论文：Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge, England: Cambridge University Press.
%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
一维梯度下降
下面以x^2为例
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)
通过改变学习率，来看一下效果
show_trace(gd(0.05))
#epoch 10, x: 3.4867844009999995
#学习率太小了，梯度下降还没有下降完全
show_trace(gd(1.1))
#epoch 10, x: 61.917364224000096
梯度太大，有点爆炸
局部极小值是梯度下降需要解决的问题
以f(x)=xcoscx 为例
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
多维梯度下降
以f(x)=x1^2+2*x2^2为例
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))
#epoch 20, x1 -0.057646, x2 -0.000073
自适应方法【上面的研究中，学习率的设置有主关因素】
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())
#epoch 10, x: 0.0
来看看牛顿法能不能解决局部最小值的问题
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)
def hessf(x):
return - 2 * c * np.sin(c * x) - x * c**2 * np.cos(c * x)
show_trace(newton())
#epoch 10, x: 26.83413291324767
#再降低一下学习率，就可以，所以说有时候学习率还是需要调的
show_trace(newton(0.5))
#epoch 10, x: 7.269860168684531
收敛性分析
二阶收敛
x=x-学习率*梯度/[heissan矩阵对角线的值]
梯度下降与线性搜索（共轭梯度法）
随机梯度下降
随机梯度下降参数更新
随机梯度下降是针对每一个样本都进行梯度更新，刚开始讲的梯度下降是针对所以样本一起进行梯度更新
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))
#epoch 50, x1 -0.027566, x2 0.137605
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))
#epoch 1000, x1 -0.677947, x2 -0.089379
def polynomial():
global ctr
ctr += 1
return (1 + 0.1 * ctr)**(-0.5)
ctr = 1
lr = polynomial # Set up learning rate
show_trace_2d(f, train_2d(sgd, steps=50))
#epoch 50, x1 -0.095244, x2 -0.041674
小批量随机梯度下降
读取数据
链接:https://archive.ics.uci.edu/ml/datasets/Airfoil+Self-Noise
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()
features.shape
#torch.Size([1500, 5])
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
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)
调差对比
train_sgd(1, 1500, 6)
#loss: 0.244373, 0.009881 sec per epoch
train_sgd(0.005, 1)
#loss: 0.245968, 0.463836 sec per epoch
train_sgd(0.05, 10)
#loss: 0.243900, 0.065017 sec per epoch
简洁实现
# 本函数与原书不同的是这里第一个参数优化器函数而不是优化器的名字
# 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)
#loss: 0.243770, 0.047664 sec per epoch
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