批量歸一化（BatchNormalization）

對(duì)輸入的標(biāo)準(zhǔn)化（淺層模型）

處理后的任意一個(gè)特征在數(shù)據(jù)集中所有樣本上的均值為0丰滑、標(biāo)準(zhǔn)差為1顾犹。
標(biāo)準(zhǔn)化處理輸入數(shù)據(jù)使各個(gè)特征的分布相近

批量歸一化（深度模型）

利用小批量上的均值和標(biāo)準(zhǔn)差，不斷調(diào)整神經(jīng)網(wǎng)絡(luò)中間輸出吨枉，從而使整個(gè)神經(jīng)網(wǎng)絡(luò)在各層的中間輸出的數(shù)值更穩(wěn)定蹦渣。

1.對(duì)全連接層做批量歸一化

位置：全連接層中的仿射變換和激活函數(shù)之間。
全連接：

2.對(duì)卷積層做批量歸?化

位置：卷積計(jì)算之后、應(yīng)?激活函數(shù)之前认臊。
如果卷積計(jì)算輸出多個(gè)通道圃庭，我們需要對(duì)這些通道的輸出分別做批量歸一化，且每個(gè)通道都擁有獨(dú)立的拉伸和偏移參數(shù)失晴。 計(jì)算：對(duì)單通道剧腻，batchsize=m,卷積計(jì)算輸出=pxq 對(duì)該通道中m×p×q個(gè)元素同時(shí)做批量歸一化,使用相同的均值和方差。

3.預(yù)測(cè)時(shí)的批量歸?化

訓(xùn)練：以batch為單位,對(duì)每個(gè)batch計(jì)算均值和方差涂屁。
預(yù)測(cè)：用移動(dòng)平均估算整個(gè)訓(xùn)練數(shù)據(jù)集的樣本均值和方差书在。
從零實(shí)現(xiàn)

#目前GPU算力資源預(yù)計(jì)17日上線，在此之前本代碼只能使用CPU運(yùn)行拆又。
#考慮到本代碼中的模型過(guò)大儒旬，CPU訓(xùn)練較慢，
#我們還將代碼上傳了一份到 https://www.kaggle.com/boyuai/boyu-d2l-deepcnn
#如希望提前使用gpu運(yùn)行請(qǐng)至kaggle帖族。

import time
import torch
from torch import nn, optim
import torch.nn.functional as F
import torchvision
import sys
sys.path.append("/home/kesci/input/") 
import d2lzh1981 as d2l
device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')

def batch_norm(is_training, X, gamma, beta, moving_mean, moving_var, eps, momentum):
    # 判斷當(dāng)前模式是訓(xùn)練模式還是預(yù)測(cè)模式
    if not is_training:
        # 如果是在預(yù)測(cè)模式下栈源，直接使用傳入的移動(dòng)平均所得的均值和方差
        X_hat = (X - moving_mean) / torch.sqrt(moving_var + eps)
    else:
        assert len(X.shape) in (2, 4)
        if len(X.shape) == 2:
            # 使用全連接層的情況，計(jì)算特征維上的均值和方差
            mean = X.mean(dim=0)
            var = ((X - mean) ** 2).mean(dim=0)
        else:
            # 使用二維卷積層的情況竖般，計(jì)算通道維上（axis=1）的均值和方差甚垦。這里我們需要保持
            # X的形狀以便后面可以做廣播運(yùn)算
            mean = X.mean(dim=0, keepdim=True).mean(dim=2, keepdim=True).mean(dim=3, keepdim=True)
            var = ((X - mean) ** 2).mean(dim=0, keepdim=True).mean(dim=2, keepdim=True).mean(dim=3, keepdim=True)
        # 訓(xùn)練模式下用當(dāng)前的均值和方差做標(biāo)準(zhǔn)化
        X_hat = (X - mean) / torch.sqrt(var + eps)
        # 更新移動(dòng)平均的均值和方差
        moving_mean = momentum * moving_mean + (1.0 - momentum) * mean
        moving_var = momentum * moving_var + (1.0 - momentum) * var
    Y = gamma * X_hat + beta  # 拉伸和偏移
    return Y, moving_mean, moving_var

