DeepLearing学习笔记-改善深层神经网络(第二周作业-优化方法)

0- 背景：

本文将介绍几种常用的优化方法，用以加快神经网络的学习速度

本文需要用到的库如下：

import numpy as np
import matplotlib.pyplot as plt
import scipy.io
import math
import sklearn
import sklearn.datasets

from opt_utils import load_params_and_grads, initialize_parameters, forward_propagation, backward_propagation
from opt_utils import compute_cost, predict, predict_dec, plot_decision_boundary, load_dataset
from testCases import *

%matplotlib inline
plt.rcParams['figure.figsize'] = (7.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

1- 梯度下降法

梯度下降是每次处理完 m个样本后对参数进行一次更新操作,也叫做Batch Gradient Descent。

对于L层模型，梯度下降法对于各层参数的更新： l=1,...,L:

W[l]=W[l]−α dW[l](1)

b[l]=b[l]−α db[l](2)

L表示层数，α 是学习率。所有的这些参数都存在

parameters

字典中。注意，循环是从L1开始。

# GRADED FUNCTION: update_parameters_with_gd

def update_parameters_with_gd(parameters, grads, learning_rate):
"""
Update parameters using one step of gradient descent

Arguments:
parameters -- python dictionary containing your parameters to be updated:
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl
grads -- python dictionary containing your gradients to update each parameters:
grads['dW' + str(l)] = dWl
grads['db' + str(l)] = dbl
learning_rate -- the learning rate, scalar.

Returns:
parameters -- python dictionary containing your updated parameters
"""

L = len(parameters) // 2 # number of layers in the neural networks

# Update rule for each parameter
for l in range(L):
### START CODE HERE ### (approx. 2 lines)
parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - learning_rate * grads['dW' + str(l+1)]
parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - learning_rate * grads['db' + str(l+1)]
### END CODE HERE ###

return parameters

测试代码：

parameters, grads, learning_rate = update_parameters_with_gd_test_case()

parameters = update_parameters_with_gd(parameters, grads, learning_rate)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

测试代码运行如下：

W1 = [[ 1.63535156 -0.62320365 -0.53718766]
[-1.07799357  0.85639907 -2.29470142]]
b1 = [[ 1.74604067]
[-0.75184921]]
W2 = [[ 0.32171798 -0.25467393  1.46902454]
[-2.05617317 -0.31554548 -0.3756023 ]
[ 1.1404819  -1.09976462 -0.1612551 ]]
b2 = [[-0.88020257]
[ 0.02561572]
[ 0.57539477]]

梯度下降的一种变体是随机梯度下降法Stochastic Gradient Descent (SGD)。这等同于mini-batch中每个mini-batch只有一个样本的梯度下降法。此时，梯度下降的更新法则就变成，每个样本都要计算一次，而不是此前的对整个样本集计算一次。

两者代码如下：

(Batch) Gradient Descent:

X = data_input
Y = labels
parameters = initialize_parameters(layers_dims)
for i in range(0, num_iterations):
# Forward propagation
a, caches = forward_propagation(X, parameters)
# Compute cost.
cost = compute_cost(a, Y)
# Backward propagation.
grads = backward_propagation(a, caches, parameters)
# Update parameters.
parameters = update_parameters(parameters, grads)

Stochastic Gradient Descent:

X = data_input
Y = labels
parameters = initialize_parameters(layers_dims)
for i in range(0, num_iterations):
for j in range(0, m):
# Forward propagation
a, caches = forward_propagation(X[:,j], parameters)
# Compute cost
cost = compute_cost(a, Y[:,j])
# Backward propagation
grads = backward_propagation(a, caches, parameters)
# Update parameters.
parameters = update_parameters(parameters, grads)

在随机梯度下降中, 我们对于每个样本都更新梯度。当训练集很大时，这种方法可以明显提高运行速度，但是参数会沿着最小方向震荡，而不是平滑地收敛。



Figure 1 : SGD vs GD
“+” 表示代价最小值。SGD在收敛前出现很多震荡，但是由于每步都只有一个样本，所以每步都比梯度下降GD要来得快。 (vs. the whole batch for GD).

