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softmax回归

一、理论知识

• 回归估计一个连续值
• 分类预测一个离散类别
  回归
• 单连续数值输出
• 自然区间R
• 跟真实值的区别作为损失
  
  分类
• 通常多个输出
• 输出i是预测为第i类的置信度
  
  一般我们使用交叉熵用来衡量两个概率的区别
  
  将它作为损失
  
  其梯度是真实概率和预测概率的区别：
  
  其梯度是真实概率和预测概率的区别
  损失函数
  (1)L2 Loss
  
  (2)L1 Loss
  
  (3)Huber’s Robust Loss
  
  优点：当预测值与真实值相差较远时候，梯度还是按照均匀的变化，在比较靠近的时候，梯度绝对值会变小，保证优化比较平滑。

1.图像分类数据集

MNIST数据集是图像分类中广泛使用的数据集之一，但其较为简单，我们将使用Fashion-MNIST数据集
1.导入需要的包

%matplotlib inline
import torch
import torchvision
from torch.utils import data
from torchvision import transforms
from d2l import torch as d2ld2l.use_svg_display() #通过svg显示图片，清晰度更高

2.通过框架中的内置函数将Fashion-MNIST数据集下载并读取到内存中
Fashion-MNIST由10个类别的图像组成

# 通过ToTensor实例将图像数据从PIL类型变换成32位浮点数格式
# 并处以255使得所有像素的数值均在0到1之间
trans = transforms.ToTensor()
mnist_train = torchvision.datasets.FashionMNIST(root="../data", train=True, transform=trans,download=True)
mnist_test = torchvision.datasets.FashionMNIST(root="../data", train=False,transform=trans,download=True)len(mnist_train), len(mnist_test)

(60000, 10000)

mnist_train[0][0].shape  #第0个example

torch.Size([1, 28, 28])
3.两个可视化数据集的函数

def get_fashion_mnist_labels(labels): """返回Fashion-MNIST数据集的文本标签"""text_labels = ['t-shirt', 'trouser', 'pullover', 'dress', 'coat','sandal', 'shirt', 'sneaker', 'bag', 'ankle boot']return [text_labels[int(i)] for i in labels]def show_images(imgs, num_rows, num_cols, titles=None, scale=1.5): """绘制图像列表"""figsize = (num_cols * scale, num_rows * scale)
#     创建一个包含多个子图的图形
#     下划线表示我们对图形本身不感兴趣，只关心返回的子图像_,axes = d2l.plt.subplots(num_rows, num_cols, figsize=figsize)
#     print(_)
#     print(axes)
#     print(type(axes)) #numpy
#     使用NumPy中的flatten()函数，将axes数组从多维数组变成一维数组
# axes:原本是一个包含多个子图对象的二维数组axes = axes.flatten()for i, (ax, img) in enumerate(zip(axes, imgs)):if torch.is_tensor(img):# 图片张量ax.imshow(img.numpy())else:# PIL图片ax.imshow(img)
#       设置子图的X轴和Y轴刻度标签不可见ax.axes.get_xaxis().set_visible(False)ax.axes.get_yaxis().set_visible(False)if titles:
#             print(titles)
#           给每个子图设置相应的标题ax.set_title(titles[i])return axes# iter():函数生成迭代器
# next()：返回迭代器的下一个项目，一般与iter()结合使用
X, y = next(iter(data.DataLoader(mnist_train, batch_size=18)))# 拿到第一个小批量的数据X和y标签y
# print(y)
show_images(X.reshape(18, 28, 28), 2, 9, titles=get_fashion_mnist_labels(y));
# 绘制两行图片，每一行有9张图片
# titles是每张图片的标号

4.读取一小批数据，大小为batch_size

batch_size = 256def get_dataloader_workers():"""使用4个进程来读取数据"""return 4train_iter = data.DataLoader(mnist_train, batch_size, shuffle=True,num_workers=4)
# num_workers = 0代表单进程而不是没有进程timer = d2l.Timer()  # Timer函数用于测试速度
# print(train_iter)batch = next(iter(train_iter))
print(batch)# for X,y in train_iter:
# #     print(train_iter)# # for X, y in train_iter:
#     continue
f'{timer.stop():.2f} sec'  # 输出读取数据所用的秒数，精度为2位小数

‘3.51 sec’

❗❗❗

如果运行出现此种报错，一般是由于taLoader的多进程造成的，我们可以通过设置num_workers=0代表单进程加载.