class BatchNorm(nn.Module):
    def __init__(self, num_features, num_dims):
        super(BatchNorm, self).__init__()
        if num_dims == 2:
            shape = (1, num_features) #全連接層輸出神經(jīng)元
        else:
            shape = (1, num_features, 1, 1)  #通道數(shù)
        # 參與求梯度和迭代的拉伸和偏移參數(shù)，分別初始化成0和1
        self.gamma = nn.Parameter(torch.ones(shape))
        self.beta = nn.Parameter(torch.zeros(shape))
        # 不參與求梯度和迭代的變量，全在內(nèi)存上初始化成0
        self.moving_mean = torch.zeros(shape)
        self.moving_var = torch.zeros(shape)

    def forward(self, X):
        # 如果X不在內(nèi)存上艰亮，將moving_mean和moving_var復(fù)制到X所在顯存上
        if self.moving_mean.device != X.device:
            self.moving_mean = self.moving_mean.to(X.device)
            self.moving_var = self.moving_var.to(X.device)
        # 保存更新過(guò)的moving_mean和moving_var, Module實(shí)例的traning屬性默認(rèn)為true, 調(diào)用.eval()后設(shè)成false
        Y, self.moving_mean, self.moving_var = batch_norm(self.training, 
            X, self.gamma, self.beta, self.moving_mean,
            self.moving_var, eps=1e-5, momentum=0.9)
        return Y

基于LeNet的應(yīng)用

net = nn.Sequential(
            nn.Conv2d(1, 6, 5), # in_channels, out_channels, kernel_size
            BatchNorm(6, num_dims=4),
            nn.Sigmoid(),
            nn.MaxPool2d(2, 2), # kernel_size, stride
            nn.Conv2d(6, 16, 5),
            BatchNorm(16, num_dims=4),
            nn.Sigmoid(),
            nn.MaxPool2d(2, 2),
            d2l.FlattenLayer(),
            nn.Linear(16*4*4, 120),
            BatchNorm(120, num_dims=2),
            nn.Sigmoid(),
            nn.Linear(120, 84),
            BatchNorm(84, num_dims=2),
            nn.Sigmoid(),
            nn.Linear(84, 10)
        )
print(net)

Sequential(
(0): Conv2d(1, 6, kernel_size=(5, 5), stride=(1, 1))
(1): BatchNorm()
(2): Sigmoid()
(3): MaxPool2d(kernel_size=2, stride=2, padding=0, dilation=1, ceil_mode=False)
(4): Conv2d(6, 16, kernel_size=(5, 5), stride=(1, 1))
(5): BatchNorm()
(6): Sigmoid()
(7): MaxPool2d(kernel_size=2, stride=2, padding=0, dilation=1, ceil_mode=False)
(8): FlattenLayer()
(9): Linear(in_features=256, out_features=120, bias=True)
(10): BatchNorm()
(11): Sigmoid()
(12): Linear(in_features=120, out_features=84, bias=True)
(13): BatchNorm()
(14): Sigmoid()
(15): Linear(in_features=84, out_features=10, bias=True)
)

#batch_size = 256  
##cpu要調(diào)小batchsize
batch_size=16

def load_data_fashion_mnist(batch_size, resize=None, root='/home/kesci/input/FashionMNIST2065'):
    """Download the fashion mnist dataset and then load into memory."""
    trans = []
    if resize:
        trans.append(torchvision.transforms.Resize(size=resize))
    trans.append(torchvision.transforms.ToTensor())
    
    transform = torchvision.transforms.Compose(trans)
    mnist_train = torchvision.datasets.FashionMNIST(root=root, train=True, download=True, transform=transform)
    mnist_test = torchvision.datasets.FashionMNIST(root=root, train=False, download=True, transform=transform)

    train_iter = torch.utils.data.DataLoader(mnist_train, batch_size=batch_size, shuffle=True, num_workers=2)
    test_iter = torch.utils.data.DataLoader(mnist_test, batch_size=batch_size, shuffle=False, num_workers=2)

    return train_iter, test_iter
train_iter, test_iter = load_data_fashion_mnist(batch_size)

lr, num_epochs = 0.001, 5
optimizer = torch.optim.Adam(net.parameters(), lr=lr)
d2l.train_ch5(net, train_iter, test_iter, batch_size, optimizer, device, num_epochs)

