A Brief Recap ( From parts 1 and 2)
Before we commence with the nitty griity of this new article which deals wtih muti layer Neural Networks, let just revisit a few key concepts. If you haven't read Part 1 or Part 2 perhaps you should start there.
Part 1 : Perceptron Configuration ( Single layer network)
The inputs (x1,x2,x3..xm) and connection weights (w1,w2,w3..wm) in figure 4 are typically real values, both postive (+) and negative (-). If the feature of some xi tends to cause the perceptron to fire, the weight wi will be positive; if the feature xi inhibits the perceptron, the weight wi will be negative.
The perceptron itself, consists of weights, the summation processor, and an activation function, and an adjustable threshold processor (called bias here after).
For convenience the normal practice is to treat the bias, as just another input. The following diagram illustrates the revised configuration.
The bias can be thought of as the propensity (a tendency towards a particular way of behaving) of the perceptron to fire irrespective of it's inputs. The perceptron configuration network shown in Figure 5 fires if the weighted sum > 0, or if your into maths type explanations
Part 2: Multi-Layer Configuration
The multi-layer network that will solve the XOR problem will look similiar to a single layer network. We are still dealing with inputs / weights / outputs. What is new is the addition of the hidden layer.
As already explained above there is one input layer, one hidden layer and one output layer.
It is by using the inputs and weights that we are able to work out the activation for a given node. This is easily achieved for the hidden layer as it has direct links to the actual input layer.
The output layer, however, knows nothing about the input layer as it is not directly connected to it. So to work out the activation for an output node we need to make use of the output from the hidden layer nodes, which are used as inputs to the output layer nodes.
This entire process described above can be thought of as a pass forward from one layer to the next.
This still works like it did with a single layer network, the activation for any given node is still worked out as follows:
Where (wi is the weight(i), and Ii is the input(i) value)
You see it the same old stuff, no demons, smoke or magic here. Its stuff we've already covered.
So thats how the network looks, so now I guess you want to know how to go about training it.
Learning
There are essentially 2 types of learning that may be applied, to a Neural Network, which is "Reinforcement" and "Supervised"
Reinforcement
In Reinforcement learning, during training an set of inputs is presented to the Neural Network, the Output is 0.75, when the target was expecting 1.0.
The error (1.0 - 0.75) is used for training (‘wrong by 0.25’).
What if there are 2 outputs then the total error is summed to give a single number (typically sum of squared
errors). Eg “your total error on all outputs is 1.76”Note that this just tells you how wrong you were, not in which direction you were wrong.
Using this method we may never get a result, or could be hunt the needle.
Using a generic algorithm to train a muti-layer neural network, offers a Reinforcment type training arrangement. Where the mutation is responsible for ‘jiggling the weights a bit’. This is what this article is all about.
Supervised
In Supervised Learning the Neural Network is given more information.
Not just ‘how wrong’ it was, but ‘in what direction it was wrong' like ‘Hunt the needle’ but where you are told ‘North a bit’ ‘West
a bit’.So you get, and use, far more information in Supervised Learning, and this is the normal form of Neural Network learning algorithm.
This training method is normally conducted using a Back Propogation training method, which I covered in Part 2, so if this is the first article of these 3 parts, and the back propogation method is of particular interest, then you should look there.
So Now The New Stuff
From this point on anything that is being discussed relates directly to this articles code.
Ok so what is the problem we are trying to solve ? Well it's the same as it was for Part 2, its the simply XOR logic problem. In fact this articles content is really just an incremetal build, on knowledge that was covered in Part 1 and Part 2, So lets march on.
For the benefit of those that may have only read this one article, the XOR look problem looks like the following truth table
Remember with a single layer (perceptron) we cant actually achieve the XOR functionality, as its not linearly seperable. But with a multi-layer network, this is achievable.
So with this in mind how are we going to achieve this ? Well we are going to use a Genetic Algorithm (GA from this point on) to breed a population of Neural Networks, that will hopefully evolve to provide a solution to the XOR logic problem, that's the basic idea anyway.
So what does this all look like.
As can be seen from the figure above, what we are going to do is have a GA which will actually contain a population of Neural Networks. The idea being that the GA will jiggle the weights of the Neural Networks, within the population, in the hope that the jiggling of the weights will push the Neural Network population towards a solution to the XOR problem.
So how does this translate into an algorithm
The basic operation of the Microbial GA training is as follows:
o Pick 2 genotypes at random
o Compares Scores (Fitness) to come up with a Winner and Loser
o Go along genotype, at each locus (Point)
- With some probability copy from Winner to Loser (overwrite)
- With some probability mutate that locus of the Loser
So ONLY the Loser gets changed, which gives a version of Elitism for free, this ensures a best in breed remains in the population.That's it. That's is the complete algorithm.
But there are some ESSENTIAL issues to be aware of, when playing with GAs
- The geneotype will be different for a a different problem domain
- The Fitness function will be different for a a different problem domain
These 2 items MUST be developed again, when ever a new problem is specified. For example if we wanted to find a person favourite pizza toppings the genetype and fitness would be different from that which is use for this articles problem domain.
