NNDL 作业13 优化算法3D可视化 [HBU]

发布时间:2024年01月04日

老师作业原博客:【23-24 秋学期】NNDL 作业13 优化算法3D可视化-CSDN博客

NNDL 作业13 优化算法3D可视化-CSDN博客


编程实现优化算法,并3D可视化

1. 函数3D可视化

分别画出x[0]^{2}+x[1]^{2}+x[1]^{3}+x[0]*x[1]?和?x^{2} /20+y^{2}的3D图

NNDL实验 优化算法3D轨迹 鱼书例题3D版_优化算法3d展示-CSDN博客

代码:

from mpl_toolkits.mplot3d import Axes3D
import numpy as np
from matplotlib import pyplot as plt
import torch
from nndl.op import Op


# 画出x**2
class OptimizedFunction3D(Op):
    def __init__(self):
        super(OptimizedFunction3D, self).__init__()
        self.params = {'x': 0}
        self.grads = {'x': 0}

    def forward(self, x):
        self.params['x'] = x
        return x[0] ** 2 + x[1] ** 2 + x[1] ** 3 + x[0] * x[1]

    def backward(self):
        x = self.params['x']
        gradient1 = 2 * x[0] + x[1]
        gradient2 = 2 * x[1] + 3 * x[1] ** 2 + x[0]
        grad1 = torch.Tensor([gradient1])
        grad2 = torch.Tensor([gradient2])
        self.grads['x'] = torch.cat([grad1, grad2])


# 使用numpy.meshgrid生成x1,x2矩阵,矩阵的每一行为[-3, 3],以0.1为间隔的数值
x1 = np.arange(-3, 3, 0.1)
x2 = np.arange(-3, 3, 0.1)
x1, x2 = np.meshgrid(x1, x2)
init_x = torch.Tensor(np.array([x1, x2]))

model = OptimizedFunction3D()

# 绘制 f_3d函数 的 三维图像
fig = plt.figure()
ax = plt.axes(projection='3d')
X = init_x[0].numpy()
Y = init_x[1].numpy()
Z = model(init_x).numpy()
ax.plot_surface(X, Y, Z, cmap='plasma')

ax.set_xlabel('x1')
ax.set_ylabel('x2')
ax.set_zlabel('f(x1,x2)')
plt.show()


# 画出x * x / 20 + y * y
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='plasma')
    plt.show()


paint_loss_func()

结果:


2.加入优化算法,画出轨迹

分别画出x[0]^{2}+x[1]^{2}+x[1]^{3}+x[0]*x[1]?和?x^{2} /20+y^{2}的3D轨迹图

结合3D动画,用自己的语言,从轨迹、速度等多个角度讲解各个算法优缺点

NNDL实验 优化算法3D轨迹 pytorch版_nndl 实验三 将数据转换为 pytorch 张量-CSDN博客

代码为:
?

import torch
import numpy as np
import copy
from matplotlib import pyplot as plt
from matplotlib import animation
from itertools import zip_longest
from nndl.op import Op


class Optimizer(object):  # 优化器基类
    def __init__(self, init_lr, model):
        """
        优化器类初始化
        """
        # 初始化学习率,用于参数更新的计算
        self.init_lr = init_lr
        # 指定优化器需要优化的模型
        self.model = model

    def step(self):
        """
        定义每次迭代如何更新参数
        """
        pass


class SimpleBatchGD(Optimizer):
    def __init__(self, init_lr, model):
        super(SimpleBatchGD, self).__init__(init_lr=init_lr, model=model)

    def step(self):
        # 参数更新
        if isinstance(self.model.params, dict):
            for key in self.model.params.keys():
                self.model.params[key] = self.model.params[key] - self.init_lr * self.model.grads[key]


class Adagrad(Optimizer):
    def __init__(self, init_lr, model, epsilon):
        """
        Adagrad 优化器初始化
        输入:
            - init_lr: 初始学习率 - model:模型,model.params存储模型参数值  - epsilon:保持数值稳定性而设置的非常小的常数
        """
        super(Adagrad, self).__init__(init_lr=init_lr, model=model)
        self.G = {}
        for key in self.model.params.keys():
            self.G[key] = 0
        self.epsilon = epsilon

    def adagrad(self, x, gradient_x, G, init_lr):
        """
        adagrad算法更新参数,G为参数梯度平方的累计值。
        """
        G += gradient_x ** 2
        x -= init_lr / torch.sqrt(G + self.epsilon) * gradient_x
        return x, G

    def step(self):
        """
        参数更新
        """
        for key in self.model.params.keys():
            self.model.params[key], self.G[key] = self.adagrad(self.model.params[key],
                                                               self.model.grads[key],
                                                               self.G[key],
                                                               self.init_lr)


class RMSprop(Optimizer):
    def __init__(self, init_lr, model, beta, epsilon):
        """
        RMSprop优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - beta:衰减率
            - epsilon:保持数值稳定性而设置的常数
        """
        super(RMSprop, self).__init__(init_lr=init_lr, model=model)
        self.G = {}
        for key in self.model.params.keys():
            self.G[key] = 0
        self.beta = beta
        self.epsilon = epsilon

    def rmsprop(self, x, gradient_x, G, init_lr):
        """
        rmsprop算法更新参数,G为迭代梯度平方的加权移动平均
        """
        G = self.beta * G + (1 - self.beta) * gradient_x ** 2
        x -= init_lr / torch.sqrt(G + self.epsilon) * gradient_x
        return x, G