注意 SGD 共需要三个循环:

1. 最外层的迭代次数

2. m个训练样本

3. 每层参数的更新 ( (W[1],b[1]) to (W[L],b[L]))

在实际情况中，我们一般是折中，即所谓的 Mini-batch gradient descent。将整体的训练集分成子数据集，然后每个子训练集计算一次梯度下降。



Figure 2 : SGD vs Mini-Batch GD
“+” 表示最小代价值。

谨记:

gradient descent, mini-batch gradient descent 和 stochastic gradient descent之间的区别在于梯度更新所用到的样本数据量。

超参数学习率 α是需要调整获取到

mini-batch的尺寸也是调整获取到的，所以也是一个超参数。一般情况下这种方式比另外两者更好，特别是当训练集特别大的时候。

2 - Mini-Batch梯度下降

mini-batches用于训练集 (X, Y)，一般有以下两个步骤：

Shuffle(洗牌): 随机洗牌的方式将训练样本的数据顺序随机打散，注意：X和Y的随机要一致，否则Y值不能与X匹配，出现张冠李戴。随机化的洗牌操作能够将样本切分成不同的mini-batches。洗牌方式如下图所示：



Partition(分割): 将已经随机化的数据集(X, Y)分割成

mini_batch_size

(本文= 64)大小的子数据集。尾部的数据可能小于一个

mini_batch_size

，所以对于最后一个mini-batch要注意处理。



我们定义

random_mini_batches

函数来实现上述功能。在采用索引切片的时候，操作1st and 2nd mini-batches如下，其他依次。

first_mini_batch_X = shuffled_X[:, 0 : mini_batch_size]
second_mini_batch_X = shuffled_X[:, mini_batch_size : 2 * mini_batch_size]
...

当样本数无法被mini_batch_size整除的时候，最后一个mini-batch<

mini_batch_size=64

。 ⌊s⌋ 表示 s向下取整 (Python中实现：

math.floor(s)

)。所以 ⌊mmini_batch_size⌋ 个mini-batches中的样本数量是= 64，最后一个min-batch中样本数量= (m−mini_batch_size×⌊mmini_batch_size⌋)。

代码实现如下：

# GRADED FUNCTION: random_mini_batches

def random_mini_batches(X, Y, mini_batch_size = 64, seed = 0):
"""
Creates a list of random minibatches from (X, Y)

Arguments:
X -- input data, of shape (input size, number of examples)
Y -- true "label" vector (1 for blue dot / 0 for red dot), of shape (1, number of examples)
mini_batch_size -- size of the mini-batches, integer

Returns:
mini_batches -- list of synchronous (mini_batch_X, mini_batch_Y)
"""

np.random.seed(seed)            # To make your "random" minibatches the same as ours
m = X.shape[1]                  # number of training examples
#print("m=",m)
mini_batches = []

# Step 1: Shuffle (X, Y)
permutation = list(np.random.permutation(m))
shuffled_X = X[:, permutation]
shuffled_Y = Y[:, permutation].reshape((1,m))

# Step 2: Partition (shuffled_X, shuffled_Y). Minus the end case.
num_complete_minibatches = math.floor(m/mini_batch_size) # number of mini batches of size mini_batch_size in your partitionning
#print("num_complete_minibatches=",num_complete_minibatches)
for k in range(0, num_complete_minibatches):
### START CODE HERE ### (approx. 2 lines)
mini_batch_X = shuffled_X[:, k * mini_batch_size : (k+1) * mini_batch_size]
mini_batch_Y = shuffled_Y[:, k * mini_batch_size : (k+1) * mini_batch_size]
### END CODE HERE ###
mini_batch = (mini_batch_X, mini_batch_Y)
mini_batches.append(mini_batch)
#print(k)

# Handling the end case (last mini-batch < mini_batch_size)
# 尾数处理
#print(num_complete_minibatches * mini_batch_size)
if m % mini_batch_size != 0:
### START CODE HERE ### (approx. 2 lines)
mini_batch_X = shuffled_X[:, num_complete_minibatches * mini_batch_size : ]
mini_batch_Y = shuffled_Y[:, num_complete_minibatches * mini_batch_size : ]
### END CODE HERE ###
mini_batch = (mini_batch_X, mini_batch_Y)
mini_batches.append(mini_batch)

return mini_batches

代码测试如下：

X_assess, Y_assess, mini_batch_size = random_mini_batches_test_case()
mini_batches = random_mini_batches(X_assess, Y_assess, mini_batch_size)

print ("shape of the 1st mini_batch_X: " + str(mini_batches[0][0].shape))
print ("shape of the 2nd mini_batch_X: " + str(mini_batches[1][0].shape))
print ("shape of the 3rd mini_batch_X: " + str(mini_batches[2][0].shape))
print ("shape of the 1st mini_batch_Y: " + str(mini_batches[0][1].shape))
print ("shape of the 2nd mini_batch_Y: " + str(mini_batches[1][1].shape))
print ("shape of the 3rd mini_batch_Y: " + str(mini_batches[2][1].shape))
print ("mini batch sanity check: " + str(mini_batches[0][0][0][0:3]))

