过拟合与欠拟合测试小例子
过拟合、欠拟合
import math
import numpy as np
import torch
from torch import nn
from d2l import torch as d2l
使用3阶多项式生成训练集和测试数据标签
y = 5 + 1.2 x − 3.4 x 2 2 ! + 5.6 x 3 3 ! + ε w h e r e ε ∈ N ( 0 , 0. 1 2 ) y = 5 + 1.2x - 3.4 \frac{x^2}{2!} + 5.6\frac{x^3}{3!} + \varepsilon \quad where \quad \varepsilon \in N(0,0.1^2) y=5+1.2x−3.42!x2+5.63!x3+εwhereε∈N(0,0.12)
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]) #20 个 w值， 前4个给出，后面的全是0
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)
labels = np.dot(poly_features,true_w)
labels += np.random.normal(scale = 0.1, size=labels.shape)
true_w, features, poly_features, labels = [torch.tensor(x,dtype=torch.float32)
for x in [true_w,features,poly_features,labels]]
features[:2],poly_features[:2,:],labels[:2]
评估模型在给的数据集上的损失
def evaluate_loss(net,data_iter,loss):
metric = d2l.Accumulator(2)
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=1000):
loss = nn.MSELoss(reduction='none') # 均方误差
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)) # 单层线性网络
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('weight:', net[0].weight.data.numpy())
三阶多项式拟合（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:])
三阶多项式拟合（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:])
三阶多项式拟合（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)
