模型复杂度影响#

演示模型复杂度如何影响预测准确性和计算性能。

我们将使用两个数据集

用于回归的糖尿病数据集。该数据集包含从糖尿病患者身上获取的 10 项测量值。任务是预测疾病进展；
用于分类的20 个新闻组文本数据集。该数据集包含新闻组帖子。任务是预测帖子是关于哪个主题（在 20 个主题中）写的。

我们将对三个不同的估计器建模复杂度影响

SGDClassifier（用于分类数据），它实现了随机梯度下降学习；
NuSVR（用于回归数据），它实现了 Nu 支持向量回归；
GradientBoostingRegressor 以正向逐步的方式构建加性模型。请注意，从中等大小的数据集 (n_samples >= 10_000) 开始，HistGradientBoostingRegressor 比 GradientBoostingRegressor 快得多，但本例并非如此。

我们通过在我们选择的每个模型中选择相关的模型参数来改变模型复杂度。接下来，我们将测量对计算性能（延迟）和预测能力（MSE 或汉明损失）的影响。

# Authors: Eustache Diemert <[email protected]>
#          Maria Telenczuk <https://github.com/maikia>
#          Guillaume Lemaitre <[email protected]>
# License: BSD 3 clause

import time

import matplotlib.pyplot as plt
import numpy as np

from sklearn import datasets
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.linear_model import SGDClassifier
from sklearn.metrics import hamming_loss, mean_squared_error
from sklearn.model_selection import train_test_split
from sklearn.svm import NuSVR

# Initialize random generator
np.random.seed(0)

加载数据#

首先，我们加载两个数据集。

注意

我们正在使用 fetch_20newsgroups_vectorized 下载 20 个新闻组数据集。它返回可以直接使用的特征。

注意

20 个新闻组数据集的 X 是一个稀疏矩阵，而糖尿病数据集的 X 是一个 numpy 数组。

def generate_data(case):
    """Generate regression/classification data."""
    if case == "regression":
        X, y = datasets.load_diabetes(return_X_y=True)
        train_size = 0.8
    elif case == "classification":
        X, y = datasets.fetch_20newsgroups_vectorized(subset="all", return_X_y=True)
        train_size = 0.4  # to make the example run faster

    X_train, X_test, y_train, y_test = train_test_split(
        X, y, train_size=train_size, random_state=0
    )

    data = {"X_train": X_train, "X_test": X_test, "y_train": y_train, "y_test": y_test}
    return data


regression_data = generate_data("regression")
classification_data = generate_data("classification")

基准测试影响#

接下来，我们可以计算参数对给定估计器的影响。在每一轮中，我们将使用 changing_param 的新值设置估计器，并且我们将收集预测时间、预测性能和复杂度，以查看这些变化如何影响估计器。我们将使用作为参数传递的 complexity_computer 计算复杂度。

def benchmark_influence(conf):
    """
    Benchmark influence of `changing_param` on both MSE and latency.
    """
    prediction_times = []
    prediction_powers = []
    complexities = []
    for param_value in conf["changing_param_values"]:
        conf["tuned_params"][conf["changing_param"]] = param_value
        estimator = conf["estimator"](**conf["tuned_params"])

        print("Benchmarking %s" % estimator)
        estimator.fit(conf["data"]["X_train"], conf["data"]["y_train"])
        conf["postfit_hook"](estimator)
        complexity = conf["complexity_computer"](estimator)
        complexities.append(complexity)
        start_time = time.time()
        for _ in range(conf["n_samples"]):
            y_pred = estimator.predict(conf["data"]["X_test"])
        elapsed_time = (time.time() - start_time) / float(conf["n_samples"])
        prediction_times.append(elapsed_time)
        pred_score = conf["prediction_performance_computer"](
            conf["data"]["y_test"], y_pred
        )
        prediction_powers.append(pred_score)
        print(
            "Complexity: %d | %s: %.4f | Pred. Time: %fs\n"
            % (
                complexity,
                conf["prediction_performance_label"],
                pred_score,
                elapsed_time,
            )
        )
    return prediction_powers, prediction_times, complexities