测试代码运行输出结果如下：

shape of the 1st mini_batch_X: (12288, 64)
shape of the 2nd mini_batch_X: (12288, 64)
shape of the 3rd mini_batch_X: (12288, 20)
shape of the 1st mini_batch_Y: (1, 64)
shape of the 2nd mini_batch_Y: (1, 64)
shape of the 3rd mini_batch_Y: (1, 20)
mini batch sanity check: [ 0.90085595 -0.7612069   0.2344157 ]

PS：一般mini-batch size的取值是2n，如 16, 32, 64, 128等

Momentum(动量梯度下降法)

由于min-batch梯度下降法是在看过训练集的一部分子数据集之后，就开始了参数的更新，那么就会在参数更新过程中出现偏差震荡。采用动量梯度下降法可以减缓震荡的出现。

momentum方式是在参数更新时候，参考历史的参数值，以平滑参数的更新。我们以变量 v存储梯度变化的历史方向。一般情况下，这个 v值是历史梯度值的指数加权平均结果。我们可以将 v视为球下坡滚动的”velocity”。



红色箭头表示在momentum作用下每个mini-batch梯度下降的方向，而蓝色则是没有momentum作用的mini-batch梯度下降方向。

velocity值初始化：

velocity, v,在Python中是一个字典，初始为0矩阵，其尺寸与

grads

一致:

for l=1,...,L:

v["dW" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["W" + str(l+1)])
v["db" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["b" + str(l+1)])

initialize_velocity

代码实现如下：

# GRADED FUNCTION: initialize_velocity

def initialize_velocity(parameters):
"""
Initializes the velocity as a python dictionary with:
- keys: "dW1", "db1", ..., "dWL", "dbL"
- values: numpy arrays of zeros of the same shape as the corresponding gradients/parameters.
Arguments:
parameters -- python dictionary containing your parameters.
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl

Returns:
v -- python dictionary containing the current velocity.
v['dW' + str(l)] = velocity of dWl
v['db' + str(l)] = velocity of dbl
"""

L = len(parameters) // 2 # number of layers in the neural networks
v = {}
#print(parameters['W1'].shape)
# Initialize velocity
for l in range(L):
### START CODE HERE ### (approx. 2 lines)
v["dW" + str(l+1)] = np.zeros((parameters['W' + str(l+1)].shape[0], parameters['W' + str(l+1)].shape[1]))
v["db" + str(l+1)] = np.zeros((parameters['b' + str(l+1)].shape[0], parameters['b' + str(l+1)].shape[1]))
### END CODE HERE ###

return v

初始化函数测试：

parameters = initialize_velocity_test_case()

v = initialize_velocity(parameters)
print("v[\"dW1\"] = " + str(v["dW1"]))
print("v[\"db1\"] = " + str(v["db1"]))
print("v[\"dW2\"] = " + str(v["dW2"]))
print("v[\"db2\"] = " + str(v["db2"]))

测试结果如下：

v["dW1"] = [[ 0.  0.  0.]
[ 0.  0.  0.]]
v["db1"] = [[ 0.]
[ 0.]]
v["dW2"] = [[ 0.  0.  0.]
[ 0.  0.  0.]
[ 0.  0.  0.]]
v["db2"] = [[ 0.]
[ 0.]
[ 0.]]

