1. 编程实现图6-1，并观察特征

参考代码：

import numpy as np
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D


# https://blog.csdn.net/weixin_39228381/article/details/108511882

def func(x, y):
    return x * x / 20 + y * y


def paint_loss_func():
    x = np.linspace(-50, 50, 100)  # x的绘制范围是-50到50，从改区间均匀取100个数
    y = np.linspace(-50, 50, 100)  # y的绘制范围是-50到50，从改区间均匀取100个数

    X, Y = np.meshgrid(x, y)
    Z = func(X, Y)

    fig = plt.figure()  # figsize=(10, 10))
    ax = Axes3D(fig)
    plt.xlabel('x')
    plt.ylabel('y')

    ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap='rainbow')
    plt.show()


paint_loss_func()

2. 观察梯度方向

3. 编写代码实现算法，并可视化轨迹

SGD、Momentum、Adagrad、Adam

参考代码：

# coding: utf-8
import numpy as np
import matplotlib.pyplot as plt
from collections import OrderedDict


class SGD:
    """随机梯度下降法（Stochastic Gradient Descent）"""

    def __init__(self, lr=0.01):
        self.lr = lr

    def update(self, params, grads):
        for key in params.keys():
            params[key] -= self.lr * grads[key]


class Momentum:
    """Momentum SGD"""

    def __init__(self, lr=0.01, momentum=0.9):
        self.lr = lr
        self.momentum = momentum
        self.v = None

    def update(self, params, grads):
        if self.v is None:
            self.v = {}
            for key, val in params.items():
                self.v[key] = np.zeros_like(val)

        for key in params.keys():
            self.v[key] = self.momentum * self.v[key] - self.lr * grads[key]
            params[key] += self.v[key]


class Nesterov:
    """Nesterov's Accelerated Gradient (http://arxiv.org/abs/1212.0901)"""

    def __init__(self, lr=0.01, momentum=0.9):
        self.lr = lr
        self.momentum = momentum
        self.v = None

    def update(self, params, grads):
        if self.v is None:
            self.v = {}
            for key, val in params.items():
                self.v[key] = np.zeros_like(val)

        for key in params.keys():
            self.v[key] *= self.momentum
            self.v[key] -= self.lr * grads[key]
            params[key] += self.momentum * self.momentum * self.v[key]
            params[key] -= (1 + self.momentum) * self.lr * grads[key]


class AdaGrad:
    """AdaGrad"""

    def __init__(self, lr=0.01):
        self.lr = lr
        self.h = None

    def update(self, params, grads):
        if self.h is None:
            self.h = {}
            for key, val in params.items():
                self.h[key] = np.zeros_like(val)

        for key in params.keys():
            self.h[key] += grads[key] * grads[key]
            params[key] -= self.lr * grads[key] / (np.sqrt(self.h[key]) + 1e-7)


class RMSprop:
    """RMSprop"""

    def __init__(self, lr=0.01, decay_rate=0.99):
        self.lr = lr
        self.decay_rate = decay_rate
        self.h = None

    def update(self, params, grads):
        if self.h is None:
            self.h = {}
            for key, val in params.items():
                self.h[key] = np.zeros_like(val)

        for key in params.keys():
            self.h[key] *= self.decay_rate
            self.h[key] += (1 - self.decay_rate) * grads[key] * grads[key]
            params[key] -= self.lr * grads[key] / (np.sqrt(self.h[key]) + 1e-7)


class Adam:
    """Adam (http://arxiv.org/abs/1412.6980v8)"""

    def __init__(self, lr=0.001, beta1=0.9, beta2=0.999):
        self.lr = lr
        self.beta1 = beta1
        self.beta2 = beta2
        self.iter = 0
        self.m = None
        self.v = None

    def update(self, params, grads):
        if self.m is None:
            self.m, self.v = {}, {}
            for key, val in params.items():
                self.m[key] = np.zeros_like(val)
                self.v[key] = np.zeros_like(val)

        self.iter += 1
        lr_t = self.lr * np.sqrt(1.0 - self.beta2 ** self.iter) / (1.0 - self.beta1 ** self.iter)

        for key in params.keys():
            self.m[key] += (1 - self.beta1) * (grads[key] - self.m[key])
            self.v[key] += (1 - self.beta2) * (grads[key] ** 2 - self.v[key])

            params[key] -= lr_t * self.m[key] / (np.sqrt(self.v[key]) + 1e-7)


def f(x, y):
    return x ** 2 / 20.0 + y ** 2


def df(x, y):
    return x / 10.0, 2.0 * y


init_pos = (-7.0, 2.0)
params = {}
params['x'], params['y'] = init_pos[0], init_pos[1]
grads = {}
grads['x'], grads['y'] = 0, 0

