Multilayer Perceptron

Non-linear Hypotheses

线性回归和逻辑回归在特征很多时，计算量会很大。
一个简单的三层神经网络模型： \[a_i^{(j)} = \text{"activation" of unit $i$ in layer $j$}\]\[\Theta^{(j)} = \text{matrix of weights controlling function mapping from layer $j$ to layer $j+1$}\]
其中：\[a_1^{(2)} = g(\Theta_{10}^{(1)}x_0 + \Theta_{11}^{(1)}x_1 + \Theta_{12}^{(1)}x_2 + \Theta_{13}^{(1)}x_3)\]\[a_2^{(2)} = g(\Theta_{20}^{(1)}x_0 + \Theta_{21}^{(1)}x_1 + \Theta_{22}^{(1)}x_2 + \Theta_{23}^{(1)}x_3)\]\[a_3^{(2)} = g(\Theta_{30}^{(1)}x_0 + \Theta_{31}^{(1)}x_1 + \Theta_{32}^{(1)}x_2 + \Theta_{33}^{(1)}x_3)\]\[h_\Theta(x) = a_1^{(3)} = g(\Theta_{10}^{(2)}a_0^{(2)} + \Theta_{11}^{(2)}a_1^{(2)} + \Theta_{12}^{(2)}a_2^{(2)} + \Theta_{13}^{(2)}a_3^{(2)})\]

vectorized implementation

将上面公式中函数\(g\)中的东西用\(z\)代替： \[a_1^{(2)} = g(z_1^{(2)})\]\[a_2^{(2)} = g(z_2^{(2)})\]\[a_3^{(2)} = g(z_3^{(2)})\] 令\(x=a^{(1)}\)： \[z^{(j)} = \Theta^{(j-1)}a^{(j-1)}\] 得到： \[ \begin{aligned}z^{(j)} = \begin{bmatrix}z_1^{(j)} \\ z_2^{(j)} \\ \cdots \\z_n^{(j)}\end{bmatrix}\end{aligned} \]

这块的记号比较多，用例子梳理下：
实现一个逻辑与的神经网络：

那么：


所以有：

再来一个多层的，实现XNOR功能（两输入都为0或都为1，输出才为1）：

基本的神经元：

• 逻辑与
  
• 逻辑或
  
• 逻辑非
  
  先构造一个表示后半部分的神经元：
  这样的：
  
  接着将前半部分组合起来：
  

Multiclass Classification

在这里插入图片描述

Motivation

Implementation

import torch
from torchvision.datasets import FashionMNIST
from torch.utils.data import DataLoader
import torchvision.transforms as transforms
import numpy as np


def load_data():
    # My data is in ../data/FashionMNIST/raw & ../data/FashionMNIST/processed
    train_data = FashionMNIST(root='../data', train=True, download=False, transform=transforms.ToTensor())
    test_data = FashionMNIST(root='../data', train=False, download=False, transform=transforms.ToTensor())
    train_iter = DataLoader(train_data, batch_size=256, shuffle=True, num_workers=0)
    test_iter = DataLoader(test_data, batch_size=256, shuffle=False, num_workers=0)
    return train_iter, test_iter


num_inputs, num_outputs, num_hiddens = 784, 10, 256
W1 = torch.tensor(np.random.normal(0, 0.01, (num_inputs, num_hiddens)), dtype=torch.float)
b1 = torch.zeros(num_hiddens, dtype=torch.float)
W2 = torch.tensor(np.random.normal(0, 0.01, (num_hiddens, num_outputs)), dtype=torch.float)
b2 = torch.zeros(num_outputs, dtype=torch.float)
# W1 = nn.Parameter(torch.randn(num_inputs, num_hiddens, requires_grad=True) * 0.01)
# b1 = nn.Parameter(torch.zeros(num_hiddens, requires_grad=True))
# W2 = nn.Parameter(torch.randn(num_hiddens, num_outputs, requires_grad=True) * 0.01)
# b2 = nn.Parameter(torch.zeros(num_outputs, requires_grad=True))
params = [W1, b1, W2, b2]
for param in params:
    param.requires_grad_(requires_grad=True)


def relu(X):
    return torch.max(input=X, other=torch.zeros(X.shape))


def softmax(X):
    X_exp = torch.exp(X)
    partition = X_exp.sum(1, keepdim=True)
    return X_exp / partition


def net(X):
    X = X.reshape((-1, num_inputs))
    H = relu(torch.matmul(X, W1) + b1)
    output = torch.matmul(H, W2) + b2
    return softmax(output)