选择参数#

我们通过创建一个包含所有必要值的字典来为每个估计器选择参数。changing_param 是将在每个估计器中变化的参数的名称。复杂度将由 complexity_label 定义，并使用 complexity_computer 计算。另请注意，根据估计器类型，我们传递不同的数据。

def _count_nonzero_coefficients(estimator):
    a = estimator.coef_.toarray()
    return np.count_nonzero(a)


configurations = [
    {
        "estimator": SGDClassifier,
        "tuned_params": {
            "penalty": "elasticnet",
            "alpha": 0.001,
            "loss": "modified_huber",
            "fit_intercept": True,
            "tol": 1e-1,
            "n_iter_no_change": 2,
        },
        "changing_param": "l1_ratio",
        "changing_param_values": [0.25, 0.5, 0.75, 0.9],
        "complexity_label": "non_zero coefficients",
        "complexity_computer": _count_nonzero_coefficients,
        "prediction_performance_computer": hamming_loss,
        "prediction_performance_label": "Hamming Loss (Misclassification Ratio)",
        "postfit_hook": lambda x: x.sparsify(),
        "data": classification_data,
        "n_samples": 5,
    },
    {
        "estimator": NuSVR,
        "tuned_params": {"C": 1e3, "gamma": 2**-15},
        "changing_param": "nu",
        "changing_param_values": [0.05, 0.1, 0.2, 0.35, 0.5],
        "complexity_label": "n_support_vectors",
        "complexity_computer": lambda x: len(x.support_vectors_),
        "data": regression_data,
        "postfit_hook": lambda x: x,
        "prediction_performance_computer": mean_squared_error,
        "prediction_performance_label": "MSE",
        "n_samples": 15,
    },
    {
        "estimator": GradientBoostingRegressor,
        "tuned_params": {
            "loss": "squared_error",
            "learning_rate": 0.05,
            "max_depth": 2,
        },
        "changing_param": "n_estimators",
        "changing_param_values": [10, 25, 50, 75, 100],
        "complexity_label": "n_trees",
        "complexity_computer": lambda x: x.n_estimators,
        "data": regression_data,
        "postfit_hook": lambda x: x,
        "prediction_performance_computer": mean_squared_error,
        "prediction_performance_label": "MSE",
        "n_samples": 15,
    },
]

运行代码并绘制结果#

我们定义了运行基准测试所需的所有函数。现在，我们将循环遍历我们之前定义的不同配置。随后，我们可以分析从基准测试中获得的图：放宽 SGD 分类器中的 L1 惩罚会降低预测误差，但会导致训练时间增加。我们可以对训练时间进行类似的分析，训练时间随着 Nu-SVR 中支持向量数量的增加而增加。但是，我们观察到存在一个最佳的支持向量数量，可以减少预测误差。事实上，支持向量太少会导致模型欠拟合，而支持向量太多会导致模型过拟合。对于梯度提升模型，可以得出完全相同的结论。与 Nu-SVR 的唯一区别是，在集成中拥有太多树木并不像那样有害。

def plot_influence(conf, mse_values, prediction_times, complexities):
    """
    Plot influence of model complexity on both accuracy and latency.
    """

    fig = plt.figure()
    fig.subplots_adjust(right=0.75)

    # first axes (prediction error)
    ax1 = fig.add_subplot(111)
    line1 = ax1.plot(complexities, mse_values, c="tab:blue", ls="-")[0]
    ax1.set_xlabel("Model Complexity (%s)" % conf["complexity_label"])
    y1_label = conf["prediction_performance_label"]
    ax1.set_ylabel(y1_label)

    ax1.spines["left"].set_color(line1.get_color())
    ax1.yaxis.label.set_color(line1.get_color())
    ax1.tick_params(axis="y", colors=line1.get_color())

