本文实现两个分类器: softmax分类器和感知机分类器

(资料图)

Softmax分类是一种常用的多类别分类算法，它可以将输入数据映射到一个概率分布上。Softmax分类首先将输入数据通过线性变换得到一个向量，然后将向量中的每个元素进行指数函数运算，最后将指数运算结果归一化得到一个概率分布。这个概率分布可以被解释为每个类别的概率估计。

定义一个softmax分类器类:

class SoftmaxClassifier(nn.Module):    def __init__(self,input_size,output_size):        # 调用父类的__init__()方法进行初始化        super(SoftmaxClassifier,self).__init__()        # 定义一个nn.Linear对象，用于将输入特征映射到输出类别        self.linear = nn.Linear(input_size,output_size)    def forward(self,x):        x = self.linear(x) # 传递给线性层        return nn.functional.softmax(x,dim=1) # 得到概率分布    def compute_accuracy(self,output,labels):        preds = torch.argmax(output,dim=1) # 获取每个样本的预测标签        correct = torch.sum(preds == labels).item() # 计算正确预测的数量        accuracy = correct / len(labels) # 除以总样本数得到准确率        return accuracy

如上定义三个方法：

生成训练数据:

import numpy as np# 生成随机样本(包含训练数据和测试数据)def generate_rand_samples(dot_num=100):    x_p = np.random.normal(3., 1, dot_num)    y_p = np.random.normal(3., 1, dot_num)    y = np.zeros(dot_num)    C1 = np.array([x_p, y_p, y]).T    x_n = np.random.normal(7., 1, dot_num)    y_n = np.random.normal(7., 1, dot_num)    y = np.ones(dot_num)    C2 = np.array([x_n, y_n, y]).T    x_n = np.random.normal(3., 1, dot_num)    y_n = np.random.normal(7., 1, dot_num)    y = np.ones(dot_num)*2    C3 = np.array([x_n, y_n, y]).T    x_n = np.random.normal(7, 1, dot_num)    y_n = np.random.normal(3, 1, dot_num)    y = np.ones(dot_num)*3    C4 = np.array([x_n, y_n, y]).T    data_set = np.concatenate((C1, C2, C3, C4), axis=0)    np.random.shuffle(data_set)    return data_set[:,:2].astype(np.float32),data_set[:,2].astype(np.int32)X_train,y_train = generate_rand_samples()y_train[y_train == -1] = 0

设置训练前的前置参数，并初始化分类器

num_inputs = 2  # 输入维度大小num_outputs = 4  # 输出维度大小learning_rate = 0.01  # 学习率num_epochs = 2000 # 训练周期数# 归一化数据 将数据特征减去均值再除以标准差X_train = (X_train - X_train.mean(axis=0)) / X_train.std(axis=0)y_train = y_train.astype(np.compat.long)# 创建model并初始化model = SoftmaxClassifier(num_inputs, num_outputs)criterion = nn.CrossEntropyLoss() # 交叉熵损失optimizer = optim.SGD(model.parameters(), lr=learning_rate)  # SGD优化器

训练:

# 遍历训练周期数for epoch in range(num_epochs):    outputs = model(torch.tensor(X_train))  # 前向传递计算    loss = criterion(outputs,torch.tensor(y_train))  # 计算预测输出和真实标签之间的损失    train_accuracy = model.compute_accuracy(outputs,torch.tensor(y_train))  # 计算模型当前训练周期中准确率    optimizer.zero_grad()  # 清楚优化器中梯度    loss.backward()  # 计算损失对模型参数的梯度    optimizer.step()    # 打印信息    if (epoch + 1) % 10 == 0:        print(f"Epoch [{epoch+1}/{num_epochs}], Loss: {loss.item():.4f}, Accuracy: {train_accuracy:.4f}")

运行：

Epoch [1820/2000], Loss: 0.9947, Accuracy: 0.9575Epoch [1830/2000], Loss: 0.9940, Accuracy: 0.9600Epoch [1840/2000], Loss: 0.9932, Accuracy: 0.9600Epoch [1850/2000], Loss: 0.9925, Accuracy: 0.9600Epoch [1860/2000], Loss: 0.9917, Accuracy: 0.9600....

生成测试并测试:

X_test, y_test = generate_rand_samples()  # 生成测试数据X_test = (X_test- np.mean(X_test)) / np.std(X_test)  # 归一化y_test = y_test.astype(np.compat.long)predicts = model(torch.tensor(X_test))  # 获取模型输出accuracy = model.compute_accuracy(predicts,torch.tensor(y_test))  # 计算准确度print(f"Test Accuracy: {accuracy:.4f}")

输出：

Test Accuracy: 0.9725

绘制图像:

# 绘制图像x_min, x_max = X_test[:, 0].min() - 1, X_test[:, 0].max() + 1y_min, y_max = X_test[:, 1].min() - 1, X_test[:, 1].max() + 1xx, yy = np.meshgrid(np.arange(x_min, x_max, 0.1), np.arange(y_min, y_max, 0.1))Z = model(torch.tensor(np.c_[xx.ravel(), yy.ravel()], dtype=torch.float32)).argmax(dim=1).numpy()Z = Z.reshape(xx.shape)plt.contourf(xx, yy, Z, alpha=0.4)plt.scatter(X_test[:, 0], X_test[:, 1], c=y_test, s=20, edgecolor="k")plt.show()

