梯度下降
代码语言：javascript复制%matplotlib inline
import numpy as np
import torch
import time
from torch import nn, optim
import math
import sys
sys.path.append('/home/input')
import d2lzh1981 as d2l一维梯度下降
证明：沿梯度反方向移动自变量可以减小函数值
泰勒展开：
代入沿梯度方向的移动量
：
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)
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)
学习率
代码语言：javascript复制show_trace(gd(0.05))
局部极小值
代码语言：javascript复制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))
多维梯度下降
代码语言：javascript复制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): # 目标函数
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))
自适应方法
牛顿法
在
处泰勒展开：
最小值点处满足:
, 即我们希望
, 对上式关于
求导，忽略高阶无穷小，有：
c = 0.5
def f(x):
return np.cosh(c * x) # Objective
def gradf(x):
return c * np.sinh(c * x) # Derivative
def hessf(x):
return c**2 * np.cosh(c * x) # Hessian
# Hide learning rate for now
def newton(eta=1):
x = 10
results = [x]
for i in range(10):
x -= eta * gradf(x) / hessf(x)
results.append(x)
print('epoch 10, x:', x)
return results
show_trace(newton())
收敛性分析
只考虑在函数为凸函数, 且最小值点上
时的收敛速度：
令
为第
次迭代后
的值，
表示
到最小值点
的距离，由
:
两边除以
, 有：
代入更新方程 , 得到
当
时，有:
随机梯度下降参数更新
对于有
个样本对训练数据集，设
是第
个样本的损失函数, 则目标函数为:
其梯度为:
使用该梯度的一次更新的时间复杂度为
随机梯度下降更新公式
:
且有：
def f(x1, x2):
return x1 ** 2 2 * x2 ** 2 # Objective
def gradf(x1, x2):
return (2 * x1, 4 * x2) # Gradient
def sgd(x1, x2): # Simulate noisy gradient
global lr # Learning rate scheduler
(g1, g2) = gradf(x1, x2) # Compute gradient
(g1, g2) = (g1 np.random.normal(0.1), g2 np.random.normal(0.1))
eta_t = eta * lr() # Learning rate at time t
return (x1 - eta_t * g1, x2 - eta_t * g2) # Update variables
eta = 0.1
lr = (lambda: 1) # Constant learning rate
show_trace_2d(f, train_2d(sgd, steps=50))
动态学习率
代码语言：javascript复制def exponential():
global ctr
ctr = 1
return math.exp(-0.1 * ctr)
ctr = 1
lr = exponential # Set up learning rate
show_trace_2d(f, train_2d(sgd, steps=1000))
小批量随机梯度下降
读取数据
读取数据
代码语言：javascript复制def get_data_ch7(): # 本函数已保存在d2lzh_pytorch包中方便以后使用
data = np.genfromtxt('/home/kesci/input/airfoil4755/airfoil_self_noise.dat', delimiter='t')
data = (data - data.mean(axis=0)) / data.std(axis=0) # 标准化
return torch.tensor(data[:1500, :-1], dtype=torch.float32),
torch.tensor(data[:1500, -1], dtype=torch.float32) # 前1500个样本(每个样本5个特征)
features, labels = get_data_ch7()
import pandas as pd
df = pd.read_csv('/home/kesci/input/airfoil4755/airfoil_self_noise.dat', delimiter='t', header=None)
df.head(10)
从零开始实现
代码语言：javascript复制def sgd(params, states, hyperparams):
for p in params:
p.data -= hyperparams['lr'] * p.grad.data
# 本函数已保存在d2lzh_pytorch包中方便以后使用
def train_ch7(optimizer_fn, states, hyperparams, features, labels,
batch_size=10, num_epochs=2):
# 初始化模型
net, loss = d2l.linreg, d2l.squared_loss
w = torch.nn.Parameter(torch.tensor(np.random.normal(0, 0.01, size=(features.shape[1], 1)), dtype=torch.float32),
requires_grad=True)
b = torch.nn.Parameter(torch.zeros(1, dtype=torch.float32), requires_grad=True)
def eval_loss():
return loss(net(features, w, b), labels).mean().item()
ls = [eval_loss()]
data_iter = torch.utils.data.DataLoader(
torch.utils.data.TensorDataset(features, labels), batch_size, shuffle=True)
for _ in range(num_epochs):
start = time.time()
for batch_i, (X, y) in enumerate(data_iter):
l = loss(net(X, w, b), y).mean() # 使用平均损失
# 梯度清零
if w.grad is not None:
w.grad.data.zero_()
b.grad.data.zero_()
l.backward()
optimizer_fn([w, b], states, hyperparams) # 迭代模型参数
if (batch_i 1) * batch_size % 100 == 0:
ls.append(eval_loss()) # 每100个样本记录下当前训练误差
# 打印结果和作图
print('loss: %f, %f sec per epoch' % (ls[-1], time.time() - start))
d2l.set_figsize()
d2l.plt.plot(np.linspace(0, num_epochs, len(ls)), ls)
d2l.plt.xlabel('epoch')
d2l.plt.ylabel('loss')
def train_sgd(lr, batch_size, num_epochs=2):
train_ch7(sgd, None, {'lr': lr}, features, labels, batch_size, num_epochs)
train_sgd(1, 1500, 6)
简洁实现
代码语言：javascript复制# 例如: optimizer_fn=torch.optim.SGD, optimizer_hyperparams={"lr": 0.05}
def train_pytorch_ch7(optimizer_fn, optimizer_hyperparams, features, labels,
batch_size=10, num_epochs=2):
# 初始化模型
net = nn.Sequential(
nn.Linear(features.shape[-1], 1)
)
loss = nn.MSELoss()
optimizer = optimizer_fn(net.parameters(), **optimizer_hyperparams)
def eval_loss():
return loss(net(features).view(-1), labels).item() / 2
ls = [eval_loss()]
data_iter = torch.utils.data.DataLoader(
torch.utils.data.TensorDataset(features, labels), batch_size, shuffle=True)
for _ in range(num_epochs):
start = time.time()
for batch_i, (X, y) in enumerate(data_iter):
# 除以2是为了和train_ch7保持一致, 因为squared_loss中除了2
l = loss(net(X).view(-1), y) / 2
optimizer.zero_grad()
l.backward()
optimizer.step()
if (batch_i 1) * batch_size % 100 == 0:
ls.append(eval_loss())
# 打印结果和作图
print('loss: %f, %f sec per epoch' % (ls[-1], time.time() - start))
d2l.set_figsize()
d2l.plt.plot(np.linspace(0, num_epochs, len(ls)), ls)
d2l.plt.xlabel('epoch')
d2l.plt.ylabel('loss')
train_pytorch_ch7(optim.SGD, {"lr": 0.05}, features, labels, 10)
参考文献
[1]《动手深度学习》李沐
[2]伯禹教育课程
未完待续，如果有什么不懂的欢迎随时交流！