5.定义load_data_fashion_mnist函数
为了方便后续使用，我们可以把上面的各个实现数据读取的操作，写在函数中

def load_data_fashion_mnist(batch_size, resize=None):"""下载Fashion-MNIST数据集，然后将其加载到内存中"""trans = [transforms.ToTensor()]if resize:trans.inserts(0, transforms.Resize(resize))trans = transforms.Compose(trans)mnist_train = torchvision.datasets.FashionMNIST(root="../data",train=True,transform=trans,download=True)mnist_test = torchvision.datasets.FashionMNIST(root="../data",train=True,transform=trans,download=True)return (data.DataLoader(mnist_train, batch_size, shuffle=True,num_workers=get_dataloader_workers()),data.DataLoader(mnist_test, batch_size, shuffle=True,num_workers=get_dataloader_workers()))

2.softmax回归的从零开始实现

1.导入相关包

import torch
from IPython import display
from d2l import torch as d2lbatch_size = 256
# 返回训练数据和测试数据的迭代器
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)

2.将展平每个图像，将它们视为长度为784的向量。因为数据有10个类别，所以网格输出维度为

# 28*28 = 784
num_inputs = 784
num_outputs = 10W = torch.normal(0, 0.01, size=(num_inputs, num_outputs), requires_grad=True)
b = torch.zeros(num_outputs, requires_grad=True)

3.给定一个矩阵X,对所有元素求和

X = torch.tensor([[1.0, 2.0, 3.0], [4.0,5.0,6.0]])
# 按照维度为0进行，将行数变为1，按照维度为1进行求和，把列数变为1，keepdim表示保持维度
X.sum(0, keepdim=True), X.sum(1, keepdim=True)

(tensor([[5., 7., 9.]]),tensor([[ 6.],[15.]]))

4.实现softmax

def softmax(X):X_exp = torch.exp(X)
#     按行求和print('----------------X_exp------------')print(X_exp)partition = X_exp.sum(1, keepdim=True)print('-------------partition------------')print(partition)
#     运用广播机制print('----------X_exp / partition--------')print(X_exp / partition)return X_exp / partition

5.将每个元素变成一个非负数，依据概率原理，每行总和为1

X = torch.normal(0, 1, (2, 5))
print(X)
X_prob = softmax(X)
X_prob, X_prob.sum(1)

tensor([[-0.0477, -0.8353,  0.6251, -0.0551, -1.3545],[ 1.9616, -0.2592, -1.7372,  1.6476, -0.1598]])
----------------X_exp------------
tensor([[0.9534, 0.4337, 1.8684, 0.9464, 0.2581],[7.1107, 0.7717, 0.1760, 5.1943, 0.8523]])
-------------partition------------
tensor([[ 4.4600],[14.1051]])
----------X_exp / partition--------
tensor([[0.2138, 0.0972, 0.4189, 0.2122, 0.0579],[0.5041, 0.0547, 0.0125, 0.3683, 0.0604]])
(tensor([[0.2138, 0.0972, 0.4189, 0.2122, 0.0579],[0.5041, 0.0547, 0.0125, 0.3683, 0.0604]]),tensor([1., 1.]))
1

6.实现softmax回归模型

def net(X):return softmax(torch.matmul(X.reshape((-1, W.shape[0])), W) + b)

7.创建一个数据y_hat,其中包含2个样本在3个类别的预测概率，使用y作为y_hat中概率的索引

y = torch.tensor([0, 2])
y_hat = torch.tensor([[0.1, 0.3, 0.6], [0.3, 0.2, 0.5]])
# y0 = 0 = 0.1   y1 = 2 = 0.5
y_hat[[0, 1], y]

tensor([0.1000, 0.5000])

