2D可视化实验
被优化函数：X的平方
import torch
import numpy as np
from matplotlib import pyplot as plt
from abc import abstractmethod
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
# 优化函数
class OptimizedFunction(Op):
def __init__(self, w):
super(OptimizedFunction, self).__init__()
self.w = w
self.params = {'x': 0}
self.grads = {'x': 0}
def forward(self, x):
self.params['x'] = x
return torch.matmul(self.w.T, torch.tensor(torch.square(self.params['x']), dtype=torch.float32))
def backward(self):
self.grads['x'] = 2 * torch.multiply(self.w.T, self.params['x'])
# 训练函数，记录梯度下降过程中每轮的参数x和损失
import copy
def train_f(model, optimizer, x_init, epoch):
"""
训练函数
输入：
- model：被优化函数
- optimizer：优化器
- x_init：x初始值
- epoch：训练回合数
"""
x = x_init
all_x = []
losses = []
for i in range(epoch):
all_x.append(copy.copy(x.numpy()))
loss = model(x)
losses.append(loss)
model.backward()
optimizer.step()
x = model.params['x']
print(all_x)
return torch.tensor(all_x), losses
# 可视化函数，用于绘制更新轨迹
class Visualization(object):
def __init__(self):
"""
初始化可视化类
"""
# 只画出参数x1和x2在区间[-5, 5]的曲线部分
x1 = np.arange(-5, 5, 0.1)
x2 = np.arange(-5, 5, 0.1)
x1, x2 = np.meshgrid(x1, x2)
self.init_x = torch.tensor([x1, x2])
def plot_2d(self, model, x, fig_name):
"""
可视化参数更新轨迹
"""
fig, ax = plt.subplots(figsize=(10, 6))
cp = ax.contourf(self.init_x[0], self.init_x[1], model(self.init_x.transpose(0, 1)),
colors=['#e4007f', '#f19ec2', '#e86096', '#eb7aaa', '#f6c8dc', '#f5f5f5', '#000000'])
c = ax.contour(self.init_x[0], self.init_x[1], model(self.init_x.transpose(0, 1)), colors='black')
cbar = fig.colorbar(cp)
ax.plot(x[:, 0], x[:, 1], '-o', color='#000000')
ax.plot(0, 'r*', markersize=18, color='#fefefe')
ax.set_xlabel('$x1$')
ax.set_ylabel('$x2$')
ax.set_xlim((-2, 5))
ax.set_ylim((-2, 5))
plt.savefig(fig_name)
# 训练模型并可视化参数更新轨迹
import numpy as np
def train_and_plot_f(model, optimizer, epoch, fig_name):
"""
训练模型并可视化参数更新轨迹
"""
# 设置x的初始值
x_init = torch.tensor([3, 4], dtype=torch.float32)
print('x1 initiate: {}, x2 initiate: {}'.format(x_init[0].numpy(), x_init[1].numpy()))
x, losses = train_f(model, optimizer, x_init, epoch)
print(x)
losses = np.array(losses)
# 展示x1、x2的更新轨迹
vis = Visualization()
vis.plot_2d(model, x, fig_name)
#1.SGD
from nndl.op import SimpleBatchGD
# 固定随机种子
torch.manual_seed(0)
w = torch.tensor([0.2, 2])
model = OptimizedFunction(w)
opt = SimpleBatchGD(init_lr=0.2, model=model)
train_and_plot_f(model, opt, epoch=45, fig_name='opti-vis-para.pdf')
plt.show()
#2.AdaGrad
from nndl.op import Optimizer
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)
# 固定随机种子
torch.manual_seed(0)
w = torch.tensor([0.2, 2])
model = OptimizedFunction(w)
opt = Adagrad(init_lr=0.5, model=model, epsilon=1e-7)
train_and_plot_f(model, opt, epoch=60, fig_name='opti-vis-para2.pdf')
plt.show()
#3.RMSprop
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)
# 固定随机种子
torch.manual_seed(0)
w = torch.tensor([0.2, 2])
model = OptimizedFunction(w)
opt = RMSprop(init_lr=0.1, model=model, beta=0.9, epsilon=1e-7)
train_and_plot_f(model, opt, epoch=50, fig_name='opti-vis-para3-RMSprop.pdf')
plt.show()
#4.Momentum
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)
# 固定随机种子
torch.manual_seed(0)
w = torch.tensor([0.2, 2])
model = OptimizedFunction(w)
opt = Momentum(init_lr=0.1, model=model, rho=0.9)
train_and_plot_f(model, opt, epoch=50, fig_name='opti-vis-para4-Momentum.pdf')
plt.show()
#5.Adam
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)
# 固定随机种子
torch.manual_seed(0)
w = torch.tensor([0.2, 2])
model = OptimizedFunction(w)
opt = Adam(init_lr=0.2, model=model, beta1=0.9, beta2=0.99, epsilon=1e-7)
train_and_plot_f(model, opt, epoch=60, fig_name='opti-vis-para5-Adam.pdf')
