第五章.与学习相关技巧
5.1 参数更新的最优化方法
- 神经网络学习的目的是找到使损失函数的值尽可能小的参数，这是寻找最优参数的问题，解决这个问题的过程称为最优化。
- 很多深度学习框架都实现了各种最优化方法，比如Lasagne深度学习框架，在update.py这个文件中以函数的形式集中实现了最优化方法，用户可以从中选择自己想用的最优化方法。
1.SGD
使用参数的梯度，沿着梯度方向更新参数，不断重复这个步骤多次，从而逐渐靠近最优参数，这个过程称为随机梯度下降法(SGD)
1).数学式：
- 参数说明：
①.W：待更新的权重参数
②.η：学习率
③.∂L/∂W：损失函数关于W的梯度
2).示例：
以 f(x,y)=1/20x2 + y2 为例：
-
3D曲面图：
-
最优化的梯度更新路径图：
缺点：
如果函数的形状非均向，比如呈延伸状SGD，搜索路径就会非常低效。SGD低效的根本原因：
梯度的方向并没有指向最小值的方向。
3).代码实现：
class SGD:
def __init__(self, lr):
self.lr = lr
def update(self, params, grads):
for key in params.keys():
params[key] -= self.lr * grads[key]
2.Momentum
Momentum可以解决SGD存在的问题：如果函数的形状非均向，搜索效率很低的情况。
1).数学式：
- 参数说明：
①.W：待更新的权重参数
②.η：学习率
③.∂L/∂W：损失函数关于W的梯度
④.v：对应物理上的速度，表示物体在梯度方向上的受力，在这个力的作用下，物体的速度增加这一物理量
⑤.α：对应物理上的地面摩擦力或空气阻力，α=0.9
2).示例：
以 f(x,y)=1/20x2 + y2 为例：
-
3D曲面图：
-
最优化的梯度更新路径图：
缺点：
在神经网络的学习中，η的值很重要，η值太小，学习花费更多时间；η值太大，学习发散不能正确进行。
3).代码实现：
class Momentum:
def __init__(self, lr, momentum):
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]
3.AdaGrad
AdaGrad方法可以解决Momentum存在的η难设置问题。AdaGrad方法在更新参数时，随着学习的进行，使学习率逐渐减小。
1).数学式：
- 参数说明：
①.W：待更新的权重参数
②.η：学习率
③.∂L/∂W：损失函数关于W的梯度
④.h：它保存了以前所有梯度值的平方和
2).示例：
以 f(x,y)=1/20x2 + y2 为例：
-
3D曲面图：
-
最优化的梯度更新路径图：
图像描述：
函数的取值高效地向着最小值移动，由于y轴方向上的梯度较大，因此刚开始变动较大，但是后面会根据这个较大的变动按比例进行调整，减小更新的步伐。
3).代码实现：
class AdaGrad:
def __init__(self, lr):
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)#加上1e-7是为了解决除数为0的情况
4.Adam
Adam是融合了Momentum与AdaGrad的方法。
1).示例：
以 f(x,y)=1/20x2 + y2 为例：
-
3D曲面图：
-
最优化的梯度更新路径图：
2).代码实现：
class Adam:
def __init__(self, lr, β1, β2):
self.lr = lr
self.β1 = β1
self.β2 = β2
self.iter = 0
self.h = None
self.v = None
def update(self, params, grads):
if self.h is None:
self.h, self.v = {}, {}
for key, val in params.items():
self.h[key] = np.zeros_like(val)
self.v[key] = np.zeros_like(val)
self.iter += 1
lr_t = self.lr * np.sqrt(1.0 - self.β2 ** self.iter) / (1.0 - self.β1 ** self.iter)
for key in params.keys():
self.h[key] += (1 - self.β1) * (grads[key] - self.h[key])
self.v[key] += (1 - self.β2) * (grads[key] ** 2 - self.v[key])
params[key] -= lr_t * self.h[key] / (np.sqrt(self.v[key]) + 1e-7) # 加上1e-7是为了解决除数为0的情况
5.各种最优化方法实现手写数字识别的完整代码：(直接替换上述类即可)
1).代码实现：
import numpy as np
import matplotlib.pyplot as plt
import sys, os
sys.path.append(os.pardir)
from dataset.mnist import load_mnist
from collections import OrderedDict
class Adam:
def __init__(self, lr, β1, β2):
self.lr = lr
self.β1 = β1
self.β2 = β2
self.iter = 0
self.h = None
self.v = None
def update(self, params, grads):
if self.h is None:
self.h, self.v = {}, {}
for key, val in params.items():
self.h[key] = np.zeros_like(val)
self.v[key] = np.zeros_like(val)
self.iter += 1
lr_t = self.lr * np.sqrt(1.0 - self.β2 ** self.iter) / (1.0 - self.β1 ** self.iter)
for key in params.keys():
self.h[key] += (1 - self.β1) * (grads[key] - self.h[key])
self.v[key] += (1 - self.β2) * (grads[key] ** 2 - self.v[key])
params[key] -= lr_t * self.h[key] / (np.sqrt(self.v[key]) + 1e-7) # 加上1e-7是为了解决除数为0的情况
class Affine:
def __init__(self, W, b):
self.W = W