8.实现交叉熵损失函数

def cross_entropy(y_hat, y):return -torch.log(y_hat[range(len(y_hat)), y])cross_entropy(y_hat, y)

tensor([2.3026, 0.6931])

9.将预测类别与真实y元素进行比较

def accuracy(y_hat, y):"""计算预测正确的数量""" 
#     print(y_hat.shape)         列数大于1if len(y_hat.shape) > 1 and y_hat.shape[1] > 1:
#     将每一行元素值最大的下标存下来y_hat = y_hat.argmax(axis=1)cmp = y_hat.type(y.dtype) == y
#     转成和y同样的形状并求和return float(cmp.type(y.dtype).sum())# 预测正确的样本数/y的长度
accuracy(y_hat, y) / len(y)

0.5

10.评估在任意模型net的准确率

def evaluate_accuracy(net, data_iter):"""计算在指定数据集上模型的精度"""if isinstance(net, torch.nn.Module):net.eval() # 将模型设置为评估模式metric = Accumulator(2)   # 正确预测数、预测总数for X, y in data_iter:metric.add(accuracy(net(X), y), y.numel())return metric[0] / metric[1]

11.Accumulator实例中创建了2个变量，用于分别存储正确预测的数量和预测的总数量

class Accumulator:"""在n个变量上累加"""def __init__(self, n):self.data = [0.0] * ndef add(self, *args):self.data = [a + float(b) for a, b in zip(self.data, args)]def reset(self):self.data = [0.0] * len(self.data)def __getitem__(self, idx):return self.data[idx]evaluate_accuracy(net, test_iter)

0.1285

12.Softmax回归的训练

def train_epoch_ch3(net, train_iter, loss, updater):  #@save"""训练模型一个迭代周期（定义见第3章）"""if isinstance(net, torch.nn.Module):net.train() # Accumulator(3)创建3个变量：训练损失总和、训练准确度总和、样本数metric = Accumulator(3)for X, y in train_iter:# 计算梯度并更新参数y_hat = net(X)l = loss(y_hat, y)# 判断updater是否为优化器if isinstance(updater, torch.optim.Optimizer):# 使用PyTorch内置的优化器和损失函数updater.zero_grad() #把梯度设置为0l.mean().backward() #计算梯度updater.step()		#自更新else:# 使用定制的优化器和损失函数# 自我实现的话，l出来是向量，先求和再求梯度l.sum().backward()updater(X.shape[0])metric.add(float(l.sum()), accuracy(y_hat, y), y.numel())# 返回训练损失和训练精度，metric的值由Accumulator得到return metric[0] / metric[2], metric[1] / metric[2]

13.定义一个在动画中绘制数据的使用程序类

class Animator:  """在动画中绘制数据"""def __init__(self, xlabel=None, ylabel=None, legend=None, xlim=None,ylim=None, xscale='linear', yscale='linear',fmts=('-', 'm--', 'g-.', 'r:'), nrows=1, ncols=1,figsize=(3.5, 2.5)):# 增量地绘制多条线if legend is None:legend = []d2l.use_svg_display()self.fig, self.axes = d2l.plt.subplots(nrows, ncols, figsize=figsize)if nrows * ncols == 1:self.axes = [self.axes, ]# 使用lambda函数捕获参数self.config_axes = lambda: d2l.set_axes(self.axes[0], xlabel, ylabel, xlim, ylim, xscale, yscale, legend)self.X, self.Y, self.fmts = None, None, fmtsdef add(self, x, y):# 向图表中添加多个数据点if not hasattr(y, "__len__"):y = [y]n = len(y)if not hasattr(x, "__len__"):x = [x] * nif not self.X:self.X = [[] for _ in range(n)]if not self.Y:self.Y = [[] for _ in range(n)]for i, (a, b) in enumerate(zip(x, y)):if a is not None and b is not None:self.X[i].append(a)self.Y[i].append(b)self.axes[0].cla()for x, y, fmt in zip(self.X, self.Y, self.fmts):self.axes[0].plot(x, y, fmt)self.config_axes()display.display(self.fig)display.clear_output(wait=True)