带momentum的参数更新:

更新规则如下：

for l=1,...,L:

{vdW[l]=βvdW[l]+(1−β)dW[l]W[l]=W[l]−αvdW[l](3)

{vdb[l]=βvdb[l]+(1−β)db[l]b[l]=b[l]−αvdb[l](4)

其中 L 表示层数, β 是momentum值，α 是学习率。 这些参数都存于

parameters

字典中。注意W[1] and b[1] 是从第1层开始的。

update_parameters_with_momentum

函数代码实现如下：

# GRADED FUNCTION: update_parameters_with_momentum

def update_parameters_with_momentum(parameters, grads, v, beta, learning_rate):
"""
Update parameters using Momentum

Arguments:
parameters -- python dictionary containing your parameters:
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl
grads -- python dictionary containing your gradients for each parameters:
grads['dW' + str(l)] = dWl
grads['db' + str(l)] = dbl
v -- python dictionary containing the current velocity:
v['dW' + str(l)] = ...
v['db' + str(l)] = ...
beta -- the momentum hyperparameter, scalar
learning_rate -- the learning rate, scalar

Returns:
parameters -- python dictionary containing your updated parameters
v -- python dictionary containing your updated velocities
"""

L = len(parameters) // 2 # number of layers in the neural networks

# Momentum update for each parameter
for l in range(L):

### START CODE HERE ### (approx. 4 lines)
# compute velocities
v["dW" + str(l+1)] = beta * v["dW" + str(l+1)] + (1-beta) * grads['dW' + str(l+1)]
v["db" + str(l+1)] = beta * v["db" + str(l+1)] + (1-beta) * grads['db' + str(l+1)]
# update parameters
parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - learning_rate * v["dW" + str(l+1)]
parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - learning_rate * v["db" + str(l+1)]
### END CODE HERE ###

return parameters, v

函数测试代码：

parameters, grads, v = update_parameters_with_momentum_test_case()

parameters, v = update_parameters_with_momentum(parameters, grads, v, beta = 0.9, learning_rate = 0.01)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))
print("v[\"dW1\"] = " + str(v["dW1"]))
print("v[\"db1\"] = " + str(v["db1"]))
print("v[\"dW2\"] = " + str(v["dW2"]))
print("v[\"db2\"] = " + str(v["db2"]))

测试结果如下：

W1 = [[ 1.62544598 -0.61290114 -0.52907334]
[-1.07347112  0.86450677 -2.30085497]]
b1 = [[ 1.74493465]
[-0.76027113]]
W2 = [[ 0.31930698 -0.24990073  1.4627996 ]
[-2.05974396 -0.32173003 -0.38320915]
[ 1.13444069 -1.0998786  -0.1713109 ]]
b2 = [[-0.87809283]
[ 0.04055394]
[ 0.58207317]]
v["dW1"] = [[-0.11006192  0.11447237  0.09015907]
[ 0.05024943  0.09008559 -0.06837279]]
v["db1"] = [[-0.01228902]
[-0.09357694]]
v["dW2"] = [[-0.02678881  0.05303555 -0.06916608]
[-0.03967535 -0.06871727 -0.08452056]
[-0.06712461 -0.00126646 -0.11173103]]
v["db2"] = [[ 0.02344157]
[ 0.16598022]
[ 0.07420442]]

注意 :

velocity初始化为zeros，所以算法需要迭代一定次数以建立起速度，实现每次迭代的bigger steps。

当β=0，则退化成标准的梯度下降法。

β值得选取：

β 越大，历史梯度值引入到当前值的权重越大，更新就会越平滑。但是如果 β太大，则会导致更新平滑过度。

β一般取值在0.8 到 0.999之间，常取β=0.9。

可以通过尝试几个β值，然后看哪个值在降低cost function J效果最好，来获取最优值。

4 - Adam算法

Adam算法应该是目前在神经网络领域最有效的优化算法了，该算法联合了RMSProp算法和Momentum算法。

Adam算法流程：

1.先计算历史梯度的指数加权平均值，存于变量 v ，vcorrected 表示校正后的值。

2. 计算历史梯度平方值的指数加权平均值，存于变量 s ，scorrected表示校正后的值。

3. 联合”1” and “2”更新参数

更新规则如下, for l=1,...,L:

⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪vdW[l]=β1vdW[l]+(1−β1)∂J∂W[l]vcorrecteddW[l]=vdW[l]1−(β1)tsdW[l]=β2sdW[l]+(1−β2)(∂J∂W[l])2scorrecteddW[l]=sdW[l]1−(β1)tW[l]=W[l]−αvcorrecteddW[l]scorrecteddW[l]√+ε

其中：

t 表示迭代的次数

L 表示层数

β1 and β2都是超参数，控制指数加权的权重

α 是学习率

ε 是一个很小的值，为了避免除0操作

变量 v,s 的初始化如下:

for l=1,...,L:

v["dW" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["W" + str(l+1)])
v["db" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["b" + str(l+1)])
s["dW" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["W" + str(l+1)])
s["db" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["b" + str(l+1)])

Adam的初始化代码：

# GRADED FUNCTION: initialize_adam

def initialize_adam(parameters) :
"""
Initializes v and s as two python dictionaries with:
- keys: "dW1", "db1", ..., "dWL", "dbL"
- values: numpy arrays of zeros of the same shape as the corresponding gradients/parameters.

