L20 梯度下降、随机梯度下降和小批量梯度下降
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提取码：dwjq
梯度下降
（Boyd & Vandenberghe, 一书中）
%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
代码解释一维梯度下降
证明：沿梯度反方向移动自变量可以减小函数值
泰勒展开：
f(x+\epsilon)=f(x)+\epsilon f^{\prime}(x)+\mathcal{O}\left(\epsilon^{2}\right)
代入沿梯度方向的移动量 \eta f^{\prime}(x)：
f\left(x-\eta f^{\prime}(x)\right)=f(x)-\eta f^{\prime 2}(x)+\mathcal{O}\left(\eta^{2} f^{\prime 2}(x)\right)
f\left(x-\eta f^{\prime}(x)\right) \lesssim f(x)
x \leftarrow x-\eta f^{\prime}(x)
e.g.
f(x) = x^2
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)
代码解释 epoch 10, x: 0.06046617599999997
代码解释 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)
代码解释
学习率
show_trace(gd(0.05))
代码解释 epoch 10, x: 3.4867844009999995
代码解释
show_trace(gd(1.1))
代码解释 epoch 10, x: 61.917364224000096
代码解释
局部极小值
e.g.
f(x) = x\cos cx
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))
代码解释 epoch 10, x: -1.528165927635083
代码解释
多维梯度下降
梯度向量\nabla f(\mathbf{x})在点\mathbf{x}处由各个偏导数组成的向量组成
f(\\x + \\varepsilon) = f(\\x) + \\varepsilon^T \\nabla f(\\x) + \\mathcal{O}(||\\varepsilon||^2)
\mathbf{x} \leftarrow \mathbf{x}-\eta \nabla f(\mathbf{x})
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')
代码解释f(x) = x_1^2 + 2x_2^2
eta = 0.1
def f_2d(x1, x2): # 目标函数
return x1 ** 2 + 2 * x2 *
def gd_2d(x1, x2):
return (x1 - eta * 2 * x1, x2 - eta * 4 * x2)
show_trace_2d(f_2d, train_2d(gd_2d))
代码解释 epoch 20, x1 -0.057646, x2 -0.000073
代码解释
自适应方法
牛顿法
在 x + \epsilon 处泰勒展开：
f(\mathbf{x}+\epsilon)可表示为f(\mathbf{x})加上\epsilon沿梯度方向的一阶近似项以及\epsilon的平方项乘以Hessian矩阵后的二次型再加上高阶无穷小项\mathcal{O}\left(\|\epsilon\|^{3}\right).
在极小值点处满足: \nabla f(\mathbf{x})=0时, 我们期望在扰动后的位置\mathbf{x}+\epsilon仍能满足该条件, 因此对上述方程两边关于\epsilon求导数, 忽略高阶无穷小项的影响, 得到以下结果：
The gradient of function f at point \mathbf{x} plus the Hessian matrix of f multiplied by vector \boldsymbol{\epsilon} equals zero, which implies that the solution for vector \boldsymbol{\epsilon} is equal to negative the inverse of the Hessian matrix of f multiplied by the gradient of f evaluated at point \mathbf{x}
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())
代码解释 epoch 10, x: 0.0
代码解释
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())
代码解释 epoch 10, x: 26.83413291324767
代码解释
show_trace(newton(0.5))
代码解释 epoch 10, x: 7.269860168684531
代码解释
收敛性分析
只考虑在函数为凸函数, 且最小值点上 f''(x^*) > 0 时的收敛速度：
令 x_k 为第 k 次迭代后 x 的值， e_{k}:=x_{k}-x^{*} 表示 x_k 到最小值点 x^{*} 的距离，由 f'(x^{*}) = 0:
其中 0 = f'(x_ k − e_ k ) = f'( x_ k ) − e_ k f''( x_ k ) + \frac{1}{2 } e_ k ^ 2 f'''( ξ _ k ) ，其中 ξ _ k ∈ [ x _ k − e _ k , x _ k ].
