# Archive for May, 2016

In the previous installment, I introduced a mystery class `NeuralNetwork` which is capable of calculating different results depending on the data that you train it with. In this article, we’ll take a closer look at this neural network and crunch a few numbers.

## Neural networks overview

A neural network is a graph of neurons made of successive layers. The graph is typically split in three parts: the leftmost column of neurons is called the “input layer”, the rightmost columns of neurons is the “output layer” and all the neurons in-between are the “hidden” layer. This hidden layer is the most important part of your graph since it’s responsible for making the calculations. There can be any numbers of hidden layers and any number of neurons in each of them (note that the Kotlin class I wrote for this series of articles only uses one hidden layer).

Each edge that connects two neurons has a weight which is used to calculate the output of this neuron. The calculation is a simple addition of each input value multiplied by its weight. Let’s take a look at a quick example:

This network has two input values, one hidden layer of size two and one output. Our first calculation is therefore:

```w11-output = 2 * 0.1 + (-3) * (-0.2) = 0.8
w12-output = 2 * (-0.4) + (-3) * 0.3 = -1.7
```

We’re not quite done: the actual outputs of neurons (also called “activations”) are typically passed to a normalization function first. To get a general intuition for this function, you can think of it as a way to constrain the outputs within the [-1, 1] range, which prevents the values flowing through the network from overflowing or underflowing. Also, it’s useful in practice for this function to have additional properties connected to its derivative but I’ll skip over this part for now. This function is called the “activation function” and the implementation I used in the `NeuralNetwork` class is the hyperbolic tangent, `tanh`.

In order to remain general, I’ll just refer to the activation function as `f()`. We therefore refine our first calculations as follows:

```w11-output = f(2 * 0.1 + (-3) * (-0.2))
w12-output = f(2 * (-0.4) + (-3) * 0.3)
```

There are a few additional details to this calculation in actual neural networks but I’ll skip those for now.

Now that we have all our activations for the hidden layer, we are ready to move to the next layer, which happens to be the ouput layer, so we’re almost done:

```output = f(0.1 * w11-output - 0.2 * w12-output
= 0.42
```

As you can see, calculating the output of a neural network is fairly straightforward and fast, much faster than actually training that network. Once you have created your networks and you are satisfied with its results, you can just pass around the characteristics of that network (weights, sizes, …) and any device (even phones) can then use that network.

## Revisiting the xor network

Let’s go back to the `xor` network we created in the first episode. I created this network as follows:

```NeuralNetwork(inputSize = 2, hiddenSize = 2, outputSize = 1)
```

We only need two inputs (the two bits) and one output (the result of `a xor b`). These two values are fixed. What is not fixed is the size of the hidden layer, and I decided to pick 2 here, for no particular reason. It’s interesting to tweak these values and see whether your neural network performs better of worse based on these values and there is actually a great deal of both intuition and arbitrary choices that go into these decisions. These values that you use to configure your network before you run it are called “hyper parameters”, in contrast to the other values which get updated while your network runs (e.g. the weights).

Let’s now take a look at the weights that our `xor` network came up with, which you can display by running the Kotlin application with --log 2:

```Input weights:
-1.21 -3.36
-1.20 -3.34
1.12 1.67

Output weights:
3.31
-2.85
```

Let’s put these values on the visual representation of our graph to get a better idea:

You will notice that the network above contains a neuron called “bias” that I haven’t introduced yet. And I’m not going to just yet besides saying that this bias helps the network avoid edge cases and learn more rapidly. For now, just accept it as an additional neuron whose output is not influenced by the previous layers.

Let’s run the graph manually on the input (1,0), which should produce 1:

```hidden1-1 = 1 * -1.21
hidden1-2 = 0 * -1.20
bias1     = 1 * 1.12

output1 = tanh(-1.21 + 1.12) = -0.09

hidden2-1 = 1 * -3.36
hidden2-2 = 0 * -3.34
bias2     = 1 * 1.67

output2 = tanh(-3.36 + 1.6) = -0.94

// Now that we have the outputs of the hidden layer, we can caculate
// our final result by combining them with the output weights:

finalOutput = tanh(output1 * 3.31 + output2 * (-2.85))
= 0.98
```

We have just verified that if we input `(0,1)` into the network, we’ll receive `0.98` in output. Feel free to calculate the other three inputs yourself or maybe just run the `NeuralNetwork` class with a log level of 2, which will show you all these calculations.