optimizers = OrderedDict()
optimizers["SGD"] = SGD(lr=0.95)
optimizers["Momentum"] = Momentum(lr=0.1)
optimizers["AdaGrad"] = AdaGrad(lr=1.5)
optimizers["Adam"] = Adam(lr=0.3)

idx = 1

for key in optimizers:
    optimizer = optimizers[key]
    x_history = []
    y_history = []
    params['x'], params['y'] = init_pos[0], init_pos[1]

    for i in range(30):
        x_history.append(params['x'])
        y_history.append(params['y'])

        grads['x'], grads['y'] = df(params['x'], params['y'])
        optimizer.update(params, grads)

    x = np.arange(-10, 10, 0.01)
    y = np.arange(-5, 5, 0.01)

    X, Y = np.meshgrid(x, y)
    Z = f(X, Y)
    # for simple contour line
    mask = Z > 7
    Z[mask] = 0

    # plot
    plt.subplot(2, 2, idx)
    idx += 1
    plt.plot(x_history, y_history, 'o-', color="red")
    plt.contour(X, Y, Z)  # 绘制等高线
    plt.ylim(-10, 10)
    plt.xlim(-10, 10)
    plt.plot(0, 0, '+')
    plt.title(key)
    plt.xlabel("x")
    plt.ylabel("y")

plt.subplots_adjust(wspace=0, hspace=0)  # 调整子图间距
plt.show()

4. 分析上图，说明原理（选做）

为什么SGD会走“之字形”？其它算法为什么会比较平滑？
Momentum、AdaGrad对SGD的改进体现在哪里？速度？方向？在图上有哪些体现？
仅从轨迹来看，Adam似乎不如AdaGrad效果好，是这样么？
四种方法分别用了多长时间？是否符合预期？
调整学习率、动量等超参数，轨迹有哪些变化？

5. 总结SGD、Momentum、AdaGrad、Adam的优缺点（选做）

6. Adam这么好，SGD是不是就用不到了？（选做）

7. 增加RMSprop、Nesterov算法。（选做）

对比Momentum与Nesterov、AdaGrad与RMSprop。

8. 基于MNIST数据集的更新方法的比较（选做）

在原图基础上，增加RMSprop、Nesterov算法。

编程实现，并谈谈自己的看法。

优化算法代码可参考前面的内容。

MNIST数据集的更新方法的比较：

# coding: utf-8
import os
import sys
sys.path.append(os.pardir)  # 为了导入父目录的文件而进行的设定
import matplotlib.pyplot as plt
from dataset.mnist import load_mnist
from common.util import smooth_curve
from common.multi_layer_net import MultiLayerNet
from common.optimizer import *


# 0:读入MNIST数据==========
(x_train, t_train), (x_test, t_test) = load_mnist(normalize=True)

train_size = x_train.shape[0]
batch_size = 128
max_iterations = 2000


# 1:进行实验的设置==========
optimizers = {}
optimizers['SGD'] = SGD()
optimizers['Momentum'] = Momentum()
optimizers['AdaGrad'] = AdaGrad()
optimizers['Adam'] = Adam()
#optimizers['RMSprop'] = RMSprop()

networks = {}
train_loss = {}
for key in optimizers.keys():
    networks[key] = MultiLayerNet(
        input_size=784, hidden_size_list=[100, 100, 100, 100],
        output_size=10)
    train_loss[key] = []    


# 2:开始训练==========
for i in range(max_iterations):
    batch_mask = np.random.choice(train_size, batch_size)
    x_batch = x_train[batch_mask]
    t_batch = t_train[batch_mask]
    
    for key in optimizers.keys():
        grads = networks[key].gradient(x_batch, t_batch)
        optimizers[key].update(networks[key].params, grads)
    
        loss = networks[key].loss(x_batch, t_batch)
        train_loss[key].append(loss)
    
    if i % 100 == 0:
        print( "===========" + "iteration:" + str(i) + "===========")
        for key in optimizers.keys():
            loss = networks[key].loss(x_batch, t_batch)
            print(key + ":" + str(loss))


# 3.绘制图形==========
markers = {"SGD": "o", "Momentum": "x", "AdaGrad": "s", "Adam": "D"}
x = np.arange(max_iterations)
for key in optimizers.keys():
    plt.plot(x, smooth_curve(train_loss[key]), marker=markers[key], markevery=100, label=key)
plt.xlabel("iterations")
plt.ylabel("loss")
plt.ylim(0, 1)
plt.legend()
plt.show()

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