self.b = b
self.x = None
self.original_x_shape = None
# 权重和偏置参数的导数
self.dW = None
self.db = None
# 向前传播
def forward(self, x):
self.original_x_shape = x.shape
x = x.reshape(x.shape[0], -1)
self.x = x
out = np.dot(self.x, self.W) + self.b
return out
# 反向传播
def backward(self, dout):
dx = np.dot(dout, self.W.T)
self.dW = np.dot(self.x.T, dout)
self.db = np.sum(dout, axis=0)
dx = dx.reshape(*self.original_x_shape) # 还原输入数据的形状（对应张量）
return dx
class ReLU:
def __init__(self):
self.mask = None
def forward(self, x):
self.mask = (x <= 0)
out = x.copy()
out[self.mask] = 0
return out
def backward(self, dout):
dout[self.mask] = 0
dx = dout
return dx
class SoftmaxWithLoss:
def __init__(self):
self.loss = None
self.y = None
self.t = None
# 输出层函数：softmax
def softmax(self, x):
if x.ndim == 2:
x = x.T
x = x - np.max(x, axis=0)
y = np.exp(x) / np.sum(np.exp(x), axis=0)
return y.T
x = x - np.max(x) # 溢出对策
y = np.exp(x) / np.sum(np.exp(x))
return y
# 误差函数：交叉熵误差
def cross_entropy_error(self, y, t):
if y.ndim == 1:
y = y.reshape(1, y.size)
t = t.reshape(1, t.size)
# 监督数据是one_hot_label的情况下，转换为正确解标签的索引
if t.size == y.size:
t = t.argmax(axis=1)
batch_size = y.shape[0]
return -np.sum(np.log(y[np.arange(batch_size), t] + 1e-7)) / batch_size
def forward(self, x, t):
self.t = t
self.y = self.softmax(x)
self.loss = self.cross_entropy_error(self.y, self.t)
return self.loss
def backward(self, dout=1):
batch_size = self.t.shape[0]
if self.t.size == self.y.size:
dx = (self.y - self.t) / batch_size
else:
dx = self.y.copy()
dx[np.arange(batch_size), self.t] -= 1
dx = dx / batch_size
return dx
class TwoLayerNet:
# 初始化
def __init__(self, input_size, hidden_size, output_size, weight_init_std=0.01):
# 初始化权重
self.params = {}
self.params['W1'] = weight_init_std * np.random.randn(input_size, hidden_size)
self.params['b1'] = np.zeros(hidden_size)
self.params['W2'] = weight_init_std * np.random.randn(hidden_size, output_size)
self.params['b2'] = np.zeros(output_size)
# 生成层
self.layers = OrderedDict()
self.layers['Affine1'] = Affine(self.params['W1'], self.params['b1'])
self.layers['ReLU'] = ReLU()
self.layers['Affine2'] = Affine(self.params['W2'], self.params['b2'])
self.lastLayer = SoftmaxWithLoss()
def predict(self, x):
for layer in self.layers.values():
x = layer.forward(x)
return x
def loss(self, x, t):
y = self.predict(x)
loss = self.lastLayer.forward(y, t)
return loss
def accuracy(self, x, t):
y = self.predict(x)
y = np.argmax(y, axis=1)
if t.ndim != 1: t = np.argmax(t, axis=1)
accuracy = np.sum(y == t) / float(t.shape[0])
return accuracy
# 微分函数
def numerical_gradient1(self, f, x):
h = 1e-4
grad = np.zeros_like(x)
it = np.nditer(x, flags=['multi_index'], op_flags=['readwrite'])
while not it.finished:
idx = it.multi_index
tmp_val = x[idx]
x[idx] = float(tmp_val) + h
fxh1 = f(x) # f(x+h)
x[idx] = tmp_val - h
fxh2 = f(x) # f(x-h)
grad[idx] = (fxh1 - fxh2) / (2 * h)
x[idx] = tmp_val # 还原值
it.iternext()
return grad
# 通过数值微分计算关于权重参数的梯度
def numerical_gradient(self, x, t):
loss_W = lambda W: self.loss(x, t)
