9.深度学习练习：Optimization Methods

本文节选自吴恩达老师《深度学习专项课程》编程作业，在此表示感谢。

课程链接：https://www.deeplearning.ai/deep-learning-specialization/

1 - Gradient Descent

2 - Mini-Batch Gradient descent

3 - Momentum

4 - Adam

5 - Model with different optimization algorithms

5.1 - Mini-batch Gradient descent

5.2 - Mini-batch gradient descent with momentum

5.3 - Mini-batch with Adam mode

Until now, you've always used Gradient Descent to update the parameters and minimize the cost. In this notebook, you will learn more advanced optimization methods that can speed up learning and perhaps even get you to a better final value for the cost function. Having a good optimization algorithm can be the difference between waiting days vs. just a few hours to get a good result.

Gradient descent goes "downhill" on a cost function ?

. Think of it as trying to do this:

At each step of the training, you update your parameters following a certain direction to try to get to the lowest possible point.

Notations: As usual, $\frac{\partial J}{\partial a } = da$ for any variable a.

To get started, run the following code to import the libraries you will need.

import numpy as np
import matplotlib.pyplot as plt
import scipy.io
import math
import sklearn
import sklearn.datasetsfrom 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 - Gradient Descent

A simple optimization method in machine learning is gradient descent (GD). When you take gradient steps with respect to all ?

examples on each step, it is also called Batch Gradient Descent.

Warm-up exercise: Implement the gradient descent update rule. The gradient descent rule is, for $l = 1, ..., L$

$W^{[l]} = W^{[l]} - \alpha \text{ } dW^{[l]} \tag{1}$

$b^{[l]} = b^{[l]} - \alpha \text{ } db^{[l]}$

where L is the number of layers and ? is the learning rate. All parameters should be stored in the parameters dictionary. Note that the iterator l starts at 0 in the for loop while the first parameters are $W^{[l]}$ and $b^{[l]}$ . You need to shift l to l+1 when coding.

def update_parameters_with_gd(parameters, grads, learning_rate):"""Update parameters using one step of gradient descentArguments:parameters -- python dictionary containing your parameters to be updated:parameters['W' + str(l)] = Wlparameters['b' + str(l)] = blgrads -- python dictionary containing your gradients to update each parameters:grads['dW' + str(l)] = dWlgrads['db' + str(l)] = dbllearning_rate -- the learning rate, scalar.Returns:parameters -- python dictionary containing your updated parameters """L = len(parameters) // 2 # number of layers in the neural networksfor l in range(L):parameters['W' + str(l+1)] -= learning_rate * grads['dW' + str(l+1)]parameters['b' + str(l+1)] -= learning_rate * grads['db' + str(l+1)]return parameters

A variant of this is Stochastic Gradient Descent (SGD), which is equivalent to mini-batch gradient descent where each mini-batch has just 1 example. The update rule that you have just implemented does not change. What changes is that you would be computing gradients on just one training example at a time, rather than on the whole training set. The code examples below illustrate the difference between stochastic gradient descent and (batch) gradient descent.

(Batch) Gradient Descent:

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

In Stochastic Gradient Descent, you use only 1 training example before updating the gradients. When the training set is large, SGD can be faster. But the parameters will "oscillate" toward the minimum rather than converge smoothly. Here is an illustration of this:

Note also that implementing SGD requires 3 for-loops in total:

Over the number of iterations
Over the ? training examples
Over the layers (to update all parameters, from $(W^{[1]},b^{[1]})$ to $(W^{[L]},b^{[L]})$ 。

In practice, you'll often get faster results if you do not use neither the whole training set, nor only one training example, to perform each update. Mini-batch gradient descent uses an intermediate number of examples for each step. With mini-batch gradient descent, you loop over the mini-batches instead of looping over individual training examples.

**What you should remember**: - The difference between gradient descent, mini-batch gradient descent and stochastic gradient descent is the number of examples you use to perform one update step. - You have to tune a learning rate hyperparameter ?. - With a well-turned mini-batch size, usually it outperforms either gradient descent or stochastic gradient descent (particularly when the training set is large).

2 - Mini-Batch Gradient descent

Let's learn how to build mini-batches from the training set (X, Y).

There are two steps:

Shuffle: Create a shuffled version of the training set (X, Y) as shown below. Each column of X and Y represents a training example. Note that the random shuffling is done synchronously between X and Y. Such that after the shuffling the $i^{th}$ column of X is the example corresponding to the $i^{th}$ label in Y. The shuffling step ensures that examples will be split randomly into different mini-batches.