Arguments:
parameters -- python dictionary containing your parameters.
parameters["W" + str(l)] = Wl
parameters["b" + str(l)] = bl

Returns:
v -- python dictionary that will contain the exponentially weighted average of the gradient.
v["dW" + str(l)] = ...
v["db" + str(l)] = ...
s -- python dictionary that will contain the exponentially weighted average of the squared gradient.
s["dW" + str(l)] = ...
s["db" + str(l)] = ...

"""

L = len(parameters) // 2 # number of layers in the neural networks
v = {}
s = {}

# Initialize v, s. Input: "parameters". Outputs: "v, s".
for l in range(L):
### START CODE HERE ### (approx. 4 lines)
v["dW" + str(l+1)] = np.zeros((parameters['W' + str(l+1)].shape[0], parameters['W' + str(l+1)].shape[1]))
v["db" + str(l+1)] = np.zeros((parameters['b' + str(l+1)].shape[0], parameters['b' + str(l+1)].shape[1]))
s["dW" + str(l+1)] = np.zeros((parameters['W' + str(l+1)].shape[0], parameters['W' + str(l+1)].shape[1]))
s["db" + str(l+1)] = np.zeros((parameters['b' + str(l+1)].shape[0], parameters['b' + str(l+1)].shape[1]))
### END CODE HERE ###

return v, s

代码测试：

parameters = initialize_adam_test_case()

v, s = initialize_adam(parameters)
print("v[\"dW1\"] = " + str(v["dW1"]))
print("v[\"db1\"] = " + str(v["db1"]))
print("v[\"dW2\"] = " + str(v["dW2"]))
print("v[\"db2\"] = " + str(v["db2"]))
print("s[\"dW1\"] = " + str(s["dW1"]))
print("s[\"db1\"] = " + str(s["db1"]))
print("s[\"dW2\"] = " + str(s["dW2"]))
print("s[\"db2\"] = " + str(s["db2"]))

测试代码输出：

v["dW1"] = [[ 0.  0.  0.]
[ 0.  0.  0.]]
v["db1"] = [[ 0.]
[ 0.]]
v["dW2"] = [[ 0.  0.  0.]
[ 0.  0.  0.]
[ 0.  0.  0.]]
v["db2"] = [[ 0.]
[ 0.]
[ 0.]]
s["dW1"] = [[ 0.  0.  0.]
[ 0.  0.  0.]]
s["db1"] = [[ 0.]
[ 0.]]
s["dW2"] = [[ 0.  0.  0.]
[ 0.  0.  0.]
[ 0.  0.  0.]]
s["db2"] = [[ 0.]
[ 0.]
[ 0.]]

Adam算法实现：

# GRADED FUNCTION: update_parameters_with_adam

def update_parameters_with_adam(parameters, grads, v, s, t, learning_rate = 0.01,
beta1 = 0.9, beta2 = 0.999,  epsilon = 1e-8):
"""
Update parameters using Adam

Arguments:
parameters -- python dictionary containing your parameters:
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl
grads -- python dictionary containing your gradients for each parameters:
grads['dW' + str(l)] = dWl
grads['db' + str(l)] = dbl
v -- Adam variable, moving average of the first gradient, python dictionary
s -- Adam variable, moving average of the squared gradient, python dictionary
learning_rate -- the learning rate, scalar.
beta1 -- Exponential decay hyperparameter for the first moment estimates
beta2 -- Exponential decay hyperparameter for the second moment estimates
epsilon -- hyperparameter preventing division by zero in Adam updates

Returns:
parameters -- python dictionary containing your updated parameters
v -- Adam variable, moving average of the first gradient, python dictionary
s -- Adam variable, moving average of the squared gradient, python dictionary
"""

