**2 years ago**, some information may be outdated!

This is my note for the course (Improving Deep Neural Networks: Hyperparameter tuning, Regularization and Optimization). The codes in this note are rewritten to be more clear and concise.

👉 **Course 1** -- Neural Networks and Deep Learning.

👉 **Course 2** -- Improving Deep Neural Networks: Hyperparameter tuning, Regularization and Optimization.

👉 **Course 3** -- Structuring Machine Learning Projects.

👉 **Course 4** -- Convolutional Neural Networks.

👉 **Course 5** -- Sequence Models.

This course will teach you the "magic" of getting deep learning to work well. Rather than the deep learning process being a black box, you will understand what drives performance, and be able to more systematically get good results. You will also learn TensorFlow.

## Initialization step

`layers_dims`

contains the size of each layer from $0$ to $L$.

### zero initialization

`parameters['W'+str(l)] = np.zeros((layers_dims[l], layers_dims[l-1]))`

parameters['b'+str(l)] = np.zeros((layers_dims[l], 1))

- The performance is really
**bad**, and the**cost**does**not**really**decrease**. - initializing all the weights to zero ⇒ failing to break symmetry ⇒ every neuron in each layer will learn the same thing ⇒ $n^{[l]}=1$ for every layer ⇒ no more powerful than a linear classifier such as logistic regression.

### Random initialization

To break symmetry, lets intialize the weights randomly.

`parameters['W'+str(l)] = np.random.randn(layers_dims[l], layers_dims[l-1]) * 10 # <- LARGE (just an example of SHOULDN'T)`

parameters['b'+str(l)] = np.zeros((layers_dims[l], 1))

- High initial weights ⇒ The cost starts very high (near 0 or 1 or infinity).
- Poor initialization ⇒ vanishing/exploding gradients ⇒ slows down the optimization algorithm.
- If you train this network longer ⇒ better results, BUT initializing with overly large random numbers ⇒ slows down the optimization.

### He initialization

Multiply randomly initial $W$ with $\sqrt{\frac{2}{n^{[l-1]}}}$. It's similar to **Xavier initialization** in which multipler factor is $\sqrt{\frac{1}{n^{[l-1]}}}$

`parameters['W' + str(l)] = np.random.randn(layers_dims[l], layers_dims[l-1]) * np.sqrt(2./layers_dims[l-1])`

parameters['b' + str(l)] = np.zeros((layers_dims[l], 1))

## Regularization step

To reduce the **overfitting problem**.

### L2 regularization

`L2_regularization_cost = 0`

for l in range(1, L+1):

L2_regularization_cost += 1/m * lambd/2 * (np.sum(np.square(W[l]))

The standard way. Modify cost function from,

$J = -\frac{1}{m} \sum\limits_{i = 1}^{m} \large{(}\small y^{(i)}\log\left(a^{[L](i)}\right) + (1-y^{(i)})\log\left(1- a^{[L](i)}\right) \large{)}$ to

$J_{\text{regularized}} = \small \underbrace{-\frac{1}{m} \sum\limits_{i = 1}^{m} \large{(}\small y^{(i)}\log\left(a^{[L](i)}\right) + (1-y^{(i)})\log\left(1- a^{[L](i)}\right) \large{)} }_\text{cross-entropy cost} + \underbrace{\frac{1}{m} \frac{\lambda}{2} \sum\limits_l\sum\limits_k\sum\limits_j W_{k,j}^{[l]2} }_\text{L2 regularization cost}$ The value of $\lambda$ is a hyperparameter that you can tune using a dev set.

L2 regularization makes your decision boundary smoother. If $\lambda$ is too large, it is also possible to "oversmooth", resulting in a model with high bias.

### Dropout

`# [Forward] An example at layer 3`

D3 = np.random.rand(A3.shape(0), A3.shape(1)) < keep_drop

A3 *= D3

A3 /= keep_drop

# [Backprop]

dA3 *= D3

dA3 /= keep_drop

- Dropout is a widely used regularization technique that is specific to deep learning.
- Randomly shuts down some neurons in each iteration.
- When you shut some neurons down, you actually modify your model. The idea behind drop-out is that at each iteration, you train a different model that uses only a subset of your neurons.
- With dropout, your neurons thus become less sensitive to the activation of one other specific neuron, because that other neuron might be shut down at any time.
- Don't apply dropout to the input layer or output layer.
- Use dropout during training, not during test time.
- Apply dropout both during forward and backward propagation.