Partition: Partition the shuffled (X, Y) into mini-batches of size mini_batch_size (here 64). Note that the number of training examples is not always divisible by mini_batch_size. The last mini batch might be smaller, but you don't need to worry about this. When the final mini-batch is smaller than the full mini_batch_size, it will look like this:

Exercise: Implement random_mini_batches. We coded the shuffling part for you. To help you with the partitioning step, we give you the following code that selects the indexes for the 1?? and 2??

mini-batches:

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

Note that the last mini-batch might end up smaller than mini_batch_size=64. Let ⌊?⌋ represents ? rounded down to the nearest integer (this is math.floor(s) in Python). If the total number of examples is not a multiple of mini_batch_size=64 then there will be $\lfloor \frac{m}{mini\_batch\_size}\rfloor$ mini-batches with a full 64 examples, and the number of examples in the final mini-batch will be

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, integerReturns:mini_batches -- list of synchronous (mini_batch_X, mini_batch_Y)"""np.random.seed(seed)            # To make your "random" minibatches the same as oursm = X.shape[1]                  # number of training examplesmini_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 partitionningfor k in range(0, num_complete_minibatches):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]mini_batch = (mini_batch_X, mini_batch_Y)mini_batches.append(mini_batch)# Handling the end case (last mini-batch < mini_batch_size)if m % mini_batch_size != 0:mini_batch_X = shuffled_X[:, num_complete_minibatches * mini_batch_size: ]mini_batch_Y = shuffled_Y[:, num_complete_minibatches * mini_batch_size: ]mini_batch = (mini_batch_X, mini_batch_Y)mini_batches.append(mini_batch)return mini_batches

**What you should remember**: - Shuffling and Partitioning are the two steps required to build mini-batches - Powers of two are often chosen to be the mini-batch size, e.g., 16, 32, 64, 128.

3 - Momentum

Because mini-batch gradient descent makes a parameter update after seeing just a subset of examples, the direction of the update has some variance, and so the path taken by mini-batch gradient descent will "oscillate" toward convergence. Using momentum can reduce these oscillations.

Momentum takes into account the past gradients to smooth out the update. We will store the 'direction' of the previous gradients in the variable ?. Formally, this will be the exponentially weighted average of the gradient on previous steps. You can also think of ? as the "velocity" of a ball rolling downhill, building up speed (and momentum) according to the direction of the gradient/slope of the hill.

Exercise: Initialize the velocity. The velocity, ?, is a python dictionary that needs to be initialized with arrays of zeros. Its keys are the same as those in the grads dictionary, that is: 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)])

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)] = Wlparameters['b' + str(l)] = blReturns:v -- python dictionary containing the current velocity.(和dW，db维度一致)v['dW' + str(l)] = velocity of dWlv['db' + str(l)] = velocity of dbl"""L = len(parameters) // 2 # number of layers in the neural networksv = {}# Initialize velocityfor l in range(L):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]))return v

Exercise: Now, implement the parameters update with momentum. The momentum update rule is, for $l = 1, ..., L$

$\begin{cases} v_{dW^{[l]}} = \beta v_{dW^{[l]}} + (1 - \beta) dW^{[l]} \\ W^{[l]} = W^{[l]} - \alpha v_{dW^{[l]}} \end{cases}\tag{3}$$ $$\begin{cases} v_{db^{[l]}} = \beta v_{db^{[l]}} + (1 - \beta) db^{[l]} \\ b^{[l]} = b^{[l]} - \alpha v_{db^{[l]}} \end{cases}\tag{4}$

def update_parameters_with_momentum(parameters, grads, v, beta, learning_rate):"""Update parameters using MomentumArguments:parameters -- python dictionary containing your parameters:parameters['W' + str(l)] = Wlparameters['b' + str(l)] = blgrads -- python dictionary containing your gradients for each parameters:grads['dW' + str(l)] = dWlgrads['db' + str(l)] = dblv -- python dictionary containing the current velocity:v['dW' + str(l)] = ...v['db' + str(l)] = ...beta -- the momentum hyperparameter, scalarlearning_rate -- the learning rate, scalarReturns: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 parameterfor l in range(L):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)]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)]return parameters, v

Note that:

The velocity is initialized with zeros. So the algorithm will take a few iterations to "build up" velocity and start to take bigger steps.
If ?=0, then this just becomes standard gradient descent without momentum.