L = len(parameters) // 2                 # number of layers in the neural networks
v_corrected = {}                         # Initializing first moment estimate, python dictionary
s_corrected = {}                         # Initializing second moment estimate, python dictionary

# Perform Adam update on all parameters
for l in range(L):
# Moving average of the gradients. Inputs: "v, grads, beta1". Output: "v".
### START CODE HERE ### (approx. 2 lines)
v["dW" + str(l+1)] = beta1 * v["dW" + str(l+1)] + (1-beta1) * grads['dW' + str(l+1)]
v["db" + str(l+1)] = beta1 * v["db" + str(l+1)] + (1-beta1) * grads['db' + str(l+1)]
### END CODE HERE ###

# Compute bias-corrected first moment estimate. Inputs: "v, beta1, t". Output: "v_corrected".
### START CODE HERE ### (approx. 2 lines)
v_corrected["dW" + str(l+1)] = v["dW" + str(l+1)]/(1-np.power(beta1,t))
v_corrected["db" + str(l+1)] = v["db" + str(l+1)]/(1-np.power(beta1,t))
### END CODE HERE ###

# Moving average of the squared gradients. Inputs: "s, grads, beta2". Output: "s".
### START CODE HERE ### (approx. 2 lines)
s["dW" + str(l+1)] = beta2 * s["dW" + str(l+1)] + (1-beta2) * np.power(grads['dW' + str(l+1)],2)
s["db" + str(l+1)] = beta2 * s["db" + str(l+1)] + (1-beta2) * np.power(grads['db' + str(l+1)],2)
### END CODE HERE ###

# Compute bias-corrected second raw moment estimate. Inputs: "s, beta2, t". Output: "s_corrected".
### START CODE HERE ### (approx. 2 lines)
s_corrected["dW" + str(l+1)] = s["dW" + str(l+1)]/(1-np.power(beta2,t))
s_corrected["db" + str(l+1)] = s["db" + str(l+1)]/(1-np.power(beta2,t))
### END CODE HERE ###

# Update parameters. Inputs: "parameters, learning_rate, v_corrected, s_corrected, epsilon". Output: "parameters".
### START CODE HERE ### (approx. 2 lines)
parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - learning_rate * v_corrected["dW" + str(l+1)]/(np.sqrt(s_corrected["dW" + str(l+1)])+epsilon)
parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - learning_rate * v_corrected["db" + str(l+1)]/(np.sqrt(s_corrected["db" + str(l+1)])+epsilon)
### END CODE HERE ###

return parameters, v, s

Adam算法测试：

parameters, grads, v, s = update_parameters_with_adam_test_case()
parameters, v, s  = update_parameters_with_adam(parameters, grads, v, s, t = 2)

print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))
print("v[\"dW1\"] = " + str(v["dW1"]))
print("v[\"db1\"] = " + str(v["db1"]))
print("v[\"dW2\"] = " + str(v["dW2"]))
print("v[\"db2\"] = " + str(v["db2"]))
print("s[\"dW1\"] = " + str(s["dW1"]))
print("s[\"db1\"] = " + str(s["db1"]))
print("s[\"dW2\"] = " + str(s["dW2"]))
print("s[\"db2\"] = " + str(s["db2"]))

测试代码运行如下：

W1 = [[ 1.63178673 -0.61919778 -0.53561312]
[-1.08040999  0.85796626 -2.29409733]]
b1 = [[ 1.75225313]
[-0.75376553]]
W2 = [[ 0.32648046 -0.25681174  1.46954931]
[-2.05269934 -0.31497584 -0.37661299]
[ 1.14121081 -1.09244991 -0.16498684]]
b2 = [[-0.88529979]
[ 0.03477238]
[ 0.57537385]]
v["dW1"] = [[-0.11006192  0.11447237  0.09015907]
[ 0.05024943  0.09008559 -0.06837279]]
v["db1"] = [[-0.01228902]
[-0.09357694]]
v["dW2"] = [[-0.02678881  0.05303555 -0.06916608]
[-0.03967535 -0.06871727 -0.08452056]
[-0.06712461 -0.00126646 -0.11173103]]
v["db2"] = [[ 0.02344157]
[ 0.16598022]
[ 0.07420442]]
s["dW1"] = [[ 0.00121136  0.00131039  0.00081287]
[ 0.0002525   0.00081154  0.00046748]]
s["db1"] = [[  1.51020075e-05]
[  8.75664434e-04]]
s["dW2"] = [[  7.17640232e-05   2.81276921e-04   4.78394595e-04]
[  1.57413361e-04   4.72206320e-04   7.14372576e-04]
[  4.50571368e-04   1.60392066e-07   1.24838242e-03]]
s["db2"] = [[  5.49507194e-05]
[  2.75494327e-03]
[  5.50629536e-04]]

