【深度学习】-线性回归

学习来源：李沐老师

线性回归

• 线性模型
• 单层神经网络
• 神经网络源于神经科学
• 衡量预测质量
• 损失函数
• 训练数据
• 参数学习
• 显示解

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• 线性回归是对n维输入的加权，外加偏差
• 使用平方损失来衡量预测值和真实值的差异
• 线性回归有显示解
• 线性回归可以看做是单层神经网络

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基础优化方法

• 梯度下降
• 选择学习率
• 小批量随机梯度下降
• 选择批量大小

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• 梯度下降通过不断沿着反梯度方向更新参数求解
• 小批量随机梯度下降是深度学习默认的求解算法
• 两个重要的超参数是批量大小和学习率

线性回归的从零开始实现

%matplotlib inline
import random
import torch
from d2l import torch as d2l

根据带有噪声的线性模型构造一个人造数据集

def synthetic_data(w, b, num_examples):
    """生成y=Xw+b+噪音"""
    X = torch.normal(0, 1, (num_examples, len(w)))
    y = torch.matmul(X, w) + b
    y += torch.normal(0, 0.01, y.shape)
    return X, y.reshape((-1, 1))


true_w = torch.tensor([2, -3.4])
true_b = 4.2
features, labels = synthetic_data(true_w, true_b, 1000)

print('features:', features[0], '\nlabel:', labels[0])

features: tensor([1.2000, 0.0916]) 
label: tensor([6.2911])

d2l.set_figsize()
d2l.plt.scatter(features[:, (1)].detach().numpy(), labels.detach().numpy(), 1)

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def data_iter(batch_size, features, labels):
    num_examples = len(features)
    indices = list(range(num_examples))
    # 这些样本是随机读取的，没有特定的顺序
    random.shuffle(indices)
    for i in range(0, num_examples, batch_size):
        batch_indices = torch.tensor(indices[i:min(i +
                                                   batch_size, num_examples)])
        yield features[batch_indices], labels[batch_indices]

batch_size = 10

for X, y in data_iter(batch_size, features, labels):
    print(X, '\n', y)
    break

tensor([[-0.4661,  0.6564],
        [-1.0357,  1.5640],
        [-1.1484, -0.2926],
        [-0.1746,  0.1806],
        [-1.1770, -0.5096],
        [ 0.6306, -0.5774],
        [-1.7197, -0.2219],
        [ 0.1825, -0.4533],
        [-1.2249,  0.7895],
        [ 1.4173, -0.3152]]) 
 tensor([[ 1.0428],
        [-3.1820],
        [ 2.8842],
        [ 3.2444],
        [ 3.5915],
        [ 7.4257],
        [ 1.4984],
        [ 6.1103],
        [-0.9367],
        [ 8.1198]])

定义初始化模型参数

w = torch.normal(0, 0.01, size=(2, 1), requires_grad=True)
b = torch.zeros(1, requires_grad=True)

定义模型

def linreg(X, w, b):  #@save
    """线性回归模型"""
    return torch.matmul(X, w) + b

定义损失函数

def squared_loss(y_hat, y):  #@save
    """均方损失"""
    return (y_hat - y.reshape(y_hat.shape))**2 / 2

定义优化算法

def sgd(params, lr, batch_size):  #@save
    """小批量随机梯度下降"""
    with torch.no_grad():
        for param in params:
            param -= lr * param.grad / batch_size
            param.grad.zero_()

训练过程

lr = 0.03
num_epochs = 3
net = linreg
loss = squared_loss

for epoch in range(num_epochs):
    for X, y in data_iter(batch_size, features, labels):
        l = loss(net(X, w, b), y)  # X和y的小批量损失
        # 因为l形状是(batch_size,1)，而不是一个标量。l中的所有元素被加到一起，
        # 并以此计算关于[w,b]的梯度
        l.sum().backward()
        sgd([w, b], lr, batch_size)  # 使用参数的梯度更新参数
    with torch.no_grad():
        train_l = loss(net(features, w, b), labels)
        print(f'epoch {epoch + 1}, loss {float(train_l.mean()):f}')

epoch 1, loss 0.042694
epoch 2, loss 0.000164
epoch 3, loss 0.000049

print(f'w的估计误差: {true_w - w.reshape(true_w.shape)}')
print(f'b的估计误差: {true_b - b}')

w的估计误差: tensor([ 0.0005, -0.0004], grad_fn=)
b的估计误差: tensor([-8.9169e-05], grad_fn=)

线性回归的简洁实现

生成数据集

import numpy as np
import torch
from torch.utils import data
from d2l import torch as d2l

true_w = torch.tensor([2, -3.4])
true_b = 4.2
features, labels = d2l.synthetic_data(true_w, true_b, 1000)

调用框架中现有的API来读取数据

def load_array(data_arrays, batch_size, is_train=True):  #@save
    """构造一个PyTorch数据迭代器"""
    dataset = data.TensorDataset(*data_arrays)
    return data.DataLoader(dataset, batch_size, shuffle=is_train)

batch_size = 10
data_iter = load_array((features, labels), batch_size)

next(iter(data_iter))

[tensor([[-0.0408,  1.0486],
         [-1.6154, -0.5745],
         [ 0.4219, -0.7431],
         [-1.0670,  0.0349],
         [-0.3468,  0.9207],
         [-0.2918,  0.2436],
         [ 1.5633, -0.4152],
         [-0.4183,  0.8276],
         [-0.7691,  0.5424],
         [-1.2084,  0.7723]]),
 tensor([[ 0.5456],
         [ 2.9275],
         [ 7.5761],
         [ 1.9399],
         [ 0.3734],
         [ 2.7938],
         [ 8.7315],
         [ 0.5322],
         [ 0.8215],
         [-0.8483]])]

使用框架的预定义好的层

# nn是神经网络的缩写
from torch import nn

net = nn.Sequential(nn.Linear(2, 1))

初始化模型参数

net[0].weight.data.normal_(0, 0.01)
net[0].bias.data.fill_(0)

tensor([0.])

计算均方误差使用的是MSELoss类，也称为平方范数

loss = nn.MSELoss()

实例化SGD实例

trainer = torch.optim.SGD(net.parameters(), lr=0.03)

训练过程代码与我们从零开始实现时所做的非常相似

num_epochs = 3
for epoch in range(num_epochs):
    for X, y in data_iter:
        l = loss(net(X) ,y)
        trainer.zero_grad()
        l.backward()
        trainer.step()
    l = loss(net(features), labels)
    print(f'epoch {epoch + 1}, loss {l:f}')

epoch 1, loss 0.000321
epoch 2, loss 0.000092
epoch 3, loss 0.000093