How do you choose ? ?

The larger the momentum ? is, the smoother the update because the more we take the past gradients into account. But if ? is too big, it could also smooth out the updates too much.

Common values for ? range from 0.8 to 0.999. If you don't feel inclined to tune this, ?=0.9 is often a reasonable default.
Tuning the optimal ? for your model might need trying several values to see what works best in term of reducing the value of the cost function ?.

**What you should remember**: - Momentum takes past gradients into account to smooth out the steps of gradient descent. It can be applied with batch gradient descent, mini-batch gradient descent or stochastic gradient descent. - You have to tune a momentum hyperparameter ? and a learning rate ?.

4 - Adam

Adam is one of the most effective optimization algorithms for training neural networks. It combines ideas from RMSProp (described in lecture) and Momentum.

How does Adam work?

It calculates an exponentially weighted average of past gradients, and stores it in variables ? (before bias correction) and $v^{corrected}$ (with bias correction)..
It calculates an exponentially weighted average of the squares of the past gradients, and stores it in variables ? (before bias correction) and $s^{corrected}$ (with bias correction).
It updates parameters in a direction based on combining information from "1" and "2".

The update rule is, for $$l = 1, ..., L$:$

$\begin{cases} v_{dW^{[l]}} = \beta_1 v_{dW^{[l]}} + (1 - \beta_1) \frac{\partial \mathcal{J} }{ \partial W^{[l]} } \\ v^{corrected}_{dW^{[l]}} = \frac{v_{dW^{[l]}}}{1 - (\beta_1)^t} \\ s_{dW^{[l]}} = \beta_2 s_{dW^{[l]}} + (1 - \beta_2) (\frac{\partial \mathcal{J} }{\partial W^{[l]} })^2 \\ s^{corrected}_{dW^{[l]}} = \frac{s_{dW^{[l]}}}{1 - (\beta_1)^t} \\ W^{[l]} = W^{[l]} - \alpha \frac{v^{corrected}_{dW^{[l]}}}{\sqrt{s^{corrected}_{dW^{[l]}}} + \varepsilon} \end{cases}$

where:

t counts the number of steps taken of Adam
L is the number of layers
?1 and ?2 are hyperparameters that control the two exponentially weighted averages.
? is the learning rate
? is a very small number to avoid dividing by zero

As usual, we will store all parameters in the parameters dictionary.

Exercise: Initialize the Adam variables ?,? which keep track of the past information.

Instruction: The variables ?,? are python dictionaries that need to be initialized with arrays of zeros. Their keys are the same as for grads, that is: for ?=1,...,?:

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)] = Wlparameters["b" + str(l)] = blReturns: 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 networksv = {}s = {}# Initialize v, s. Input: "parameters". Outputs: "v, s".for l in range(L):]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]))return v, s

Exercise: Now, implement the parameters update with Adam. Recall the general update rule is, for ?=1,...,?:

$\begin{cases} v_{W^{[l]}} = \beta_1 v_{W^{[l]}} + (1 - \beta_1) \frac{\partial J }{ \partial W^{[l]} } \\ v^{corrected}_{W^{[l]}} = \frac{v_{W^{[l]}}}{1 - (\beta_1)^t} \\ s_{W^{[l]}} = \beta_2 s_{W^{[l]}} + (1 - \beta_2) (\frac{\partial J }{\partial W^{[l]} })^2 \\ s^{corrected}_{W^{[l]}} = \frac{s_{W^{[l]}}}{1 - (\beta_2)^t} \\ W^{[l]} = W^{[l]} - \alpha \frac{v^{corrected}_{W^{[l]}}}{\sqrt{s^{corrected}_{W^{[l]}}}+\varepsilon} \end{cases}$

