深度学习线性回归（pytorch)

线性回归

• 模型
  $y = wx + b$y=wx+b

• 损失函数
  $\ell(w_1, w_2, b) =\frac{1}{n} \sum_{i=1}^n \ell^{(i)}(w_1, w_2, b) =\frac{1}{n} \sum_{i=1}^n \frac{1}{2}\left(x_1^{(i)} w_1 + x_2^{(i)} w_2 + b - y^{(i)}\right)^2.$ℓ(w1​,w2​,b)=n1​i=1∑n​ℓ(i)(w1​,w2​,b)=n1​i=1∑n​21​(x1(i)​w1​+x2(i)​w2​+b−y(i))2.

• 优化函数
  $\begin{aligned} w_1 &\leftarrow w_1 - \frac{\eta}{|\mathcal{B}|} \sum_{i \in \mathcal{B}} \frac{ \partial \ell^{(i)}(w_1, w_2, b) }{\partial w_1} = w_1 - \frac{\eta}{|\mathcal{B}|} \sum_{i \in \mathcal{B}}x_1^{(i)} \left(x_1^{(i)} w_1 + x_2^{(i)} w_2 + b - y^{(i)}\right),\\ w_2 &\leftarrow w_2 - \frac{\eta}{|\mathcal{B}|} \sum_{i \in \mathcal{B}} \frac{ \partial \ell^{(i)}(w_1, w_2, b) }{\partial w_2} = w_2 - \frac{\eta}{|\mathcal{B}|} \sum_{i \in \mathcal{B}}x_2^{(i)} \left(x_1^{(i)} w_1 + x_2^{(i)} w_2 + b - y^{(i)}\right),\\ b &\leftarrow b - \frac{\eta}{|\mathcal{B}|} \sum_{i \in \mathcal{B}} \frac{ \partial \ell^{(i)}(w_1, w_2, b) }{\partial b} = b - \frac{\eta}{|\mathcal{B}|} \sum_{i \in \mathcal{B}}\left(x_1^{(i)} w_1 + x_2^{(i)} w_2 + b - y^{(i)}\right). \end{aligned}$w1​w2​b​←w1​−∣B∣η​i∈B∑​∂w1​∂ℓ(i)(w1​,w2​,b)​=w1​−∣B∣η​i∈B∑​x1(i)​(x1(i)​w1​+x2(i)​w2​+b−y(i)),←w2​−∣B∣η​i∈B∑​∂w2​∂ℓ(i)(w1​,w2​,b)​=w2​−∣B∣η​i∈B∑​x2(i)​(x1(i)​w1​+x2(i)​w2​+b−y(i)),←b−∣B∣η​i∈B∑​∂b∂ℓ(i)(w1​,w2​,b)​=b−∣B∣η​i∈B∑​(x1(i)​w1​+x2(i)​w2​+b−y(i)).​

• 神经网络图（单层神经网络）

线性回归的pytorch实现

生成数据集

import torch

num_inputs = 2
num_examples = 1000
true_w = [2, -3.4]
true_b = 4.2
features = torch.randn(num_examples, num_inputs)
labels = true_w[0] * features[:, 0] + true_w[1] * features[:, 1] + true_b
labels += torch.normal(mean=torch.zeros(labels.shape), std=0.01)

读取数据

from torch.utils import data as tdata

batch_size = 10
# 将训练数据的特征和标签组合
dataset = tdata.TensorDataset(features, labels)
# 随机读取小批量
data_iter = tdata.DataLoader(dataset, batch_size, shuffle=True)

for X, y in data_iter:
    print(X, y)
    break

tensor([[ 0.2115,  1.4861],
        [-0.2630,  0.8898],
        [ 0.8301, -2.6101],
        [ 1.5199, -0.5050],
        [-0.4478,  0.6990],
        [ 1.4203,  1.1574],
        [ 1.3185,  1.1949],
        [ 2.0129,  0.8379],
        [ 1.1585, -0.1882],
        [ 0.9050,  0.0398]]) tensor([-0.4229,  0.6551, 14.7292,  8.9412,  0.9225,  3.1058,  2.7778,  5.3856,
         7.1542,  5.8629])

定义模型

先导入nn模块。实际上，“nn”是neural networks（神经网络）的缩写。顾名思义，该模块定义了大量神经网络的层。我们先定义一个模型变量net，它是一个Sequential实例。在nn中，Sequential实例可以看作是一个串联各个层的容器。在构造模型时，我们在该容器中依次添加层。当给定输入数据时，容器中的每一层将依次计算并将输出作为下一层的输入。

from torch import nn

net = nn.Sequential()

 # 全连接层是一个线性层，特征数为2，输出个数为1
net.add_module('linear', nn.Linear(2, 1)) 

初始化模型参数

这里主要是初始化线性回归模型中的权重与偏差。使用nn.init模块
如nn.init.normal(tensor, std=0.01)指定随机初始化将随机采样均值为0、标准差为0.01的正态分布。偏差初始化默认为0

from torch.nn import init

def params_init(model):
    if isinstance(model, nn.Linear):
        init.normal_(tensor=model.weight.data, std=0.01)
        init.constant_(tensor=model.bias.data, val=0)
        
net.apply(params_init)

Sequential(
  (linear): Linear(in_features=2, out_features=1, bias=True)
)

定义损失函数与优化算法

loss = nn.MSELoss() # 均方误差损失函数

from torch import optim

optimizer = optim.SGD(net.parameters(), lr=0.03)  # lr为学习率

训练模型

num_epochs = 3 # 初始化训练周期
for epoch in range(1, num_epochs + 1):
    for X, y in data_iter:
        net.zero_grad()
        l = loss(net(X), y.reshape(batch_size, -1))  # -1表示自动计算列
        l.backward()
        optimizer.step()
        
    with torch.no_grad():
        l = loss(net(features), labels.reshape(num_examples, -1))
        print('epoch %d, loss: %f' % (epoch, l.data.numpy()))

epoch 1, loss: 0.000241
epoch 2, loss: 0.000099
epoch 3, loss: 0.000099

linear = net[0]
true_w, linear.weight.data

([2, -3.4], tensor([[ 2.0001, -3.4001]]))

true_b, linear.bias.data

(4.2, tensor([4.2001]))

net

Sequential(
  (linear): Linear(in_features=2, out_features=1, bias=True)
)