## Gradient checking

To answer "

*Give me a proof that your backpropagation is actually working!"*We are confident on computing $J$ but $\frac{\partial J}{\partial\theta}$.

Use $J$ to compute an approximation of $\frac{\partial J}{\partial\theta}$ and compare with $\frac{\partial J}{\partial\theta}$.

$\frac{\partial J}{\partial \theta} = \lim_{\varepsilon \to 0} \frac{J(\theta + \varepsilon) - J(\theta - \varepsilon)}{2 \varepsilon}$

## Optimization algorithms

**Intuition**:

**Gradient Descent**: go down the hill.**Momentum**/**RMSprop**/**Adam**: which direction?

### Mini-batch gradient descent

**Problem**: NN works great on big data but many data leads to slow the training ⇒ We need to optimize!**Solution**: Divide into smaller "mini-batches" (for example, from 5M to 5K of 1K each).

*Different between mini-batch and normal batch on the cost function. It's oscillated for mini-batch because the cost may be large for this mini-batch but small for the others. Image from the course.*

#### Notations

- $X^{(i)}$: $i$th training example.
- $z^{[l]}$: $z$ value in $l$th layer.
- $X^{\{t\}}, Y^{\{t\}}$: index of different mini-batches.

#### Algorithm

`X = data_input`

Y = labels

parameters = initialize_parameters(layers_dims)

for i in range(0, num_iterations): # loop through epoches: to get the convergence

for t in range(0, num_batches): # loop through the batches

# Forward propagation

a, caches = forward_propagation(X[:,t], parameters)

# Compute cost

cost += compute_cost(a, Y[:,t])

# Backward propagation

grads = backward_propagation(a, caches, parameters)

# Update parameters.

parameters = update_parameters(parameters, grads)

#### How to build mini-batches?

We need 2 steps:

**Shuffle**: shuffle columns (training examples) correspondingly between $X$ and $Y$. The shuffling step ensures that examples will be split randomly into different mini-batches.**Partition**: choose a batch size and take mini-batches. Note that, the last batch may be smaller than the others.

#### Type of mini-batch

There are 3 types based on the size of batches:

**Batch Gradient Descent**($n_t = m$) : entire training examples, i.e. $(X^{\{1\}}, Y^{\{1\}}) = (X,Y)$.**Stochastic Gradient Descent**($n_t = 1$) : every training example is it own a mini-batch ($m$ mini batches).- $1<n_t<m$.

*Different between 3 types of mini-batch. Image from the course.*

**Guideline**:

- If small training set ($m \le 2000$): using batch gradient descent.
- Typical mini-batch sizes: $64, 128, 256, 512, \ldots$
- Make sure mibi-batch size in CPU/GPU memory!

### Exponentially weighted averages

It's faster than Gradient Descent!

**Example**(temperature in London):$\theta_t$: the temperature on day $t$.

$v_t$: the average temp of each day. It's called exponential average over $\frac{1}{1-\beta}$ days temperature.

$v_t = \beta v_{t-1} + (1-\beta)\theta_t$ E.g. $\beta=0.9 \Rightarrow v_t \simeq 10$ days temperature; $\beta=0.98 \Rightarrow v_t \simeq 50$ days temperature.

$\beta$ larger ⇒ smoother average line because we consider more days. However, curve is now shifted further to the right.

*Exponentially weighted average curves: red line ($\beta=0.9$), green line ($\beta=0.98$). Image from the course.*When $\beta$ is so large ⇒ $v_t$ adapts slowly to the changes of temperature (more latency).

Why we call "exponentially"?

$\begin{aligned}v_{100} &= 0.9\times v_{99} + 0.1\times \theta_{100}\\&= 0.1\times \theta_{100} + 0.1\times 0.99\times\theta_{99} + 0.1\times 0.99^2 \times\theta_{99} + \ldots\end{aligned}$

#### Bias correction

**Problem**: the value of $v_t$ at the beginning of exp ave curves may be lower than what we expect. For example, with $v_0=0$, we have $v_1 = 0.02\theta_1$ instead of $v_1 = v_0 + 0.02\theta_1$.**Solution**: Instead of using $v_t$, we take$\dfrac{v_t}{1-\beta_t}$ When $t$ is large ⇒ $\beta^t \simeq 0 \Rightarrow \dfrac{v_t}{1-\beta_t} \simeq v_t$

*Bias correction for the green line, it's effective at the beginning of the line, with bigger $t$, green and violet are overlapped. Image from the course.***In practice**, we don't really see people bothering with bias correction!

