线性神经网络4-softmax回归
一、分类问题
以不同的整数表示不同的类别，如以y ∈ {1, 2, 3}分别代表{狗, 猫, 鸡}； 独热编码(one‐hot encoding)：独热编码是一个向量，它的分量和类别一样多。类别对 应的分量设置为1，其他所有分量设置为0。如标签定义y为三维向量，其中(1, 0, 0)对应于 “猫”、(0, 1, 0)对应于“鸡”、(0, 0, 1)对应于“狗”。
二、网络架构
三、softmax运算
四、交叉熵损失函数
五、分类问题采用交叉熵而不是均方差的为损失函数的解释
六、softmax函数上溢的解决方案
七、softmax分类的实现
import torchfrom torch.utils import dataimport torchvisionfrom torchvision import transformsimport matplotlib.pyplot as pltimport typingfrom matplotlib_inline import backend_inlinefrom IPython import displayimport plotShow# 定义一个累加器，实现对n个变量进行累加class Accumulator:'''在n个变量上累加'''def __init__(self, n:int):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]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)):# 增量地绘制多条线条if legend is None:legend = []backend_inline.set_matplotlib_formats('svg')self.fig, self.axes = plt.subplots(nrows, ncols, figsize=figsize)if nrows*ncols==1:self.axes = [self.axes, ]# 使用lambda函数捕获参数self.configAxes = lambda: plotShow.setAxes(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.configAxes()display.display(self.fig)plt.draw()plt.pause(0.001)display.clear_output(wait=True)def show(self):display.display(self.fig)def getFashionMnistLabels(labels:typing.Sequence):'''返回FashionMnist数据集的文本标签'''textLabels = ['t-shirt', 'trouser', 'pullever', 'dress', 'coat','sandal', 'shirt', 'sneaker', 'bag', 'ankle boot']return [textLabels[int(i)] for i in labels]def showImages(imgs:list, numRows:int, numCols:int, titles:list=None, scale:float=1.5):figsize = (numCols*scale, numRows*scale)_, axes = plt.subplots(numRows, numCols, figsize=figsize)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)ax.axes.get_xaxis().set_visible(False)ax.axes.get_yaxis().set_visible(False)if titles:ax.set_title(titles[i])return axesdef getDataloaderWorkers()->int:'''使用4个进程来读取数据'''return 4# 读取数据集def loadDataFashionMnist(batchSize:int, resize=None)->tuple:'''下载Fashion-MNIST数据集，然后将其加载到内存中'''trans = [transforms.ToTensor()]if resize:trans.insert(0, transforms.Resize(resize))trans = transforms.Compose(trans)mnistTrain = torchvision.datasets.FashionMNIST(root='../data', train=True, transform=trans, download=True)mnistTest = torchvision.datasets.FashionMNIST(root='../data', train=False, transform=trans, download=True)return (data.DataLoader(mnistTrain, batchSize, shuffle=True, num_workers=getDataloaderWorkers()),data.DataLoader(mnistTest, batchSize, shuffle=False, num_workers=getDataloaderWorkers()))# softmax函数def softmax(X:torch.Tensor)->torch.Tensor:'''1. 对每个项求幂(使用exp);2. 对每一行求和(小批量中每个样本是一行), 得到每个样本的规范化常数；3. 将每一行除以其规范化常数, 确保结果的和为1。'''XExp:torch.Tensor = torch.exp(X)partition:torch.Tensor = XExp.sum(1, keepdim=True)return XExp/partition# 定义分类精度def accuracy(yHat:torch.Tensor, y:torch.Tensor)->float:'''计算预测正确的数量'''if len(yHat.shape)>1 and yHat.shape[1]>1:yHat = yHat.argmax(axis=1)cmp = yHat.type(y.dtype)==yreturn float(cmp.type(y.dtype).sum())def evaluateAccuracy(net:torch.nn.Module, dataIter:typing.Tuple[torch.Tensor])->float:'''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) # 统计正确预测数、预测总数with torch.no_grad():for X, y in dataIter:metric.add(accuracy(net(X), y), y.numel())return metric[0]/metric[1]# 定义训练模型def trainEpochCh3(net:torch.nn.Module,trainIter:typing.Tuple[torch.Tensor],loss:typing.Callable[[torch.Tensor, torch.Tensor], torch.Tensor],updater:torch.optim.Optimizer)->typing.Tuple[float]:'''训练模型一个周期'''# 