Varying regularization in Multi-layer Perceptron

http://scikit-learn.org/stable/auto_examples/neural_networks/plot_mlp_alpha.html#sphx-glr-auto-examples-neural-networks-plot-mlp-alpha-py

此範例是比較不同的正歸化參數’alpha’，對於使用scikit-learn的資料產生器
，所產生的circles、 moon
和random n-class classification，三種資料集的成效。

PS:正規化為一種處理無限大、發散以及一些不合理表示式的方法，透過引入一項輔助性的概念——正規子(regulator)，去限制函數使得函數不會發散

此處的Alpha參數即為正規子，目的是去限制權重(Weight,W)的大小，以防萬一overfitting與underfitting的問題，增加alpha值可能可以處理overfitting，反之減小alpha可能可以解決underfitting的問題，至於權重大小，如何影響輸出請看圖1:

圖1:比較同樣輸入，對於不同大小權重值，對於輸出的影響左圖為權重為5時，當輸入變動0.1時，輸出增加0.5，即輸出改變10%，右圖為權重為1時，當輸入變動0.1時，輸出增加0.1，即輸出改變2%，通常模型對於input較不敏感，模型表現較好
結果將顯示出:使用不同alpha值去限制權重產生出的決策邊界

(一)引入函式庫

print(__doc__)
# Author: Issam H. Laradji
# License: BSD 3 clause
import numpy as np
from matplotlib import pyplot as plt
from matplotlib.colors import ListedColormap
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import make_moons, make_circles, make_classification
from sklearn.neural_network import MLPClassifier

(二)設定模型參數與產生資料

h = .02  # step size in the mesh
alphas = np.logspace(-5, 3, 5)#
names = []
for i in alphas:
    names.append('alpha ' + str(i))
classifiers = []
for i in alphas:
    classifiers.append(MLPClassifier(alpha=i, random_state=1))
X, y = make_classification(n_features=2, n_redundant=0, n_informative=2,
                           random_state=0, n_clusters_per_class=1)
rng = np.random.RandomState(2)
X += 2 * rng.uniform(size=X.shape)
linearly_separable = (X, y)
datasets = [make_moons(noise=0.3, random_state=0),
            make_circles(noise=0.2, factor=0.5, random_state=1),
            linearly_separable]
figure = plt.figure(figsize=(17, 9))
i = 1
# iterate over datasets
for X, y in datasets:
    # preprocess dataset, split into training and test part
    X = StandardScaler().fit_transform(X)
    X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=.4)
    x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5
    y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5
    xx, yy = np.meshgrid(np.arange(x_min, x_max, h),
                         np.arange(y_min, y_max, h))

(三)繪製圖形

    # just plot the dataset first
    cm = plt.cm.RdBu
    cm_bright = ListedColormap(['#FF0000', '#0000FF'])
    ax = plt.subplot(len(datasets), len(classifiers) + 1, i)
    # Plot the training points
    ax.scatter(X_train[:, 0], X_train[:, 1], c=y_train, cmap=cm_bright)
    # and testing points
    ax.scatter(X_test[:, 0], X_test[:, 1], c=y_test, cmap=cm_bright, alpha=0.6)
    ax.set_xlim(xx.min(), xx.max())
    ax.set_ylim(yy.min(), yy.max())
    ax.set_xticks(())
    ax.set_yticks(())
    i += 1
    # iterate over classifiers
    for name, clf in zip(names, classifiers):
        ax = plt.subplot(len(datasets), len(classifiers) + 1, i)
        clf.fit(X_train, y_train)
        score = clf.score(X_test, y_test)
        # Plot the decision boundary. For that, we will assign a color to each
        # point in the mesh [x_min, x_max]x[y_min, y_max].
        if hasattr(clf, "decision_function"):
            Z = clf.decision_function(np.c_[xx.ravel(), yy.ravel()])
        else:
            Z = clf.predict_proba(np.c_[xx.ravel(), yy.ravel()])[:, 1]
        # Put the result into a color plot
        Z = Z.reshape(xx.shape)
        ax.contourf(xx, yy, Z, cmap=cm, alpha=.8)
        # Plot also the training points
        ax.scatter(X_train[:, 0], X_train[:, 1], c=y_train, cmap=cm_bright)
        # and testing points
        ax.scatter(X_test[:, 0], X_test[:, 1], c=y_test, cmap=cm_bright,
                   alpha=0.6)
        ax.set_xlim(xx.min(), xx.max())
        ax.set_ylim(yy.min(), yy.max())
        ax.set_xticks(())
        ax.set_yticks(())
        ax.set_title(name)
        ax.text(xx.max() - .3, yy.min() + .3, ('%.2f' % score).lstrip('0'),
                size=15, horizontalalignment='right')
        i += 1
figure.subplots_adjust(left=.02, right=.98)
plt.show()

圖2:不同alpha結果圖，每張子圖右下角是分辨率，alpha值很大，模型的結果明顯underfitting