DeepLearning.ai深度学习课程笔记
  • Introduction
  • 第一门课 神经网络和深度学习(Neural-Networks-and-Deep-Learning)
    • 第一周：深度学习引言(Introduction to Deep Learning)
      • 1.1 神经网络的监督学习(Supervised Learning with Neural Networks)
      • 1.2 为什么神经网络会流行？(Why is Deep Learning taking off?)
    • 第二周：神经网络的编程基础(Basics of Neural Network programming)
      • 2.1 二分类(Binary Classification)
      • 2.2 逻辑回归(Logistic Regression)
      • 2.3 逻辑回归的代价函数（Logistic Regression Cost Function）
      • 2.4 逻辑回归的梯度下降（Logistic Regression Gradient Descent）
      • 2.5 梯度下降的例子(Gradient Descent on m Examples)
      • 2.6 向量化 logistic 回归的梯度输出（Vectorizing Logistic Regression’s Gradient Output）
      • 2.7 （选修）logistic 损失函数的解释（Explanation of logistic regression cost function ）
      • Logistic Regression with a Neural Network mindset 代码
      • lr_utils.py
    • 第三周：浅层神经网络(Shallow neural networks)
      • 3.1 神经网络概述（Neural Network Overview）
      • 3.2 神经网络的表示（Neural Network Representation ）
      • 3.3 计算一个神经网络的输出（Computing a Neural Network's output ）
      • 3.4 多样本向量化（Vectorizing across multiple examples ）
      • 3.5 激活函数（Activation functions）
      • 3.6 为什么需要（ 非线性激活函数？（why need a nonlinear activation function?）
      • 3.7 激活函数的导数（Derivatives of activation functions ）
      • 3.8 神经网络的梯度下降（Gradient descent for neural networks）
      • 3.9 （选修）直观理解反向传播（Backpropagation intuition ）
      • 3.10 随机初始化（Random+Initialization）
      • Planar data classification with one hidden layer
      • planar_utils.py
      • testCases.py
    • 第四周：深层神经网络(Deep Neural Networks)
      • 4.1 深层神经网络（Deep L-layer neural network）
      • 4.2 前向传播和反向传播（Forward and backward propagation）
      • 4.3 深层网络中的前向传播（Forward propagation in a Deep Network ）
      • 4.4 为什么使用深层表示？（Why deep representations?）
      • 4.5 搭建神经网络块（Building blocks of deep neural networks）
      • 4.6 参数 VS 超参数（Parameters vs Hyperparameters）
      • Building your Deep Neural Network Step by Step
      • dnn_utils.py
      • testCases.py
      • Deep Neural Network Application
      • dnn_app_utils.py
  • 第二门课 改善深层神经网络：超参数调试、 正 则 化 以 及 优 化 (Improving Deep Neural Networks:Hyperparameter tuning, Regulariza
    • 第二门课 改善深层神经网络：超参数调试、正则化以及优化(Improving Deep Neural Networks:Hyperparameter tuning, Regularization and
      • 第一周：深度学习的实用层面(Practical aspects of Deep Learning)
        • 1.1 训练，验证，测试集（Train / Dev / Test sets）
        • 1.2 偏差，方差（Bias /Variance）
        • 1.3 机器学习基础（Basic Recipe for Machine Learning）
        • 1.4 正则化（Regularization）
        • 1.5 为什么正则化有利于预防过拟合呢？（Why regularization reduces overfitting?）
        • 1.6 dropout 正则化（Dropout Regularization）
        • 1.7 理解 dropout（Understanding Dropout）
        • 1.8 其他正则化方法（Other regularization methods）
        • 1.9 归一化输入（Normalizing inputs）
        • 1.10 梯度消失/梯度爆炸（Vanishing / Exploding gradients）
        • 1.11 神经网络的权重初始化（Weight Initialization for Deep Networks）
        • 1.12 梯度的数值逼近（Numerical approximation of gradients）
        • 1.13 梯度检验（Gradient checking）
        • 1.14 梯度检验应用的注意事项（Gradient Checking Implementation Notes）
        • Initialization
        • Gradient Checking
        • Regularization
        • reg_utils.py
        • testCases.py
      • 第二周：优化算法 (Optimization algorithms)
        • 2.1 Mini-batch 梯度下降（Mini-batch gradient descent）
        • 2.2 理解 mini-batch 梯度下降法（Understanding mini-batch gradient descent）
        • 2.3 指数加权平均数（Exponentially weighted averages）
        • 2.4 理解指数加权平均数（Understanding exponentially weighted averages ）
        • 2.5 指 数 加 权 平 均 的 偏 差 修 正 （ Bias correction in exponentially weighted averages ）
        • 2.6 动量梯度下降法（Gradient descent with Momentum ）
        • 2.7 RMSprop( root mean square prop)