## Revisiting the parity network

So the calculations hold up but it’s still a bit hard to understand where these weights come from and why they interact in the way they do. Elucidating this will be the topic of the next installment but before I wrap up this article, I’d like to take a quick look at the parity network because its content might look a bit more intuitive to the human eye, while the `xor` network detailed above still seems mysterious.

If you train the parity network and you ask the `NeuralNetwotk` class to dump its output, here are the weights that you’ll get:

```Input weights:
0.36 -0.36
0.10 -0.09
0.30 -0.30
-2.23 -1.04
0.57 -0.58

Output weights:
-1.65
-1.64
```

If you pay attention, you will notice an interesting detail about these numbers: the weights of the first three neurons of our hidden layer cancel each other out while the two inputs of the fourth neuron reinforce each other. It’s pretty clear that the network has learned that when you are testing the parity of a number in binary format, the only bit that really matters is the least significant one (the last one). All the others can simply be ignored, so the network has learned from the training data that the best way to get close to the desired output is to only pay attention to that last bit and cancel all the other ones out so they don’t take part in the final result.

## Wrapping up

This last result is quite remarkable if you think about it, because it really looks like the network learned how to test parity at the formal level (“The output of the network should be the value of the least significant bit, ignore all the others”), inferring that result just from the numeric data we trained it with. Understanding how the network learned how to modify itself to reach this level will be the topic of the next installment of the series.

It’s hard not to hear about machine learning and neural networks these days since the practice is being applied to an ever increasingly wide variety of problems. Neural networks can be intimidating and look downright magical to the untrained (ah!) eye, so I’m going to attempt to dispel these fears by demonstrating how these mysterious networks operate. And since there are already so many tutorials on the subject, I’m going to take a different approach and go from top to bottom.

## Goal

In this first series of articles, I will start by running a very simple network on two simple problems, show you that they work and then walk through the network to explain what happened. Then I’ll backtrack to deconstruct the logic behind the network and why it works.

The neural network I’ll be using in this article is a simple one I wrote. No TensorFlow, no Torch, no Theano. Just some basic Kotlin code. The original version was about 230 lines but it’s a bit bigger now that I broke it up in separate classes and added comments. The whole project can be found on github under the temporary “nnk” name. In particular, here is the source of the neural network we’ll be using.

I will be glossing over a lot of technical terms in this introduction in order to focus on the numeric aspect but I’m hoping to be able to get into more details as we slowly peel the layers. For now, we’ll just look at the network as a black box that get fed input values and which outputs values.

The main characteristic of a neural network is that it starts completely empty but it can be taught to solve problems. We do this by feeding it values and telling it what the expected output is. We iterate over this approach many times, changing these inputs/expected parameters and as we do that, the network updates its knowledge to come up with answers that are as close to the expected answers as possible. This phase is called “training” the network. Once we think the network is trained enough, we can then feed it new values that it hasn’t seen yet and compare its answer to the one we’re expecting.

## The problems

Let’s start with a very simple example: `xor`.

This is a trivial and fundamental binary arithmetic operation which returns 1 if the two inputs are different and 0 if they are equal. We will train the network by feeding it all four possible combinations and telling it what the expected outcome is. With the Kotlin implementation of the Neural Network, the code looks like this:

```with(NeuralNetwork(inputSize = 2, hiddenSize = 2, outputSize = 1)) {
val trainingValues = listOf(
NetworkData.create(listOf(0, 0), listOf(0)),
NetworkData.create(listOf(0, 1), listOf(1)),
NetworkData.create(listOf(1, 0), listOf(1)),
NetworkData.create(listOf(1, 1), listOf(0)))

train(trainingValues)
test(trainingValues)
}
```

Let’s ignore the parameters given to `NeuralNetwork` for now and focus on the rest. Each line of `NetworkData` contains the inputs (each combination of 0 and 1: (0,0), (0,1), (1,0), (1,1)) and the expected output. In this example, the output is just a single value (the result of the operation) so it’s a list of one value, but networks can return an arbitrary number of outputs.