    # second axes (latency)
    ax2 = fig.add_subplot(111, sharex=ax1, frameon=False)
    line2 = ax2.plot(complexities, prediction_times, c="tab:orange", ls="-")[0]
    ax2.yaxis.tick_right()
    ax2.yaxis.set_label_position("right")
    y2_label = "Time (s)"
    ax2.set_ylabel(y2_label)
    ax1.spines["right"].set_color(line2.get_color())
    ax2.yaxis.label.set_color(line2.get_color())
    ax2.tick_params(axis="y", colors=line2.get_color())

    plt.legend(
        (line1, line2), ("prediction error", "prediction latency"), loc="upper center"
    )

    plt.title(
        "Influence of varying '%s' on %s"
        % (conf["changing_param"], conf["estimator"].__name__)
    )


for conf in configurations:
    prediction_performances, prediction_times, complexities = benchmark_influence(conf)
    plot_influence(conf, prediction_performances, prediction_times, complexities)
plt.show()

Benchmarking SGDClassifier(alpha=0.001, l1_ratio=0.25, loss='modified_huber',
              n_iter_no_change=2, penalty='elasticnet', tol=0.1)
Complexity: 4948 | Hamming Loss (Misclassification Ratio): 0.2675 | Pred. Time: 0.060812s

Benchmarking SGDClassifier(alpha=0.001, l1_ratio=0.5, loss='modified_huber',
              n_iter_no_change=2, penalty='elasticnet', tol=0.1)
Complexity: 1847 | Hamming Loss (Misclassification Ratio): 0.3264 | Pred. Time: 0.047284s

Benchmarking SGDClassifier(alpha=0.001, l1_ratio=0.75, loss='modified_huber',
              n_iter_no_change=2, penalty='elasticnet', tol=0.1)
Complexity: 997 | Hamming Loss (Misclassification Ratio): 0.3383 | Pred. Time: 0.038960s

Benchmarking SGDClassifier(alpha=0.001, l1_ratio=0.9, loss='modified_huber',
              n_iter_no_change=2, penalty='elasticnet', tol=0.1)
Complexity: 802 | Hamming Loss (Misclassification Ratio): 0.3582 | Pred. Time: 0.034147s

Benchmarking NuSVR(C=1000.0, gamma=3.0517578125e-05, nu=0.05)
Complexity: 18 | MSE: 5558.7313 | Pred. Time: 0.000186s

Benchmarking NuSVR(C=1000.0, gamma=3.0517578125e-05, nu=0.1)
Complexity: 36 | MSE: 5289.8022 | Pred. Time: 0.000259s

Benchmarking NuSVR(C=1000.0, gamma=3.0517578125e-05, nu=0.2)
Complexity: 72 | MSE: 5193.8353 | Pred. Time: 0.000411s

Benchmarking NuSVR(C=1000.0, gamma=3.0517578125e-05, nu=0.35)
Complexity: 124 | MSE: 5131.3279 | Pred. Time: 0.000740s

Benchmarking NuSVR(C=1000.0, gamma=3.0517578125e-05)
Complexity: 178 | MSE: 5149.0779 | Pred. Time: 0.000961s

Benchmarking GradientBoostingRegressor(learning_rate=0.05, max_depth=2, n_estimators=10)
Complexity: 10 | MSE: 4066.4812 | Pred. Time: 0.000185s

Benchmarking GradientBoostingRegressor(learning_rate=0.05, max_depth=2, n_estimators=25)
Complexity: 25 | MSE: 3551.1723 | Pred. Time: 0.000212s

Benchmarking GradientBoostingRegressor(learning_rate=0.05, max_depth=2, n_estimators=50)
Complexity: 50 | MSE: 3445.2171 | Pred. Time: 0.000241s

Benchmarking GradientBoostingRegressor(learning_rate=0.05, max_depth=2, n_estimators=75)
Complexity: 75 | MSE: 3433.0358 | Pred. Time: 0.000528s

Benchmarking GradientBoostingRegressor(learning_rate=0.05, max_depth=2)
Complexity: 100 | MSE: 3456.0602 | Pred. Time: 0.000333s

结论#

总之，我们可以推断出以下见解

更复杂（或更具表现力）的模型将需要更长的训练时间；
更复杂的模型并不能保证减少预测误差。

这些方面与模型泛化以及避免模型欠拟合或过拟合有关。

**脚本总运行时间：**（0 分钟 20.491 秒）