### Gradient Descent with Momentum

It's faster than Gradient Descent!

**Why**: when we use mini-batch, there are oscillation, momentum helps use reduce this.**One sentence**: compute the exponential weighted average of your gradient ⇒ use that gradient to update your weights instead.**Idea**: Momentum takes into account the past gradients to smooth out the update. We will store the 'direction' of the previous gradients in the variable $v$ . Formally, this will be the exponentially weighted average of the gradient on previous steps.**Intuition**: 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.- $dW, db$ like "acceleration".
- $VdW, Vdb$ like "velocity".
- $\beta$ likes "friction".

*We want slower learning in vertial direction and faster in horizontal direction. Image from the course.***Algorithm**: on iteration $t$:- Compute $dW, db$ on current mini-batch.
- $VdW = \beta VdW + (1-\beta)dW$.
- $Vdb = \beta Vdb + (1-\beta)db$.
- $W:=W-\alpha VdW$.
- $b:=b-\alpha Vdb$.

**Implementation**:- Try to tune between $[0.8; 0.999]$, commonly use $\beta=0.9$.
- Don't bother bias correction, NO NEED.
- Don't need $(1-\beta)$ in the formulas but Andrew prefer to keep it!
- Bigger $\beta$, smaller in vertical direction.

### RMSprop

- It's "Root Mean Square propagation".
**Algorithm**: on iteration $t$,- Compute $dW, db$ on current element-wise mini-batch.
- $SdW = \beta SdW + (1-\beta)dW^2$.
- $Sdb = \beta Sdb + (1-\beta)db^2$.
- $W:=W -\alpha \frac{dW}{\sqrt{SdW}+\epsilon}$.
- $b:=b-\alpha \frac{db}{\sqrt{SdW} + \epsilon}$.

- We choose $\epsilon=10^{-8}$ if $\sqrt{SdW}$ is too small, otherwise $\epsilon=0$.
**In practice**: $dW, db$ are very high dimensional vectors.

### Adam Optimization

- It's "Adaptive Moment Estimation".
- One of the most effective optimization algorithm for training NN. It's commonly used and proven to be very effective for many different NN of a very wide variety of architectures.
- Adam = Momentum + RMSprop.
**Implementation**: on iteration $t$,- Compute $dW, db$ using current mini-batch.
- (Monentum) $VdW = \beta_1 VdW + (1-\beta_1)dW$; $Vdb = \beta_1 Vdb+(1-\beta_1)db$.
- (RMSprop) $SdW = \beta_2 SdW + (1-\beta_2)dW^2$; $Sdb = \beta_2Sdb +(1-\beta_2)db^2$.
- $V_{dW}^{\text{corrected}} = \dfrac{VdW}{1-\beta_1^t}$; $V_{db}^{\text{corrected}} = \dfrac{Vdb}{1-\beta_1^t}$.
- $S_{dW}^{\text{corrected}} = \dfrac{SdW}{1-\beta_2^t}$; $S_{db}^{\text{corrected}} = \dfrac{Sdb}{1-\beta_2^t}$.
- $W:=W-\alpha \dfrac{V_{dW}^{\text{corrected}}}{\sqrt{S_{dW}^{\text{corrected}}} + \epsilon}$; $b:=b-\alpha \dfrac{V_{db}^{\text{corrected}}}{\sqrt{S_{db}^{\text{corrected}}} + \epsilon}$.

**Initialization**of the velocity is zero, i.e. $VdW=SdW=Vdb=Sdb=0$.- If $\beta=0$, it's standard gradient descent without momentum.
**Hyperparameter choices**:- $\alpha$ = needs to be tuned, very important!
- $\beta_1 = 0.9$ ($dW$), first moment.
- $\beta_2 = 0.999$ ($dW^2$), second mement.
- $\epsilon = 10^{-8}$.

## Learning rate decay

**Idea**: slowly reduce learning rate over time, it's learning rate decay.**Why**? Below figure showes that, we need slower rate $\alpha$ (smaller step) at the area near the center.*Example of learning rate decay. Image from the course.*Recall that, 1 epoch = 1 pass through data.