簡(jiǎn)潔實(shí)現(xiàn)

net = nn.Sequential(
            nn.Conv2d(1, 6, 5), # in_channels, out_channels, kernel_size
            nn.BatchNorm2d(6),
            nn.Sigmoid(),
            nn.MaxPool2d(2, 2), # kernel_size, stride
            nn.Conv2d(6, 16, 5),
            nn.BatchNorm2d(16),
            nn.Sigmoid(),
            nn.MaxPool2d(2, 2),
            d2l.FlattenLayer(),
            nn.Linear(16*4*4, 120),
            nn.BatchNorm1d(120),
            nn.Sigmoid(),
            nn.Linear(120, 84),
            nn.BatchNorm1d(84),
            nn.Sigmoid(),
            nn.Linear(84, 10)
        )

optimizer = torch.optim.Adam(net.parameters(), lr=lr)
d2l.train_ch5(net, train_iter, test_iter, batch_size, optimizer, device, num_epochs)

殘差網(wǎng)絡(luò)（ResNet）

深度學(xué)習(xí)的問(wèn)題：深度CNN網(wǎng)絡(luò)達(dá)到一定深度后再一味地增加層數(shù)并不能帶來(lái)進(jìn)一步地分類性能提高闭翩，反而會(huì)招致網(wǎng)絡(luò)收斂變得更慢，準(zhǔn)確率也變得更差迄埃。

殘差塊（Residual Block）

恒等映射：
左邊：f(x)=x
右邊：f(x)-x=0 （易于捕捉恒等映射的細(xì)微波動(dòng)）

在殘差塊中疗韵，輸?可通過(guò)跨層的數(shù)據(jù)線路更快 地向前傳播。

class Residual(nn.Module):  # 本類已保存在d2lzh_pytorch包中方便以后使用
    #可以設(shè)定輸出通道數(shù)调俘、是否使用額外的1x1卷積層來(lái)修改通道數(shù)以及卷積層的步幅伶棒。
    def __init__(self, in_channels, out_channels, use_1x1conv=False, stride=1):
        super(Residual, self).__init__()
        self.conv1 = nn.Conv2d(in_channels, out_channels, kernel_size=3, padding=1, stride=stride)
        self.conv2 = nn.Conv2d(out_channels, out_channels, kernel_size=3, padding=1)
        if use_1x1conv:
            self.conv3 = nn.Conv2d(in_channels, out_channels, kernel_size=1, stride=stride)
        else:
            self.conv3 = None
        self.bn1 = nn.BatchNorm2d(out_channels)
        self.bn2 = nn.BatchNorm2d(out_channels)

    def forward(self, X):
        Y = F.relu(self.bn1(self.conv1(X)))
        Y = self.bn2(self.conv2(Y))
        if self.conv3:
            X = self.conv3(X)
        return F.relu(Y + X)

blk = Residual(3, 3)
X = torch.rand((4, 3, 6, 6))
blk(X).shape # torch.Size([4, 3, 6, 6])

out7:

torch.Size([4, 3, 6, 6])

blk = Residual(3, 6, use_1x1conv=True, stride=2)
blk(X).shape # torch.Size([4, 6, 3, 3])

out8:

torch.Size([4, 6, 3, 3])

ResNet模型

卷積(64,7x7,3)
批量一體化
最大池化(3x3,2)

殘差塊x4 (通過(guò)步幅為2的殘差塊在每個(gè)模塊之間減小高和寬)