    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.G[key] = self.rmsprop(self.model.params[key],
                                                               self.model.grads[key],
                                                               self.G[key],
                                                               self.init_lr)


class Momentum(Optimizer):
    def __init__(self, init_lr, model, rho):
        """
        Momentum优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - rho:动量因子
        """
        super(Momentum, self).__init__(init_lr=init_lr, model=model)
        self.delta_x = {}
        for key in self.model.params.keys():
            self.delta_x[key] = 0
        self.rho = rho

    def momentum(self, x, gradient_x, delta_x, init_lr):
        """
        momentum算法更新参数,delta_x为梯度的加权移动平均
        """
        delta_x = self.rho * delta_x - init_lr * gradient_x
        x += delta_x
        return x, delta_x

    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.delta_x[key] = self.momentum(self.model.params[key],
                                                                      self.model.grads[key],
                                                                      self.delta_x[key],
                                                                      self.init_lr)


class Nesterov(Optimizer):
    def __init__(self, init_lr, model, rho):
        """
        Nesterov优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - rho:动量因子
        """
        super(Nesterov, self).__init__(init_lr=init_lr, model=model)
        self.delta_x = {}
        for key in self.model.params.keys():
            self.delta_x[key] = 0
        self.rho = rho

    def nesterov(self, x, gradient_x, delta_x, init_lr):
        """
        Nesterov算法更新参数,delta_x为梯度的加权移动平均
        """
        delta_x_prev = delta_x
        delta_x = self.rho * delta_x - init_lr * gradient_x
        x += -self.rho * delta_x_prev + (1 + self.rho) * delta_x
        return x, delta_x

    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.delta_x[key] = self.nesterov(self.model.params[key],
                                                                      self.model.grads[key],
                                                                      self.delta_x[key],
                                                                      self.init_lr)


class Adam(Optimizer):
    def __init__(self, init_lr, model, beta1, beta2, epsilon):
        """
        Adam优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - beta1, beta2:移动平均的衰减率
            - epsilon:保持数值稳定性而设置的常数
        """
        super(Adam, self).__init__(init_lr=init_lr, model=model)
        self.beta1 = beta1
        self.beta2 = beta2
        self.epsilon = epsilon
        self.M, self.G = {}, {}
        for key in self.model.params.keys():
            self.M[key] = 0
            self.G[key] = 0
        self.t = 1

    def adam(self, x, gradient_x, G, M, t, init_lr):
        """
        adam算法更新参数
        输入:
            - x:参数
            - G:梯度平方的加权移动平均
            - M:梯度的加权移动平均
            - t:迭代次数
            - init_lr:初始学习率
        """
        M = self.beta1 * M + (1 - self.beta1) * gradient_x
        G = self.beta2 * G + (1 - self.beta2) * gradient_x ** 2
        M_hat = M / (1 - self.beta1 ** t)
        G_hat = G / (1 - self.beta2 ** t)
        t += 1
        x -= init_lr / torch.sqrt(G_hat + self.epsilon) * M_hat
        return x, G, M, t

    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.G[key], self.M[key], self.t = self.adam(self.model.params[key],
                                                                                 self.model.grads[key],
                                                                                 self.G[key],
                                                                                 self.M[key],
                                                                                 self.t,
                                                                                 self.init_lr)


class OptimizedFunction3D(Op):
    def __init__(self):
        super(OptimizedFunction3D, self).__init__()
        self.params = {'x': 0}
        self.grads = {'x': 0}

    def forward(self, x):
        self.params['x'] = x
        return x[0] ** 2 + x[1] ** 2 + x[1] ** 3 + x[0] * x[1]

    def backward(self):
        x = self.params['x']
        gradient1 = 2 * x[0] + x[1]
        gradient2 = 2 * x[1] + 3 * x[1] ** 2 + x[0]
        grad1 = torch.Tensor([gradient1])
        grad2 = torch.Tensor([gradient2])
        self.grads['x'] = torch.cat([grad1, grad2])


class Visualization3D(animation.FuncAnimation):
    """    绘制动态图像,可视化参数更新轨迹    """

    def __init__(self, *xy_values, z_values, labels=[], colors=[], fig, ax, interval=600, blit=True, **kwargs):
        """
        初始化3d可视化类
        输入:
            xy_values:三维中x,y维度的值
            z_values:三维中z维度的值
            labels:每个参数更新轨迹的标签
            colors:每个轨迹的颜色
            interval:帧之间的延迟(以毫秒为单位)
            blit:是否优化绘图
        """
        self.fig = fig
        self.ax = ax
        self.xy_values = xy_values
        self.z_values = z_values

        frames = max(xy_value.shape[0] for xy_value in xy_values)
        self.lines = [ax.plot([], [], [], label=label, color=color, lw=2)[0]
                      for _, label, color in zip_longest(xy_values, labels, colors)]
        super(Visualization3D, self).__init__(fig, self.animate, init_func=self.init_animation, frames=frames,
                                              interval=interval, blit=blit, **kwargs)

    def init_animation(self):
        # 数值初始化
        for line in self.lines:
            line.set_data([], [])
            # line.set_3d_properties(np.asarray([]))  # 源程序中有这一行,加上会报错。 Edit by David 2022.12.4
        return self.lines

    def animate(self, i):
        # 将x,y,z三个数据传入,绘制三维图像
        for line, xy_value, z_value in zip(self.lines, self.xy_values, self.z_values):
            line.set_data(xy_value[:i, 0], xy_value[:i, 1])
            line.set_3d_properties(z_value[:i])
        return self.lines


def train_f(model, optimizer, x_init, epoch):
    x = x_init
    all_x = []
    losses = []
    for i in range(epoch):
        all_x.append(copy.deepcopy(x.numpy()))  # 浅拷贝 改为 深拷贝, 否则List的原值会被改变。 Edit by David 2022.12.4.
        loss = model(x)
        losses.append(loss)
        model.backward()
        optimizer.step()
        x = model.params['x']
    return torch.Tensor(np.array(all_x)), losses