        • 2.8 Adam 优化算法(Adam optimization algorithm)
        • 2.9 学习率衰减(Learning rate decay)
        • 2.10 局部最优的问题(The problem of local optima)
        • Optimization
        • opt_utils.py
        • testCases.py
      • 第 三 周 超 参 数 调 试 、 Batch 正 则 化 和 程 序 框 架 （Hyperparameter tuning）
        • 3.1 调试处理（Tuning process）
        • 3.2 为超参数选择合适的范围（Using an appropriate scale to pick hyperparameters）
        • 3.3 超参数训练的实践： Pandas VS Caviar（Hyperparameters tuning in practice: Pandas vs. Caviar）
        • 3.4 归一化网络的激活函数（ Normalizing activations in a network）
        • 3.5 将 Batch Norm 拟合进神经网络（Fitting Batch Norm into a neural network）
        • 3.6 Batch Norm 为什么奏效？（Why does Batch Norm work?）
        • 3.7 测试时的 Batch Norm（Batch Norm at test time）
        • 3.8 Softmax 回归（Softmax regression）
        • 3.9 训练一个 Softmax 分类器（Training a Softmax classifier）
        • tensorflow tutorial
        • improv_utils.py
        • tf_utils.py
  • 第三门课 结构化机器学习项目（Structuring Machine Learning Projects）
    • 第三门课 结构化机器学习项目（Structuring Machine Learning Projects）
      • 第一周 机器学习（ML）策略（1）（ML strategy（1））
        • 1.1 为什么是 ML 策略？（Why ML Strategy?）
        • 1.2 正交化（Orthogonalization）
        • 1.3 单一数字评估指标（Single number evaluation metric）
        • 1.4 满足和优化指标（Satisficing and optimizing metrics）
        • 1.5 训练/开发/测试集划分（Train/dev/test distributions）
        • 1.6 开发集和测试集的大小（Size of dev and test sets）
        • 1.7 什么时候该改变开发/测试集和指标？（When to change dev/test sets and metrics）
        • 1.8 为什么是人的表现？（ Why human-level performance?）
        • 1.9 可避免偏差（Avoidable bias）
        • 1.10 理解人的表现（Understanding human-level performance）
        • 1.11 超过人的表现（Surpassing human- level performance）
        • 1.12 改善你的模型的表现（Improving your model performance）
      • 第二周：机器学习策略（2）(ML Strategy (2))
        • 2.1 进行误差分析（Carrying out error analysis）
        • 2.2 清楚标注错误的数据（Cleaning up Incorrectly labeled data）
        • 2.3 快速搭建你的第一个系统，并进行迭代（Build your first system quickly, then iterate）
        • 2.4 在不同的划分上进行训练并测试（Training and testing on different distributions）
        • 2.5 不匹配数据划分的偏差和方差（Bias and Variance with mismatched data distributions）
        • 2.6 定位数据不匹配（Addressing data mismatch）
        • 2.7 迁移学习（Transfer learning）
        • 2.8 多任务学习（Multi-task learning）
        • 2.9 什么是端到端的深度学习？（What is end-to-end deep learning?）
        • 2.10 是否要使用端到端的深度学习？（Whether to use end-to-end learning?）
  • 第四门课 卷积神经网络（Convolutional Neural Networks）
    • 第四门课 卷积神经网络（Convolutional Neural Networks）
      • 第一周 卷积神经网络（Foundations of Convolutional Neural Networks）
        • 1.1 计算机视觉（Computer vision）
        • 1.2 边缘检测示例（Edge detection example）
        • 1.3 更多边缘检测内容（More edge detection）
        • 1.4 Padding
        • 1.5 卷积步长（Strided convolutions）
        • 1.6 三维卷积（Convolutions over volumes）
        • 1.7 单层卷积网络（One layer of a convolutional network）
        • 1.8 简单卷积网络示例（A simple convolution network example）
        • 1.9 池化层（Pooling layers）
        • 1.10 卷积神经网络示例（Convolutional neural network example）
        • 1.11 为什么使用卷积？（Why convolutions?）
        • Convolution model Step by Step
        • Convolutional Neural Networks: Application
        • cnn_utils
      • 第二周 深度卷积网络：实例探究（Deep convolutional models: case studies）
        • 2.1 经典网络（Classic networks）
        • 2.2 残差网络（Residual Networks (ResNets)）
        • 2.3 残差网络为什么有用？（Why ResNets work?）
        • 2.4 网络中的网络以及 1×1 卷积（Network in Network and 1×1 convolutions）
        • 2.5 谷歌 Inception 网络简介（Inception network motivation）
        • 2.6 Inception 网络（Inception network）
        • 2.7 迁移学习（Transfer Learning）
        • 2.8 数据扩充（Data augmentation）
        • 2.9 计算机视觉现状（The state of computer vision）
        • Residual Networks