5 - Model with different optimization algorithms

对于上述几种优化算法的测试，在这里我们采用 “moons” 数据集。

数据加载：

train_X, train_Y = load_dataset()

对于一个3层的神经网络，我们将采用下述三种优化算法来训练：

Mini-batch Gradient Descent: 通过调用:

update_parameters_with_gd()

来实现

Mini-batch Momentum: 通过调用:

initialize_velocity()

和

update_parameters_with_momentum()

来实现。

Mini-batch Adam: 通过调用:

initialize_adam()

和

update_parameters_with_adam()

来实现

模型代码：

def model(X, Y, layers_dims, optimizer, learning_rate = 0.0007, mini_batch_size = 64, beta = 0.9,
beta1 = 0.9, beta2 = 0.999,  epsilon = 1e-8, num_epochs = 10000, print_cost = True):
"""
3-layer neural network model which can be run in different optimizer modes.

Arguments:
X -- input data, of shape (2, number of examples)
Y -- true "label" vector (1 for blue dot / 0 for red dot), of shape (1, number of examples)
layers_dims -- python list, containing the size of each layer
learning_rate -- the learning rate, scalar.
mini_batch_size -- the size of a mini batch
beta -- Momentum hyperparameter
beta1 -- Exponential decay hyperparameter for the past gradients estimates
beta2 -- Exponential decay hyperparameter for the past squared gradients estimates
epsilon -- hyperparameter preventing division by zero in Adam updates
num_epochs -- number of epochs
print_cost -- True to print the cost every 1000 epochs

Returns:
parameters -- python dictionary containing your updated parameters
"""

L = len(layers_dims)             # number of layers in the neural networks
costs = []                       # to keep track of the cost
t = 0                            # initializing the counter required for Adam update
seed = 10                        # For grading purposes, so that your "random" minibatches are the same as ours

# Initialize parameters
parameters = initialize_parameters(layers_dims)

# Initialize the optimizer
if optimizer == "gd":
pass # no initialization required for gradient descent
elif optimizer == "momentum":
v = initialize_velocity(parameters)
elif optimizer == "adam":
v, s = initialize_adam(parameters)

# Optimization loop
for i in range(num_epochs):

# Define the random minibatches. We increment the seed to reshuffle differently the dataset after each epoch
seed = seed + 1
minibatches = random_mini_batches(X, Y, mini_batch_size, seed)

for minibatch in minibatches:

# Select a minibatch
(minibatch_X, minibatch_Y) = minibatch

# Forward propagation
a3, caches = forward_propagation(minibatch_X, parameters)

# Compute cost
cost = compute_cost(a3, minibatch_Y)

# Backward propagation
grads = backward_propagation(minibatch_X, minibatch_Y, caches)

# Update parameters
if optimizer == "gd":
parameters = update_parameters_with_gd(parameters, grads, learning_rate)
elif optimizer == "momentum":
parameters, v = update_parameters_with_momentum(parameters, grads, v, beta, learning_rate)
elif optimizer == "adam":
t = t + 1 # Adam counter
parameters, v, s = update_parameters_with_adam(parameters, grads, v, s,
t, learning_rate, beta1, beta2,  epsilon)

# Print the cost every 1000 epoch
if print_cost and i % 1000 == 0:
print ("Cost after epoch %i: %f" %(i, cost))
if print_cost and i % 100 == 0:
costs.append(cost)

# plot the cost
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('epochs (per 100)')
plt.title("Learning rate = " + str(learning_rate))
plt.show()

return parameters

5-1 Mini-batch Gradient descent

代码如下：

# train 3-layer model
layers_dims = [train_X.shape[0], 5, 2, 1]
parameters = model(train_X, train_Y, layers_dims, optimizer = "gd")

# Predict
predictions = predict(train_X, train_Y, parameters)

# Plot decision boundary
plt.title("Model with Gradient Descent optimization")
axes = plt.gca()
axes.set_xlim([-1.5,2.5])
axes.set_ylim([-1,1.5])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