plt.show()
nndl.py
import torch
import numpy as np
import copy
def make_moons(n_samples=1000, shuffle=True, noise=None):
"""
生成半月形数据分布的模拟数据
"""
n_samples_out = n_samples // 2
n_samples_in = n_samples - n_samples_out
# 使用torch.linspace生成线性间距的张量，模拟角度值
outer_circ_x = torch.cos(torch.linspace(0, np.pi, n_samples_out))
outer_circ_y = torch.sin(torch.linspace(0, np.pi, n_samples_out))
inner_circ_x = 1 - torch.cos(torch.linspace(0, np.pi, n_samples_in))
inner_circ_y = 1 - torch.sin(torch.linspace(0, np.pi, n_samples_in)) - 0.5
# 拼接坐标张量，构建特征数据X
X = torch.stack([torch.cat([outer_circ_x, inner_circ_x]),
torch.cat([outer_circ_y, inner_circ_y])],
dim=1)
# 构建对应的标签数据y
y = torch.cat([torch.zeros(n_samples_out), torch.ones(n_samples_in)])
if shuffle:
# 生成随机索引，用于打乱数据顺序
idx = torch.randperm(X.shape[0])
X_new = copy.deepcopy(X)
y_new = copy.deepcopy(y)
for i in range(X.shape[0]):
X_new[i] = X[idx[i]]
y_new[i] = y[idx[i]]
X = X_new
y = y_new
if noise is not None:
# 给特征数据添加噪声
X += torch.normal(mean=0.0, std=noise, size=X.shape)
return X, y
def accuracy(preds, labels):
"""
计算预测准确率
"""
# 判断是二分类任务还是多分类任务
if preds.shape[1] == 1:
# 二分类时，根据概率阈值判断类别
preds = (preds >= 0.5).float()
else:
# 多分类时，取预测概率最大的类别作为预测结果
preds = torch.argmax(preds, dim=1).long()
correct = (preds == labels.squeeze()).float().sum()
return correct / len(labels)
class RunnerV3(object):
def __init__(self, model, optimizer, loss_fn, metric, **kwargs):
self.model = model
self.optimizer = optimizer
self.loss_fn = loss_fn
self.metric = metric
# 记录训练过程中的评价指标变化情况
self.dev_scores = []
# 记录训练过程中的损失函数变化情况
self.train_epoch_losses = []
self.train_step_losses = []
self.dev_losses = []
# 记录全局最优指标
self.best_score = 0
def train(self, train_loader, dev_loader=None, **kwargs):
# 将模型切换为训练模式
self.model.train()
# 获取训练轮数，默认值为0
num_epochs = kwargs.get("num_epochs", 0)
# 获取log打印频率，默认值为100
log_steps = kwargs.get("log_steps", 100)
# 评价频率
eval_steps = kwargs.get("eval_steps", 0)
# 获取模型保存路径，默认值为"best_model.pth"
save_path = kwargs.get("save_path", "best_model.pth")
custom_print_log = kwargs.get("custom_print_log", None)
# 训练总的步数
num_training_steps = num_epochs * len(train_loader)
if eval_steps:
if self.metric is None:
raise RuntimeError('Error: Metric can not be None!')
if dev_loader is None:
raise RuntimeError('Error: dev_loader can not be None!')
# 当前运行的step数目
global_step = 0
# 进行num_epochs轮训练
for epoch in range(num_epochs):
# 用于统计训练集的损失
total_loss = 0
for step, data in enumerate(train_loader):
X, y = data
# 获取模型预测
logits = self.model(X)
loss = self.loss_fn(logits, y) # 默认求平均损失
total_loss += loss.item()
# 训练过程中，每个step的loss进行保存
self.train_step_losses.append((global_step, loss.item()))
if log_steps and global_step % log_steps == 0:
print(f"[Train] epoch: {epoch}/{num_epochs}, step: {global_step}/{num_training_steps}, loss: {loss.item():.5f}")
# 梯度反向传播，计算每个参数的梯度值
loss.backward()
if custom_print_log:
custom_print_log(self)
# 小批量梯度下降进行参数更新
self.optimizer.step()
# 梯度归零
self.optimizer.zero_grad()
# 判断是否需要评价
if eval_steps > 0 and global_step!= 0 and \
(global_step % eval_steps == 0 or global_step == (num_training_steps - 1)):
dev_score, dev_loss = self.evaluate(dev_loader, global_step=global_step)
print(f"[Evaluate] dev score: {dev_score:.5f}, dev_loss: {dev_loss:.5f}")
# 将模型切换为训练模式
self.model.train()
# 如果当前指标为最优指标，保存该模型
if dev_score > self.best_score:
self.save_model(save_path)
print(f"[Evaluate] best accuracy performence has been updated: {self.best_score:.5f} --> {dev_score:.5f}")
self.best_score = dev_score
global_step += 1
# 当前epoch训练loss累计值
trn_loss = (total_loss / len(train_loader))
# epoch粒度的训练loss保存
self.train_epoch_losses.append(trn_loss)
print("[Train] Training done!")
