Recent Tutorials

24th March 2015 at 22:07 · By Lee Jacobson

Solving the Traveling Salesman Problem Using Google Maps and Genetic Algorithms

An ideal way to explore the potentials and pitfalls of genetic algorithms is by applying them to real world data. Perhaps one of the easiest ways to do this is by using the Google Maps API to implement a solution to the traveling salesman problem.

If you're unfamiliar with genetic algorithms, or the traveling salesman problem, then you might find the following links useful:
Creating a genetic algorithm for beginners
Applying a genetic algorithm to the traveling salesman problem

Below I've created a very simple route optimizer which uses distance and duration data from the Google Maps API to find the quickest route.

One interesting thing you might observe is how in real life the quickest route is often neither the most direct, or the most obvious route you might think of taking. In the real world there are many restrictions such as road speed and traffic conditions, which are usually equal in importance to the direction when searching for the quickest route. This becomes clear when optimizing the route for driving, to walking or cycling. If you spend a lot of your day behind the wheel, or on foot, you may even be pleasantly surprised to find yourself a quicker daily route. =)

This demo has been built completely in HTML and JavaScript, using just the Google Maps API and the JQuery library. You're welcome to explore the code using the download link at the bottom of the post.

Loading...
Configuration
Travel Mode:
Avoid Highways:
Population Size:
Mutation Rate:
Crossover Rate:
Elitism:
Max Generations:
Debug Info
Destinations Count: 0

Download from GitHub: https://github.com/leejacobson/googlemaps-tsp-ga

Comments


26th March 2014 at 21:16 · By Lee Jacobson

Introduction to Artificial Neural Networks Part 2 - Learning

Welcome to part 2 of the introduction to my artificial neural networks series, if you haven't yet read part 1 you should probably go back and read that first!

Introduction

In part 1 we were introduced to what artificial neural networks are and we learnt the basics on how they can be used to solve problems. In this tutorial we will begin to find out how artificial neural networks can learn, why learning is so useful and what the different types of learning are. We will specifically be looking at training single-layer perceptrons with the perceptron learning rule.

Before we begin, we should probably first define what we mean by the word learning in the context of this tutorial. It is still unclear whether machines will ever be able to learn in the sense that they will have some kind of metacognition about what they are learning like humans. However, they can learn how to perform tasks better with experience. So here, we define learning simply as being able to perform better at a given task, or a range of tasks with experience.

Learning in Artificial Neural Networks

One of the most impressive features of artificial neural networks is their ability to learn. You may recall from the previous tutorial that artificial neural networks are inspired by the biological nervous system, in particular, the human brain. One of the most interesting characteristics of the human brain is it's ability to learn. We should note that our understanding of how exactly the brain does this is still very primitive, although we do still have a basic understanding of the process. It is believed that during the learning process the brain's neural structure is altered, increasing or decreasing the strength of it's synaptic connections depending on their activity. This is why more relevant information is easier to recall than information that hasn't been recalled for a long time. More relevant information will have stronger synaptic connections and less relevant information will gradually have it's synaptic connections weaken, making it harder to recall.

Although simplified, artificial neural networks can model this learning process by adjusting the weighted connections found between neurons in the network. This effectively emulates the strengthening and weakening of the synaptic connections found in our brains. This strengthening and weakening of the connections is what enables the network to learn.

Learning algorithms are extremely useful when it comes to certain problems that either can't be practically written by a programmer or can be done more efficiently by a learning algorithm. Facial recognition would be an example of a problem extremely hard for a human to accurately convert into code. A problem that could be solved better by a learning algorithm, would be a loan granting application which could use past loan data to classify future loan applications. Although a human could write rules to do this, a learning algorithm can better pick up on subtleties in the data which may be hard to code for.

Learning Types

There are many different algorithms that can be used when training artificial neural networks, each with their own separate advantages and disadvantages. The learning process within artificial neural networks is a result of altering the network's weights, with some kind of learning algorithm. The objective is to find a set of weight matrices which when applied to the network should - hopefully - map any input to a correct output. In this tutorial, the learning type we will be focusing on is supervised learning. But before we begin, lets take a quick look at the three major learning paradigms.

