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)