def evaluate(self, dev_loader, **kwargs):
assert self.metric is not None
# 将模型设置为评估模式
self.model.eval()
global_step = kwargs.get("global_step", -1)
# 用于统计训练集的损失
total_loss = 0
# 重置评价指标相关统计量
self.metric.reset()
# 遍历验证集每个批次
with torch.no_grad():
for batch_id, data in enumerate(dev_loader):
X, y = data
# 计算模型输出
logits = self.model(X)
# 计算损失函数
loss = self.loss_fn(logits, y).item()
# 累积损失
total_loss += loss
# 累积评价指标相关统计量
self.metric.update(logits, y)
dev_loss = (total_loss / len(dev_loader))
self.dev_losses.append((global_step, dev_loss))
dev_score = self.metric.accumulate()
self.dev_scores.append(dev_score)
return dev_score, dev_loss
def predict(self, x, **kwargs):
# 将模型设置为评估模式
self.model.eval()
# 运行模型前向计算，得到预测值
with torch.no_grad():
logits = self.model(x)
return logits
def save_model(self, save_path):
torch.save(self.model.state_dict(), save_path)
def load_model(self, model_path):
model_state_dict = torch.load(model_path)
self.model.load_state_dict(model_state_dict)
class Accuracy:
def __init__(self, is_logist=True):
"""
输入：
- is_logist: outputs是logits还是激活后的值
"""
# 用于统计正确的样本个数
self.num_correct = 0
# 用于统计样本的总数
self.num_count = 0
self.is_logist = is_logist
def update(self, outputs, labels):
"""
输入：
- outputs: 预测值, shape=[N,class_num]
- labels: 标签值, shape=[N,1]
"""
# 判断是二分类任务还是多分类任务
if outputs.shape[1] == 1:
outputs = outputs.squeeze()
if self.is_logist:
# logits判断是否大于0
preds = (outputs >= 0).float()
else:
# 如果不是logits，根据概率阈值判断类别
preds = (outputs >= 0.5).float()
else:
# 多分类时，取预测概率最大的类别作为预测结果
preds = torch.argmax(outputs, dim=1).long()
# 获取本批数据中预测正确的样本个数
labels = labels.squeeze()
batch_correct = (preds == labels).float().sum().item()
batch_count = len(labels)
# 更新num_correct和num_count
self.num_correct += batch_correct
self.num_count += batch_count
def accumulate(self):
# 使用累计的数据，计算总的指标
if self.num_count == 0:
return 0
return self.num_correct / self.num_count
def reset(self):
self.num_correct = 0
self.num_count = 0
def name(self):
return "Accuracy"
op.py
import torch
from abc import abstractmethod
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
# 优化器基类
class Optimizer(object):
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
@abstractmethod
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.init_lr * self.model.grads[key]
class BatchGD(Optimizer):
def __init__(self, init_lr, model):
super(BatchGD, self).__init__(init_lr=init_lr, model=model)
def step(self):
# 参数更新
for layer in self.model.layers: # 遍历所有层
if isinstance(layer.params, dict):
for key in layer.params.keys():
layer.params[key] -= self.init_lr * layer.grads[key]
class Linear(Op):
def __init__(self, in_features, out_features, name, weight_init=torch.randn, bias_init=torch.zeros):
self.params = {}
self.params['W'] = weight_init((in_features, out_features))
self.params['b'] = bias_init((1, out_features))
self.inputs = None
self.grads = {}
self.name = name
def forward(self, inputs):
self.inputs = inputs
outputs = torch.matmul(self.inputs, self.params['W']) + self.params['b']
return outputs
def backward(self, grads):
"""
输入：
- grads：损失函数对当前层输出的导数
输出：
- 损失函数对当前层输入的导数
"""
self.grads['W'] = torch.matmul(self.inputs.T, grads)
self.grads['b'] = torch.sum(grads, dim=0, keepdim=True)
return torch.matmul(grads, self.params['W'].T)
class Logistic(Op):
def __init__(self):
self.inputs = None
self.outputs = None
self.params = None
def forward(self, inputs):
outputs = 1.0 / (1.0 + torch.exp(-inputs))
self.outputs = outputs
return outputs
def backward(self, outputs_grads):