14.训练函数

def train_ch3(net, train_iter, test_iter, loss, num_epochs, updater):  """训练模型（定义见第3章）"""animator = Animator(xlabel='epoch', xlim=[1, num_epochs], ylim=[0.3, 0.9],legend=['train loss', 'train acc', 'test acc'])# num_epochs：训练次数for epoch in range(num_epochs):# train_epoch_ch3：训练模型，返回准确率和错误度train_metrics = train_epoch_ch3(net, train_iter, loss, updater)# 在测试数据集上评估精度test_acc = evaluate_accuracy(net, test_iter)animator.add(epoch + 1, train_metrics + (test_acc,))train_loss, train_acc = train_metrics

15.小批量随机梯度下降来优化模型的损失函数

lr = 0.1def updater(batch_size):return d2l.sgd([W, b], lr, batch_size)

16.模型训练10个迭代周期

num_epochs = 10if __name__ == "__main__":train_ch3(net, train_iter, test_iter, cross_entropy, num_epochs, updater)

17.预测

def predict_ch3(net, test_iter, n=6):  """预测标签"""for X, y in test_iter:breaktrues = d2l.get_fashion_mnist_labels(y) # 实际标签preds = d2l.get_fashion_mnist_labels(net(X).argmax(axis=1)) #预测标签，取最大化概率titles = [true +'\n' + pred for true, pred in zip(trues, preds)]d2l.show_images(X[0:n].reshape((n, 28, 28)), 1, n, titles=titles[0:n])# predict_ch3(net, test_iter)if __name__ == "__main__":predict_ch3(net, test_iter)

3.Softmax简洁实现

import torch
from torch import nn
from d2l import torch as d2lbatch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)
train_iter.num_workers = 0
test_iter.num_workers = 0

1.softmax回归的输出层是一个全连接层

# PyTorch不会隐式地调整输入的形状
# 因此，我们定义了展平层 在线形层前调整网络输入的形状
net = nn.Sequential(nn.Flatten(), nn.Linear(784, 10))def init_weights(m):if type(m) == nn.Linear:nn.init.normal_(m.weight, std=0.01)net.apply(init_weights);

2.在交叉熵损失函数中传递未归一化的预测，并同时计算softmax及其对数

loss = nn.CrossEntropyLoss(reduction='none')

3.使用学习率为0.1的小批量随机梯度下降作为优化算法

trainer = torch.optim.SGD(net.parameters(), lr=0.1)
num_epoches = 10
train_ch3(net, train_iter, test_iter, loss, num_epoches, trainer)

【相关总结】

torch.sum()

(1)torch.sum(input, *, dtype=None):返回输入张量input所有元素的和。
(2)torch.sum(input,dim,keepdim=False, *,dtype=None):返回指定维度进行求和

import torch
x = torch.ones((2, 3))x_sum =  torch.sum(a)
x_0 =  torch.sum(a, dim=0)
x_1 =  torch.sum(a, dim=1)print(x)
print(x_0)
print(x_1)

tensor([[1., 1., 1.],
[1., 1., 1.]])
tensor([2., 2., 2.])
tensor([3., 3.])

torch.argmax()

import torch
x = torch.rand(3, 2)
print(x)
y0 = torch.argmax(x, dim=0)  #dim=0,返回每一列最大值的索引
print(y0)
y1 = torch.argmax(x, dim=1)  #dim=1,返回每一行最大值的索引
print(y1)

tensor([[0.9407, 0.8543],
[0.4057, 0.6790],
[0.0154, 0.3698]])
tensor([0, 0])
tensor([0, 1, 1])

isinstance():[python]

函数isinstance()可以判断一个变量的类型。