运行结果如下：

Cost after epoch 0: 0.690736
Cost after epoch 1000: 0.685273
Cost after epoch 2000: 0.647072
Cost after epoch 3000: 0.619525
Cost after epoch 4000: 0.576584
Cost after epoch 5000: 0.607243
Cost after epoch 6000: 0.529403
Cost after epoch 7000: 0.460768
Cost after epoch 8000: 0.465586
Cost after epoch 9000: 0.464518

Accuracy: 0.796666666667

c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.maximum.reduce will be axis=0, not the current None, to match np.maximum.reduce. Explicitly pass 0 or None to silence this warning.
return self.reduce(a)
c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.minimum.reduce will be axis=0, not the current None, to match np.minimum.reduce. Explicitly pass 0 or None to silence this warning.
return self.reduce(a)

5-2 Mini-batch gradient descent with momentum

代码如下：

# train 3-layer model
layers_dims = [train_X.shape[0], 5, 2, 1]
parameters = model(train_X, train_Y, layers_dims, beta = 0.9, optimizer = "momentum")

# Predict
predictions = predict(train_X, train_Y, parameters)

# Plot decision boundary
plt.title("Model with Momentum optimization")
axes = plt.gca()
axes.set_xlim([-1.5,2.5])
axes.set_ylim([-1,1.5])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

运行结果如下：

Cost after epoch 0: 0.690741
Cost after epoch 1000: 0.685341
Cost after epoch 2000: 0.647145
Cost after epoch 3000: 0.619594
Cost after epoch 4000: 0.576665
Cost after epoch 5000: 0.607324
Cost after epoch 6000: 0.529476
Cost after epoch 7000: 0.460936
Cost after epoch 8000: 0.465780
Cost after epoch 9000: 0.464740

Accuracy: 0.796666666667

c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.maximum.reduce will be axis=0, not the current None, to match np.maximum.reduce. Explicitly pass 0 or None to silence this warning.
return self.reduce(a)
c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.minimum.reduce will be axis=0, not the current None, to match np.minimum.reduce. Explicitly pass 0 or None to silence this warning.
return self.reduce(a)

5-3 Mini-batch with Adam mode

代码如下：

# train 3-layer model
layers_dims = [train_X.shape[0], 5, 2, 1]
parameters = model(train_X, train_Y, layers_dims, optimizer = "adam")

# Predict
predictions = predict(train_X, train_Y, parameters)

# Plot decision boundary
plt.title("Model with Adam optimization")
axes = plt.gca()
axes.set_xlim([-1.5,2.5])
axes.set_ylim([-1,1.5])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

运行结果如下：

Cost after epoch 0: 0.690552
Cost after epoch 1000: 0.185567
Cost after epoch 2000: 0.150852
Cost after epoch 3000: 0.074454
Cost after epoch 4000: 0.125936
Cost after epoch 5000: 0.104235
Cost after epoch 6000: 0.100552
Cost after epoch 7000: 0.031601
Cost after epoch 8000: 0.111709
Cost after epoch 9000: 0.197648

Accuracy: 0.94

c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.maximum.reduce will be axis=0, not the current None, to match np.maximum.reduce. Explicitly pass 0 or None to silence this warning.
return self.reduce(a)
c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.minimum.reduce will be axis=0, not the current None, to match np.minimum.reduce. Explicitly pass 0 or None to silence this warning.
return self.reduce(a)

5-4 总结：

optimization method	accuracy	cost shape
Gradient descent	79.7%	oscillations
Momentum	79.7%	oscillations
Adam	94%	smoother

Momentum一般都是有助于提升速度，但是当学习率较小，数据集相对简单的时候，其性能的优越性没有太明显。我们在优化算法中看到的那些较大的震荡是由于一些minibatches 相对更加复杂所造成的。

从运行结果可以看出，Adam算法比mini-batch gradient descent 和 Momentum都要显得优越。对于model如果在简单数据集上，迭代次数更多的话，这三种优化算法都会产生较好的结果，但是我们也可以看出，Adam算法收敛得更快些。

Adam算法的优点:

内存要求低 (尽管比gradient descent 和 gradient descent with momentum要高些)

一般微调超参数就可以获得较好的结果(除了α)