从零开始学Pytorch（十三）之梯度下降

墨明棋妙27

发布于 2022-09-23 10:54:27

3030

发布于 2022-09-23 10:54:27

文章被收录于专栏：1996

梯度下降

%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

\eta f^{\prime}(x)

：

f\left(x-\eta f^{\prime}(x)\right) \lesssim f(x)

x \leftarrow x-\eta f^{\prime}(x)

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

show_trace(gd(0.05))

局部极小值

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))

多维梯度下降

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

x + \epsilon

处泰勒展开：

最小值点处满足:

\nabla f(\mathbf{x})=0

, 即我们希望

\nabla f(\mathbf{x} + \epsilon)=0

, 对上式关于

\epsilon

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

f''(x^*) > 0

时的收敛速度：

令

x_k

为第

次迭代后

的值，

e_{k}:=x_{k}-x^{*}

表示

x_k

到最小值点

x^{*}

的距离，由

f'(x^{*}) = 0

两边除以

f''(x_k)

, 有：

代入更新方程 , 得到

当

\frac{1}{2} f^{\prime \prime \prime}\left(\xi_{k}\right) / f^{\prime \prime}\left(x_{k}\right) \leq c

时，有:

e_{k+1} \leq c e_{k}^{2}

随机梯度下降参数更新

对于有

个样本对训练数据集，设

f_i(x)

是第

个样本的损失函数, 则目标函数为:

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：

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

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

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

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

# 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

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