  • Supervised Learning
    The learning algorithm would fall under this category if the desired output for the network is also provided with the input while training the network. By providing the neural network with both an input and output pair it is possible to calculate an error based on it's target output and actual output. It can then use that error to make corrections to the network by updating it's weights.
  • Unsupervised Learning
    In this paradigm the neural network is only given a set of inputs and it's the neural network's responsibility to find some kind of pattern within the inputs provided without any external aid. This type of learning paradigm is often used in data mining and is also used by many recommendation algorithms due to their ability to predict a user's preferences based on the preferences of other similar users it has grouped together.
  • Reinforcement Learning
    Reinforcement learning is similar to supervised learning in that some feedback is given, however instead of providing a target output a reward is given based on how well the system performed. The aim of reinforcement learning is to maximize the reward the system receives through trial-and-error. This paradigm relates strongly with how learning works in nature, for example an animal might remember the actions it's previously taken which helped it to find food (the reward).

Implementing Supervised Learning

As mentioned earlier, supervised learning is a technique that uses a set of input-output pairs to train the network. The idea to provide the network with examples of inputs and outputs then to let it find a function that can correctly map the data we provided to a correct output. If the network has been trained with a good range of training data when the network has finished learning we should even be able to give it a new, unseen input and the network should be able to map it correctly to an output.

There are many different supervised learning algorithms we could use but the most popular, and the one we will be looking at in more detail is backpropagation. Before we look at why backpropagation is needed to train multi-layered networks, let's first look at how we can train single-layer networks, or as they're otherwise known, perceptrons.

The Perceptron Learning rule

The perceptron learning ruleworks by finding out what went wrong in the network and making slight corrections to hopefully prevent the same errors happening again. Here's how it works... First we take the network's actual output and compare it to the target output in our training set. If the network's actual output and target output don't match we know something went wrong and we can update the weights based on the amount of error. Lets run through the algorithm step by step to understand how exactly it works.

First, we need to calculate the perceptron's output for each output node. As you should remember from the previous tutorial we can do this by:

output = f( input1 × weight1 + input2 × weight2 + ... )
- or -
$o = f(\sum\limits_{i=1}^n x_iw_i)$

Now we have the actual output we can compare it to the target output to find the error:

error = target output - output
- or -
E = t - o

Now we want to use the perceptron's error to adjust the weights.

weight change = learning rate × error × input
- or -
Δwi = r E x

We want to ensure only small changes are made to the weights on each iteration, so to do this we apply a small learning rate (r). If the learning rate is too high the perceptron can jump too far and miss the solution, if it's too low, it can take an unreasonably long time to train.

This gives us a final weight update equation of:

weight change = learning rate × (target output - actual output) × input
- or -
Δwi = r ( t - o ) xi


Here's an example of how this would work with the AND function...

Learning rate = 0.1
Expected output = 1
Actual output =  0
Error = 1

Weight Update:
wi = r E x + wi
w1 = 0.1 x 1 x 1 + w1
w2 = 0.1 x 1 x 1 + w2

New Weights:
w1 = 0.4
w2 = 0.4


Learning rate = 0.1
Expected output = 1
Actual output =  0
Error = 1

Weight Update:
wi = r E x + wi
w1 = 0.1 x 1 x 1 + w1
w2 = 0.1 x 1 x 1 + w2

New Weights:
w1 = 0.5
w2 = 0.5


Learning rate = 0.1
Expected output = 1
Actual output =  1
Error = 0

No error,
training complete.

Implementing The Perceptron Learning Rule

To help fully understand what's happening let's implement a basic example in Java.

First, we initiate our network's threshold, learning rate and weights. We could initiate the weights with a small random starting weight, however for simplicity here we'll just set them to 0.

double threshold = 1;
double learningRate = 0.1;
double[] weights = {0.0, 0.0};

Next, we need to create our training data to train our perceptron. In this example our perceptron will be learning the AND function.

// AND function Training data
int[][][] trainingData = {
    {{0, 0}, {0}},
    {{0, 1}, {0}},
    {{1, 0}, {0}},
    {{1, 1}, {1}},
};

Now, we need to create a loop that we can break from later if our network completes a cycle of the training data without any errors. Then, we need a second loop that will iterate over each input in the training data.

// Start training loop
while(true){
    int errorCount = 0;
    // Loop over training data
    for(int i=0; i < trainingData.length; i++){
        System.out.println("Starting weights: " + Arrays.toString(weights));
    }

From here we can calculate the weighted sum of the inputs and get the output.

