1103过拟合欠拟合

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import math
import numpy as np
import torch
from torch import nn
from d2l import torch as d2l

# 生成数据集
max_degree = 20  # 多项式的最大阶数
n_train, n_test = 100, 100  # 训练和测试数据集大小
true_w = np.zeros(max_degree)  # 分配大量的空间
true_w[0:4] = np.array([5, 1.2, -3.4, 5.6])
# 特征从 𝑥𝑖 调整为 𝑥𝑖/𝑖! 的原因:避免非常大的梯度值或损失值。
features = np.random.normal(size=(n_train + n_test, 1))
np.random.shuffle(features)
poly_features = np.power(features, np.arange(max_degree).reshape(1, -1))
for i in range(max_degree):
    poly_features[:, i] /= math.gamma(i + 1)  # gamma(n)=(n-1)!
# labels的维度:(n_train+n_test,)
labels = np.dot(poly_features, true_w)
labels += np.random.normal(scale=0.1, size=labels.shape)

# NumPy ndarray转换为tensor
true_w, features, poly_features, labels = [torch.tensor(x, dtype=
    torch.float32) for x in [true_w, features, poly_features, labels]]

print(features[:2], poly_features[:2, :], labels[:2])

# 对模型进行训练和测试
# 实现一个函数来评估模型在给定数据集上的损失
def evaluate_loss(net, data_iter, loss):  #@save
    """评估给定数据集上模型的损失"""
    metric = d2l.Accumulator(2)  # 损失的总和,样本数量
    for X, y in data_iter:
        out = net(X)
        y = y.reshape(out.shape)
        l = loss(out, y)
        metric.add(l.sum(), l.numel())
    return metric[0] / metric[1]

# 定义训练函数
def train(train_features, test_features, train_labels, test_labels,
          num_epochs=400):
    loss = nn.MSELoss(reduction='none')
    input_shape = train_features.shape[-1]
    # 不设置偏置，因为我们已经在多项式中实现了它
    net = nn.Sequential(nn.Linear(input_shape, 1, bias=False))
    batch_size = min(10, train_labels.shape[0])
    train_iter = d2l.load_array((train_features, train_labels.reshape(-1,1)),
                                batch_size)
    test_iter = d2l.load_array((test_features, test_labels.reshape(-1,1)),
                               batch_size, is_train=False)
    trainer = torch.optim.SGD(net.parameters(), lr=0.01)
    # animator = d2l.Animator(xlabel='epoch', ylabel='loss', yscale='log',
    #                         xlim=[1, num_epochs], ylim=[1e-3, 1e2],
    #                         legend=['train', 'test'])
    for epoch in range(num_epochs):
        d2l.train_epoch_ch3(net, train_iter, loss, trainer)
        if epoch == 0 or (epoch + 1) % 20 == 0:
            # animator.add(epoch + 1, (evaluate_loss(net, train_iter, loss),
            #                          evaluate_loss(net, test_iter, loss)))
            print(evaluate_loss(net, train_iter, loss))
            print(evaluate_loss(net, test_iter, loss))
            print('---------------------------------')
    print('weight:', net[0].weight.data.numpy())

# 正常
# 从多项式特征中选择前4个维度，即1,x,x^2/2!,x^3/3!
train(poly_features[:n_train, :4], poly_features[n_train:, :4],
      labels[:n_train], labels[n_train:])

# 欠拟合
# 从多项式特征中选择前2个维度，即1和x
train(poly_features[:n_train, :2], poly_features[n_train:, :2],
      labels[:n_train], labels[n_train:])

# 过拟合
# 从多项式特征中选取所有维度
train(poly_features[:n_train, :], poly_features[n_train:, :],
      labels[:n_train], labels[n_train:], num_epochs=1500)