動手學深度學習-18 梯度下降

論文：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

 一維梯度下降

下面以x^2爲例

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)

 

通過改變學習率，來看一下效果

show_trace(gd(0.05))
#epoch 10, x: 3.4867844009999995
#學習率太小了，梯度下降還沒有下降完全

show_trace(gd(1.1))
#epoch 10, x: 61.917364224000096
梯度太大，有點爆炸

 

局部極小值是梯度下降需要解決的問題

以f(x)=xcoscx 爲例

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

 

多維梯度下降

以f(x)=x1^2+2*x2^2爲例

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):  # 目標函數
    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

 

自適應方法【上面的研究中，學習率的設置有主關因素】

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

 

來看看牛頓法能不能解決局部最小值的問題

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

 

#再降低一下學習率，就可以，所以說有時候學習率還是需要調的
show_trace(newton(0.5))
#epoch 10, x: 7.269860168684531

 

 收斂性分析

二階收斂

 

x=x-學習率*梯度/[heissan矩陣對角線的值]

梯度下降與線性搜索（共軛梯度法）

隨機梯度下降

隨機梯度下降參數更新

隨機梯度下降是針對每一個樣本都進行梯度更新，剛開始講的梯度下降是針對所以樣本一起進行梯度更新

 

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

小批量隨機梯度下降

讀取數據

鏈接:https://archive.ics.uci.edu/ml/datasets/Airfoil+Self-Noise

def get_data_ch7():  # 本函數已保存在d2lzh_pytorch包中方便以後使用
    data = np.genfromtxt('/home/kesci/input/airfoil4755/airfoil_self_noise.dat', delimiter='\t')
    data = (data - data.mean(axis=0)) / data.std(axis=0) # 標準化
    return torch.tensor(data[:1500, :-1], dtype=torch.float32), \
           torch.tensor(data[:1500, -1], dtype=torch.float32) # 前1500個樣本(每個樣本5個特徵)

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)

 

從零開始實現

def sgd(params, states, hyperparams):
    for p in params:
        p.data -= hyperparams['lr'] * p.grad.data

# 本函數已保存在d2lzh_pytorch包中方便以後使用
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()  # 使用平均損失
            
            # 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)  # 迭代模型參數
            if (batch_i + 1) * batch_size % 100 == 0:
                ls.append(eval_loss())  # 每100個樣本記錄下當前訓練誤差
    # 打印結果和作圖
    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)

調差對比

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

 

 簡潔實現

# 本函數與原書不同的是這裏第一個參數優化器函數而不是優化器的名字
# 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是爲了和train_ch7保持一致, 因爲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())
    # 打印結果和作圖
    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

更多知識 關注公衆號 'AI算法與數學之美'