将模型设置为训练模型if isinstance(net, torch.nn.Module):net.train()# 训练损失总和、训练准确度总和、样本数metric = Accumulator(3)for X, y in trainIter:yHat:torch.Tensor = net(X)l:torch.Tensor = loss(yHat, y)if isinstance(updater, torch.optim.Optimizer):# 使用PyTorch内置的优化器和损失函数updater.zero_grad()l.mean().backward()updater.step()else:# 使用定制的优化器和损失函数l.sum().backward()updater(X.shape[0])metric.add(float(l.sum()), accuracy(yHat, y), y.numel())# 返回训练损失和训练精度return metric[0]/metric[2], metric[1]/metric[2]def trainCh3(net:torch.nn.Module,trainIter:data.DataLoader,testIter:data.DataLoader,loss:typing.Callable[[torch.Tensor, torch.Tensor], torch.Tensor],numEpochs: int,updater: torch.optim.Optimizer)->None:'''训练模型'''animator = Animator(xlabel='epoch', xlim=[1, numEpochs], ylim=[0.3,0.9],legend=['train loss', 'train acc', 'test acc'])for epoch in range(numEpochs):trainMetrics = trainEpochCh3(net, trainIter, loss, updater)testAcc = evaluateAccuracy(net, testIter)animator.add(epoch+1, trainMetrics+(testAcc, ))trainLoss, trainAcc = trainMetricsassert trainLoss<0.5, trainLossassert trainAcc<=1 and trainAcc>0.7, trainAccassert testAcc<=1 and testAcc>0.7, testAcc# 定义优化算法def SGD(params:list[torch.Tensor], lr: float, batchSize:int):'''小批量随机梯度下降法params: 模型参数集 合lr: 学习率，确定每一步更新的大小batchSize: 批量样本大小'''with torch.no_grad():for param in params:param -= lr*param.grad/batchSizeparam.grad.zero_()# 预测def predictCh3(net, testIter, n=6):'''预测标签'''for X, y in testIter:breaktrues = getFashionMnistLabels(y)preds = getFashionMnistLabels(net(X).argmax(axis=1))titles = [true + '\n' + pred for true, pred in zip(trues, preds)]showImages(X[0:n].reshape((n, 28, 28)), 1, n, titles=titles[0:n])plt.show()
# softmax回归的从零开始实现if __name__ == '__main__':# 1. 读取数据集batchSize: int = 256trainIter, testIter = loadDataFashionMnist(batchSize)# 2. 模型参数初始化numInputs: int = 784 # 原始数据集每个样本是28*28的图像，因此以构造一个长度为784的向量来表示每一个像素点numOutPuts: int = 10 # 数据集中有10个类别W = torch.normal(0.0, 0.01, size=(numInputs, numOutPuts), requires_grad=True) # 权重项b = torch.zeros(numOutPuts, requires_grad=True) # 偏置项# 3. 定义softmax模型def net(X:torch.Tensor)->torch.Tensor:return softmax(torch.matmul(X.reshape((-1, W.shape[0])), W) + b)# 4. 定义损失函数def crossEntropy(yHat:torch.Tensor, y:torch.Tensor)->torch.Tensor:'''yHat: 预测概率y: 标签向量'''return -torch.log(yHat[range(len(yHat)), y])# 5. 定义优化函数lr = 0.1def updater(batchSize:int):return SGD([W,b], lr, batchSize)# 6. 模型训练numEpochs:int = 10trainCh3(net, trainIter, testIter, crossEntropy, numEpochs, updater)# 7. 预测predictCh3(net, testIter, n=10)
# softmax回归的简洁实现if __name__ == '__main__':# 1. 读取数据集batchSize:int = 256trainIter, testIter = loadDataFashionMnist(batchSize)# 2. 初始化模型参数# PyTorch不会隐式地调整输入的形状。因此，# 我们在线性层前定义了展平层（flatten），来调整网络输入的形状net = torch.nn.Sequential(torch.nn.Flatten(), torch.nn.Linear(784, 10))def initWeight(m):if type(m)==torch.nn.Linear:torch.nn.init.normal_(m.weight, std=0.01)net.apply(initWeight)# 3. 定义损失函数loss = torch.nn.CrossEntropyLoss(reduction='none')# 4. 定义优化算法trainer = torch.optim.SGD(net.parameters(), lr=0.1)# 5. 训练numEpochs:int = 10trainCh3(net, trainIter, testIter, loss, numEpochs, trainer)
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首次发布时间：2025-08-26
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