Learning rate decay can be chosen 1 of below,

### Problem of local optima

*Local optima problem: local & right optima (left) and saddle point (right). Image from the course.*

- In high dimension, you likely see saddle points than local optimum.
**Problem of plateau**: a region where derivative is close to zero for a long time.- Unlikely get stuck in a bad local optimal.
- Plateau can make learning slow: use Momentum, RMSprop, Adam.

### Batch GD makes learning too long?

- Try better random initialization for weights.
- Try mini-batch GD.
- Try using Adam
- Try tuning learning rate $\alpha$.

## Hyperparameter tuning

### Tuning process

There are many hyperparameters but some are more important than others!

Learning rate $\alpha$ (most important), #hiddien units, $\beta$, mini-batch size (2nd important), #layers, learning decay,...

Don't use grid, use random!

*Tuning process. Don't use grid (left), use random (right). Image from the course.***Coarse to fine**: find an area containing effective values ⇒ zoom in and take more points in that area,*Coarse to fine: first try on a big square, then focus on the smaller one (blue). Image from the course.*Choose randomly but NOT mean uniform scale! We can choose uniformly on #hidden units, #layers, but not for the others (e.g. $\alpha$).

For $\alpha$, for example, we need to divide into equal "large" spaces and then use uniform.

*Appropriate scale for hyperparameters. Image from the course.*Hyperparameters for

**exponentially weighted averages**:We cannot try with values between $[0.9, 0.999]$ because,

- $\beta: 0.9000 \to 0.9005$ : no much changes,
- $\beta: 0.999 \to 0.995$ : huge impact!

Consider $1-\beta \in [10^{-1}, 10^{-3}]$ instead!

$\begin{aligned}r &\in [-3, -1] \\1-\beta = 10^r &\Leftrightarrow \beta = 1-10^r\end{aligned}$

### In practice: Panda vs Caviar

- How to organize your hyperparameter search?
**Advice**: Re-testing/Re-evaluating your hyperparameters at least once every several months.- 2 ways:
**Babysitting one model**(Panda): when we have huge data but weak CPU/GPU $\Rightarrow$ try very small number of models at a time. Check the performance step by step (cost function reduces...)- In some domains like advertising, computer vision apps,...
- We call "panda" because panda has very few number of babies at a time (and in their life) $\Rightarrow$ try to keep them alike once at a time.

**Training many models in parallel**(Caviar): when we don't work on huge data + strong CPU/GPU. $\Rightarrow$ Try many models in parallel and choose the best performance!- We call "Caviar" because of intuition.

## Batch Normalization

- Make NN much more robust to the choice of hyperparameters. $\Leftarrow$ doesn't work for all NN but if it does, make training faster!
- One of the most important ideas in the rise of Deep Learning.
- Like we wanna normalize input to speed up learning, in this case, we wanna normalize $Z$ (in the hidden layers)

Given some initial values in NN $Z^{[l](1)},\ldots, Z^{[l](m)}$,

- $\mu = \dfrac{1}{m} \sum_i Z^{[l](i)}$
- $\sigma^2 = \dfrac{1}{m}\sum_i (Z^{[l](i)} - \mu)^2$
- $Z^{[l](i)}_{\text{norm}} = \dfrac{Z^{[l](i)} - \mu}{\sqrt{\sigma^2} + \epsilon}$ to get mean $\mu=0$ and STD $\sigma=1$.
- $\tilde{Z}^{[l](i)} = \gamma Z^{[l](i)}_{\text{norm}} + \beta$ to have different other normal distribution.

Now, $\gamma, \beta$ are learnable parameters of the model.

- If we choose different $\beta, \gamma$ $\Rightarrow$ hidden units have other means & variances.
- Instead of using $Z^{[l](1)}, \ldots, Z^{[l](m)}$, we use $\tilde{Z}^{[l](i)}$.
- Difference between normalizing input $X$ and normalizing in hidden units:
- $X$: after normalizing, $\mu=0, \sigma=1$.
- $Z$: after normalizing, various $\mu, \sigma$.

Note that, $\beta$ in this case is different from $\beta$ in Adam optimization!

