- 基本原理
- 梯度下降
- 1.步骤
- 2.小批量梯度下降
- 线性回归实现
- 1.生成数据集
- 2.数据可视化
- 3.读取数据
- 4.定义模型
- 5.初始化参数
- 6.定义损失函数
- 7.定义优化算法
- 8.训练模型
⭐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：
ML1 单变量线性回归_什么都只会一点的博客-CSDN博客
dl3.2线性回归 | Kaggle
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x = [ x 1 , x 2 , x 3 , x 4 , ⋯ ⋯ , x n ] ⊤ w = [ w 1 , w 2 , ⋯ ⋯ , w n ] ⊤ y = w 1 x 1 + w 2 x 2 + ⋯ ⋯ + w n x n + b y = ⟨ w , x ⟩ + b D ( y ) = 1 2 ( y − y ^ ) 2 l ( x , y , w , b ) = 1 2 n ∑ i = 1 n ( y i − y ^ ) 2 = 1 2 n ∑ i = 1 n ( y i − ⟨ w , x ⟩ − b ) 2 l ( x , y , w , b ) → min → w ∗ , b ∗ \begin{array}{l} x=\left[x_{1}, x_{2}, x_{3}, x_{4} ,\cdots \cdots, x_{n}\right]^{\top}\ w=\left[w_{1}, w_{2}, \cdots \cdots, w_{n}\right]^{\top}\ y=w_{1} x_{1}+w_{2} x_{2}+\cdots \cdots+w_{n} x_{n}+b\ y=\langle w, x\rangle+b\ D(y)=\frac{1}{2}(y-\hat{y})^{2}\ l(x, y, w, b)=\frac{1}{2 n} \sum_{i=1}^{n}\left(y_{i}-\hat{y} \right)^{2}\ =\frac{1}{2 n} \sum_{i=1}^{n}\left(y_{i}-\langle w, x\rangle-b\right)^{2}\ l(x, y, w, b) \rightarrow \min\ \rightarrow w^{*}, b^{*} \end{array} x=[x1,x2,x3,x4,⋯⋯,xn]⊤w=[w1,w2,⋯⋯,wn]⊤y=w1x1+w2x2+⋯⋯+wnxn+by=⟨w,x⟩+bD(y)=21(y−y^)2l(x,y,w,b)=2n1∑i=1n(yi−y^)2=2n1∑i=1n(yi−⟨w,x⟩−b)2l(x,y,w,b)→min→w∗,b∗
梯度下降 1.步骤-
挑选一个初始值 w 0 w_0 w0
-
重复迭代参数：t= 1，2，3
w t = w t − 1 − η ∂ l ∂ w t − 1 \mathbf{w}_{t}=\mathbf{w}_{t-1}-\eta \frac{\partial l}{\partial \mathbf{w}_{t-1}} wt=wt−1−η∂wt−1∂l
-
沿梯度方向将增加损失函数值
w 0 − w 1 ： η ∂ l ∂ w t − 1 w_0-w_1：\eta \frac{\partial l}{\partial \mathbf{w}_{t-1}} w0−w1：η∂wt−1∂l
-
🍔学习率 n：步长的参数
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
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…
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i_1,i_2,i_3,………i_b
i1,i2,i3,………ib以便近似损失
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\frac{1}{b} \sum_{i \in I_{b}} l^{2}\left(\mathbf{x}_{i}, y_{i}, \mathbf{w}\right)
b1i∈Ib∑l2(xi,yi,w)
- 🍟
b是选取的训练样本多少，不宜过大或过小
import torch
import random
import numpy as np
import matplotlib_inline.backend_inline
from matplotlib import pyplot as plt
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
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,
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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,