# 构建6个模型,分别配备不同的优化器
model1 = OptimizedFunction3D()
opt_gd = SimpleBatchGD(init_lr=0.01, model=model1)

model2 = OptimizedFunction3D()
opt_adagrad = Adagrad(init_lr=0.5, model=model2, epsilon=1e-7)

model3 = OptimizedFunction3D()
opt_rmsprop = RMSprop(init_lr=0.1, model=model3, beta=0.9, epsilon=1e-7)

model4 = OptimizedFunction3D()
opt_momentum = Momentum(init_lr=0.01, model=model4, rho=0.9)

model5 = OptimizedFunction3D()
opt_adam = Adam(init_lr=0.1, model=model5, beta1=0.9, beta2=0.99, epsilon=1e-7)

model6 = OptimizedFunction3D()
opt_Nesterov = Nesterov(init_lr=0.1, model=model6, rho=0.9)

models = [model1, model2, model3, model4, model5, model6]
opts = [opt_gd, opt_adagrad, opt_rmsprop, opt_momentum, opt_adam, opt_Nesterov]

x_all_opts = []
z_all_opts = []

# 使用不同优化器训练

for model, opt in zip(models, opts):
    x_init = torch.FloatTensor([2, 3])
    x_one_opt, z_one_opt = train_f(model, opt, x_init, 150)  # epoch
    # 保存参数值
    x_all_opts.append(x_one_opt.numpy())
    z_all_opts.append(np.squeeze(z_one_opt))

# 使用numpy.meshgrid生成x1,x2矩阵,矩阵的每一行为[-3, 3],以0.1为间隔的数值
x1 = np.arange(-3, 3, 0.1)
x2 = np.arange(-3, 3, 0.1)
x1, x2 = np.meshgrid(x1, x2)
init_x = torch.Tensor(np.array([x1, x2]))

model = OptimizedFunction3D()

# 绘制 f_3d函数 的 三维图像
fig = plt.figure()
ax = plt.axes(projection='3d')
X = init_x[0].numpy()
Y = init_x[1].numpy()
Z = model(init_x).numpy()  # 改为 model(init_x).numpy() David 2022.12.4
ax.plot_surface(X, Y, Z, cmap='plasma')

ax.set_xlabel('x1')
ax.set_ylabel('x2')
ax.set_zlabel('f(x1,x2)')

labels = ['SGD', 'AdaGrad', 'RMSprop', 'Momentum', 'Adam', 'Nesterov']
colors = ['#8B0000', '#0000FF', '#000000', '#008B00', '#FF0000']

animator = Visualization3D(*x_all_opts, z_values=z_all_opts, labels=labels, colors=colors, fig=fig, ax=ax)
ax.legend(loc='upper left')

plt.show()
animator.save('animation.gif')

(一直整不出来动态图,先攒着,等过了期末考试再回来研究,最近实在是太忙了)

代码:

import torch
import numpy as np
import copy
from matplotlib import pyplot as plt
from matplotlib import animation
from itertools import zip_longest
from matplotlib import cm
 
 
class Op(object):
    def __init__(self):
        pass
 
    def __call__(self, inputs):
        return self.forward(inputs)
 
    # 输入:张量inputs
    # 输出:张量outputs
    def forward(self, inputs):
        # return outputs
        raise NotImplementedError
 
    # 输入:最终输出对outputs的梯度outputs_grads
    # 输出:最终输出对inputs的梯度inputs_grads
    def backward(self, outputs_grads):
        # return inputs_grads
        raise NotImplementedError
 
 
class Optimizer(object):  # 优化器基类
    def __init__(self, init_lr, model):
        """
        优化器类初始化
        """
        # 初始化学习率,用于参数更新的计算
        self.init_lr = init_lr
        # 指定优化器需要优化的模型
        self.model = model
 
    def step(self):
        """
        定义每次迭代如何更新参数
        """
        pass
 
 
class SimpleBatchGD(Optimizer):
    def __init__(self, init_lr, model):
        super(SimpleBatchGD, self).__init__(init_lr=init_lr, model=model)
 
    def step(self):
        # 参数更新
        if isinstance(self.model.params, dict):
            for key in self.model.params.keys():
                self.model.params[key] = self.model.params[key] - self.init_lr * self.model.grads[key]
 
 
class Adagrad(Optimizer):
    def __init__(self, init_lr, model, epsilon):
        """
        Adagrad 优化器初始化
        输入:
            - init_lr: 初始学习率 - model:模型,model.params存储模型参数值  - epsilon:保持数值稳定性而设置的非常小的常数
        """
        super(Adagrad, self).__init__(init_lr=init_lr, model=model)
        self.G = {}
        for key in self.model.params.keys():
            self.G[key] = 0
        self.epsilon = epsilon
 
    def adagrad(self, x, gradient_x, G, init_lr):
        """
        adagrad算法更新参数,G为参数梯度平方的累计值。
        """
        G += gradient_x ** 2
        x -= init_lr / torch.sqrt(G + self.epsilon) * gradient_x
        return x, G
 