def cross_entropy(y_predict, y):
    return -torch.log(y_predict[range(len(y_predict)), y])


def sgd(params, lr, batch_size):
    for param in params:
        param.data -= lr * param.grad / batch_size


def evaluate_accuracy(net, data_iter):
    acc_sum, n = 0.0, 0
    for X, y in data_iter:
        acc_sum += (net(X).argmax(dim=1) == y).float().sum().item()
        n += y.shape[0]
    return acc_sum / n


def train(net, train_iter, test_iter, loss, num_epochs, batch_size=256, lr=0.1, params=None, optimizer=None):
    for epoch in range(num_epochs):
        train_loss_sum, train_acc_sum, n = 0.0, 0.0, 0
        for X, y in train_iter:
            y_predict = net(X)
            l = cross_entropy(y_predict, y).sum()
            l.backward()
            sgd(params, lr, batch_size)

            W1.grad.data.zero_()
            b1.grad.data.zero_()
            W2.grad.data.zero_()
            b2.grad.data.zero_()

            train_loss_sum += l.item()
            train_acc_sum += (y_predict.argmax(dim=1) == y).sum().item()
            n += y.shape[0]
        test_acc = evaluate_accuracy(net, test_iter)
        print('epoch %d, loss %.4f, train_acc %.4f, test_acc %.4f'
              % (epoch + 1, train_loss_sum / n, train_acc_sum / n, test_acc))


if __name__ == "__main__":
    # device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
    train_iter, test_iter = load_data()
    train(net, train_iter, test_iter, cross_entropy, 10, lr=0.1, params=params)

import torch
import torch.nn as nn
import torch.nn.init as init
from torchvision.datasets import FashionMNIST
from torch.utils.data import DataLoader
import torchvision.transforms as transforms


def load_data():
    # My data is in ../data/FashionMNIST/raw & ../data/FashionMNIST/processed
    train_data = FashionMNIST(root='../data', train=True, download=False, transform=transforms.ToTensor())
    test_data = FashionMNIST(root='../data', train=False, download=False, transform=transforms.ToTensor())
    train_iter = DataLoader(train_data, batch_size=256, shuffle=True, num_workers=0)
    test_iter = DataLoader(test_data, batch_size=256, shuffle=False, num_workers=0)
    return train_iter, test_iter


class Net(nn.Module):
    def __init__(self, num_inputs, num_outputs, num_hiddens):
        super(Net, self).__init__()
        self.l1 = nn.Linear(num_inputs, num_hiddens)
        self.relu1 = nn.ReLU()
        self.l2 = nn.Linear(num_hiddens, num_outputs)

    def forward(self, X):
        X = X.view(X.shape[0], -1)
        o1 = self.relu1(self.l1(X))
        o2 = self.l2(o1)
        return o2

    def init_params(self):
        for param in self.parameters():
            init.normal_(param, mean=0, std=0.01)


def evaluate_accuracy(net, data_iter):
    acc_sum, n = 0.0, 0
    for X, y in data_iter:
        acc_sum += (net(X).argmax(dim=1) == y).float().sum().item()
        n += y.shape[0]
    return acc_sum / n


def train(net, train_iter, test_iter, loss, num_epochs, batch_size=256, lr=0.1, params=None, optimizer=None):
    for epoch in range(num_epochs):
        train_loss_sum, train_acc_sum, n = 0.0, 0.0, 0
        for X, y in train_iter:
            y_predict = net(X)
            l = loss(y_predict, y).sum()
            l.backward()

            optimizer.step()
            optimizer.zero_grad()

            train_loss_sum += l.item()
            train_acc_sum += (y_predict.argmax(dim=1) == y).sum().item()
            n += y.shape[0]
        test_acc = evaluate_accuracy(net, test_iter)
        print('epoch %d, loss %.4f, train_acc %.4f, test_acc %.4f'
              % (epoch + 1, train_loss_sum / n, train_acc_sum / n, test_acc))


num_inputs, num_outputs, num_hiddens = 784, 10, 256
net = Net(num_inputs, num_outputs, num_hiddens)
net.init_params()

loss = nn.CrossEntropyLoss()
optimizer = torch.optim.SGD(net.parameters(), lr=0.1)


if __name__ == "__main__":
    train_iter, test_iter = load_data()
    train(net, train_iter, test_iter, loss, 10, optimizer=optimizer)