多层感知机原理

隐藏层

严格一点来讲：我们需要隐藏层是因为线性是一个很强的假设，线性模型在有些情况会不适用或者出错。

• 一个形象的例子：

就如同上面图片中展示的XOR问题,如果我们现在想要将绿和红球分开，如果只用一条"线性"，我们会发现我们是做不到的，起码要两条及以上的"线性"

激活函数

简单来说激活函数的作用是将隐藏层的输出从线性转换为非线性
一般来说，有了激活函数，就不可能再将我们的多层感知机退化成线性模型

多层感知机的从零开始实现

import torch
from torch import nn
from d2l import torch as d2l
from IPython import display



# 可视化
class Accumulator:  #@save
    """在n个变量上累加"""
    def __init__(self, n):
        self.data = [0.0] * n

    def 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]
    
def accuracy(y_hat, y):  #@save
    """计算预测正确的数量"""
    if len(y_hat.shape) > 1 and y_hat.shape[1] > 1:
        y_hat = y_hat.argmax(axis=1)
    cmp = y_hat.type(y.dtype) == y
    return float(cmp.type(y.dtype).sum())

def evaluate_accuracy(net, data_iter):  #@save
    """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
    with torch.no_grad():
        for X, y in data_iter:
            metric.add(accuracy(net(X), y), y.numel())
    return metric[0] / metric[1]


class Animator:  #@save
    """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, fmts

    def 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] * n
        if 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)

# 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）"""
    # 将模型设置为训练模式
    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 train_iter:
        # 计算梯度并更新参数
        y_hat = net(X) # X.shape:torch.Size([256, 1, 28, 28])，y_hat得到的是一个256x10的矩阵
        l = loss(y_hat, y) #y.shape:torch.Size([256]),l.shape:torch.Size([256])
        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(y_hat, y), y.numel())
    # 返回训练损失和训练精度
    return metric[0] / metric[2], metric[1] / metric[2]

def train_ch3(net, train_iter, test_iter, loss, num_epochs, updater):  #@save
    """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'])
    for epoch in range(num_epochs):
        train_metrics = train_epoch_ch3(net, train_iter, loss, updater)
        test_acc = evaluate_accuracy(net, test_iter)
        animator.add(epoch + 1, train_metrics + (test_acc,))
    train_loss, train_acc = train_metrics
    assert train_loss < 0.5, train_loss
    assert train_acc <= 1 and train_acc > 0.7, train_acc
    assert test_acc <= 1 and test_acc > 0.7, test_acc








#获取数据
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)

#初始化模型参数
num_inputs, num_outputs, num_hiddens = 784, 10, 256

W1 = nn.Parameter(torch.randn(
    num_hiddens,num_inputs ,requires_grad=True) * 0.01)
b1 = nn.Parameter(torch.zeros(num_hiddens, requires_grad=True))
W2 = nn.Parameter(torch.randn(
    num_outputs,num_hiddens ,requires_grad=True) * 0.01)
b2 = nn.Parameter(torch.zeros(num_outputs, requires_grad=True))

params = [W1, b1, W2, b2]

# 定义激活函数
def relu(X):
    # torch.zeros_like是一个模仿X的shape但是将其中元素全部填充为0
    a = torch.zeros_like(X)
    return torch.max(X, a)

# 定义模型
def net(X):
    # 这里X一来照样是X.shape:torch.Size([256, 1, 28, 28])
    # 然后通过下面的操作变成[256,784]
    X = X.reshape((-1, num_inputs))
    # b是一个向量，W1我设置的是256x784 这里要转置一下，X@W1.t()为256x256
    H = relu(X@W1.t() + b1)  # 这里“@”代表矩阵乘法
    return (H@W2.t() + b2)

# 定义损失函数
loss = nn.CrossEntropyLoss(reduction='none')

# 定义优化函数
num_epochs, lr = 10, 0.1
updater = torch.optim.SGD(params, lr=lr)

# 训练
train_ch3(net, train_iter, test_iter, loss, num_epochs, updater)