XGBoost：表格資料的重要基準模型

總目標 = 資料損失 + 各樹的複雜度懲罰

Ω(一棵樹) = γT + ½λΣⱼwⱼ²
T：葉數
wⱼ：第j葉的分數
γ、λ ≥ 0

gᵢ = ∂L(yᵢ,Fᵢ)/∂Fᵢ
hᵢ = ∂²L(yᵢ,Fᵢ)/∂Fᵢ²

本輪相關部分 ≈ Σᵢ[gᵢf(xᵢ) + ½hᵢf(xᵢ)²] + Ω(f)

Gⱼ = Σᵢ∈Iⱼ gᵢ
Hⱼ = Σᵢ∈Iⱼ hᵢ

Gⱼwⱼ + ½(Hⱼ+λ)wⱼ²

wⱼ* = −Gⱼ / (Hⱼ+λ)

Obj* = −½Σⱼ Gⱼ²/(Hⱼ+λ) + γT

Net Gain = ½[G左²/(H左+λ) + G右²/(H右+λ)
             − (G左+G右)²/(H左+H右+λ)] − γ

Net Gain = ½[9/3 + 9/2 − 0/4] − 0.1 = 3.65
父葉Obj* = 0.1
分兩葉Obj* = −3.55
改善 = 0.1 − (−3.55) = 3.65

gradients = [-2.0, -1.0, 3.0]
hessians = [1.0, 1.0, 1.0]
lam, gamma = 1.0, 0.1
G_left, H_left = sum(gradients[:2]), sum(hessians[:2])
G_right, H_right = sum(gradients[2:]), sum(hessians[2:])
G_parent, H_parent = sum(gradients), sum(hessians)
w_left = -G_left / (H_left + lam)
w_right = -G_right / (H_right + lam)
parent_obj = -0.5 * G_parent ** 2 / (H_parent + lam) + gamma
split_obj = -0.5 * (G_left ** 2 / (H_left + lam)
                    + G_right ** 2 / (H_right + lam)) + 2 * gamma
print("leaf weights:", w_left, w_right)
print("parent/split objectives:", round(parent_obj, 6), round(split_obj, 6))
print("net gain:", round(parent_obj - split_obj, 6))

python -m pip install "numpy==1.26.3" "scikit-learn==1.5.1" "xgboost==2.1.0"

import numpy as np
import xgboost as xgb
from sklearn.datasets import make_friedman1
from sklearn.dummy import DummyRegressor
from sklearn.metrics import mean_squared_error
from sklearn.model_selection import train_test_split

X, y = make_friedman1(n_samples=1000, n_features=10, noise=1.0, random_state=42)
X_dev, X_test, y_dev, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
X_train, X_valid, y_train, y_valid = train_test_split(
    X_dev, y_dev, test_size=0.25, random_state=42)
model = xgb.XGBRegressor(
    objective="reg:squarederror", n_estimators=1000, max_depth=3,
    learning_rate=0.05, subsample=0.8, colsample_bytree=0.8,
    reg_lambda=1.0, tree_method="hist", eval_metric="rmse",
    early_stopping_rounds=20, random_state=42, n_jobs=1)
model.fit(X_train, y_train, eval_set=[(X_valid, y_valid)], verbose=False)
baseline = DummyRegressor(strategy="mean").fit(X_train, y_train)
rmse = lambda actual, predicted: float(np.sqrt(mean_squared_error(actual, predicted)))
print("xgboost:", xgb.__version__)
print("train/valid/test:", len(y_train), len(y_valid), len(y_test))
print("best round (1-based):", model.best_iteration + 1)
print("stored rounds:", model.get_booster().num_boosted_rounds())
print("validation RMSE:", round(rmse(y_valid, model.predict(X_valid)), 6))
print("test RMSE:", round(rmse(y_test, model.predict(X_test)), 6))
print("constant baseline test RMSE:", round(rmse(y_test, baseline.predict(X_test)), 6))

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