全局平均池化

全連接

net = nn.Sequential(
        nn.Conv2d(1, 64, kernel_size=7, stride=2, padding=3),
        nn.BatchNorm2d(64), 
        nn.ReLU(),
        nn.MaxPool2d(kernel_size=3, stride=2, padding=1))

def resnet_block(in_channels, out_channels, num_residuals, first_block=False):
    if first_block:
        assert in_channels == out_channels # 第一個(gè)模塊的通道數(shù)同輸入通道數(shù)一致
    blk = []
    for i in range(num_residuals):
        if i == 0 and not first_block:
            blk.append(Residual(in_channels, out_channels, use_1x1conv=True, stride=2))
        else:
            blk.append(Residual(out_channels, out_channels))
    return nn.Sequential(*blk)

net.add_module("resnet_block1", resnet_block(64, 64, 2, first_block=True))
net.add_module("resnet_block2", resnet_block(64, 128, 2))
net.add_module("resnet_block3", resnet_block(128, 256, 2))
net.add_module("resnet_block4", resnet_block(256, 512, 2))

net.add_module("global_avg_pool", d2l.GlobalAvgPool2d()) # GlobalAvgPool2d的輸出: (Batch, 512, 1, 1)
net.add_module("fc", nn.Sequential(d2l.FlattenLayer(), nn.Linear(512, 10))) 

X = torch.rand((1, 1, 224, 224))
for name, layer in net.named_children():
    X = layer(X)
    print(name, ' output shape:\t', X.shape)
0  output shape:     torch.Size([1, 64, 112, 112])
1  output shape:     torch.Size([1, 64, 112, 112])
2  output shape:     torch.Size([1, 64, 112, 112])
3  output shape:     torch.Size([1, 64, 56, 56])
resnet_block1  output shape:     torch.Size([1, 64, 56, 56])
resnet_block2  output shape:     torch.Size([1, 128, 28, 28])
resnet_block3  output shape:     torch.Size([1, 256, 14, 14])
resnet_block4  output shape:     torch.Size([1, 512, 7, 7])
global_avg_pool  output shape:   torch.Size([1, 512, 1, 1])
fc  output shape:    torch.Size([1, 10])

lr, num_epochs = 0.001, 5
optimizer = torch.optim.Adam(net.parameters(), lr=lr)
d2l.train_ch5(net, train_iter, test_iter, batch_size, optimizer, device, num_epochs)

稠密連接網(wǎng)絡(luò)（DenseNet）

def conv_block(in_channels, out_channels):
    blk = nn.Sequential(nn.BatchNorm2d(in_channels), 
                        nn.ReLU(),
                        nn.Conv2d(in_channels, out_channels, kernel_size=3, padding=1))
    return blk

class DenseBlock(nn.Module):
    def __init__(self, num_convs, in_channels, out_channels):
        super(DenseBlock, self).__init__()
        net = []
        for i in range(num_convs):
            in_c = in_channels + i * out_channels
            net.append(conv_block(in_c, out_channels))
        self.net = nn.ModuleList(net)
        self.out_channels = in_channels + num_convs * out_channels # 計(jì)算輸出通道數(shù)

    def forward(self, X):
        for blk in self.net:
            Y = blk(X)
            X = torch.cat((X, Y), dim=1)  # 在通道維上將輸入和輸出連結(jié)
        return X

blk = DenseBlock(2, 3, 10)
X = torch.rand(4, 3, 8, 8)
Y = blk(X)
Y.shape # torch.Size([4, 23, 8, 8])

out:

torch.Size([4, 23, 8, 8])

過(guò)渡層

1X1 卷積層：來(lái)減小通道數(shù)
步幅為2的平均池化層：減半高和寬

def transition_block(in_channels, out_channels):
    blk = nn.Sequential(
            nn.BatchNorm2d(in_channels), 
            nn.ReLU(),
            nn.Conv2d(in_channels, out_channels, kernel_size=1),
            nn.AvgPool2d(kernel_size=2, stride=2))
    return blk

blk = transition_block(23, 10)
blk(Y).shape # torch.Size([4, 10, 4, 4])

out

torch.Size([4, 10, 4, 4])