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 AdamArguments:parameters -- python dictionary containing your parameters:parameters['W' + str(l)] = Wlparameters['b' + str(l)] = blgrads -- python dictionary containing your gradients for each parameters:grads['dW' + str(l)] = dWlgrads['db' + str(l)] = dblv -- Adam variable, moving average of the first gradient, python dictionarys -- Adam variable, moving average of the squared gradient, python dictionarylearning_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 updatesReturns:parameters -- python dictionary containing your updated parameters v -- Adam variable, moving average of the first gradient, python dictionarys -- Adam variable, moving average of the squared gradient, python dictionary"""L = len(parameters) // 2                 # number of layers in the neural networksv_corrected = {}                         # Initializing first moment estimate, python dictionarys_corrected = {}                         # Initializing second moment estimate, python dictionary# Perform Adam update on all parametersfor l in range(L):# Moving average of the gradients. Inputs: "v, grads, beta1". Output: "v".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)]# Compute bias-corrected first moment estimate. Inputs: "v, beta1, t". Output: "v_corrected".v_corrected["dW" + str(l+1)] = v["dW" + str(l+1)] / (1 - beta1 ** t)v_corrected["db" + str(l+1)] = v["db" + str(l+1)] / (1 - beta1 ** t)# Moving average of the squared gradients. Inputs: "s, grads, beta2". Output: "s".s["dW" + str(l+1)] = beta2 * s["dW" + str(l+1)] + (1 - beta2) * (grads['dW' + str(l+1)] ** 2)s["db" + str(l+1)] = beta2 * s["db" + str(l+1)] + (1 - beta2) * (grads['db' + str(l+1)] ** 2)# Compute bias-corrected second raw moment estimate. Inputs: "s, beta2, t". Output: "s_corrected".s_corrected["dW" + str(l+1)] = s["dW" + str(l+1)] / (1 - beta2 ** t)s_corrected["db" + str(l+1)] = s["db" + str(l+1)] / (1 - beta2 ** t)# Update parameters. Inputs: "parameters, learning_rate, v_corrected, s_corrected, epsilon". Output: "parameters".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)return parameters, v, s

5 - Model with different optimization algorithms

Lets use the following "moons" dataset to test the different optimization methods. (The dataset is named "moons" because the data from each of the two classes looks a bit like a crescent-shaped moon.)

train_X, train_Y = load_dataset()

We have already implemented a 3-layer neural network. You will train it with:

Mini-batch Gradient Descent: it will call your function:
update_parameters_with_gd()
Mini-batch Momentum: it will call your functions:
initialize_velocity() and update_parameters_with_momentum()
Mini-batch Adam: it will call your functions:
initialize_adam() and update_parameters_with_adam()

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 layerlearning_rate -- the learning rate, scalar.mini_batch_size -- the size of a mini batchbeta -- Momentum hyperparameterbeta1 -- 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 updatesnum_epochs -- number of epochsprint_cost -- True to print the cost every 1000 epochsReturns:parameters -- python dictionary containing your updated parameters """L = len(layers_dims)             # number of layers in the neural networkscosts = []                       # to keep track of the costt = 0                            # initializing the counter required for Adam updateseed = 10                        # For grading purposes, so that your "random" minibatches are the same as ours# Initialize parametersparameters = initialize_parameters(layers_dims)# Initialize the optimizerif optimizer == "gd":pass # no initialization required for gradient descentelif optimizer == "momentum":v = initialize_velocity(parameters)elif optimizer == "adam":v, s = initialize_adam(parameters)# Optimization loopfor i in range(num_epochs):# Define the random minibatches. We increment the seed to reshuffle differently the dataset after each epochseed = seed + 1minibatches = random_mini_batches(X, Y, mini_batch_size, seed)for minibatch in minibatches:# Select a minibatch(minibatch_X, minibatch_Y) = minibatch# Forward propagationa3, caches = forward_propagation(minibatch_X, parameters)# Compute costcost = compute_cost(a3, minibatch_Y)# Backward propagationgrads = backward_propagation(minibatch_X, minibatch_Y, caches)# Update parametersif 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 counterparameters, v, s = update_parameters_with_adam(parameters, grads, v, s,t, learning_rate, beta1, beta2,  epsilon)# Print the cost every 1000 epochif 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 costplt.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

# 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)

5.2 - Mini-batch gradient descent with momentum

Run the following code to see how the model does with momentum. Because this example is relatively simple, the gains from using momemtum are small; but for more complex problems you might see bigger gains.

# 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)

5.3 - Mini-batch with Adam mode

# 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)

Momentum usually helps, but given the small learning rate and the simplistic dataset, its impact is almost negligeable. Also, the huge oscillations you see in the cost come from the fact that some minibatches are more difficult thans others for the optimization algorithm.

Adam on the other hand, clearly outperforms mini-batch gradient descent and Momentum. If you run the model for more epochs on this simple dataset, all three methods will lead to very good results. However, you've seen that Adam converges a lot faster.

Some advantages of Adam include:

Relatively low memory requirements (though higher than gradient descent and gradient descent with momentum)
Usually works well even with little tuning of hyperparameters (except ?)