We can use gradient descent to update $\beta$ and

**even use**Adam/RMSprop/Momentum to update params $\gamma, \beta$, not just for Gradient Descent.**In practice**, we won't have to implement Batch Norm step by step by ourself, programming framework (like Tensorflow) will do!**In practice**, Batch Norm is usually applied with mini-batch of your training set.**Parameters**: $W^{[l]}, \beta^{[l]}, \gamma^{[l]}$. We don't need to consider $b^{[l]}$ becase it will be subtracted out in the process of normalization!**Fitting Batch Norm into a NN**: for $t$ goes through the number of mini-batches,- Compute forward prop on $X^{\{t\}}$.
- In each hidden layer, use Batch Norm to reparameter $Z^{[l]}$ to $\tilde{Z}^{[l]}$.
- Use backprop to compute $dW^{[l]}, d\beta^{[l]}, d\gamma^{[l]}$.
- Update params (we can use Momentum / RMSprop / Adam):

$\begin{aligned}W^{[l]} &:= W^{[l]} - \alpha dW^{[l]}, \\\beta^{[l]} &:= \beta^{[l]} - \alpha d\beta^{[l]}, \\\gamma^{[l]} &:= \gamma^{[l]} - \alpha d\gamma^{[l]}.\end{aligned}$ Sometimes, BN has a 2nd effect as a regularization technique but it's unintended! We don't use it for the purpose of regularization, use L1, L2 or dropout instead.

(Recall) **Regularization**: techniques that lower the complexity of a NN during training, thus prevent the overfitting.

### Why BN works?

- Make weights in later / deeper layers be more robust to changing to the weights in the earlier layers.
**Covariate shift problem**: suppose we have $X \to Y$. If $X$'s distribution changes, it changes the result in $Y$ much. We have to re-train our model.- Example: "cat vs non-cat" problem. If we apply params from the model of "black cat vs non-cat" to the problem of "colored-cat vs non-cat", it won't work because distribution in "black cat" is different from "colored cat".

*Covariate problem. Image from the course.*

*Why BN works?. Image from the course.*

In the perspective of layer 3, it depends only on layer 2 $\Rightarrow$ If layers before layer 2 changes $\Rightarrow$ distribution of layer 2 changes $\Rightarrow$ covariate shift problem for layer 3 $\Rightarrow$ Batch Norm makes sure that mean and variance in layer 2 is always robust before going to layer 3!

### Batch Norm in test time

- BN processes our data one min-batch at a time. However, in test time, you need to process the examples at a time. $\Rightarrow$ Need to adapt your network to do that.
**Idea**: calculate $\mu, \sigma^2$ using*exponentially weighted average*(across mini-batches). Other words,- In the training time, we calculate (and store) also the $\mu^{\{t\}[l]}, \sigma^{\{t\}[l]}$ in each mini-batch.
- Find $\mu, \sigma^2$ (exponentially weighted average) of all mini-batches.
- Use this $\mu, \sigma^2$ to find $Z_{\text{norm}}$ and $\tilde{Z}$ (at each example $i$).

- Don't worry, it's easy to use with Deep Learning Frameworks.

## Tensorflow introduction

Writing and running programs in TensorFlow has the following steps:

- Create Tensors (variables) that are not yet executed/evaluated.
- Write operations between those Tensors.
- Initialize your Tensors.
- Create a Session.
- Run the Session. This will run the operations you'd written above.

`# create placeholders`

x = tf.placeholder(tf.int64, name = 'x')

X = tf.placeholder(tf.float32, [n_x, None], name="X")

# initialize

W1 = tf.get_variable("W1", [25,12288], initializer = tf.contrib.layers.xavier_initializer(seed = 1))

b1 = tf.get_variable("b1", [25,1], initializer = tf.zeros_initializer())

There are two typical ways to create and use sessions in tensorflow:

- Method 1:

`sess = tf.Session()`

# Run the variables initialization (if needed), run the operations

result = sess.run(..., feed_dict = {...})

sess.close() # Close the session

- Method 2:

`with tf.Session() as sess:`

# run the variables initialization (if needed), run the operations

result = sess.run(..., feed_dict = {...})

# This takes care of closing the session for you :)

**What you should remember**:

- Tensorflow is a programming framework used in deep learning
- The two main object classes in tensorflow are Tensors and Operators.
- When you code in tensorflow you have to take the following steps:
- Create a graph containing Tensors (Variables, Placeholders ...) and Operations (
`tf.matmul`

,`tf.add`

, ...) - Create a session
- Initialize the session
- Run the session to execute the graph

- Create a graph containing Tensors (Variables, Placeholders ...) and Operations (
- You can execute the graph multiple times as you've seen in model()
- The backpropagation and optimization is automatically done when running the session on the "optimizer" object.

👉 Check more details about the codes in the notebook.