in range(1, m+1)])
                p = np.vstack((self.base_learner_total[0].predict(X_val), p))
                stage_pred = np.sum(p, axis=0)
                if self.method == "regression":
                    later_loss = np.sqrt(mean_squared_error(stage_pred, y_val))
                if self.method == "classification":
                    stage_pred = np.where(logistic(stage_pred) >= 0.5, 1, -1)
                    later_loss = zero_one_loss(stage_pred, y_val)

                if later_loss > (former_loss + self.tol):
                    count += 1
                    self.tol = self.tol / 2
                    print(self.tol)
                else:
                    count = 0
                    self.tol = tol_init

                if count == 2:
                    self.M = m - 20
                    print("early stopping in round {}, best round is {}, M = {}".format(m, m - 20, self.M))
                    break
                former_loss = later_loss

        return self

    def predict(self, X):
        pred = np.array([self.base_learner_total[m].predict(X) * self.learning_rate for m in range(1, self.M + 1)])
        pred = np.vstack((self.base_learner_total[0].predict(X), pred))    # 初始值 + 各基学习器
        if self.method == "regression":
            pred_final = np.sum(pred, axis=0)
        elif self.method == "classification":
            if self.loss == "modified_huber":
                p = np.sum(pred, axis=0)
                pred_final = np.where(modified_huber(p) >= 0.5, 1, -1)
            elif self.loss == "logistic":
                p = np.sum(pred, axis=0)
                pred_final = np.where(logistic(p) >= 0.5, 1, -1)
        return pred_final

class GBRegression(GradientBoosting):
    def __init__(self, M, base_learner, learning_rate, method="regression", loss="square",tol=None, subsample=None, alpha=0.9):
        super(GBRegression, self).__init__(M=M, base_learner=base_learner, learning_rate=learning_rate, method=method,
                                            loss=loss, tol=tol, subsample=subsample, alpha=alpha)

class GBClassification(GradientBoosting):
    def __init__(self, M, base_learner, learning_rate, method="classification", loss="logistic", tol=None, subsample=None):
        super(GBClassification, self).__init__(M=M, base_learner=base_learner, learning_rate=learning_rate, method=method,
                                                loss=loss, tol=tol, subsample=subsample)

if __name__ == "__main__":
	# 创建数据集进行测试
    X, y = datasets.make_regression(n_samples=20000, n_features=10, n_informative=4, noise=1.1, random_state=1)
    X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)
    model = GBRegression(M=1000, base_learner=DecisionTreeRegressor(max_depth=2, random_state=1), learning_rate=0.1,
                         loss="huber")
    model.fit(X_train, y_train)
    pred = model.predict(X_test)
    rmse = np.sqrt(mean_squared_error(y_test, pred))
    print('RMSE: ', rmse)

    X, y = datasets.make_classification(n_samples=20000, n_features=10, n_informative=4, flip_y=0.1,
                                    n_clusters_per_class=1, n_classes=2, random_state=1)
    y[y==0] = -1
    X_train, X_test, y_train, y_test = train_test_split(X, y)
    model = GBClassification(M=1000, base_learner=DecisionTreeRegressor(max_depth=1, random_state=1), learning_rate=1.0,
                             method="classification", loss="logistic")
    model.fit(X_train, y_train)
    pred = model.predict(X_test)
    acc = np.zeros(pred.shape)
    acc[np.where(pred == y_test)] = 1
    accuracy = np.sum(acc) / len(pred)
    print('accuracy logistic score: ', accuracy)

    model = GBClassification(M=1000, base_learner=DecisionTreeRegressor(max_depth=1, random_state=1), learning_rate=1.0,
                             method="classification", loss="modified_huber")
    model.fit(X_train, y_train)
    pred = model.predict(X_test)
    acc = np.zeros(pred.shape)
    acc[np.where(pred == y_test)] = 1
    accuracy = np.sum(acc) / len(pred)
    print('accuracy modified_huber score: ', accuracy)

输出结果：

RMSE:  8.454462867923157
accuracy logistic score:  0.9434
accuracy modified_huber score:  0.9402

回归：

X, y = datasets.make_regression(n_samples=20000, n_features=20, n_informative=10, noise=100, random_state=1)  # 数据集

下图比较了回归问题中使用平方损失和Huber损失的差别以及各自的early stopping point：

分类：

在分类问题中将上一篇中的 AdaBoost 和本篇中的GBDT作比较，仍使用之前的数据集，其中GBDT分别使用了logistic loss和 这篇文章 最后提到的modified huber loss：

下面换一个噪音较大的数据集，用PCA降到二维进行可视化：

X, y = datasets.make_classification(n_samples=20000, n_features=10, n_informative=4, flip_y=0.3, n_clusters_per_class=1, n_classes=2, random_state=1)

这一次modified loss比logistic loss表现好，但都不如Real AdaBoost。

/