python - xgboost 中的泊松回归在低频时失败

Question

我正在尝试在 xgboost 中实现增强型泊松回归模型，但我发现结果在低频时有偏差。为了说明这一点，这里有一些我认为可以重现该问题的最小 Python 代码:

import numpy as np
import pandas as pd
import xgboost as xgb

def get_preds(mult):
    # generate toy dataset for illustration
    # 4 observations with linearly increasing frequencies
    # the frequencies are scaled by `mult`
    dmat = xgb.DMatrix(data=np.array([[0, 0], [0, 1], [1, 0], [1, 1]]),
                       label=[i*mult for i in [1, 2, 3, 4]],
                       weight=[1000, 1000, 1000, 1000])

    # train a poisson booster on the toy data
    bst = xgb.train(
        params={"objective": "count:poisson"},
        dtrain=dmat,
        num_boost_round=100000,
        early_stopping_rounds=5,
        evals=[(dmat, "train")],
        verbose_eval=False)

    # return fitted frequencies after reversing scaling
    return bst.predict(dmat)/mult

# test multipliers in the range [10**(-8), 10**1]
# display fitted frequencies 
mults = [10**i for i in range(-8, 1)]
df = pd.DataFrame(np.round(np.vstack([get_preds(m) for m in mults]), 0))
df.index = mults
df.columns = ["(0, 0)", "(0, 1)", "(1, 0)", "(1, 1)"]
df

# --- result ---
#               (0, 0)   (0, 1)   (1, 0)   (1, 1)
#1.000000e-08  11598.0  11598.0  11598.0  11598.0
#1.000000e-07   1161.0   1161.0   1161.0   1161.0
#1.000000e-06    118.0    118.0    118.0    118.0
#1.000000e-05     12.0     12.0     12.0     12.0
#1.000000e-04      2.0      2.0      3.0      3.0
#1.000000e-03      1.0      2.0      3.0      4.0
#1.000000e-02      1.0      2.0      3.0      4.0
#1.000000e-01      1.0      2.0      3.0      4.0
#1.000000e+00      1.0      2.0      3.0      4.0

请注意，在低频下，预测似乎会爆炸。这可能与 Poisson lambda * 权重下降到 1 以下有关(事实上，将权重增加到 1000 以上确实会将“放大”转移到较低频率)，但我仍然希望预测接近均值训练频率(2.5)。此外(上面的示例中未显示)，减少 eta 似乎会增加预测中的偏差量。

什么会导致这种情况发生？是否有可以减轻影响的可用参数？

Answer 1

经过一番挖掘，我找到了解决方案。在这里记录以防其他人遇到同样的问题。事实证明，我需要添加一个等于平均频率的(自然)对数的偏移项。如果这不是很明显，那是因为初始预测从 0.5 的频率开始，并且需要许多提升迭代才能将预测重新调整为平均频率。

请参阅下面的代码以获取玩具示例的更新。正如我在原始问题中所建议的那样，预测现在接近较低尺度的平均频率 (2.5)。

import numpy as np
import pandas as pd
import xgboost as xgb

def get_preds(mult):
    # generate toy dataset for illustration
    # 4 observations with linearly increasing frequencies
    # the frequencies are scaled by `mult`
    dmat = xgb.DMatrix(data=np.array([[0, 0], [0, 1], [1, 0], [1, 1]]),
                       label=[i*mult for i in [1, 2, 3, 4]],
                       weight=[1000, 1000, 1000, 1000])

    ## adding an offset term equal to the log of the mean frequency
    offset = np.log(np.mean([i*mult for i in [1, 2, 3, 4]]))
    dmat.set_base_margin(np.repeat(offset, 4))

    # train a poisson booster on the toy data
    bst = xgb.train(
        params={"objective": "count:poisson"},
        dtrain=dmat,
        num_boost_round=100000,
        early_stopping_rounds=5,
        evals=[(dmat, "train")],
        verbose_eval=False)

    # return fitted frequencies after reversing scaling
    return bst.predict(dmat)/mult

# test multipliers in the range [10**(-8), 10**1]
# display fitted frequencies 
mults = [10**i for i in range(-8, 1)]
## round to 1 decimal point to show the result approaches 2.5
df = pd.DataFrame(np.round(np.vstack([get_preds(m) for m in mults]), 1))
df.index = mults
df.columns = ["(0, 0)", "(0, 1)", "(1, 0)", "(1, 1)"]
df

# --- result ---
#              (0, 0)  (0, 1)  (1, 0)  (1, 1)
#1.000000e-08     2.5     2.5     2.5     2.5
#1.000000e-07     2.5     2.5     2.5     2.5
#1.000000e-06     2.5     2.5     2.5     2.5
#1.000000e-05     2.5     2.5     2.5     2.5
#1.000000e-04     2.4     2.5     2.5     2.6
#1.000000e-03     1.0     2.0     3.0     4.0
#1.000000e-02     1.0     2.0     3.0     4.0
#1.000000e-01     1.0     2.0     3.0     4.0
#1.000000e+00     1.0     2.0     3.0     4.0