        • Keras tutorial - the Happy House
        • kt_utils.py
      • 第三周 目标检测（Object detection）
        • 3.1 目标定位（Object localization）
        • 3.2 特征点检测（Landmark detection）
        • 3.3 目标检测（Object detection）
        • 3.4 卷积的滑动窗口实现（Convolutional implementation of sliding windows）
        • 3.5 Bounding Box预测（Bounding box predictions）
        • 3.6 交并比（Intersection over union）
        • 3.7 非极大值抑制（Non-max suppression）
        • 3.8 Anchor Boxes
        • 3.9 YOLO 算法（Putting it together: YOLO algorithm）
        • 3.10 候选区域（选修）（Region proposals (Optional)）
        • Autonomous driving application - Car detection
        • yolo_utils.py
      • 第四周 特殊应用：人脸识别和神经风格转换（Special applications: Face recognition &Neural style transfer）
        • 4.1 什么是人脸识别？（What is face recognition?）
        • 4.2 One-Shot学习（One-shot learning）
        • 4.3 Siamese 网络（Siamese network）
        • 4.4 Triplet 损失（Triplet 损失）
        • 4.5 面部验证与二分类（Face verification and binary classification）
        • 4.6 什么是深度卷积网络？（What are deep ConvNets learning?）
        • 4.7 代价函数（Cost function）
        • 4.8 内容代价函数（Content cost function）
        • 4.9 风格代价函数（Style cost function）
        • 4.10 一维到三维推广（1D and 3D generalizations of models）
        • Art Generation with Neural Style Transfer
        • nst_utils.py
        • Face Recognition for the Happy House
        • fr_utils.py
        • inception_blocks.py
  • 第五门课 序列模型(Sequence Models)
    • 第五门课 序列模型(Sequence Models)
      • 第一周 循环序列模型（Recurrent Neural Networks）
        • 1.1 为什么选择序列模型？（Why Sequence Models?）
        • 1.2 数学符号（Notation）
        • 1.3 循环神经网络模型（Recurrent Neural Network Model）
        • 1.4 通过时间的反向传播（Backpropagation through time）
        • 1.5 不同类型的循环神经网络（Different types of RNNs）
        • 1.6 语言模型和序列生成（Language model and sequence generation）
        • 1.7 对新序列采样（Sampling novel sequences）
        • 1.8 循环神经网络的梯度消失（Vanishing gradients with RNNs）
        • 1.9 GRU单元（Gated Recurrent Unit（GRU））
        • 1.10 长短期记忆（LSTM（long short term memory）unit）
        • 1.11 双向循环神经网络（Bidirectional RNN）
        • 1.12 深层循环神经网络（Deep RNNs）
        • Building your Recurrent Neural Network
        • rnn_utils.py
        • Dinosaurus Island -- Character level language model final
        • utils.py
        • shakespeare_utils.py
        • Improvise a Jazz Solo with an LSTM Network
      • 第二周 自然语言处理与词嵌入（Natural Language Processing and Word Embeddings）
        • 2.1 词汇表征（Word Representation）
        • 2.2 使用词嵌入（Using Word Embeddings）
        • 2.3 词嵌入的特性（Properties of Word Embeddings）
        • 2.4 嵌入矩阵（Embedding Matrix）
        • 2.5 学习词嵌入（Learning Word Embeddings）
        • 2.6 Word2Vec
        • 2.7 负采样（Negative Sampling）
        • 2.8 GloVe 词向量（GloVe Word Vectors）
        • 2.9 情感分类（Sentiment Classification）
        • 2.10 词嵌入除偏（Debiasing Word Embeddings）
        • Operations on word vectors
        • w2v_utils.py
        • Emojify
        • emo_utils.py
      • 第三周 序列模型和注意力机制（Sequence models & Attention mechanism）
        • 3.1 基础模型（Basic Models）
        • 3.2 选择最可能的句子（Picking the most likely sentence）
        • 3.3 集束搜索（Beam Search）
        • 3.4 改进集束搜索（Refinements to Beam Search）
        • 3.5 集束搜索的误差分析（Error analysis in beam search）
        • 3.6 Bleu 得分（选修）（Bleu Score (optional)）
        • 3.7 注意力模型直观理解（Attention Model Intuition）
        • 3.8注意力模型（Attention Model）
        • 3.9语音识别（Speech recognition）
        • 3.10触发字检测（Trigger Word Detection）
        • Neural machine translation with attention
        • nmt_utils.py
        • Trigger word detection
        • td_utils.py
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On this page
  • 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
  • 5.4 - Summary

