⭐本文内容：多变量线性回归数学推导，梯度下降，基于Pytorch的代码实现
💌参考链接：
3.3. 线性回归的简洁实现 — 动手学深度学习 2.0.0-beta0 documentation (d2l.ai)
基本原理
梯度下降
1.步骤
- 挑选一个初始值 w 0
- 重复迭代参数：t= 1，2，3
- 沿梯度方向将增加损失函数值
- 🍔学习率 n：步长的参数
2.小批量梯度下降
小批量梯度下降是深度学习默认的求解方法
为了节约训练的时间和数据，我们可以随机抽取 b 个样本
i 1 , i 2 , i 3 , … … … i b
以便近似损失
- 🍟b 是选取的训练样本多少，不宜过大或过小
线性回归实现
import torch import random import numpy as np import matplotlib_inline.backend_inline from matplotlib import pyplot as plt
1.生成数据集
定义权重和偏差
num_inputs = 2 #输入两个参数：w、b num_examples = 1000 #样本个数 true_w = [2, -3.4] true_b = 4.2
随机生成正态分布的输入 fatures，feature包括两个影响因素（参数），w和b
features = torch.tensor(np.random.normal(0, 1, (num_examples,num_inputs)), dtype=torch.float)
根据输入 features ,生成输出 labels
labels = true_w[0] * features[:, 0] + true_w[1] * features[:, 1] +true_b labels = labels+torch.tensor(np.random.normal(0, 0.01,size = labels.size()), dtype=torch.float) #在真实数据的基础上，加上噪声
2.数据可视化
def use_svg_display(): # 用⽮量图显示 matplotlib_inline.backend_inline.set_matplotlib_formats('svg') def set_figsize(figsize=(3.5, 2.5)): use_svg_display() # 设置图的尺⼨ plt.rcParams['figure.figsize'] = figsize set_figsize() plt.scatter(features[:, 1].numpy(), labels.numpy(), 1);
3.读取数据
PyTorch提供了了 data 包来读取数据
batch_size = 10，随机读取包含10个数据样本的小批量
import torch.utils.data as Data batch_size = 10
将训练数据的特征和标签组合
features,labels
(tensor([[ 2.1282, -0.8920], [-1.0093, 0.2924], [ 1.5258, -0.5783], ..., [-1.6918, -0.7286], [ 0.2524, 1.0183], [-0.7605, -1.3882]]), tensor([ 1.1486e+01, 1.1739e+00, 9.2210e+00, -3.1652e+00, 1.3957e+00, 1.4540e+00, 1.0448e+01, -3.0799e+00, -6.0249e+00, 6.2256e+00, 1.8337e+00, -2.2532e+00, 9.8570e+00, 4.3742e+00, 1.9240e+00, 9.9293e+00, 4.0584e+00, 8.3627e+00, 1.0823e+00, 7.2722e-01, -7.6379e-01, 5.2927e+00, 6.2904e+00, 5.2404e+00, 3.8104e+00, 6.5634e+00, 4.5559e-01, 1.1481e+00, 6.5535e-01, 5.5602e+00, -4.5899e-01, -4.4074e+00, 3.7406e+00, 7.3968e+00, 3.1700e+00, 8.9007e+00, 9.8127e+00, 1.3216e+00, 6.4583e+00, 8.1877e+00, 1.1374e+00, 1.4746e+00, 6.3436e+00, 3.4523e+00, 7.4513e+00, -5.2496e+00, 2.1701e+00, 6.2465e+00, 3.6356e-02, 3.9021e+00, 9.7487e+00, 1.1606e+01, -5.7617e+00, 1.1398e+00, 2.4662e+00, 6.8591e+00, 2.6556e+00, 1.3848e+00, 8.7197e+00, 8.0591e+00, 3.8578e+00, -3.5013e+00, 2.3531e+00, 5.4477e+00, 1.1095e+00, 3.5379e+00, 4.1187e+00, 5.8690e+00, 8.4104e-01, 3.7816e+00, 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将训练数据的特征和标签组合
dataset = Data.TensorDataset(features,labels) dataset
<torch.utils.data.dataset.TensorDataset at 0x7f1cf8f69250>
