ML/Deep Learning Note1-杂记
Published:
机器学习的一点笔记（其实只是拿来练一下公式输入）
1、sigmoid activation function
\[f(x)= \frac1 {1+e^x}\]2、RELU activation function
\[f(x) = max(0,x)\]通常 $x$ 用线性函数 $y=wx+b$代替
即上述函数可以化为 \(f(x)= \frac{1}{e^{wx+b}+1}\) \(f(x) = max(0,wx+b)\)
tip：RELU用得比较多，因为计算量少，理论上一次$e^x$的计算，就相当于百多次乘法运算，重复使用对算力要求比较高。
3、梯度下降（Gradient Descent）的通式
repeat \(w_i = w_i - \alpha\frac{\partial J(\vec{w},b)}{\partial w_i}\) \(b = b-\alpha\frac{\partial J(\vec{w},b)}{\partial b}\) 其中，$\alpha$为学习率，需要手动选择，这类参数称为超参数
4、sigmoid 函数的 loss function
$L(f_{ {\vec{w},b} }(\vec{x}^i,y^i)) = \begin{cases} -\log(f_{ {\vec{w},b} }(\vec{x}^i)) & y^i = 1 \\ -\log(1-f_{\vec{w},b}(\vec{x}^i)) & y^i = 0 \end{cases}$
可以化简为：
\[L(f_{ {\vec{w},b}}(\vec{x}^i,y^i))=-y^ilog(f_{ {\vec{w},b} }(\vec{x}^i))-(1-y^i)log(1-f_{\vec{w},b}(\vec{x}^i))\]5、sigmoid 函数的cost function
\[J(\vec{w},b) = \frac{1}{m}\sum_{i=1}^m[L(f_{\vec{w},b},y^i)]=-\frac{1}{m}\sum_{i=1}^\infty[y^ilog(f_{\vec{w},b}(\vec{x}^i))+(1-y^i)log(1-f_{\vec{w},b}(\vec{x}^i))]\]6、多类分类函数 Softmax \(\hat{y}=softmax(o)\)
\(\hat{y}_i=\frac{e^{o_i}}{\sum_ke^{o_k}}\) 可以将每个类转化为0-1的概率分布(预测置信度)，且和为1
概率$y$和$\hat{y}$的区别作为损失
通常用交叉熵来衡量两个概率的区别 \(H(p,q)=\sum_i-p_ilog(q_i)\) 故$y$与$\hat{y}$的损失为 \(l(y,\hat{y})=-\sum_iy_ilog(\hat{y_i})=-log\hat{y_y}\)
其梯度是真实概率和预测概率的区别 \(\partial_{o_i}l(y,\hat{y})=softmax(o)_i-y_i\)
softmax（直接调用）代码
import torch
from d2l import torch as d2l
from torch import nn
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)
net = nn.Sequential(nn.Flatten(), nn.Linear(784, 10))
#Flatten函数将图像格式从二维数组转换为一维数组，输入形状是(批量大小，通道，高，宽)
def init_weights(m):
if type(m) == nn.Linear:
nn.init.normal_(m.weight, std=0.01)
net.apply(init_weights)
loss = nn.CrossEntropyLoss()
trainer = torch.optim.SGD(net.parameters(), lr=0.1)
num_epochs = 10
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, trainer)
softmax实现代码
import torch
from IPython import display
from d2l import torch as d2l
from d2l.torch import Accumulator, Animator
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size)
#把图像拉成一条向量，28*28 = 784
num_inputs = 784
#分类有十个类
num_outputs = 10
#W为权重，b为偏差
W = torch.normal(0, 0.01, size=(num_inputs, num_outputs), requires_grad=True)
b = torch.zeros(num_outputs, requires_grad=True)
def softmax(X):
#0是行，1是列，sum里面是哪个就把哪一个压缩
X_exp = torch.exp(X)
partition = X_exp.sum(1, keepdim=True)
return X_exp / partition # 这里应用了广播
def net(X):
return softmax(torch.matmul(X.reshape((-1, W.shape[0])), W) + b)
def cross_entropy(y_hat, y):
return - torch.log(y_hat[range(len(y_hat)), y])
def accuracy(y_hat, y):
"""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):
"""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]
def train_epoch_ch3(net, train_iter, loss, updater):
"""训练模型一个迭代周期"""
# 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)
l = loss(y_hat, 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(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):
"""训练模型"""
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
lr = 0.1
def updater(batch_size):
return d2l.sgd([W, b], lr, batch_size)
num_epochs = 10
train_ch3(net, train_iter, test_iter, cross_entropy, num_epochs, updater)
def show_images(imgs, num_rows, num_cols, titles=None, scale=1.5):
"""Plot a list of images.
Defined in :numref:`sec_fashion_mnist`"""
figsize = (num_cols * scale, num_rows * scale)
_, axes = d2l.plt.subplots(num_rows, num_cols, figsize=figsize)
axes = axes.flatten()
for i, (ax, img) in enumerate(zip(axes, imgs)):
if torch.is_tensor(img):
# Tensor Image
ax.imshow(img.numpy())
else:
# PIL Image
ax.imshow(img)
ax.axes.get_xaxis().set_visible(False)
ax.axes.get_yaxis().set_visible(False)
if titles:
ax.set_title(titles[i])
d2l.plt.show()
return axes
def predict_ch3(net, test_iter, n=6):
"""预测标签"""
for X, y in test_iter:
break
trues = d2l.get_fashion_mnist_labels(y)
preds = d2l.get_fashion_mnist_labels(net(X).argmax(axis=1))
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)