grad = {}
grad['W1'] = self.numerical_gradient1(loss_W, self.params['W1'])
grad['b1'] = self.numerical_gradient1(loss_W, self.params['b1'])
grad['W2'] = self.numerical_gradient1(loss_W, self.params['W2'])
grad['b2'] = self.numerical_gradient1(loss_W, self.params['b2'])
return grad
# 通过误差反向传播法计算权重参数的梯度误差
def gradient(self, x, t):
# 正向传播
self.loss(x, t)
# 反向传播
dout = 1
dout = self.lastLayer.backward(dout)
layers = list(self.layers.values())
layers.reverse()
for layer in layers:
dout = layer.backward(dout)
# 设定
grads = {}
grads['W1'] = self.layers['Affine1'].dW
grads['b1'] = self.layers['Affine1'].db
grads['W2'] = self.layers['Affine2'].dW
grads['b2'] = self.layers['Affine2'].db
return grads
# 读入数据
def get_data():
(x_train, t_train), (x_test, t_test) = load_mnist(normalize=True, one_hot_label=True)
return (x_train, t_train), (x_test, t_test)
# 读入数据
(x_train, t_train), (x_test, t_test) = get_data()
network = TwoLayerNet(input_size=784, hidden_size=50, output_size=10)
# 超参数
iter_num = 10000
train_size = x_train.shape[0]
batch_size = 100
lr = 0.01
momentum = 0.9
β1 = 0.9
β2 = 0.999
iter_per_epoch = max(train_size / batch_size, 1)
train_loss_list = []
train_acc_list = []
test_acc_list = []
# 可替换的类
optimizer = Adam(lr, β1, β2)
for i in range(iter_num):
batch_mask = np.random.choice(train_size, batch_size)
x_batch = x_train[batch_mask]
t_batch = t_train[batch_mask]
grads = network.gradient(x_batch, t_batch)
params = network.params
optimizer.update(params, grads)
loss = network.loss(x_batch, t_batch)
train_loss_list.append(loss)
if i % iter_per_epoch == 0:
train_acc = network.accuracy(x_train, t_train)
train_acc_list.append(train_acc)
test_acc = network.accuracy(x_test, t_test)
test_acc_list.append(test_acc)
print('train_acc,test_acc|', str(train_acc) + ',' + str(test_acc))
# 绘制识别精度图像
plt.rcParams['font.sans-serif'] = ['SimHei'] # 解决中文乱码
plt.rcParams['axes.unicode_minus'] = False # 解决负号不显示的问题
plt.figure(figsize=(8, 4))
plt.subplot(1, 2, 1)
x_data = np.arange(0, len(train_acc_list))
plt.plot(x_data, train_acc_list, 'b')
plt.plot(x_data, test_acc_list, 'r')
plt.xlabel('epoch')
plt.ylabel('accuracy')
plt.ylim(0.0, 1.0)
plt.title('训练数据和测试数据的识别精度')
plt.legend(['train_acc', 'test_acc'])
plt.subplot(1, 2, 2)
x_data = np.arange(0, len(train_loss_list))
plt.plot(x_data, train_loss_list, 'g')
plt.xlabel('iters_num')
plt.ylabel('loss')
plt.title('损失函数')
plt.show()
6.各种最优化方法绘制梯度更新路径图的代码
1).代码实现：
import sys, os
sys.path.append(os.pardir) # 为了导入父目录的文件而进行的设定
import numpy as np
import matplotlib.pyplot as plt
from collections import OrderedDict
class SGD:
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:
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 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 Adam:
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
plt.figure(figsize=(12, 10))
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)
plt.ylim(-10, 10)
plt.xlim(-10, 10)
plt.plot(0, 0, '+')
plt.title(key)
plt.xlabel("x")
plt.ylabel("y")
plt.show()