    def step(self):
        """
        参数更新
        """
        for key in self.model.params.keys():
            self.model.params[key], self.G[key] = self.adagrad(self.model.params[key],
                                                               self.model.grads[key],
                                                               self.G[key],
                                                               self.init_lr)
 
 
class RMSprop(Optimizer):
    def __init__(self, init_lr, model, beta, epsilon):
        """
        RMSprop优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - beta:衰减率
            - epsilon:保持数值稳定性而设置的常数
        """
        super(RMSprop, self).__init__(init_lr=init_lr, model=model)
        self.G = {}
        for key in self.model.params.keys():
            self.G[key] = 0
        self.beta = beta
        self.epsilon = epsilon
 
    def rmsprop(self, x, gradient_x, G, init_lr):
        """
        rmsprop算法更新参数,G为迭代梯度平方的加权移动平均
        """
        G = self.beta * G + (1 - self.beta) * gradient_x ** 2
        x -= init_lr / torch.sqrt(G + self.epsilon) * gradient_x
        return x, G
 
    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.G[key] = self.rmsprop(self.model.params[key],
                                                               self.model.grads[key],
                                                               self.G[key],
                                                               self.init_lr)
 
 
class Momentum(Optimizer):
    def __init__(self, init_lr, model, rho):
        """
        Momentum优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - rho:动量因子
        """
        super(Momentum, self).__init__(init_lr=init_lr, model=model)
        self.delta_x = {}
        for key in self.model.params.keys():
            self.delta_x[key] = 0
        self.rho = rho
 
    def momentum(self, x, gradient_x, delta_x, init_lr):
        """
        momentum算法更新参数,delta_x为梯度的加权移动平均
        """
        delta_x = self.rho * delta_x - init_lr * gradient_x
        x += delta_x
        return x, delta_x
 
    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.delta_x[key] = self.momentum(self.model.params[key],
                                                                      self.model.grads[key],
                                                                      self.delta_x[key],
                                                                      self.init_lr)
class Nesterov(Optimizer):
    def __init__(self, init_lr, model, rho):
        """
        Nesterov优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - rho:动量因子
        """
        super(Nesterov, self).__init__(init_lr=init_lr, model=model)
        self.delta_x = {}
        for key in self.model.params.keys():
            self.delta_x[key] = 0
        self.rho = rho
 
    def nesterov(self, x, gradient_x, delta_x, init_lr):
        """
        Nesterov算法更新参数,delta_x为梯度的加权移动平均
        """
        delta_x_prev = delta_x
        delta_x = self.rho * delta_x - init_lr * gradient_x
        x += -self.rho * delta_x_prev + (1 + self.rho) * delta_x
        return x, delta_x
 
    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.delta_x[key] = self.nesterov(self.model.params[key],
                                                                      self.model.grads[key],
                                                                      self.delta_x[key],
                                                                      self.init_lr)
 
 
class Adam(Optimizer):
    def __init__(self, init_lr, model, beta1, beta2, epsilon):
        """
        Adam优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - beta1, beta2:移动平均的衰减率
            - epsilon:保持数值稳定性而设置的常数
        """
        super(Adam, self).__init__(init_lr=init_lr, model=model)
        self.beta1 = beta1
        self.beta2 = beta2
        self.epsilon = epsilon
        self.M, self.G = {}, {}
        for key in self.model.params.keys():
            self.M[key] = 0
            self.G[key] = 0
        self.t = 1
 
    def adam(self, x, gradient_x, G, M, t, init_lr):
        """
        adam算法更新参数
        输入:
            - x:参数
            - G:梯度平方的加权移动平均
            - M:梯度的加权移动平均
            - t:迭代次数
            - init_lr:初始学习率
        """
        M = self.beta1 * M + (1 - self.beta1) * gradient_x
        G = self.beta2 * G + (1 - self.beta2) * gradient_x ** 2
        M_hat = M / (1 - self.beta1 ** t)
        G_hat = G / (1 - self.beta2 ** t)
        t += 1
        x -= init_lr / torch.sqrt(G_hat + self.epsilon) * M_hat
        return x, G, M, t
 
    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.G[key], self.M[key], self.t = self.adam(self.model.params[key],
                                                                                 self.model.grads[key],
                                                                                 self.G[key],
                                                                                 self.M[key],
                                                                                 self.t,
                                                                                 self.init_lr)
 
class OptimizedFunction3D(Op):
    def __init__(self):
        super(OptimizedFunction3D, self).__init__()
        self.params = {'x': 0}
        self.grads = {'x': 0}
 
    def forward(self, x):
        self.params['x'] = x
        return x[0] * x[0] / 20 + x[1] * x[1] / 1
        # return x[0] ** 2 + x[1] ** 2 + x[1] ** 3 + x[0] * x[1]
 
    def backward(self):
        x = self.params['x']
        gradient1 = 2 * x[0] / 20
        gradient2 = 2 * x[1] / 1
        grad1 = torch.Tensor([gradient1])
        grad2 = torch.Tensor([gradient2])
        self.grads['x'] = torch.cat([grad1, grad2])
 
 
class Visualization3D(animation.FuncAnimation):
    """    绘制动态图像,可视化参数更新轨迹    """
 
    def __init__(self, *xy_values, z_values, labels=[], colors=[], fig, ax, interval=100, blit=True, **kwargs):
        """
        初始化3d可视化类
        输入:
            xy_values:三维中x,y维度的值
            z_values:三维中z维度的值
            labels:每个参数更新轨迹的标签
            colors:每个轨迹的颜色
            interval:帧之间的延迟(以毫秒为单位)
            blit:是否优化绘图
        """
        self.fig = fig
        self.ax = ax
        self.xy_values = xy_values
        self.z_values = z_values
 
