Until now, I’ve always used Gradient Descent to update the parameters and minimize the cost. In this aricle, I will show more advanced optimization methods that can speed up learning and perhaps even get 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 $J$. 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.

A simple optimization method in machine learning is gradient descent (GD). When you take gradient steps with respect to all $m$ examples on each step, it is also called Batch Gradient Descent.

This section will implement the gradient descent update rule. The gradient descent rule is, for $l = 1, …, L$:

where L is the number of layers 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 $W^{[1]}$ and $b^{[1]}$. We need to shift l to l+1 when coding:

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 I have just implemented does not change. What changes is that I 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:
• Stochastic Gradient Descent:

In Stochastic Gradient Descent, I 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:

Figure 2 : 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).

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

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

In practice, we’ll often get faster results if we 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, we 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 our optimization algorithm often leads to faster optimization.

What we should remember:

• The difference between gradient descent, mini-batch gradient descent and stochastic gradient descent is the number of examples we use to perform one update step.
• We have to tune a learning rate hyperparameter $\alpha$.
• With a well-turned mini-batch size, usually it outperforms either gradient descent or stochastic gradient descent (particularly when the training set is large).

## 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:

This section will implement random_mini_batches. We coded the shuffling part . For the partitioning step, we will use the following code that selects the indexes for the $1^{st}$ and $2^{nd}$ mini-batches:

We should note that the last mini-batch might end up smaller than mini_batch_size=64. Let $\lfloor s \rfloor$ 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 $\lfloor \frac{m}{mini_batch_size}\rfloor$ mini-batches with a full 64 examples, and the number of examples in the final mini-batch will be ($m-mini_batch_size \times \lfloor \frac{m}{mini_batch_size}\rfloor$):

What we 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.

## 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 $v$. 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 4: 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 vv and then take a step in the direction of vv.

So let’s initialize the velocity. The velocity, $v$, 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 $l =1,…,L$:

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.

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

where L is the number of layers, $\beta$ 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 $W^{[1]}$ and $b^{[1]}$ (that’s a “one” on the superscript). So you will need to shift l to l+1 when coding.

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

How do you choose $\beta$?

• The larger the momentum $\beta$ is, the smoother the update because the more we take the past gradients into account. But if $\beta$ is too big, it could also smooth out the updates too much.
• Common values for $\beta$ range from 0.8 to 0.999. If you don’t feel inclined to tune this, $\beta = 0.9$ is often a reasonable default.
• Tuning the optimal $\beta$ for your model might need trying several values to see what works best in term of reducing the value of the cost function $J$.

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 $\beta$ and a learning rate $\alpha$.

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?

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

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

where:

• t counts the number of steps taken of Adam
• L is the number of layers
• $\beta_1$ and $\beta_2$ are hyperparameters that control the two exponentially weighted averages.
• $\alpha$ is the learning rate
• $\varepsilon$ is a very small number to avoid dividing by zero

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

Firstly we need to initialize the Adam variables $v, s$ which keep track of the past information.

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

Now, we can implement the parameters update with Adam. Recall the general update rule is, for $l = 1, …, L$:

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

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.

## 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.)

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()

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

Output:

Output:

Output:

## Summary

optimization method accuracy cost shape
Gradient descent 79.7% oscillations
Momentum 79.7% oscillations

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.

• Usually works well even with little tuning of hyperparameters (except $\alpha$)