线性模型

线性模型介绍

线性模型是很常见的机器学习模型，通常通过线性的公式来拟合训练数据集。训练集包括(x,y)，x为特征,y为目标。如下图：

将真实值和预测值用于构建损失函数，训练的目标是最小化这个函数，从而更新w。当损失函数达到最小时(理想上,实际情况可能会陷入局部最优)，此时的模型为最优模型，线性模型常见的的损失函数：

线性模型例子

下面通过一个例子可以观察不同权重(w)对模型损失函数的影响。

#author:yuquanle#data:2018.2.5#Study of Linear Modelimport numpy as npimport matplotlib.pyplot as pltx_data = [1.0, 2.0, 3.0]y_data = [2.0, 4.0, 6.0]def forward(x): return x * wdef loss(x, y): y_pred = forward(x) return (y_pred - y)*(y_pred - y)w_list = []mse_list = []for w in np.arange(0.0, 4.1, 0.1): print("w=", w) l_sum = 0 for x_val, y_val in zip(x_data, y_data): # error l = loss(x_val, y_val) l_sum += l print("MSE=", l_sum/3) w_list.append(w) mse_list.append(l_sum/3)plt.plot(w_list, mse_list)plt.ylabel("Loss")plt.xlabel("w")plt.show()输出结果：w= 0.0MSE= 18.6666666667w= 0.1MSE= 16.8466666667w= 0.2MSE= 15.12w= 0.3MSE= 13.4866666667w= 0.4MSE= 11.9466666667w= 0.5MSE= 10.5w= 0.6MSE= 9.14666666667

调整w，loss变化图：

可以发现当w=2时，loss最小。但是现实中最常见的情况是，我们知道数据集，定义好损失函数之后(loss)，我们并不会从0到n去设置w的值，然后求loss，最后选取使得loss最小的w作为最佳模型的参数。更常见的做法是，首先随机初始化w的值，然后根据loss函数定义对w求梯度，然后通过w的梯度来更新w的值，这就是经典的梯度下降法思想。

梯度下降法

梯度的本意是一个向量，表示某一函数在该点处的方向导数沿着该方向取得最大值，即函数在该点处沿着该方向（此梯度的方向）变化最快，变化率最大（为该梯度的模）。

梯度下降是迭代法的一种,可以用于求解最小二乘问题(线性和非线性都可以)。在求解机器学习算法的模型参数，即无约束优化问题时，梯度下降（Gradient Descent）是最常采用的方法之一，另一种常用的方法是最小二乘法。在求解损失函数的最小值时，可以通过梯度下降法来一步步的迭代求解，得到最小化的损失函数和模型参数值。即每次更新参数w减去其梯度(通常会乘以学习率)。

#author:yuquanle#data:2018.2.5#Study of SGDx_data = [1.0, 2.0, 3.0]y_data = [2.0, 4.0, 6.0]# any random valuew = 1.0# forward passdef forward(x): return x * wdef loss(x, y): y_pred = forward(x) return (y_pred - y)*(y_pred - y)# compute gradient (loss对w求导)def gradient(x, y): return 2*x*(x*w - y)# Before trainingprint("predict (before training)", 4, forward(4))# Training loopfor epoch in range(20): for x, y in zip(x_data, y_data): grad = gradient(x, y) w = w - 0.01 * grad print("\t grad: ",x, y, grad) l = loss(x, y) print("progress:", epoch, l)# After trainingprint("predict (after training)", 4, forward(4))输出结果：predict (before training) 4 4.0 grad: 1.0 2.0 -2.0 grad: 2.0 4.0 -7.84 grad: 3.0 6.0 -16.2288progress: 0 4.919240100095999 grad: 1.0 2.0 -1.478624 grad: 2.0 4.0 -5.796206079999999 grad: 3.0 6.0 -11.998146585599997progress: 1 2.688769240265834 grad: 1.0 2.0 -1.093164466688 grad: 2.0 4.0 -4.285204709416961 grad: 3.0 6.0 -8.87037374849311progress: 2 1.4696334962911515 grad: 1.0 2.0 -0.8081896081960389 grad: 2.0 4.0 -3.1681032641284723 grad: 3.0 6.0 -6.557973756745939progress: 3 0.8032755585999681 grad: 1.0 2.0 -0.59750427561463 grad: 2.0 4.0 -2.3422167604093502 grad: 3.0 6.0 -4.848388694047353progress: 4 0.43905614881022015 grad: 1.0 2.0 -0.44174208101320334 grad: 2.0 4.0 -1.7316289575717576 grad: 3.0 6.0 -3.584471942173538progress: 5 0.2399802903801062 grad: 1.0 2.0 -0.3265852213980338 grad: 2.0 4.0 -1.2802140678802925 grad: 3.0 6.0 -2.650043120512205progress: 6 0.1311689630744999 grad: 1.0 2.0 -0.241448373202223 grad: 2.0 4.0 -0.946477622952715 grad: 3.0 6.0 -1.9592086795121197progress: 7 0.07169462478267678 grad: 1.0 2.0 -0.17850567968888198 grad: 2.0 4.0 -0.6997422643804168 grad: 3.0 6.0 -1.4484664872674653progress: 8 0.03918700813247573 grad: 1.0 2.0 -0.13197139106214673 grad: 2.0 4.0 -0.5173278529636143 grad: 3.0 6.0 -1.0708686556346834progress: 9 0.021418922423117836predict (after training) 4 7.804863933862125