        frames = max(xy_value.shape[0] for xy_value in xy_values)
 
        self.lines = [ax.plot([], [], [], label=label, color=color, lw=2)[0]
                      for _, label, color in zip_longest(xy_values, labels, colors)]
        self.points = [ax.plot([], [], [], color=color, markeredgewidth=1, markeredgecolor='black', marker='o')[0]
                       for _, color in zip_longest(xy_values, colors)]
        # print(self.lines)
        super(Visualization3D, self).__init__(fig, self.animate, init_func=self.init_animation, frames=frames,
                                              interval=interval, blit=blit, **kwargs)
 
    def init_animation(self):
        # 数值初始化
        for line in self.lines:
            line.set_data_3d([], [], [])
        for point in self.points:
            point.set_data_3d([], [], [])
        return self.points + self.lines
 
    def animate(self, i):
        # 将x,y,z三个数据传入,绘制三维图像
        for line, xy_value, z_value in zip(self.lines, self.xy_values, self.z_values):
            line.set_data_3d(xy_value[:i, 0], xy_value[:i, 1], z_value[:i])
        for point, xy_value, z_value in zip(self.points, self.xy_values, self.z_values):
            point.set_data_3d(xy_value[i, 0], xy_value[i, 1], z_value[i])
        return self.points + self.lines
 
 
def train_f(model, optimizer, x_init, epoch):
    x = x_init
    all_x = []
    losses = []
    for i in range(epoch):
        all_x.append(copy.deepcopy(x.numpy()))  # 浅拷贝 改为 深拷贝, 否则List的原值会被改变。 Edit by David 2022.12.4.
        loss = model(x)
        losses.append(loss)
        model.backward()
        optimizer.step()
        x = model.params['x']
    return torch.Tensor(np.array(all_x)), losses
 
 
# 构建6个模型,分别配备不同的优化器
model1 = OptimizedFunction3D()
opt_gd = SimpleBatchGD(init_lr=0.95, model=model1)
 
model2 = OptimizedFunction3D()
opt_adagrad = Adagrad(init_lr=1.5, model=model2, epsilon=1e-7)
 
model3 = OptimizedFunction3D()
opt_rmsprop = RMSprop(init_lr=0.05, model=model3, beta=0.9, epsilon=1e-7)
 
model4 = OptimizedFunction3D()
opt_momentum = Momentum(init_lr=0.1, model=model4, rho=0.9)
 
model5 = OptimizedFunction3D()
opt_adam = Adam(init_lr=0.3, model=model5, beta1=0.9, beta2=0.99, epsilon=1e-7)
 
model6 = OptimizedFunction3D()
opt_Nesterov = Nesterov(init_lr=0.1, model=model6, rho=0.9)  # 将 model4 改为 model6
 
models = [model1, model2, model3, model4, model5, model6]
opts = [opt_gd, opt_adagrad, opt_rmsprop, opt_momentum, opt_adam, opt_Nesterov]
 
x_all_opts = []
z_all_opts = []
 
# 使用不同优化器训练
for model, opt in zip(models, opts):
    x_init = torch.FloatTensor([-7, 2])
    x_one_opt, z_one_opt = train_f(model, opt, x_init, 100)  # epoch
    # 保存参数值
    x_all_opts.append(x_one_opt.numpy())
    z_all_opts.append(np.squeeze(z_one_opt))
 
 
 
# 使用numpy.meshgrid生成x1,x2矩阵,矩阵的每一行为[-3, 3],以0.1为间隔的数值
x1 = np.arange(-10, 10, 0.01)
x2 = np.arange(-5, 5, 0.01)
x1, x2 = np.meshgrid(x1, x2)
init_x = torch.Tensor(np.array([x1, x2]))
 
model = OptimizedFunction3D()
 
# 绘制 f_3d函数 的 三维图像
fig = plt.figure()
ax = plt.axes(projection='3d')
X = init_x[0].numpy()
Y = init_x[1].numpy()
Z = model(init_x).numpy()  # 改为 model(init_x).numpy() David 2022.12.4
surf = ax.plot_surface(X, Y, Z, edgecolor='grey', cmap=cm.coolwarm)
# fig.colorbar(surf, shrink=0.5, aspect=1)
# ax.set_zlim(-3, 2)
ax.set_xlabel('x1')
ax.set_ylabel('x2')
ax.set_zlabel('f(x1,x2)')
 
labels = ['SGD', 'AdaGrad', 'RMSprop', 'Momentum', 'Adam', 'Nesterov']
colors = ['#8B0000', '#0000FF', '#000000', '#008B00', '#FF0000']
 
animator = Visualization3D(*x_all_opts, z_values=z_all_opts, labels=labels, colors=colors, fig=fig, ax=ax)
ax.legend(loc='upper right')
 
plt.show()
# animator.save('teaser' + '.gif', writer='imagemagick',fps=10) # 效果不好,估计被挡住了…… 有待进一步提高 Edit by David 2022.12.4

图像结果:


3.复现CS231经典动画

结合3D动画,用自己的语言,从轨迹、速度等多个角度讲解各个算法优缺点

NNDL实验 优化算法3D轨迹 复现cs231经典动画_深度学习 优化算法 动画展示-CSDN博客

Animations that may help your intuitions about the learning process dynamics.?