DenseNet模型

net = nn.Sequential(
        nn.Conv2d(1, 64, kernel_size=7, stride=2, padding=3),
        nn.BatchNorm2d(64), 
        nn.ReLU(),
        nn.MaxPool2d(kernel_size=3, stride=2, padding=1))

num_channels, growth_rate = 64, 32  # num_channels為當(dāng)前的通道數(shù)
num_convs_in_dense_blocks = [4, 4, 4, 4]

for i, num_convs in enumerate(num_convs_in_dense_blocks):
    DB = DenseBlock(num_convs, num_channels, growth_rate)
    net.add_module("DenseBlosk_%d" % i, DB)
    # 上一個(gè)稠密塊的輸出通道數(shù)
    num_channels = DB.out_channels
    # 在稠密塊之間加入通道數(shù)減半的過(guò)渡層
    if i != len(num_convs_in_dense_blocks) - 1:
        net.add_module("transition_block_%d" % i, transition_block(num_channels, num_channels // 2))
        num_channels = num_channels // 2

net.add_module("BN", nn.BatchNorm2d(num_channels))
net.add_module("relu", nn.ReLU())
net.add_module("global_avg_pool", d2l.GlobalAvgPool2d()) # GlobalAvgPool2d的輸出: (Batch, num_channels, 1, 1)
net.add_module("fc", nn.Sequential(d2l.FlattenLayer(), nn.Linear(num_channels, 10))) 

X = torch.rand((1, 1, 96, 96))
for name, layer in net.named_children():
    X = layer(X)
    print(name, ' output shape:\t', X.shape)

結(jié)果

0  output shape:     torch.Size([1, 64, 48, 48])
1  output shape:     torch.Size([1, 64, 48, 48])
2  output shape:     torch.Size([1, 64, 48, 48])
3  output shape:     torch.Size([1, 64, 24, 24])
DenseBlosk_0  output shape:  torch.Size([1, 192, 24, 24])
transition_block_0  output shape:    torch.Size([1, 96, 12, 12])
DenseBlosk_1  output shape:  torch.Size([1, 224, 12, 12])
transition_block_1  output shape:    torch.Size([1, 112, 6, 6])
DenseBlosk_2  output shape:  torch.Size([1, 240, 6, 6])
transition_block_2  output shape:    torch.Size([1, 120, 3, 3])
DenseBlosk_3  output shape:  torch.Size([1, 248, 3, 3])
BN  output shape:    torch.Size([1, 248, 3, 3])
relu  output shape:  torch.Size([1, 248, 3, 3])
global_avg_pool  output shape:   torch.Size([1, 248, 1, 1])
fc  output shape:    torch.Size([1, 10])

#batch_size = 256
batch_size=16
# 如出現(xiàn)“out of memory”的報(bào)錯(cuò)信息，可減小batch_size或resize
train_iter, test_iter =load_data_fashion_mnist(batch_size, resize=96)
lr, num_epochs = 0.001, 5
optimizer = torch.optim.Adam(net.parameters(), lr=lr)
d2l.train_ch5(net, train_iter, test_iter, batch_size, optimizer, device, num_epochs)

優(yōu)化與深度學(xué)習(xí)

優(yōu)化與估計(jì)

盡管優(yōu)化方法可以最小化深度學(xué)習(xí)中的損失函數(shù)值骇钦，但本質(zhì)上優(yōu)化方法達(dá)到的目標(biāo)與深度學(xué)習(xí)的目標(biāo)并不相同宛渐。

優(yōu)化方法目標(biāo)：訓(xùn)練集損失函數(shù)值
深度學(xué)習(xí)目標(biāo)：測(cè)試集損失函數(shù)值（泛化性）

%matplotlib inline
import sys
sys.path.append('/home/kesci/input')
import d2lzh1981 as d2l
from mpl_toolkits import mplot3d # 三維畫(huà)圖
import numpy as np

def f(x): return x * np.cos(np.pi * x)
def g(x): return f(x) + 0.2 * np.cos(5 * np.pi * x)

d2l.set_figsize((5, 3))
x = np.arange(0.5, 1.5, 0.01)
fig_f, = d2l.plt.plot(x, f(x),label="train error")
fig_g, = d2l.plt.plot(x, g(x),'--', c='purple', label="test error")
fig_f.axes.annotate('empirical risk', (1.0, -1.2), (0.5, -1.1),arrowprops=dict(arrowstyle='->'))
fig_g.axes.annotate('expected risk', (1.1, -1.05), (0.95, -0.5),arrowprops=dict(arrowstyle='->'))
d2l.plt.xlabel('x')
d2l.plt.ylabel('risk')
d2l.plt.legend(loc="upper right")

out2:

<matplotlib.legend.Legend at 0x7fc092436080>

優(yōu)化在深度學(xué)習(xí)中的挑戰(zhàn)

• 局部最小值
• 鞍點(diǎn)
• 梯度消失

局部最小值

def f(x):
    return x * np.cos(np.pi * x)

d2l.set_figsize((4.5, 2.5))
x = np.arange(-1.0, 2.0, 0.1)
fig,  = d2l.plt.plot(x, f(x))
fig.axes.annotate('local minimum', xy=(-0.3, -0.25), xytext=(-0.77, -1.0),
                  arrowprops=dict(arrowstyle='->'))
fig.axes.annotate('global minimum', xy=(1.1, -0.95), xytext=(0.6, 0.8),
                  arrowprops=dict(arrowstyle='->'))
d2l.plt.xlabel('x')
d2l.plt.ylabel('f(x)');

鞍點(diǎn)

x = np.arange(-2.0, 2.0, 0.1)
fig, = d2l.plt.plot(x, x**3)
fig.axes.annotate('saddle point', xy=(0, -0.2), xytext=(-0.52, -5.0),
                  arrowprops=dict(arrowstyle='->'))
d2l.plt.xlabel('x')
d2l.plt.ylabel('f(x)');

x, y = np.mgrid[-1: 1: 31j, -1: 1: 31j]
z = x**2 - y**2

d2l.set_figsize((6, 4))
ax = d2l.plt.figure().add_subplot(111, projection='3d')
ax.plot_wireframe(x, y, z, **{'rstride': 2, 'cstride': 2})
ax.plot([0], [0], [0], 'ro', markersize=10)
ticks = [-1,  0, 1]
d2l.plt.xticks(ticks)
d2l.plt.yticks(ticks)
ax.set_zticks(ticks)
d2l.plt.xlabel('x')
d2l.plt.ylabel('y');

x = np.arange(-2.0, 5.0, 0.01)
fig, = d2l.plt.plot(x, np.tanh(x))
d2l.plt.xlabel('x')
d2l.plt.ylabel('f(x)')
fig.axes.annotate('vanishing gradient', (4, 1), (2, 0.0) ,arrowprops=dict(arrowstyle='->'))

out:

Text(2, 0.0, 'vanishing gradient')

凸性 （Convexity）

基礎(chǔ)

集合

函數(shù)

def f(x):
    return 0.5 * x**2  # Convex

def g(x):
    return np.cos(np.pi * x)  # Nonconvex

def h(x):
    return np.exp(0.5 * x)  # Convex

x, segment = np.arange(-2, 2, 0.01), np.array([-1.5, 1])
d2l.use_svg_display()
_, axes = d2l.plt.subplots(1, 3, figsize=(9, 3))

for ax, func in zip(axes, [f, g, h]):
    ax.plot(x, func(x))
    ax.plot(segment, func(segment),'--', color="purple")
    # d2l.plt.plot([x, segment], [func(x), func(segment)], axes=ax)

Jensen 不等式

性質(zhì)

1. 無(wú)局部極小值
2. 與凸集的關(guān)系
3. 二階條件

無(wú)局部最小值

與凸集的關(guān)系

x, y = np.meshgrid(np.linspace(-1, 1, 101), np.linspace(-1, 1, 101),
                   indexing='ij')

z = x**2 + 0.5 * np.cos(2 * np.pi * y)

# Plot the 3D surface
d2l.set_figsize((6, 4))
ax = d2l.plt.figure().add_subplot(111, projection='3d')
ax.plot_wireframe(x, y, z, **{'rstride': 10, 'cstride': 10})
ax.contour(x, y, z, offset=-1)
ax.set_zlim(-1, 1.5)