// Calculate weighted sum of inputs
double weightedSum = 0;
for(int ii=0; ii < trainingData[i][0].length; ii++) {
    weightedSum += trainingData[i][0][ii] * weights[ii];
}

// Calculate output
int output = 0;
if(threshold <= weightedSum){
    output = 1;
}

System.out.println("Target output: " + trainingData[i][1][0] + ", "
    + "Actual Output: " + output);

The next step is to calculate the error and adjust the weights...

// Calculate error
int error = trainingData[i][1][0] - output;

// Increase error count for incorrect output
if(error != 0){
    errorCount++;
}

// Update weights
for(int ii=0; ii < trainingData[i][0].length; ii++) {
    weights[ii] += learningRate * error * trainingData[i][0][ii];
}

System.out.println("New weights: " + Arrays.toString(weights));
System.out.println();

Finally, break if a solution is found and close loop

    // If there are no errors, stop
    if(errorCount == 0){
        System.out.println("Final weights: " + Arrays.toString(weights));
        System.exit(0);
    }
}

And if we put it all together...

package perceptron;
import java.util.Arrays;

public class PerceptronLearningRule {
    public static void main(String args[]){
        double threshold = 1;
        double learningRate = 0.1;
        // Init weights
        double[] weights = {0.0, 0.0};
        
        // AND function Training data
        int[][][] trainingData = {
            {{0, 0}, {0}},
            {{0, 1}, {0}},
            {{1, 0}, {0}},
            {{1, 1}, {1}},
        };
        
        // Start training loop
        while(true){
            int errorCount = 0;
            // Loop over training data
            for(int i=0; i < trainingData.length; i++){
                System.out.println("Starting weights: " + Arrays.toString(weights));
                // Calculate weighted input
                double weightedSum = 0;
                for(int ii=0; ii < trainingData[i][0].length; ii++) {
                    weightedSum += trainingData[i][0][ii] * weights[ii];
                }

                // Calculate output
                int output = 0;
                if(threshold <= weightedSum){
                    output = 1;
                }

                System.out.println("Target output: " + trainingData[i][1][0] + ", "
                        + "Actual Output: " + output);
                                
                // Calculate error
                int error = trainingData[i][1][0] - output;
                
                // Increase error count for incorrect output
                if(error != 0){
                    errorCount++;
                }
                
                // Update weights
                for(int ii=0; ii < trainingData[i][0].length; ii++) {
                    weights[ii] += learningRate * error * trainingData[i][0][ii];
                }

                System.out.println("New weights: " + Arrays.toString(weights));
                System.out.println();
            }

            // If there are no errors, stop
            if(errorCount == 0){
                System.out.println("Final weights: " + Arrays.toString(weights));
                System.exit(0);
            }
        }
    }
}
Perceptron learning rule source

Bias units

In our last example we set our threshold to 1, this means our weighted input needs to equal or exceed 1 to give us an output of 1. This is okay when learning the AND function because we know we only need an output when both inputs will be set, allowing (with the correct weights) for the threshold to be reached or exceeded. In the case of the NOR function however, the network should only output 1 if both inputs are off. This means if we have a threshold of 1 there isn't a combination of weights that will ever make the following true,
x1 = 0
x2 = 0
1 <= x1w1 + x2w2


There's a simple fix to this though, a bias unit. A bias unit is simply a neuron with a constant output, typically of 1. Bias units are weighted just like other units in the network, the only difference is that they will always output 1 regardless of the input from the previous layer, this is where they get their name! So why are they important? Bias inputs effectively allow the neuron to learn a threshold value. Consider our previous equation, with a bias input (x0) added we can change it to,
x0 = 1
x1 = 0
x2 = 0
1 <= x0w0 + x1w1 + x2w2


Now we can satisfy that equation. You can try this yourself by updating your perceptron training set to train for the NOR function. Just add a bias input to the training data and also an additional weight for the new bias input. Here is the updated code:

// Init weights
double[] weights = {0.0, 0.0, 0.0};

// NOR function training data
int[][][] trainingData = {
    {{1, 0, 0}, {1}},
    {{1, 0, 1}, {0}},
    {{1, 1, 0}, {0}},
    {{1, 1, 1}, {0}},
};

To be continued...

Hopefully you should now have a clearer understanding about the types of learning we can apply to neural networks and the process in which a simple, single layer perceptrons can be trained. In the next tutorial we will be learning how to implement the back propagation algorithm and why it's needed when working with multi-layer networks.

Comments


5th December 2013 at 7:42 · By Lee Jacobson

Introduction to Artificial Neural Networks - Part 1

This is the first part of a three part introductory tutorial on artificial neural networks. In this first tutorial we will discover what neural networks are, why they're useful for solving certain types of tasks and finally how they work.