Left: Contours of a loss surface and time evolution of different optimization algorithms. Notice the "overshooting" behavior of momentum-based methods, which make the optimization look like a ball rolling down the hill.?

import torch
import numpy as np
import copy
from matplotlib import pyplot as plt
from matplotlib import animation
from itertools import zip_longest
from matplotlib import cm
 
 
class Op(object):
    def __init__(self):
        pass
 
    def __call__(self, inputs):
        return self.forward(inputs)
 
    # 输入:张量inputs
    # 输出:张量outputs
    def forward(self, inputs):
        # return outputs
        raise NotImplementedError
 
    # 输入:最终输出对outputs的梯度outputs_grads
    # 输出:最终输出对inputs的梯度inputs_grads
    def backward(self, outputs_grads):
        # return inputs_grads
        raise NotImplementedError
 
 
class Optimizer(object):  # 优化器基类
    def __init__(self, init_lr, model):
        """
        优化器类初始化
        """
        # 初始化学习率,用于参数更新的计算
        self.init_lr = init_lr
        # 指定优化器需要优化的模型
        self.model = model
 
    def step(self):
        """
        定义每次迭代如何更新参数
        """
        pass
 
 
class SimpleBatchGD(Optimizer):
    def __init__(self, init_lr, model):
        super(SimpleBatchGD, self).__init__(init_lr=init_lr, model=model)
 
    def step(self):
        # 参数更新
        if isinstance(self.model.params, dict):
            for key in self.model.params.keys():
                self.model.params[key] = self.model.params[key] - self.init_lr * self.model.grads[key]
 
 
class Adagrad(Optimizer):
    def __init__(self, init_lr, model, epsilon):
        """
        Adagrad 优化器初始化
        输入:
            - init_lr: 初始学习率 - model:模型,model.params存储模型参数值  - epsilon:保持数值稳定性而设置的非常小的常数
        """
        super(Adagrad, self).__init__(init_lr=init_lr, model=model)
        self.G = {}
        for key in self.model.params.keys():
            self.G[key] = 0
        self.epsilon = epsilon
 
    def adagrad(self, x, gradient_x, G, init_lr):
        """
        adagrad算法更新参数,G为参数梯度平方的累计值。
        """
        G += gradient_x ** 2
        x -= init_lr / torch.sqrt(G + self.epsilon) * gradient_x
        return x, G
 
    def step(self):
        """
        参数更新
        """
        for key in self.model.params.keys():
            self.model.params[key], self.G[key] = self.adagrad(self.model.params[key],
                                                               self.model.grads[key],
                                                               self.G[key],
                                                               self.init_lr)
 
 
class RMSprop(Optimizer):
    def __init__(self, init_lr, model, beta, epsilon):
        """
        RMSprop优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - beta:衰减率
            - epsilon:保持数值稳定性而设置的常数
        """
        super(RMSprop, self).__init__(init_lr=init_lr, model=model)
        self.G = {}
        for key in self.model.params.keys():
            self.G[key] = 0
        self.beta = beta
        self.epsilon = epsilon
 
    def rmsprop(self, x, gradient_x, G, init_lr):
        """
        rmsprop算法更新参数,G为迭代梯度平方的加权移动平均
        """
        G = self.beta * G + (1 - self.beta) * gradient_x ** 2
        x -= init_lr / torch.sqrt(G + self.epsilon) * gradient_x
        return x, G
 
    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.G[key] = self.rmsprop(self.model.params[key],
                                                               self.model.grads[key],
                                                               self.G[key],
                                                               self.init_lr)
 
 
class Momentum(Optimizer):
    def __init__(self, init_lr, model, rho):
        """
        Momentum优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - rho:动量因子
        """
        super(Momentum, self).__init__(init_lr=init_lr, model=model)
        self.delta_x = {}
        for key in self.model.params.keys():
            self.delta_x[key] = 0
        self.rho = rho
 
    def momentum(self, x, gradient_x, delta_x, init_lr):
        """
        momentum算法更新参数,delta_x为梯度的加权移动平均
        """
        delta_x = self.rho * delta_x - init_lr * gradient_x
        x += delta_x
        return x, delta_x
 
    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.delta_x[key] = self.momentum(self.model.params[key],
                                                                      self.model.grads[key],
                                                                      self.delta_x[key],
                                                                      self.init_lr)
 
class Nesterov(Optimizer):
    def __init__(self, init_lr, model, rho):
        """
        Nesterov优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - rho:动量因子
        """
        super(Nesterov, self).__init__(init_lr=init_lr, model=model)
        self.delta_x = {}
        for key in self.model.params.keys():
            self.delta_x[key] = 0
        self.rho = rho
 
    def nesterov(self, x, gradient_x, delta_x, init_lr):
        """
        Nesterov算法更新参数,delta_x为梯度的加权移动平均
        """
        delta_x_prev = delta_x
        delta_x = self.rho * delta_x - init_lr * gradient_x
        x += -self.rho * delta_x_prev + (1 + self.rho) * delta_x
        return x, delta_x
 