# Adjust labels
for func in [d2l.plt.xticks, d2l.plt.yticks, ax.set_zticks]:
    func([-1, 0, 1])

凸函數(shù)與二階導(dǎo)數(shù)

def f(x):
    return 0.5 * x**2

x = np.arange(-2, 2, 0.01)
axb, ab = np.array([-1.5, -0.5, 1]), np.array([-1.5, 1])

d2l.set_figsize((3.5, 2.5))
fig_x, = d2l.plt.plot(x, f(x))
fig_axb, = d2l.plt.plot(axb, f(axb), '-.',color="purple")
fig_ab, = d2l.plt.plot(ab, f(ab),'g-.')

fig_x.axes.annotate('a', (-1.5, f(-1.5)), (-1.5, 1.5),arrowprops=dict(arrowstyle='->'))
fig_x.axes.annotate('b', (1, f(1)), (1, 1.5),arrowprops=dict(arrowstyle='->'))
fig_x.axes.annotate('x', (-0.5, f(-0.5)), (-1.5, f(-0.5)),arrowprops=dict(arrowstyle='->'))

out

Text(-1.5, 0.125, 'x')

限制條件

拉格朗日乘子法

懲罰項(xiàng)

投影

梯度下降

介紹梯度下降、隨機(jī)梯度下降和小批量梯度下降的原理及實(shí)現(xiàn)

%matplotlib inline
import numpy as np
import torch
import time
from torch import nn, optim
import math
import sys
sys.path.append('/home/kesci/input')
import d2lzh1981 as d2l

一維梯度下降

證明：沿梯度反方向移動(dòng)自變量可以減小函數(shù)值

泰勒展開(kāi)：

def f(x):
    return x**2  # Objective function

def gradf(x):
    return 2 * x  # Its derivative

def gd(eta):
    x = 10
    results = [x]
    for i in range(10):
        x -= eta * gradf(x)
        results.append(x)
    print('epoch 10, x:', x)
    return results

res = gd(0.2)

epoch 10, x: 0.06046617599999997

def show_trace(res):
    n = max(abs(min(res)), abs(max(res)))
    f_line = np.arange(-n, n, 0.01)
    d2l.set_figsize((3.5, 2.5))
    d2l.plt.plot(f_line, [f(x) for x in f_line],'-')
    d2l.plt.plot(res, [f(x) for x in res],'-o')
    d2l.plt.xlabel('x')
    d2l.plt.ylabel('f(x)')
    

show_trace(res)

學(xué)習(xí)率

show_trace(gd(0.05))

epoch 10, x: 3.4867844009999995

show_trace(gd(1.1)

epoch 10, x: 61.917364224000096

局部極小值

c = 0.15 * np.pi

def f(x):
    return x * np.cos(c * x)

def gradf(x):
    return np.cos(c * x) - c * x * np.sin(c * x)

show_trace(gd(2))

epoch 10, x: -1.528165927635083

多維梯度下降

def train_2d(trainer, steps=20):
    x1, x2 = -5, -2
    results = [(x1, x2)]
    for i in range(steps):
        x1, x2 = trainer(x1, x2)
        results.append((x1, x2))
    print('epoch %d, x1 %f, x2 %f' % (i + 1, x1, x2))
    return results

def show_trace_2d(f, results): 
    d2l.plt.plot(*zip(*results), '-o', color='#ff7f0e')
    x1, x2 = np.meshgrid(np.arange(-5.5, 1.0, 0.1), np.arange(-3.0, 1.0, 0.1))
    d2l.plt.contour(x1, x2, f(x1, x2), colors='#1f77b4')
    d2l.plt.xlabel('x1')
    d2l.plt.ylabel('x2')

eta = 0.1

def f_2d(x1, x2):  # 目標(biāo)函數(shù)
    return x1 ** 2 + 2 * x2 ** 2

def gd_2d(x1, x2):
    return (x1 - eta * 2 * x1, x2 - eta * 4 * x2)

show_trace_2d(f_2d, train_2d(gd_2d))

epoch 20, x1 -0.057646, x2 -0.000073

自適應(yīng)方法

牛頓法