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6.2324e+00, 1.2647e+01, 7.2575e+00, 6.2891e+00, 4.9514e+00,
6.2109e+00, 5.0092e+00, -9.9337e-01, 5.1607e+00, 6.7718e+00,
9.1802e+00, 6.9370e+00, 2.4028e+00, 4.2869e+00, 3.7471e+00,
8.8147e+00, 6.0244e+00, 6.3032e+00, 2.6607e+00, 3.3901e+00,
6.7301e+00, 5.5077e+00, 1.7717e+00, 6.5372e-01, 1.0034e+00,
2.8830e+00, 1.1419e+01, 1.1749e+00, 1.0106e-02, 8.7469e+00,
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-3.9768e+00, 2.6253e+00, 5.3503e+00, 4.5756e+00, 1.3480e+00,
5.1243e+00, 8.4004e+00, 3.4222e+00, 6.4114e+00, 5.3183e-01,
9.8974e+00, 2.7052e+00, 3.0212e+00, 2.8368e+00, 8.3128e+00,
4.6546e+00, -4.9879e+00, -4.3837e+00, 2.0918e+00, 4.9817e+00,
-2.1596e+00, 1.6440e+00, -2.1060e+00, 6.8430e+00, 2.1973e+00,
4.6677e+00, 5.2363e+00, -3.8503e+00, 9.0410e+00, 1.0080e+01,
5.9184e+00, 1.8692e+01, 4.9923e+00, 6.0264e+00, 8.2590e-01,
8.0111e+00, 1.3049e+01, 1.0785e+01, 7.4764e+00, 3.9071e+00,
1.5800e+00, 7.7218e-01, -9.2418e+00, 4.4402e+00, 5.6216e+00,
6.0654e+00, 3.6753e+00, 1.1367e+01, 7.0856e+00, 2.8123e+00,
3.8476e+00, 5.7843e+00, -2.6613e+00, 8.5492e+00, 1.0268e+01,
3.4798e+00, 6.6199e+00, 1.2376e+01, 5.1228e+00, 1.3886e+00,
2.0064e+00, -2.6576e+00, 6.8330e+00, 3.6461e+00, 6.6444e+00,
1.3514e+01, 6.6098e+00, 1.6626e+00, 7.5031e+00, 3.7306e+00,
4.1361e+00, -1.0839e+00, 4.7839e+00, 1.1011e+00, 9.5930e+00,
9.0655e+00, 6.2866e+00, -3.4704e-02, 2.1236e+00, 3.6567e+00,
1.1865e+01, 5.2644e+00, 3.9271e+00, -3.2964e+00, -1.0615e+00,
7.9056e+00, 3.3556e+00, 6.9934e+00, 3.8433e+00, 5.2684e+00,
1.0572e+01, 5.6345e+00, 5.2405e+00, 7.0610e+00, 8.5568e+00,
9.2425e+00, 3.1511e+00, 1.2075e+01, -1.6131e+00, 2.7143e+00,
1.2283e+01, 2.9278e+00, 1.6995e+00, 4.1698e+00, 6.1994e+00,
-2.4863e+00, 1.0766e+01, 8.6130e+00, -1.4727e+00, 9.1569e+00,
2.3191e+00, 9.3571e+00, 3.9649e+00, 6.5649e+00, 5.1595e-01,
4.0500e+00, 4.2225e+00, 8.2217e+00, -6.3868e-01, 7.4402e+00,
-2.4692e+00, -3.7820e+00, 1.8397e+00, -3.7670e-01, -2.8920e+00,
3.5418e+00, -1.1415e+00, 8.7870e+00, 4.9213e+00, 9.9247e+00,
6.6192e+00, 1.7358e+00, -3.6307e-01, -3.6719e+00, 4.1485e+00,
5.8974e+00, 5.5616e+00, 8.7071e+00, 6.3648e+00, 1.2180e+01,
1.5238e+00, 7.0139e+00, 2.3830e+00, 4.3800e-03, -1.9390e+00,
4.2378e+00, 1.9782e+00, 7.2340e+00, 6.6344e+00, 3.1206e+00,