These 2 essential elements of a GA (for this article problem domain) are specified below
1. The Geneotype
For this article the problem domain states that we had a population of Neural Networks. So I created a single dimension array of NeuralNetwork objects this can be seen from the Constructor code within the GA_Trainer_XOR object
//ANN's private NeuralNetwork[] networks; public GA_Trainer_XOR() { networks = new NeuralNetwork[POPULATION]; //create new ANN objects, random weights applied at start for (int i = 0; i <= networks.GetUpperBound(0); i++) { networks[i] = new NeuralNetwork(2, 2, 1); networks[i].Change += new NeuralNetwork.ChangeHandler(GA_Trainer_NN_Change); } }2. The Fitness Function
Remembering that the problem domain description stated the following truthtable is what we are trying to achieve
So how can we tell how fit (how close) the neural network is to this ? It is fairly simply really, what we do is present the entire set of inputs to the Neural Network one at a time and keep an accumulated error value, which is worked out as follows:
Within the NeuralNetwork class there is an getError(..) method like this
public double getError(double[] targets) { //storage for error double error = 0.0; //this calculation is based on something I read about weight space in //Artificial Intellegence - A Modern Approach, 2nd edition.Prentice Hall //2003. Stuart Rusell, Peter Norvig. Pg 741 error = Math.Sqrt(Math.Pow((targets[0] - outputs[0]), 2)); return error; }Then in the NN_Trainer_XOR class, there is an evaluate method, that accepts an int value, which represents the member of the population to fetch and evaluate (get fitness for). This overall fitness is then returned to the GA training method, to see which Neural Network should be the WINNER and which Neural Network should be the LOSER.
private double evaluate(int popMember) { double error = 0.0; //loop through the entire training set for (int i = 0; i <= train_set.GetUpperBound(0); i++) { //forward these new values through network //forward weights through ANN forwardWeights(popMember, getTrainSet(i)); double[] targetValues = getTargetValues(getTrainSet(i)); error += networks[popMember].getError(targetValues); } //if the Error term is < acceptableNNError value we have found //a good configuration of weights for teh NeuralNetwork, so tell //GA to stop looking if (error < acceptableNNError) { bestConfiguration = popMember; foundGoodANN = true; } //return error return error; }So how do we know when we have a trained Neural Network? In this articles code what I have done is provide a fixed limit value within the NN_Trainer_XOR class, that when reached, indicates that the training has yielded a best configured Neural Network.
If however the entire training loop is done, and there is still no well configured Neural Network, I simply return the value of the WINNER (of the last training epoch) as the overall best configured Neural Network.
This is shown in the code snippet below, this should be read in conjunction with the evaluate(..) method shown above
//check to see if there was a best configuration found, may not have done //enough training to find a good NeuralNetwork configuration, so will simply //have to return the WINNER if (bestConfiguration == -1) { bestConfiguration = WINNER; } //return the best Neural network return networks;So Finally The Code
Well the code for this article looks like the following class diagram (Its visual studio 2005 C#, .NET v2.0)
The main classes that people should take the time to look at would be :
GA_Trainer_XOR : Trains a Neural Network to solve the XOR problemm using a Microbial GA
TrainerEventArgs : Training event args, for use with a GUI
NeuralNetwork : A configurable Neural Network
NeuralNetworkEventArgs : Training event args, for use with a GUI
SigmoidActivationFunction : A static method to provide the sigmoid activation function
The rest are a GUI I constructed simply to show how it all fits together.
NOTE : the demo project contains all code, so I wont list it here Also not that most of these classes are quite simliar to those included with the Part 2 article code. I wanted to keep the code, simliar so people who had already looked at Part 2 would recognize the common pattern.
Code Demos
The DEMO application attached has 3 main areas which are described below:
LIVE RESULTS Tab
It can be seen that this has very nearly solved the XOR problem, it did however take nearly 45000 iterations (epoch) of a training loop . Remembering that epoch, we have to also present the entire training set to the network, and also do this twice, once to find a WINNER and once to find a LOSER. That is quite a lot of work, im sure you would all agree. This is why Neural Networks are not normally trained by GAs, this article is really about how to apply a GA to a problem domain. You know because the GA training took 45000 epochs to yield an acceptable result, does not mean that GAs are useless. Far from it, GAs have their place and can be used for many problems, such as
- Sudoko Solver (the popular game)
- Backpack problem (trying to optimize the use of a backpack of limited size, to get as many items in as will fit)
- Favourite pizza toppings problem (try and find out what someones favourite pizza is)
To name but a few, basically if you can come up with the geneotype and a Fitness Function, you should be able to get a GA to work out a solution. GAs have also been used to grow entire sytax trees of grammar, in order to predict which grammar is more optimal. There is more research being done in this area as I write this article, in fact there is a nice article on this topic (Gene Expression Programming) by Andrew Krillov, right here at the Codeproject if anyone wants to read further.
TRAINING RESULTS Tab
Viewing the target/outputs together
Viewing the errors
TRAINED RESULTS Tab
Viewing the target/outputs together
It is also possible to view the Neural Networks final configuration using the "View Neural Network Config" button.
What Do You Think ?
Thats it, I would just like to ask, if you liked the article please vote for it.
Points of Interest
I think AI is fairly interesting, that's why I am taking the time to publish these articles. So I hope someone else finds it interesting, and that it might help further someones knowledge, as it has my own.
Anyone that wants to look further into AI type stuff, that finds the content of this article a bit basic should check out Andrew Krillovs articles, at Andrew Krillov CP articles as his are more advanced, and very good.