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  1. 第二门课 改善深层神经网络：超参数调试、 正 则 化 以 及 优 化 (Improving Deep Neural Networks:Hyperparameter tuning, Regulariza
  2. 第二门课 改善深层神经网络：超参数调试、正则化以及优化(Improving Deep Neural Networks:Hyperparameter tuning, Regularization and
  3. 第二周：优化算法 (Optimization algorithms)

Optimization

Previous2.10 局部最优的问题(The problem of local optima)Nextopt_utils.py

Last updated 6 years ago

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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 JJJ. Think of it as trying to do this:

Figure 1 : Minimizing the cost is like finding the lowest point in a hilly landscape 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, ∂J∂a=\frac{\partial J}{\partial a } =∂a∂J​= 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.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 - Gradient Descent

# 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
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"]))
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]]

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 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)

Figure 1 : SGD vs GD "+" denotes a minimum of the cost. SGD leads to many oscillations to reach convergence. But each step is a lot faster to compute for SGD than for GD, as it uses only one training example (vs. the whole batch for GD).

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.

Figure 2 SGD vs Mini-Batch GD "+" denotes a minimum of the cost. Using mini-batches in your optimization algorithm often leads to faster optimization.

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.

  • 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:

  • 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: 

first_mini_batch_X = shuffled_X[:, 0 : mini_batch_size]
second_mini_batch_X = shuffled_X[:, mini_batch_size : 2 * mini_batch_size]
...
# 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
    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
    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].reshape((1, mini_batch_size))
        ### END CODE HERE ###
        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:
        ### START CODE HERE ### (approx. 2 lines)
        mini_batch_X = shuffled_X[:, math.floor(m/mini_batch_size)*mini_batch_size: m ]
        mini_batch_Y = shuffled_Y[:, math.floor(m/mini_batch_size)*mini_batch_size : m ].reshape((1, m - math.floor(m/mini_batch_size)*mini_batch_size))
        ### END CODE HERE ###
        mini_batch = (mini_batch_X, mini_batch_Y)
        mini_batches.append(mini_batch)

    return mini_batches
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]))
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 ]

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.