Introduction

Computers are great at solving algorithmic and math problems, but often the world can't easily be defined with a mathematical algorithm. Facial recognition and language processing are a couple of examples of problems that can't easily be quantified into an algorithm, however these tasks are trivial to humans. The key to Artificial Neural Networks is that their design enables them to process information in a similar way to our own biological brains, by drawing inspiration from how our own nervous system functions. This makes them useful tools for solving problems like facial recognition, which our biological brains can do easily.

How do they work?

First lets take a look at what a biological neuron looks like.
Biological Neuron
Our brains use extremely large interconnected networks of neurons to process information and model the world we live in. Electrical inputs are passed through this network of neurons which result in an output being produced. In the case of a biological brain this could result in contracting a muscle or signaling your sweat glands to produce sweat. A neuron collects inputs using a structure called dendrites, the neuron effectively sums all of these inputs from the dendrites and if the resulting value is greater than it's firing threshold, the neuron fires. When the neuron fires it sends an electrical impulse through the neuron's axon to it's boutons. These boutons can then be networked to thousands of other neurons via connections called synapses. There are about one hundred billion (100,000,000,000) neurons inside the human brain each with about one thousand synaptic connections. It's effectively the way in which these synapses are wired that give our brains the ability to process information the way they do.

Modeling Artificial Neurons

Artificial neuron models are at their core simplified models based on biological neurons. This allows them to capture the essence of how a biological neuron functions. We usually refer to these artificial neurons as 'perceptrons'. So now lets take a look at what a perceptron looks like.
Perceptron
As shown in the diagram above a typical perceptron will have many inputs and these inputs are all individually weighted. The perceptron weights can either amplify or deamplify the original input signal. For example, if the input is 1 and the input's weight is 0.2 the input will be decreased to 0.2. These weighted signals are then added together and passed into the activation function. The activation function is used to convert the input into a more useful output. There are many different types of activation function but one of the simplest would be step function. A step function will typically output a 1 if the input is higher than a certain threshold, otherwise it's output will be 0.

Here's an example of how this might work:
Input 1 (x1)  = 0.6
Input 2 (x2)  = 1.0

Weight 1 (w1) = 0.5
Weight 2 (w2) = 0.8

Threshold = 1.0


First we multiple the inputs by their weights and sum them:
x1w1 + x2w2 = (0.6 x 0.5) + (1 x 0.8) = 1.1


Now we compare our input total to the perceptron's activation threshold. In this example the total input (1.1) is higher than the activation threshold (1.0) so the neuron would fire.

Implementing Artificial Neural Networks

So now you're probably wondering what an artificial neural network looks like and how it uses these artificial neurons to process information. In this tutorial we're going to be looking at feedforward networks and how their design links our perceptron together creating a functioning artificial neural network. Before we begin lets take a look at what a basic feedforward network looks like:
Artificial Neural Network
Each input from the input layer is fed up to each node in the hidden layer, and from there to each node on the output layer. We should note that there can be any number of nodes per layer and there are usually multiple hidden layers to pass through before ultimately reaching the output layer. Choosing the right number of nodes and layers is important later on when optimising the neural network to work well a given problem. As you can probably tell from the diagram, it's called a feedforward network because of how the signals are passed through the layers of the neural network in a single direction. These aren't the only type of neural network though. There are also feedback networks where its architecture allows signals to travel in both directions.

Linear separability

To explain why we usually require a hidden layer to solve our problem, take a look at the following examples:
Linearly Separability
Notice how the OR function can be separated on the graph with a single straight line, this means the function is “linearly separable” and can be modelled within our neural network without implementing a hidden layer, for example, the OR function can be modeled with a single perceptron like this:
OR Function
However to model the XOR function we need to use an extra layer:
XOR Function
We call this type of neural network a 'multi layer perceptron'. In almost every case you should only ever need to use one or two hidden layers, however it make take more experimentation to find the optimal amount of nodes for the hidden layer(s).

To be continued...

So now you should have a basic understanding of some of the typical applications for neural networks and why we use them for these purposes. You should also have a rough understanding of how a basic neural network operates and how it can process data. In the next tutorial we will be looking at ways to construct a neural network and then how we can 'train' it to do the things we want it to do.

Part 2 is now available here, Introduction to Artificial Neural Networks Part 2 - Learning

Comments


Read More

1 2 3 4 Next »