The next step is to test the network. Since there are only four different inputs here and we used them all for training, let’s just use that same list of inputs but this time, we’ll display the ouput produced by the network instead of the expected one. The result of this run is as follows:

Running neural network xor()

```[0.0, 0.0] -> [0.013128957]
[0.0, 1.0] -> [0.9824073]
[1.0, 0.0] -> [0.9822749]
[1.0, 1.0] -> [-2.1314621E-4]
```

As you can see, these values are pretty decent for such a simple network and such a small training data set and you might rightfully wonder: is this just luck? Or did the network cheat and memorize the values we fed it while we were training it?

One way to find out is to see if we can train our network to learn something else, so let’s do that.

## A harder problem

This time, we are going to teach our network to determine whether a number is odd or even. Because the implementation of the graph is pretty naïve and this is just an example, we are going to train our network with binary numbers. Also, we are going to learn a first important lesson in neural networks which is to choose your training and testing data wisely.

You probably noticed in the example above that I used the same data to train and test the network. This is not a good practice but it was necessary for `xor` since there are so few cases. For better results, you usually want to train your network on a certain population of the data and then test it on data that your network hasn’t seen yet. This will guarantee that you are not “overfitting” your network and also that it is able to generalize what you taught it to input values that it hasn’t seen yet. Overfitting means that your network does great on the data you trained it with but poorly on new data. When this happens, you usually want to tweak your network so that it will possibly perform less well on the training data but it will return better results for new data.

For our parity test network, let’s settle on four bits (integers 0 – 15) and we’ll train our network on about ten numbers and test it on the remaining six:

```with(NeuralNetwork(inputSize = 4, hiddenSize = 2, outputSize = 1)) {
val trainingValues = listOf(
NetworkData.create(listOf(0, 0, 0, 0), listOf(0)),
NetworkData.create(listOf(0, 0, 0, 1), listOf(1)),
NetworkData.create(listOf(0, 0, 1, 0), listOf(0)),
NetworkData.create(listOf(0, 1, 1, 0), listOf(0)),
NetworkData.create(listOf(0, 1, 1, 1), listOf(1)),
NetworkData.create(listOf(1, 0, 1, 0), listOf(0)),
NetworkData.create(listOf(1, 0, 1, 1), listOf(1)),
NetworkData.create(listOf(1, 1, 0, 0), listOf(0)),
NetworkData.create(listOf(1, 1, 0, 1), listOf(1)),
NetworkData.create(listOf(1, 1, 1, 0), listOf(0)),
NetworkData.create(listOf(1, 1, 1, 1), listOf(1))
)
train(trainingValues)

val testValues = listOf(
NetworkData.create(listOf(0, 0, 1, 1), listOf(1)),
NetworkData.create(listOf(0, 1, 0, 0), listOf(0)),
NetworkData.create(listOf(0, 1, 0, 1), listOf(1)),
NetworkData.create(listOf(1, 0, 0, 0), listOf(0)),
NetworkData.create(listOf(1, 0, 0, 1), listOf(1))
)
test(testValues)
}
```

And here is the output of the test:

```Running neural network isOdd()

[0.0, 0.0, 1.0, 1.0] -> [0.9948013]
[0.0, 1.0, 0.0, 0.0] -> [0.0019584869]
[0.0, 1.0, 0.0, 1.0] -> [0.9950419]
[1.0, 0.0, 0.0, 0.0] -> [0.0053276513]
[1.0, 0.0, 0.0, 1.0] -> [0.9947305]
```

Notice that the network is now outputting correct results for numbers that it hadn’t seen before, just because of the way it adapted itself to the training data it was initially fed. This gives us good confidence that the network has configured itself to classify numbers from any input values and not just the one it was trained for.

## Wrapping up

I hope that this brief overview will have whetted your appetite or at least piqued your curiosity. In the next installment, I’ll dive a bit deeper into the `NeuralNetwork` class, explain the constructor parameters and we’ll walk through the inner working of the neural network that we created to demonstrate how it works.