9.1773e+00, 1.0010e+00, 6.2425e-01, 2.2324e+00, 1.4394e+00,
-1.0583e+00, 1.1862e+00, 3.3771e+00, 6.6086e+00, 1.5868e+00,
3.7503e+00, 1.9657e+00, 5.8942e+00, 5.5674e+00, 9.7752e+00,
3.9570e+00, 9.6817e+00, 5.5203e+00, 3.2171e+00, 2.0135e+00,
1.9813e+00, 7.1679e+00, 1.2223e+00, 6.4231e+00, 3.3712e+00,
6.1226e+00, 7.3023e+00, 7.1456e+00, 1.1152e+00, 4.3584e+00,
-2.1064e+00, 2.5603e+00, 9.0358e+00, 1.0727e+01, 5.5608e+00,
2.2114e+00, 1.9765e+00, 5.3305e+00, 1.1187e+00, 4.9850e+00,
6.4261e+00, 5.4080e+00, 1.8553e+00, 5.4068e+00, 1.3708e+01,
2.3931e+00, 4.4118e+00, 3.2059e+00, -3.6586e-01, 1.8181e+00,
-3.6866e-02, 4.8933e+00, 5.8703e+00, 8.9500e+00, 5.1381e+00,
5.0268e+00, 6.1895e+00, 2.0560e+01, 2.9177e+00, -1.1442e+00,
-4.0448e+00, 3.6746e+00, 4.4213e+00, 3.9709e+00, 1.0074e+01,
7.8624e+00, 4.4473e+00, 3.1089e+00, -2.1261e+00, 4.5493e+00,
6.1902e+00, 3.9441e-01, 6.3312e+00, 2.6792e+00, 2.2669e+00,
5.3156e+00, 3.5936e-01, 3.0885e+00, 4.6025e+00, 3.0167e+00,
-4.9812e-01, -5.2459e-02, 8.6781e+00, 6.2955e+00, 1.1866e+01,
3.0411e+00, 1.7559e-01, 5.5648e+00, 6.2147e+00, 7.5455e+00,
9.4367e+00, 8.8933e+00, 2.5610e+00, 4.3828e+00, 3.7813e+00,
2.0521e+00, 2.8338e+00, 1.0147e+00, 1.6295e+00, 1.7991e+00,
4.6832e+00, 5.6675e+00, -4.6209e-01, 3.3670e+00, 2.1762e+00,
5.9204e+00, 6.4009e+00, 3.7618e+00, 1.2200e+01, 2.5451e+00,
3.5890e-01, -4.5513e-01, 5.3301e+00, -2.5259e-01, 6.0445e+00,
7.6773e+00, -8.3305e+00, 5.9668e+00, 5.1585e+00, 1.4224e+00,
7.0593e+00, -7.1631e-01, 6.0731e+00, 5.7953e+00, -1.9298e-01,
2.0513e+00, 2.5159e+00, 7.1324e+00, 1.2148e+00, 2.3089e+00,
1.5061e+01, 1.3625e+00, 2.6933e+00, 6.7279e+00, -2.5925e+00,
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-3.1313e-01, 1.3205e+01, 2.0962e+00, -8.5085e-01, 2.3345e+00,
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1.6860e+00, 7.1576e+00, 5.5403e+00, 1.5703e+01, -2.4011e-01,
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4.1862e+00, 6.3072e+00, 6.5124e+00, 1.2564e+01, -2.1824e+00,
9.7297e+00, 3.1217e+00, 2.9313e+00, 6.0977e+00, 5.8781e-01,
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1.0711e+01, 1.2701e+01, 1.4397e+00, 1.4085e+00, 4.3338e+00,
2.6513e+00, 2.3139e+00, 6.7066e+00, 1.1019e+01, 4.3761e+00,
2.9457e+00, -1.6806e+00, 1.1144e+01, 6.7122e+00, 2.5283e+00,