实现与上述softmax分类器相似，此处实现sigmod感知机，采用sigmod作为分类函数，该函数可以将线性变换的结果映射为0到1之间的实数值，通常被用作神经网络中的激活函数

sigmoid感知机的学习算法与普通的感知机类似，也是采用随机梯度下降（SGD）的方式进行更新。不同之处在于，sigmoid感知机的输出是一个概率值，需要将其转化为类别标签。

通常使用阈值来决定输出值所属的类别，如将输出值大于0.5的样本归为正类，小于等于0.5的样本归为负类。

# 感知机分类器class PerceptronClassifier(nn.Module):    def __init__(self, input_size,output_size):        super(PerceptronClassifier, self).__init__()        self.linear = nn.Linear(input_size,output_size)    def forward(self, x):        logits = self.linear(x)        return torch.sigmoid(logits)    def compute_accuracy(self, pred, target):        pred = torch.where(pred >= 0.5, 1, -1)        accuracy = (pred == target).sum().item() / target.size(0)        return accuracy

给定一个输入向量(x1,x2,x3...xn)，输出为y=σ(w⋅x+b)=1/(e^−(w⋅x+b))

生成训练集:

def generate_rand_samples(dot_num=100):    x_p = np.random.normal(3., 1, dot_num)    y_p = np.random.normal(3., 1, dot_num)    y = np.ones(dot_num)    C1 = np.array([x_p, y_p, y]).T    x_n = np.random.normal(6., 1, dot_num)    y_n = np.random.normal(0., 1, dot_num)    y = np.ones(dot_num)*-1    C2 = np.array([x_n, y_n, y]).T    data_set = np.concatenate((C1, C2), axis=0)    np.random.shuffle(data_set)    return data_set[:,:2].astype(np.float32),data_set[:,2].astype(np.int32)X_train,y_train = generate_rand_samples()X_test,y_test = generate_rand_samples()

该过程与上述softmax分类器相似:

num_inputs = 2num_outputs = 1learning_rate = 0.01num_epochs = 200# 归一化数据 将数据特征减去均值再除以标准差X_train = (X_train - X_train.mean(axis=0)) / X_train.std(axis=0)# 创建model并初始化model = PerceptronClassifier(num_inputs, num_outputs)optimizer = optim.SGD(model.parameters(), lr=learning_rate)  # SGD优化器criterion = nn.functional.binary_cross_entropy

训练：

# 遍历训练周期数for epoch in range(num_epochs):    outputs = model(torch.tensor(X_train))    labels = torch.tensor(y_train).unsqueeze(1)    loss = criterion(outputs,labels.float())    train_accuracy = model.compute_accuracy(outputs, labels)    optimizer.zero_grad()    loss.backward()    optimizer.step()    if (epoch + 1) % 10 == 0:        print(f"Epoch [{epoch+1}/{num_epochs}], Loss: {loss.item():.4f}, Accuracy: {train_accuracy:.4f}")

输出:

Epoch [80/200], Loss: -0.5429, Accuracy: 0.9550Epoch [90/200], Loss: -0.6235, Accuracy: 0.9550Epoch [100/200], Loss: -0.7015, Accuracy: 0.9500Epoch [110/200], Loss: -0.7773, Accuracy: 0.9400....

X_test, y_test = generate_rand_samples() # 生成测试集X_test = (X_test - X_test.mean(axis=0)) / X_test.std(axis=0)test_inputs = torch.tensor(X_test)test_labels = torch.tensor(y_test).unsqueeze(1)with torch.no_grad():    outputs = model(test_inputs)    accuracy = model.compute_accuracy(outputs, test_labels)    print(f"Test Accuracy: {accuracy:.4f}")

绘图:

x_min, x_max = X_test[:, 0].min() - 1, X_test[:, 0].max() + 1y_min, y_max = X_test[:, 1].min() - 1, X_test[:, 1].max() + 1xx, yy = torch.meshgrid(torch.linspace(x_min, x_max, 100), torch.linspace(y_min, y_max, 100))# 预测每个点的类别Z = torch.argmax(model(torch.cat((xx.reshape(-1,1), yy.reshape(-1,1)), 1)), 1)Z = Z.reshape(xx.shape)# 绘制分类图plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral,alpha=0.0)# 绘制分界线w = model.linear.weight.detach().numpy()  # 权重b = model.linear.bias.detach().numpy()  # 偏置x1 = np.linspace(x_min, x_max, 100)x2 = (-b - w[0][0]*x1) / w[0][1]plt.plot(x1, x2, "k-")# 绘制样本点plt.scatter(X_train[:, 0], X_train[:, 1], c=y_train, cmap=plt.cm.Spectral)plt.show()

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