# 计算Logistic激活函数对输入的导数
outputs_grad_inputs = self.outputs * (1.0 - self.outputs)
return outputs_grads * outputs_grad_inputs
class ReLU(Op):
def __init__(self):
self.inputs = None
self.outputs = None
self.params = None
def forward(self, inputs):
self.inputs = inputs
return inputs * (inputs > 0).float()
def backward(self, outputs_grads):
return outputs_grads * (self.inputs > 0).float()
class MLP_3L(Op):
def __init__(self, layers_size):
self.fc1 = Linear(layers_size[0], layers_size[1], name='fc1')
# ReLU激活函数
self.act_fn1 = ReLU()
self.fc2 = Linear(layers_size[1], layers_size[2], name='fc2')
self.act_fn2 = ReLU()
self.fc3 = Linear(layers_size[2], layers_size[3], name='fc3')
self.layers = [self.fc1, self.act_fn1, self.fc2, self.act_fn2, self.fc3]
def __call__(self, X):
return self.forward(X)
def forward(self, X):
z1 = self.fc1(X)
a1 = self.act_fn1(z1)
z2 = self.fc2(a1)
a2 = self.act_fn2(z2)
z3 = self.fc3(a2)
return z3
def backward(self, loss_grad_z3):
loss_grad_a2 = self.fc3.backward(loss_grad_z3)
loss_grad_z2 = self.act_fn2.backward(loss_grad_a2)
loss_grad_a1 = self.fc2.backward(loss_grad_z2)
loss_grad_z1 = self.act_fn1.backward(loss_grad_a1)
return self.fc1.backward(loss_grad_z1)
被优化函数：1/20*X**2+Y**2
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
learningrate = [0.9, 0.3, 0.3, 0.6, 0.6, 0.6, 0.6]
optimizers = OrderedDict() #一个有序的字典
optimizers["SGD"] = SGD(lr=learningrate[0])
optimizers["Momentum"] = Momentum(lr=learningrate[1])
optimizers["Nesterov"] = Nesterov(lr=learningrate[2])
optimizers["AdaGrad"] = AdaGrad(lr=learningrate[3])
optimizers["RMSprop"] = RMSprop(lr=learningrate[4])
optimizers["Adam"] = Adam(lr=learningrate[5])
idx = 1
id_lr = 0
for key in optimizers:
optimizer = optimizers[key]
lr = learningrate[id_lr]
id_lr = id_lr + 1
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 #这个操作可以使高度大于 7 的点在绘制等高线时被忽略掉，只画出高度小于等于 7 的部分，从而得到一个简单的等高线图。
# plot
plt.subplot(2, 3, idx)
idx += 1
plt.plot(x_history, y_history, 'o-',markersize=4, color="r")
# plt.contour(X, Y, Z) # 绘制等高线
plt.contour(X, Y, Z, cmap='rainbow') # 颜色填充
plt.ylim(-10, 10)
plt.xlim(-10, 10)
plt.plot(0, 0, '+')
# plt.axis('off')
# plt.title(key+'\nlr='+str(lr), fontstyle='italic')
plt.text(0, 10, key + '\nlr=' + str(lr), fontsize=10, color="b",
verticalalignment='top', horizontalalignment='center', fontstyle='italic')
plt.xlabel("x")
plt.ylabel("y")
plt.subplots_adjust(wspace=0, hspace=0) # 调整子图间距
plt.show()
不同优化器的3D可视化对比
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='rainbow')
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')
ax.view_init(elev=90, azim=0)
plt.show()
animator.save('animation.gif')
cs231
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 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)
models = [model5, model2, model3, model4, model1]
opts = [opt_adam, opt_adagrad, opt_rmsprop, opt_momentum, opt_gd]
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 = ['Adam', 'AdaGrad', 'RMSprop', 'Momentum', 'SGD']
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()
优化算法2D轨迹 鱼书例题2D版
# 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()
优化算法3D轨迹 鱼书例题3D版
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 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 # 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5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)
models = [model1, model2, model3, model4, model5]
opts = [opt_gd, opt_adagrad, opt_rmsprop, opt_momentum, opt_adam]
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']
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()