随机读取小批量
data_iter = Data.DataLoader(dataset, batch_size, shuffle=True) data_iter
<torch.utils.data.dataloader.DataLoader at 0x7f1cf8f69d90>
打印第一个小批量样本数据
for x, y in data_iter: print(x, y) break
tensor([[ 1.8036, 0.2046], [-0.1144, 1.2728], [ 0.4237, -0.0737], [ 0.9859, -0.9073], [-1.9821, 2.7871], [-0.7970, -1.2866], [ 0.1646, -0.6856], [ 0.7002, 1.0762], [-0.9576, -1.3074], [ 0.7070, -0.1870]]) tensor([ 7.1322, -0.3659, 5.2927, 9.2499, -9.2418, 6.9704, 6.8748, 1.9477, 6.7563, 6.2395])
4.定义模型
导⼊ torch.nn 模块
import torch.nn as nn
用 nn.Sequential 来更加⽅便地搭建网络， Sequential 是一个有序的容器，网络层将按照在传入 Sequential 的顺序依次被添加到计算图中
net = nn.Sequential( nn.Linear(num_inputs, 1) # 此处还可以传⼊入其他层 ) print(net) print(net[0])
Sequential( (0): Linear(in_features=2, out_features=1, bias=True) ) Linear(in_features=2, out_features=1, bias=True)
可以通过 net.parameters() 来查看模型所有的可学习参数，此函数将返回一个生成器
for param in net.parameters(): print(param)
Parameter containing: tensor([[0.6623, 0.0845]], requires_grad=True) Parameter containing: tensor([0.1282], requires_grad=True)
5.初始化参数
from torch.nn import init
我们通过 init.normal_ 将权重参数每个元素初始化为随机采样于均值为0、标准差为0.01的正态分布。偏差会初始化为零。
init.normal_(net[0].weight,mean=0,std=0.01) init.constant_(net[0].bias,val=0) #也可以直接修改bias的data net[0].bias.data.fill_(0)
tensor([0.])
6.定义损失函数
使用均方误差损失作为模型的损失函数
loss = nn.MSELoss()
7.定义优化算法
指定学习率为0.03的⼩小批量量随机梯度下降（SGD）为优化算法
import torch.optim as optim optimizer = optim.SGD( #如果对某个参数不指定学习率，就使用最外层的默认学习率 net.parameters(),lr=0.03 ) print(optimizer)
SGD ( Parameter Group 0 dampening: 0 lr: 0.03 momentum: 0 nesterov: False weight_decay: 0 )
构建新的optimizer，动态调整学习率
for param_group in optimizer.param_groups: param_group['lr'] *= 0.1 #学习率为之前的0.1倍
8.训练模型
num_epochs = 10 #迭代的次数，次数越大最后的准确率越高 for epoch in range(1,num_epochs+1): for x,y in data_iter: output = net(x) l = loss(output,y.view(-1,1)) optimizer.zero_grad() #梯度清零，等价于net.zero_grad() l.backward() optimizer.step() print('epoch: %d,loss: %f' %(epoch,l.item()))
epoch: 1,loss: 15.321363 epoch: 2,loss: 6.421220 epoch: 3,loss: 0.666509 epoch: 4,loss: 0.269027 epoch: 5,loss: 0.071433 epoch: 6,loss: 0.027302 epoch: 7,loss: 0.005877 epoch: 8,loss: 0.001547 epoch: 9,loss: 0.000483 epoch: 10,loss: 0.000219
训练值和真实值对比
dense = net[0] print(true_w, dense.weight) print(true_b, dense.bias)
[2, -3.4] Parameter containing: tensor([[ 1.9972, -3.3939]], requires_grad=True) 4.2 Parameter containing: tensor([4.1887], requires_grad=True)
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