    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.delta_x[key] = self.nesterov(self.model.params[key],
                                                                      self.model.grads[key],
                                                                      self.delta_x[key],
                                                                      self.init_lr)
 
 
class Adam(Optimizer):
    def __init__(self, init_lr, model, beta1, beta2, epsilon):
        """
        Adam优化器初始化
        输入:
            - init_lr:初始学习率
            - model:模型,model.params存储模型参数值
            - beta1, beta2:移动平均的衰减率
            - epsilon:保持数值稳定性而设置的常数
        """
        super(Adam, self).__init__(init_lr=init_lr, model=model)
        self.beta1 = beta1
        self.beta2 = beta2
        self.epsilon = epsilon
        self.M, self.G = {}, {}
        for key in self.model.params.keys():
            self.M[key] = 0
            self.G[key] = 0
        self.t = 1
 
    def adam(self, x, gradient_x, G, M, t, init_lr):
        """
        adam算法更新参数
        输入:
            - x:参数
            - G:梯度平方的加权移动平均
            - M:梯度的加权移动平均
            - t:迭代次数
            - init_lr:初始学习率
        """
        M = self.beta1 * M + (1 - self.beta1) * gradient_x
        G = self.beta2 * G + (1 - self.beta2) * gradient_x ** 2
        M_hat = M / (1 - self.beta1 ** t)
        G_hat = G / (1 - self.beta2 ** t)
        t += 1
        x -= init_lr / torch.sqrt(G_hat + self.epsilon) * M_hat
        return x, G, M, t
 
    def step(self):
        """参数更新"""
        for key in self.model.params.keys():
            self.model.params[key], self.G[key], self.M[key], self.t = self.adam(self.model.params[key],
                                                                                 self.model.grads[key],
                                                                                 self.G[key],
                                                                                 self.M[key],
                                                                                 self.t,
                                                                                 self.init_lr)
 
 
class OptimizedFunction3D(Op):
    def __init__(self):
        super(OptimizedFunction3D, self).__init__()
        self.params = {'x': 0}
        self.grads = {'x': 0}
 
    def forward(self, x):
        self.params['x'] = x
        return - x[0] * x[0] / 2 + x[1] * x[1] / 1  # x[0] ** 2 + x[1] ** 2 + x[1] ** 3 + x[0] * x[1]
 
    def backward(self):
        x = self.params['x']
        gradient1 = - 2 * x[0] / 2
        gradient2 = 2 * x[1] / 1
        grad1 = torch.Tensor([gradient1])
        grad2 = torch.Tensor([gradient2])
        self.grads['x'] = torch.cat([grad1, grad2])
 
 
class Visualization3D(animation.FuncAnimation):
    """    绘制动态图像,可视化参数更新轨迹    """
 
    def __init__(self, *xy_values, z_values, labels=[], colors=[], fig, ax, interval=100, blit=True, **kwargs):
        """
        初始化3d可视化类
        输入:
            xy_values:三维中x,y维度的值
            z_values:三维中z维度的值
            labels:每个参数更新轨迹的标签
            colors:每个轨迹的颜色
            interval:帧之间的延迟(以毫秒为单位)
            blit:是否优化绘图
        """
        self.fig = fig
        self.ax = ax
        self.xy_values = xy_values
        self.z_values = z_values
 
        frames = max(xy_value.shape[0] for xy_value in xy_values)
 
        self.lines = [ax.plot([], [], [], label=label, color=color, lw=2)[0]
                      for _, label, color in zip_longest(xy_values, labels, colors)]
        self.points = [ax.plot([], [], [], color=color, markeredgewidth=1, markeredgecolor='black', marker='o')[0]
                       for _, color in zip_longest(xy_values, colors)]
        # print(self.lines)
        super(Visualization3D, self).__init__(fig, self.animate, init_func=self.init_animation, frames=frames,
                                              interval=interval, blit=blit, **kwargs)
 
    def init_animation(self):
        # 数值初始化
        for line in self.lines:
            line.set_data_3d([], [], [])
        for point in self.points:
            point.set_data_3d([], [], [])
        return self.points + self.lines
 
    def animate(self, i):
        # 将x,y,z三个数据传入,绘制三维图像
        for line, xy_value, z_value in zip(self.lines, self.xy_values, self.z_values):
            line.set_data_3d(xy_value[:i, 0], xy_value[:i, 1], z_value[:i])
        for point, xy_value, z_value in zip(self.points, self.xy_values, self.z_values):
            point.set_data_3d(xy_value[i, 0], xy_value[i, 1], z_value[i])
        return self.points + self.lines
 
 
def train_f(model, optimizer, x_init, epoch):
    x = x_init
    all_x = []
    losses = []
    for i in range(epoch):
        all_x.append(copy.deepcopy(x.numpy()))  # 浅拷贝 改为 深拷贝, 否则List的原值会被改变。 Edit by David 2022.12.4.
        loss = model(x)
        losses.append(loss)
        model.backward()
        optimizer.step()
        x = model.params['x']
    return torch.Tensor(np.array(all_x)), losses
 
 
# 构建5个模型,分别配备不同的优化器
model1 = OptimizedFunction3D()
opt_gd = SimpleBatchGD(init_lr=0.05, model=model1)
 
model2 = OptimizedFunction3D()
opt_adagrad = Adagrad(init_lr=0.05, model=model2, epsilon=1e-7)
 
model3 = OptimizedFunction3D()
opt_rmsprop = RMSprop(init_lr=0.05, model=model3, beta=0.9, epsilon=1e-7)
 
model4 = OptimizedFunction3D()
opt_momentum = Momentum(init_lr=0.05, model=model4, rho=0.9)
 
model5 = OptimizedFunction3D()
opt_adam = Adam(init_lr=0.05, model=model5, beta1=0.9, beta2=0.99, epsilon=1e-7)
 
model6 = OptimizedFunction3D()
opt_Nesterov = Nesterov(init_lr=0.1, model=model6, rho=0.9)
 
models = [model1, model2, model3, model4, model5, model6]
opts = [opt_gd, opt_adagrad, opt_rmsprop, opt_momentum, opt_adam, opt_Nesterov]
 
x_all_opts = []
z_all_opts = []
 