3.9789e+00, 3.0566e-01, 3.1576e+00, -1.0638e+00, 5.3148e+00,
8.6122e-01, 9.8231e+00, -1.7613e-01, 7.0995e+00, 6.3133e+00,
4.8049e+00, -1.4338e+00, 2.9295e-01, 7.9991e+00, 4.0050e+00,
4.4134e+00, 9.9134e+00, 6.9708e+00, 4.1459e+00, 5.1887e+00,
7.1081e+00, -3.4246e+00, 7.5349e+00, 3.7021e+00, 3.6239e+00,
1.0308e+01, 4.3593e+00, 6.3187e+00, 6.4794e+00, 1.4713e+01,
-2.8738e+00, 5.1832e+00, 6.5156e+00, 1.1308e+00, 3.0447e+00,
7.3644e-01, 1.3814e+00, 1.0465e+01, 1.3175e+01, -1.2418e+00,
3.7152e+00, 4.1893e+00, 8.6820e-01, 7.6717e+00, 4.2367e+00,
5.7733e+00, 4.8947e+00, 8.6885e+00, 5.7364e+00, 5.9605e+00,
4.0355e+00, 2.8231e+00, 4.3217e+00, -5.2753e+00, 7.0826e+00,
9.9783e+00, -1.8532e+00, 7.9443e+00, 3.8591e+00, 9.1033e+00,
-3.5306e+00, 5.0788e+00, 8.3283e+00, 3.4355e+00, 6.3829e+00,
5.2743e-01, -1.2355e+00, 5.8636e+00, 2.2426e+00, 1.7321e+00,
1.2331e+00, 8.1444e+00, 9.1087e-01, 6.5348e+00, 8.8130e+00,
2.4073e+00, 9.1603e+00, 5.4269e-02, 8.6125e-01, 8.5849e+00,
5.8992e+00, 1.0239e+00, 5.5160e-01, 9.0970e+00, 7.4586e+00,
4.4703e+00, 7.9342e+00, -6.7836e+00, 3.3555e+00, 5.2390e+00,
1.1013e+00, 3.4443e+00, 1.3650e+01, 4.1885e+00, 3.4643e+00,
4.0415e+00, 7.5722e+00, 5.1383e+00, 2.3401e+00, 5.2583e+00,
4.1095e+00, 8.1516e-01, 4.7037e+00, 2.3122e-02, -1.5872e+00,
2.2099e+00, 5.2872e+00, 8.3151e+00, 1.6701e+00, 3.4691e+00,
4.7135e+00, 7.3734e+00, -2.1556e+00, 1.1943e+01, -4.8788e-01,
2.6438e+00, 7.0337e+00, 6.4599e+00, 4.1977e+00, 6.7508e+00,
-5.6997e+00, 7.5895e+00, -5.8350e+00, 2.6028e+00, 2.4710e+00,
2.9316e+00, 5.5433e+00, 3.2023e+00, 6.8027e+00, 2.1754e+00,
5.1678e+00, 7.6736e+00, 3.1644e-01, 7.4585e+00, 6.7207e+00,
2.4918e+00, 5.2083e+00, 4.3911e+00, 2.1199e+00, 7.3935e+00,
3.0019e+00, 5.1438e+00, 1.6835e+00, 1.7025e+01, 6.4380e+00,
3.9269e+00, 1.3423e+00, 1.0056e+01, 6.4518e+00, 1.1786e+01,
1.4369e+01, 5.9835e+00, -4.0346e+00, 6.7038e+00, 6.3128e+00,
2.5318e+00, 5.7233e-01, -3.8579e+00, 3.5814e+00, 3.6234e+00,
3.7589e+00, 3.2349e+00, 9.7685e+00, -5.3370e+00, 3.0212e-01,
7.4470e+00, 7.9035e+00, 3.3017e+00, 4.4283e+00, -1.1999e+00,
3.4110e+00, -7.9651e-01, 3.2304e+00, -3.5655e+00, -5.1933e+00,
1.2617e+01, 3.7052e+00, 6.2573e+00, 5.5877e+00, -5.4492e-01,
4.6039e+00, 1.4071e+00, 1.0263e+01, 1.5162e+00, 1.0058e+01,