两边除以 f''(x_k), 有：
e_{k}-f'(x_k)/f''(x_k)= (1/2)e_k^2 f'''(\xi_k)/f''(x_k)
代入更新方程 x_{k+1} = x_{k} - f^{\prime}\left(x_{k}\right) / f^{\prime \prime}\left(x_{k}\right), 得到：
x sub k minus x star minus left(f prime at x sub k right) divided by left(f double prime at x sub k right) equals one-half times e sub k squared times left(f triple prime at xi sub k right) divided by left(f double prime at x sub k right)
误差项e_{k+1}等于说它可表示为\frac{1}{2} e_k^2与f'''(ξ_k)/f''(x_k)的比率。
假设满足以下条件：当\frac{1}{2} f^{\prime \prime \prime}(\xi_k) / f^{\prime \prime}(x_k) \leq c时，则存在一个常数c使得以下不等式成立。
e_{k+1} \leq c e_{k}^{2}
预处理 （Heissan阵辅助梯度下降）
变量\mathbf{x}被赋值为其自身减去步长因子乘以Hessian矩阵主对角线元素倒数与梯度向量的乘积
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
基于包含n个样本对的训练数据集
f(\mathbf{x})=\frac{1}{n} \sum_{i=1}^{n} f_{i}(\mathbf{x})
其梯度为:
\nabla f(\mathbf{x})=\frac{1}{n} \sum_{i=1}^{n} \nabla f_{i}(\mathbf{x})
使用该梯度的一次更新的时间复杂度为 \mathcal{O}(n)
随机梯度下降更新公式 \mathcal{O}(1):
\mathbf{x} \leftarrow \mathbf{x}-\eta \nabla f_{i}(\mathbf{x})
且有：
\mathbb{E}_{i} \nabla f_{i}(\mathbf{x})=\frac{1}{n} \sum_{i=1}^{n} \nabla f_{i}(\mathbf{x})=\nabla f(\mathbf{x})
e.g.
f(x_1, x_2) = x_1^2 + 2 x_2^2
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))
代码解释 epoch 50, x1 -0.027566, x2 0.137605
代码解释
动态学习率
\begin{array}{ll}{\eta(t)=\eta_{i} \text { if } t_{i} \leq t \leq t_{i+1}} & {\text { piecewise constant function over the interval }[t_i, t_{i+1}]} \\ {\eta(t)=\eta_{0} \cdot e^{-\lambda t}} & {\text { exponential decay model with parameter }\lambda} \\ {\eta(t)=\eta_{0} \cdot(\beta t+1)^{-\alpha}} & {\text { polynomial decay law characterized by parameters }\beta\text{ and }\alpha}\end{array}
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))
代码解释 epoch 1000, x1 -0.677947, x2 -0.089379
代码解释
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))
代码解释 epoch 50, x1 -0.095244, x2 -0.041674
代码解释
小批量随机梯度下降
读取数据
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
代码解释 torch.Size([1500, 5])
代码解释 import pandas as pd
df = pd.read_csv('/home/kesci/input/airfoil4755/airfoil_self_noise.dat', delimiter='\t', header=None)
df.head(10)
代码解释| 0 | 1 | 2 | 3 | 4 | 5 | |
|---|---|---|---|---|---|---|
| 0 | 800 | 0.0 | 0.3048 | 71.3 | 0.002663 | 126.201 |
| 1 | 1000 | 0.0 | 0.3048 | 71.3 | 0.002663 | 125.201 |
| 2 | 1250 | 0.0 | 0.3048 | 71.3 | 0.002663 | 125.951 |
| 3 | 1600 | 0.0 | 0.3048 | 71.3 | 0.002663 | 127.591 |
| 4 | 2000 | 0.0 | 0.3048 | 71.3 | 0.002663 | 127.461 |
| 5 | 2500 | 0.0 | 0.3048 | 71.3 | 0.002663 | 125.571 |
| 6 | 3150 | 0.0 | 0.3048 | 71.3 | 0.002663 | 125.201 |
| 7 | 4000 | 0.0 | 0.3048 | 71.3 | 0.002663 | 123.061 |
| 8 | 5000 | 0.0 | 0.3048 | 71.3 | 0.002663 | 121.301 |
| 9 | 6300 | 0.0 | 0.3048 | 71.3 | 0.002663 | 119.541 |
从零开始实现
def sgd(params, states, hyperparams):
for p in params:
p.data -= hyperparams['lr'] * p.grad.data
代码解释 # 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')
代码解释 def train_sgd(lr, batch_size, num_epochs=2):
train_ch7(sgd, None, {'lr': lr}, features, labels, batch_size, num_epochs)
代码解释对比
train_sgd(1, 1500, 6)
代码解释 loss: 0.244373, 0.009881 sec per epoch
代码解释
train_sgd(0.005, 1)
代码解释 loss: 0.245968, 0.463836 sec per epoch
代码解释
train_sgd(0.05, 10)
代码解释 loss: 0.243900, 0.065017 sec per epoch
代码解释
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')
代码解释 train_pytorch_ch7(optim.SGD, {"lr": 0.05}, features, labels, 10)
代码解释 loss: 0.243770, 0.047664 sec per epoch
代码解释