# 使用不同优化器训练
 
for model, opt in zip(models, opts):
    x_init = torch.FloatTensor([0.00001, 0.5])
    x_one_opt, z_one_opt = train_f(model, opt, x_init, 100)  # epoch
    # 保存参数值
    x_all_opts.append(x_one_opt.numpy())
    z_all_opts.append(np.squeeze(z_one_opt))
 
# 使用numpy.meshgrid生成x1,x2矩阵,矩阵的每一行为[-3, 3],以0.1为间隔的数值
x1 = np.arange(-1, 2, 0.01)
x2 = np.arange(-1, 1, 0.05)
x1, x2 = np.meshgrid(x1, x2)
init_x = torch.Tensor(np.array([x1, x2]))
 
model = OptimizedFunction3D()
 
# 绘制 f_3d函数 的 三维图像
fig = plt.figure()
ax = plt.axes(projection='3d')
X = init_x[0].numpy()
Y = init_x[1].numpy()
Z = model(init_x).numpy()  # 改为 model(init_x).numpy() David 2022.12.4
surf = ax.plot_surface(X, Y, Z, edgecolor='grey', cmap=cm.coolwarm)
# fig.colorbar(surf, shrink=0.5, aspect=1)
ax.set_zlim(-3, 2)
ax.set_xlabel('x1')
ax.set_ylabel('x2')
ax.set_zlabel('f(x1,x2)')
 
labels = ['SGD', 'AdaGrad', 'RMSprop', 'Momentum', 'Adam', 'Nesterov']
colors = ['#8B0000', '#0000FF', '#000000', '#008B00', '#FF0000']
 
 
animator = Visualization3D(*x_all_opts, z_values=z_all_opts, labels=labels, colors=colors, fig=fig, ax=ax)
ax.legend(loc='upper right')
 
plt.show()
# animator.save('teaser' + '.gif', writer='imagemagick',fps=10) # 效果不好,估计被挡住了…… 有待进一步提高 Edit by David 2022.12.4
# save不好用,不费劲了,安装个软件做gif https://pc.qq.com/detail/13/detail_23913.html


4. 结合3D动画,用自己的语言,从轨迹、速度等多个角度讲解各个算法优缺点


SGD
SGD较于其他几个算法,速度相对较慢,会呈现“之”字型的轨迹,并且在cs231经典动画中,SGD出现了陷入局部最小值,出不来的情况。

所以根据动画可以看出SGD的缺点有:

(1)容易陷入局部最优

(2)速度相对较慢且需要调整学习率

AdaGrad
可以看出,AdaGrad图中的轨迹图都是刚开始速度明显大于RMSprop和SGD算法的,偶尔比Momentum和Nesterov还要快,但是随着时间的增长,AdaGrad会成为图中速度最慢的算法。方向上,该算法的方向一直都很准确,并且明显解决了SGD的“之”字型问题,收敛稳定。

相较于SGD算法,AdaGrad的优点:

(1)自适应算法:AdaGrad算法根据每个参数的历史梯度信息来自适应地调整学习率,使得梯度不会太大或太小。

(2)“之”字形的变动程度衰减,呈现稳定的向最优点收敛

缺点:

学习率衰减过快,可能发生早停现象:随着训练的进行,AdaGrad会累积历史梯度的平方和,导致学习率不断减小。在训练后期,学习率可能会变得非常小,甚至接近于零,导致训练过早停止。

RMSprop
RMSprop的轨迹图,速度上很稳定,在前期比AdaGrad要慢,但是后期AdaGrad很慢的时候,RMSprop依然稳定前进。在轨迹方向上,基本和AdaGrad是一样的。

所以相较于AdaGrad而言,RMSprop在它的基础上进行改进,优点为:

收敛速度快解决了AdaGrad算法的早停问题: 引入了衰减率,不会一直累积梯度平方,而是通过梯度平方的指数衰减移动平均来调整学习率,解决了AdaGrad的早衰问题。

Momentum
Momentum算法在速度上,要明显快于前几个函数,跟Nesterov差不多,但是在方向上,Momentum算法每次都是去错的方向转几次,然后才能修正过来。所以Momentum的优点为:

很快的收敛速度,特别是对于类似鞍点的问题,由于动量维持了运动,能够更有效地收敛至局部最小值或平坦区域。但是方向要相对差些,之前的动量仍然会对下一次的下降造成影响,导致Momentum其实有一点大幅度的“之”字型的轨迹。

Nesterov
Nesterov算法的方向和速度效果都是很好的,速度上,它是最快的;方向上,轨迹正确性要好于Momentum,但是仍然要比AdaGrad、RMSprop要差些。Nesterov是对Momentum进行的改进,不仅仅根据当前梯度调整位置,而是根据当前动量在预期的未来位置计算梯度。它的优点为速度快且轨迹呈现出更加平滑、更有方向性的路径朝向最优点。

Adam
根据3D轨迹图:Adam算法的轨迹为稳定,快速的向最小值收敛,就速度和方向的正确性、稳定性而言,都是居中。所以Adam算法的优点就是结合了调整学习率的算法:RMSprop和梯度估计修正算法:Momentum二者的优点:稳定、快速,实用性较高。?
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这是我的另一篇博客总结的优化算法,需要期末复习的同学可以点击链接:

NNDL学期知识点总结 [HBU]-CSDN博客

动图我一直贴不上去,考完试了回来研究研究,怎么才能贴上动图。

文章来源:https://blog.csdn.net/weixin_63010525/article/details/135264616
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