L2范数惩罚项，高维线性回归

%matplotlib inline
import mxnet
from mxnet import nd,autograd
from mxnet import gluon,init
from mxnet.gluon import data as gdata,loss as gloss,nn
import gluonbook as gb

n_train, n_test, num_inputs = 20,100,200

true_w = nd.ones((num_inputs, 1)) * 0.01
true_b = 0.05

features = nd.random.normal(shape=(n_train+n_test, num_inputs))
labels = nd.dot(features,true_w) + true_b
labels += nd.random.normal(scale=0.01, shape=labels.shape)

train_feature = features[:n_train,:]
test_feature = features[n_train:,:]
train_labels = labels[:n_train]
test_labels = labels[n_train:]

#print(features,train_feature,test_feature)

# 初始化模型参数
def init_params():
    w = nd.random.normal(scale=1, shape=(num_inputs, 1))
    b = nd.zeros(shape=(1,))
    w.attach_grad()
    b.attach_grad()
    return [w,b]

# 定义，训练，测试

batch_size = 1
num_epochs = 100
lr = 0.03

train_iter = gdata.DataLoader(gdata.ArrayDataset(train_feature,train_labels),batch_size=batch_size,shuffle=True)

# 定义网络
def linreg(X, w, b):
    return nd.dot(X,w) + b

# 损失函数
def squared_loss(y_hat, y):
    """Squared loss."""
    return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2

# L2 范数惩罚
def l2_penalty(w):
    return (w**2).sum() / 2

def sgd(params, lr, batch_size):
    for param in params:
        param[:] = param - lr * param.grad / batch_size

def fit_and_plot(lambd):
    w, b = init_params()
    train_ls, test_ls = [], []
    for _ in range(num_epochs):
        for X, y in train_iter:
            with autograd.record():
                # 添加了 L2 范数惩罚项。
                l = squared_loss(linreg(X, w, b), y) + lambd * l2_penalty(w)
            l.backward()
            sgd([w, b], lr, batch_size)
        train_ls.append(squared_loss(linreg(train_feature, w, b),
                             train_labels).mean().asscalar())
        test_ls.append(squared_loss(linreg(test_feature, w, b),
                            test_labels).mean().asscalar())
    gb.semilogy(range(1, num_epochs + 1), train_ls, 'epochs', 'loss',
                range(1, num_epochs + 1), test_ls, ['train', 'test'])
    print('L2 norm of w:', w.norm().asscalar())

fit_and_plot(0)
fit_and_plot(3)

训练集太少，容易出现过拟合，即训练集loss远小于测试集loss，解决方案，权重衰减——（L2范数正则化）

例如线性回归：

loss(w1,w2,b) = 1/n * sum(x1w1 + x2w2 + b - y)^2 /2 ，平方损失函数。

权重参数 w = [w1,w2]，

新损失函数 loss(w1,w2,b) += lambd / 2n *||w||^2

迭代方程：