1.9859e+00, 6.4663e+00, 5.1987e-01, 6.8607e+00, 6.7446e+00,
1.7579e+00, -1.7222e+00, -1.0771e+00, 7.7494e-01, -1.9333e-02,
8.8316e+00, -4.7399e+00, 4.9970e+00, 9.0686e+00, -6.3899e-01,
3.5481e+00, 7.5947e+00, 4.1835e+00, 4.8132e+00, 5.4006e+00,
7.1721e+00, 1.3880e+01, 8.8979e+00, 5.6534e+00, 2.4463e+00,
-8.5860e+00, 5.2726e+00, -1.9059e+00, 9.7225e+00, 6.1324e+00,
2.7480e+00, 9.5607e+00, -5.1262e-01, -5.2414e-01, 5.8667e+00,
5.3955e+00, 7.5670e+00, 7.3300e+00, 8.9820e+00, 4.0835e+00,
7.5790e+00, 8.7937e+00, 4.8071e+00, 3.3220e+00, 1.1934e+01,
9.3510e+00, 5.2204e+00, 4.7458e+00, 9.7410e+00, 5.5066e-01,
2.2319e+00, 4.5363e+00, 8.8502e+00, -2.3816e+00, 1.1200e+01,
8.4412e+00, 6.6048e+00, 1.0004e+00, 8.3695e+00, 7.5421e+00,
5.3460e-01, 9.5623e+00, 3.5176e+00, 1.5785e+00, 6.8748e+00,
-2.5705e-01, 3.8226e+00, 1.2004e+01, 5.6965e+00, 5.1880e+00,
9.5765e+00, 7.1425e+00, 3.8174e+00, 2.7385e+00, 1.5513e+00,
-2.4735e+00, 7.1869e+00, 2.1463e+00, 2.9772e+00, -2.1224e+00,
6.4781e+00, 3.9532e-01, -2.8792e+00, -1.8212e+00, 1.3565e+01,
8.5305e+00, 7.3443e+00, 3.3902e+00, 7.2337e+00, 8.1134e+00,
1.9861e+00, 7.0784e-01, 5.0915e+00, 2.6137e+00, 2.9595e+00,
6.4367e+00, 4.6851e+00, -3.0581e+00, 6.5745e+00, -1.3078e+00,
2.3855e+00, 6.5987e+00, 3.0967e+00, 3.3623e+00, 1.5926e+00,
9.8759e+00, 3.0744e+00, 2.4427e-01, 6.9704e+00, 3.6868e+00,
-3.4440e+00, 4.8641e+00, 6.7563e+00, 8.7857e+00, 7.6898e+00,
1.3370e+01, 3.7387e+00, -2.7800e+00, -4.4384e+00, 4.4311e+00,
4.6384e+00, 2.8387e+00, 4.7712e+00, 4.1730e+00, 2.1846e+00,
3.8315e+00, 7.0497e+00, -1.2256e+00, 4.5546e+00, 3.3716e+00,
1.2256e-01, 5.6450e+00, 9.6056e+00, -5.0523e-01, 2.6049e+00,
-7.1354e-01, 5.1554e+00, 5.3114e+00, 8.1763e+00, -9.1869e+00,
-2.5365e+00, 4.2363e+00, 3.2928e+00, 1.2413e+00, 7.4122e+00]))
将训练数据的特征和标签组合
dataset = Data.TensorDataset(features,labels)
dataset
随机读取小批量
data_iter = Data.DataLoader(dataset, batch_size, shuffle=True)
data_iter
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])
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)
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.])
使用均方误差损失作为模型的损失函数
loss = nn.MSELoss()
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
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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