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)])

Note that the iterator l starts at 0 in the for loop while the first parameters are v["dW1"] and v["db1"] (that's a "one" on the superscript). This is why we are shifting l to l+1 in the for loop.

# 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 = {}

    # 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)
        v["db" + str(l+1)] = np.zeros(parameters['b' + str(l + 1)].shape)
        ### END CODE HERE ###

    return v
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"]))
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.]]
# 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
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"]))
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]]

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.

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.

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.

where:

  • t counts the number of steps taken of Adam

  • L is the number of layers

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

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)])
# 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)
        v["db" + str(l+1)] = np.zeros(parameters['b' + str(l+1)].shape)
        s["dW" + str(l+1)] = np.zeros(parameters['W' + str(l+1)].shape)
        s["db" + str(l+1)] = np.zeros(parameters['b' + str(l+1)].shape)
    ### END CODE HERE ###

    return v, s
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"]))
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.]]
# 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))

        # 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)*(grads['db' + str(l+1)]*grads['db' + str(l+1)])
        ### 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
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"]))
    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]]

You now have three working optimization algorithms (mini-batch gradient descent, Momentum, Adam). Let's implement a model with each of these optimizers and observe the difference.

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 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

You will now run this 3 layer neural network with each of the 3 optimization methods.

5.1 - Mini-batch Gradient descent

Run the following code to see how the model does with 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)
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

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)
    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

5.3 - Mini-batch with Adam mode

Run the following code to see how the model does with Adam.

# 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)
    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

5.4 - Summary

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)

References:

A simple optimization method in machine learning is gradient descent (GD). When you take gradient steps with respect to all mmm 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,...,Ll = 1, ..., Ll=1,...,L:

W[l]=W[l]−α dW[l]    (1)W^{[l]} = W^{[l]} - \alpha \text{ } dW^{[l]} \ \ \ \ (1)W[l]=W[l]−α dW[l]    (1)
b[l]=b[l]−α db[l]    (2)b^{[l]} = b^{[l]} - \alpha \text{ } db^{[l]} \ \ \ \ (2)b[l]=b[l]−α db[l]    (2)

where L is the number of layers and α\alphaα 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[1]W^{[1]}W[1] and b[1]b^{[1]}b[1]. You need to shift l to l+1 when coding.

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: 1. Over the number of iterations 2. Over the mmm training examples 3. Over the layers (to update all parameters, from (W[1],b[1])(W^{[1]},b^{[1]})(W[1],b[1]) to (W[L],b[L])(W^{[L]},b^{[L]})(W[L],b[L]))

You have to tune a learning rate hyperparameter α\alphaα.

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 ithi^{th}ith column of X is the example corresponding to the ithi^{th}ith label in Y. The shuffling step ensures that examples will be split randomly into different mini-batches.

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 1st1^{st}1st and 2nd2^{nd}2nd mini-batches:

Note that the last mini-batch might end up smaller than mini_batch_size=64. Let ⌊s⌋\lfloor s \rfloor⌊s⌋ represents $s$ 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 ⌊mmini_batch_size⌋\lfloor \frac{m}{mini\_batch\_size}\rfloor⌊mini_batch_sizem​⌋ mini-batches with a full 64 examples, and the number of examples in the final mini-batch will be (m−mini_batch_size×⌊mmini_batch_size⌋m-mini_\_batch_\_size \times \lfloor \frac{m}{mini\_batch\_size}\rfloorm−mini_​batch_​size×⌊mini_batch_sizem​⌋).

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

Figure 3: The red arrows shows the direction taken by one step of mini-batch gradient descent with momentum. The blue points show the direction of the gradient (with respect to the current mini-batch) on each step. Rather than just following the gradient, we let the gradient influence vvv and then take a step in the direction of vvv.