反向传播

但是在定义好模型之后，使用pytorch框架不需要我们手动的求导，我们可以通过反向传播将梯度往回传播。通常有二个过程，forward和backward:

#author:yuquanle#data:2018.2.6#Study of BackPagationimport torchfrom torch import nnfrom torch.autograd import Variablex_data = [1.0, 2.0, 3.0]y_data = [2.0, 4.0, 6.0]# Any random valuew = Variable(torch.Tensor([1.0]), requires_grad=True)# forward passdef forward(x): return x*w# Before trainingprint("predict (before training)", 4, forward(4))def loss(x, y): y_pred = forward(x) return (y_pred-y)*(y_pred-y)# Training: forward, backward and update weight# Training loopfor epoch in range(10): for x, y in zip(x_data, y_data): l = loss(x, y) l.backward() print("\t grad:", x, y, w.grad.data[0]) w.data = w.data - 0.01 * w.grad.data # Manually zero the gradients after running the backward pass and update w w.grad.data.zero_() print("progress:", epoch, l.data[0])# After trainingprint("predict (after training)", 4, forward(4))输出结果：predict (before training) 4 Variable containing: 4[torch.FloatTensor of size 1] grad: 1.0 2.0 -2.0 grad: 2.0 4.0 -7.840000152587891 grad: 3.0 6.0 -16.228801727294922progress: 0 7.315943717956543 grad: 1.0 2.0 -1.478623867034912 grad: 2.0 4.0 -5.796205520629883 grad: 3.0 6.0 -11.998146057128906progress: 1 3.9987640380859375 grad: 1.0 2.0 -1.0931644439697266 grad: 2.0 4.0 -4.285204887390137 grad: 3.0 6.0 -8.870372772216797progress: 2 2.1856532096862793 grad: 1.0 2.0 -0.8081896305084229 grad: 2.0 4.0 -3.1681032180786133 grad: 3.0 6.0 -6.557973861694336progress: 3 1.1946394443511963 grad: 1.0 2.0 -0.5975041389465332 grad: 2.0 4.0 -2.3422164916992188 grad: 3.0 6.0 -4.848389625549316progress: 4 0.6529689431190491 grad: 1.0 2.0 -0.4417421817779541 grad: 2.0 4.0 -1.7316293716430664 grad: 3.0 6.0 -3.58447265625progress: 5 0.35690122842788696 grad: 1.0 2.0 -0.3265852928161621 grad: 2.0 4.0 -1.2802143096923828 grad: 3.0 6.0 -2.650045394897461progress: 6 0.195076122879982 grad: 1.0 2.0 -0.24144840240478516 grad: 2.0 4.0 -0.9464778900146484 grad: 3.0 6.0 -1.9592113494873047progress: 7 0.10662525147199631 grad: 1.0 2.0 -0.17850565910339355 grad: 2.0 4.0 -0.699742317199707 grad: 3.0 6.0 -1.4484672546386719progress: 8 0.0582793727517128 grad: 1.0 2.0 -0.1319713592529297 grad: 2.0 4.0 -0.5173273086547852 grad: 3.0 6.0 -1.070866584777832progress: 9 0.03185431286692619predict (after training) 4 Variable containing: 7.8049[torch.FloatTensor of size 1]Process finished with exit code 0

以上这篇PyTorch: 梯度下降及反向传播的实例详解就是小编分享给大家的全部内容了，希望能给大家一个参考，也希望大家多多支持。