Regularization on GBDT

之前一篇文章简单地讲了XGBoost的实现与普通GBDT实现的不同之处，本文尝试总结一下GBDT运用的正则化技巧。

Early Stopping

Early Stopping是机器学习迭代式训练模型中很常见的防止过拟合技巧，维基百科里如下描述:

In machine learning, early stopping is a form of regularization used to avoid overfitting when training a learner with an iterative method, such as gradient descent.

具体的做法是选择一部分样本作为验证集，在迭代拟合训练集的过程中，如果模型在验证集里错误率不再下降，就停止训练，也就是说控制迭代的轮数（树的个数）。

XGBoost Python关于early stopping的参数设置文档非常清晰，API如下：

# code snippets from xgboost python-package training.py
def train(..., evals=(), early_stopping_rounds=None)
    """Train a booster with given parameters.
    Parameters
    ----------
    early_stopping_rounds: int
        Activates early stopping. Validation error needs to decrease at least
        every <early_stopping_rounds> round(s) to continue training.
    """

Sklearn的GBDT实现虽然可以添加early stopping，但是比较复杂。官方没有相应的文档和代码样例，必须看源码。实现的时候需要用户提供monitor回调函数，且要了解源码内部_fit_stages函数的locals，总之对新手很不友好：

#code snippets from sklearn.ensemble.gradient_boosting
class BaseGradientBoosting(six.with_metaclass(ABCMeta, BaseEnsemble,
                           _LearntSelectorMixin)):
    """Abstract base class for Gradient Boosting. """
    ...
    def fit(self, X, y, sample_weight=None, monitor=None):
        """Fit the gradient boosting model.
        Parameters
        ----------
        monitor : callable, optional
            The monitor is called after each iteration with the current
            iteration, a reference to the estimator and the local variables of
            ``_fit_stages`` as keyword arguments ``callable(i, self,
            locals())``. If the callable returns ``True`` the fitting procedure
            is stopped. The monitor can be used for various things such as
            computing held-out estimates, early stopping, model introspect, and
            snapshoting.
        """

对Sklearn感兴趣的可以看这篇文章Using Gradient Boosting (with Early Stopping)，里面有回调函数monitor的参考实现。

Shrinkage

Shrinkage就是将每棵树的输出结果乘一个因子(0<ν<10<ν<1)，其中ΣJmj=1γjmI(x∈Rjm)Σj=1JmγjmI(x∈Rjm)是第m棵的输出，而f(m)f(m)是前m棵树的ensemble:

fm(x)=fm−1(x)+ν⋅ΣJmj=1γjmI(x∈Rjm)fm(x)=fm−1(x)+ν⋅Σj=1JmγjmI(x∈Rjm)

ESL书中这样讲：

The parameter νν can be regarded as controlling the leanring rate of the boosting procedure

νν和迭代轮数M(树个数)是一个tradeoff，推荐的是νν值设置小一点(如0.1)，而M设置大一些。这样一般能有比较好的准确率，代价是训练时间变长(与M成比例)。

下面是Sklearn的实现关于该参数设置的片段，XGBoost类似：

#code snippets from sklearn.ensemble.gradient_boosting
class GradientBoostingClassifier(BaseGradientBoosting, ClassifierMixin):
    """Gradient Boosting for classification."""

    def __init__(self, ..., learning_rate=0.1, n_estimators=100, ...):
    """
    Parameters
    ----------
    learning_rate : float, optional (default=0.1)
        learning rate shrinks the contribution of each tree by `learning_rate`.
        There is a trade-off between learning_rate and n_estimators.
    n_estimators : int (default=100)
        The number of boosting stages to perform. Gradient boosting
        is fairly robust to over-fitting so a large number usually
        results in better performance
    """

Subsampling

Subsampling其实源于bootstrap averaging(bagging)思想，GBDT里的做法是在每一轮建树时，样本是从训练集合中无放回随机抽样的ηη部分，典型的ηη值是0.5。这样做既能对模型起正则作用，也能减少计算时间。

事实上，XGBoost和Sklearn的实现均借鉴了随机森林，除了有样本层次上的采样，也有特征采样。也就是说建树的时候只从随机选取的一些特征列寻找最优分裂。 下面是Sklearn里的相关参数设置的片段，

#code snippets from sklearn.ensemble.gradient_boosting
class GradientBoostingClassifier(BaseGradientBoosting, ClassifierMixin):
    """Gradient Boosting for classification."""

    def __init__(self, ..., subsample=1.0, max_features=None,...):
    """
    Parameters
    ----------
    subsample : float, optional (default=1.0)
        The fraction of samples to be used for fitting the individual base
        learners. If smaller than 1.0 this results in Stochastic Gradient
        Boosting. `subsample` interacts with the parameter `n_estimators`.
        Choosing `subsample < 1.0` leads to a reduction of variance
        and an increase in bias.
    max_features : int, float, string or None, optional (default=None)
        The number of features to consider when looking for the best split:
    """

Regularized Learning Objective

将树模型的复杂度作为正则项显式地加进优化目标里，是XGBoost实现的独到之处。

L(t)=∑i=1nl(yi,y∗(t−1)i+ft(xi))+Ω(ft)L(t)=∑i=1nl(yi,yi∗(t−1)+ft(xi))+Ω(ft)

where

Ω(f)=γT+12λ||w||2Ω(f)=γT+12λ||w||2

其中y∗(t)iyi∗(t)是第t轮第i个instance的预测值，ftft是第t轮建的树，TT是树叶结点数目，ww是树叶结点的输出，γ,λγ,λ是正则化参数。深入了解加了正则后如何推导剃度更新的可以看XGBoost的论文。

我个人的看法是将树模型的复杂度作为正则化项加在优化目标，相比自己通过参数控制每轮树的复杂度更直接，这可能是XGBoost相比普通GBDT实现效果更好的一个很重要的原因。很遗憾，Sklearn暂时无相应的实现。

Dropout

Dropout是deep learning里很常用的正则化技巧，很自然的我们会想能不能把Dropout用到GBDT模型上呢？AISTATS2015有篇文章DART: Dropouts meet Multiple Additive Regression Trees进行了一些尝试。

文中提到GBDT里会出现over-specialization的问题：

Trees added at later iterations tend to impact the prediction of only a few instances, and they make negligible contribution towards the prediction of all the remaining instances. We call this issue of subsequent trees affecting the prediction of only a small fraction of the training instances over-specialization.

也就是说前面迭代的树对预测值的贡献比较大，后面的树会集中预测一小部分样本的偏差。Shrinkage可以减轻over-specialization的问题，但不是很好。作者想通过Dropout来平衡所有树对预测的贡献，如下图的效果： 

具体的做法如下：

DART divergesfrom MART at two places. First, when computing the gradient that the next tree will fit, only a random subset of the existing ensemble is considered. The second place at which DART diverges from MART is when adding the new tree to the ensemble where DART performs a normalization step.

简单说就是每次新加一棵树，这棵树要拟合的并不是之前全部树ensemble后的残差，而是随机抽取的一些树ensemble；同时新加的树结果要规范化一下。

这种新做法对GBDT效果的提升有多明显还有待大家探索尝试。