Exercise: Initialize the velocity. The velocity, vvv, 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,...,Ll =1,...,Ll=1,...,L:

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

{vdW[l]=βvdW[l]+(1−β)dW[l]W[l]=W[l]−αvdW[l]     (3)\begin{cases} v_{dW^{[l]}} = \beta v_{dW^{[l]}} + (1 - \beta) dW^{[l]} \\ W^{[l]} = W^{[l]} - \alpha v_{dW^{[l]}} \end{cases}\ \ \ \ \ (3){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)\begin{cases} v_{db^{[l]}} = \beta v_{db^{[l]}} + (1 - \beta) db^{[l]} \\ b^{[l]} = b^{[l]} - \alpha v_{db^{[l]}} \end{cases}\ \ \ \ \ (4){vdb[l]​=βvdb[l]​+(1−β)db[l]b[l]=b[l]−αvdb[l]​​     (4)

where L is the number of layers, β\betaβ is the momentum and $\alpha$ 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[1]W^{[1]}W[1] and b[1]b^{[1]}b[1] (that's a "one" on the superscript). So you will need to shift l to l+1 when coding.

If β=0\beta = 0β=0, then this just becomes standard gradient descent without momentum.

How do you choose β\betaβ?

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

Common values for β\betaβ range from 0.8 to 0.999. If you don't feel inclined to tune this, β=0.9\beta = 0.9β=0.9 is often a reasonable default.

Tuning the optimal β\betaβ for your model might need trying several values to see what works best in term of reducing the value of the cost function JJJ.

You have to tune a momentum hyperparameter β\betaβ and a learning rate α\alphaα.

How does Adam work? 1. It calculates an exponentially weighted average of past gradients, and stores it in variables vvv (before bias correction) and vcorrectedv^{corrected}vcorrected (with bias correction). 2. It calculates an exponentially weighted average of the squares of the past gradients, and stores it in variables sss (before bias correction) and scorrecteds^{corrected}scorrected (with bias correction). 3. It updates parameters in a direction based on combining information from "1" and "2".

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

{vdW[l]=β1vdW[l]+(1−β1)∂J∂W[l]vdW[l]corrected=vdW[l]1−(β1)tsdW[l]=β2sdW[l]+(1−β2)(∂J∂W[l])2sdW[l]corrected=sdW[l]1−(β1)tW[l]=W[l]−αvdW[l]correctedsdW[l]corrected+ε\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}⎩⎨⎧​vdW[l]​=β1​vdW[l]​+(1−β1​)∂W[l]∂J​vdW[l]corrected​=1−(β1​)tvdW[l]​​sdW[l]​=β2​sdW[l]​+(1−β2​)(∂W[l]∂J​)2sdW[l]corrected​=1−(β1​)tsdW[l]​​W[l]=W[l]−αsdW[l]corrected​​+εvdW[l]corrected​​​

β1\beta_1β1​ and β2\beta_2β2​ are hyperparameters that control the two exponentially weighted averages.

α\alphaα is the learning rate

ε\varepsilonε is a very small number to avoid dividing by zero

Exercise: Initialize the Adam variables v,sv, sv,s which keep track of the past information.

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

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

{vW[l]=β1vW[l]+(1−β1)∂J∂W[l]vW[l]corrected=vW[l]1−(β1)tsW[l]=β2sW[l]+(1−β2)(∂J∂W[l])2sW[l]corrected=sW[l]1−(β2)tW[l]=W[l]−αvW[l]correctedsW[l]corrected+ε\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}⎩⎨⎧​vW[l]​=β1​vW[l]​+(1−β1​)∂W[l]∂J​vW[l]corrected​=1−(β1​)tvW[l]​​sW[l]​=β2​sW[l]​+(1−β2​)(∂W[l]∂J​)2sW[l]corrected​=1−(β2​)tsW[l]​​W[l]=W[l]−αsW[l]corrected​​+εvW[l]corrected​​​

Note that the iterator l starts at 0 in the for loop while the first parameters are W[1]W^{[1]}W[1] and b[1]b^{[1]}b[1]. You need to shift l to l+1 when coding.

Usually works well even with little tuning of hyperparameters (except α\alphaα)

Adam paper:

https://arxiv.org/pdf/1412.6980.pdf
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