## GANN-EWN / Part 5 / Applying the Neural Network trained by the Genetic Algorithm to the game EWN

If you haven’t read the previous parts, you should start at part 1.

Now that we have a Neural Network architecture and a Genetic Algorithm, we can apply them to the game “Einstein Würfelt Nicht”.

# The Parameters of the Problem

There are several parameters that need to be addressed first:

• how to feed information to the NN and how to get a result,
• how big should the NN be, how many layers, how many neurons per layer, how many connections between each layer,
• how to tune the parameters for the GA (population size, mutation rate, etc.).

There are some precedents for every of these 3 points (NNs as well as GAs have been studied), but there is no existing answer for this particular set of problems.

# Feeding Information to the NN

The answer to the first question seems trivial, but it is actually not. The first answer that comes to mind is “just feed it the board and the die and get the stone that has to be played and the move to be played”.

This is of course a valid answer, but there are many more:

• don’t feed the whole board, but rather the positions of the stones along with the die,
• feed the possible moves instead of the die,
• or you could use the network as a “value network”, in other words let the network evaluate how favorable a certain position is for the current player. In that case, the algorithm has to simulate every possible move and apply the network on every resulting board.

There are many other ways of feeding information, including feeding redundant information. For instance, you could feed the number of stones left for each player in addition to the board: that’s an information which is obviously useful and that the network could directly use rather than having to calculate it again.

# Getting the NN’s Result

The Neural Network can be used to gather very different results, also depending on which information it was given as inputs. Here are a few ways that the network can be used:

• the number of the tile to be played along with the move to be played,
• among all the possible moves, get the index of the chosen move,
• return one output number for each tile and each move, and play the tile that has the highest number along with the move that has the highest number,
• if used as a value network, just return one value: the fitness of the current position.

Again, there are many ways of using a neural network to play. We could even use two networks: one to choose the tile to play, and then a second one to choose the move. Whatever we choose, we have to keep in mind two main points here:

• the result can never be an invalid move, which is not always trivial,
• make sure that results cannot be incoherent. For instance, a valid possibility would be to have two integer outputs: one for the tile to play on the board, one for the move to play, each of them being applied the mathematical “mod” to make sure that the results are within range. But then there might be a discrepancy between the chosen tile and the chosen move. Maybe the move made perfect sense with another tile, but not with the one that was eventually selected.

# The Size of the Neural Network

I tend to reason in terms of complexity of the game to address this problem. Just think about “if I had to code a perfect player for this game, how many rules and cases would I have to take into account”.

The answer also depends on what you feed the network. If you feed it a lot of redundant information (for instance, feed it the board and the number of remaining tiles for each player), then the network will have to extract less metadata from the board.

In the case of the game “Einstein Würfelt Nicht”, I chose not to mostly not give any redundant information to the network. Given the size of the board, I believed that a simple network of just a few layers and a couple of hundred neurons would probably do the trick.

Then comes the number of connections between layers. In order to extract as much information as possible from the board, I believed that a fully connected first layer was needed – although I chose not to enforce it, but I gave a sufficient amount of connections for this to happen. So I started off with a first layer of 20 neurons, along with 500 connections between the board (which is a 25 bytes array, with an additional byte for the die). I have also tried other different variants.

# The Parameters of the Genetic Algorithm

## Population size, mutation and cross-over rates

I started off with a population of 100 individuals and made some tests with 200. In that population, I chose to keep a fair amount of the best individuals, 10 to 30%, without checking their scores. All the others are discarded and replaced with either new individuals, either top individuals that have been mutated and crossed-over.

As for the mutation rate, I made it randomly chosen between 1/1000 and ten to twenty per 1000. That is to say that to create a mutated individual, 1 to 10/20 random mutations are applied for every 1000 bytes of its DNA. Note that with a network of 10000 elements, that’s just a few mutations in the network. A mutation can be a change in an operation, a connection move or a change in parameters such as weight and offset.

As for the crossover rate, I made it from 0.01% to 1%. As we will see later, it wasn’t that successful in the first versions.

## Evaluating players

Another important parameter is the accuracy of the evaluation for every individual. In the case of a game, it can be measured by having this individual play many games against other players. Other players may be other individuals of the population and/or a fixed hard-coded player. The more games are played, the more accurate the rating of a player. And this is getting more and more critical as the player improves: at the beginning, the player just needs to improve very basic skills and moves, it is failing often anyway, so it is easy to tell the difference between a good player and a bad one. As it improves, it becomes more and more difficult to find situations in which players can gain an advantage or make a slight mistake.

In the case of EWN, as it is a highly probabilistic game, the number of matches that are required can grow exponentially. Note that there are even a large number of starting positions: 6! x 6! which is roughly 500 thousand permutations. With symmetries we can remove some of them, but there is still a large number of starting positions despite the very simple placing rules. So even if you play 100.000 games for a player, it is still not covering the wide variety of openings. What if your player handles well those 100.000 openings but is totally lame at playing the rest? Not even mentioning the number of possible positions after a few turns.

A good indicator to check whether we have played enough games to correctly rate players is the “stability” of the ranking of players as we continue playing games. As the rankings stabilize (eg for instance the top player remains at the top for quite a long time) we are getting better and better accuracy.

## Individuals selection and breeding

As I developed this and started testing it, I realized that the evolution was going very slowly: new individuals were bad in general, with only a few of them reaching the “elite” of the population. That’s because of the randomness of the alterations. We will see later how I tackled that problem.

As I was going forward in this and observing how slow the process was on a CPU, I also started planning to switch the whole evaluation process to the GPU.

## GANN-EWN / Part 4 / Developing a Genetic Algorithm from scratch

Welcome to this fourth part of building Neural Networks evolved by Genetic Algorithms. If you haven’t done so yet, I would advise to read the first, second and third parts before reading this article.

So, what exactly is a Genetic Algorithm? The name applied to Computer Science sounds both scary and mysterious.

However, it is actually quite simple and refers to Darwin’s theory of evolution which can be summed up in one simple sentence: “The fittest individuals in a given environment have a better chance than other individuals of surviving and having an offspring that will survive”. Additionally, the offspring carries a mutated crossover version of the genes of the parents.

Given this general definition, the transposition to Computer Science is the following:

• design “individuals” to solve a particular problem, whose parameters are in the form of a “chromosome”, most of the time a simple array of bits,
• create a “population” of these individuals,
• evaluate all individuals in the population and sort them by their fitness (e.g. how close they get to solving the problem well),
• create new individuals by making crossovers and mutations on the best individuals of the current generation,
• replace the worst individuals in the population by those new individuals,
• rinse and repeat: go back to evaluating all individuals in the population.

With this in mind, the choice in part 3 to store our neural networks in simple arrays comes into a new light: those arrays are the chromosomes of our individuals.

# Our Genetic Algorithm

The genetic algorithm I built went through several phases already.

Here is the first phase:

1. generate n individuals, each representing a player, note that players may also be non Neural-Network-driven players or players using different types of NNs, we can actually mix players of different types in the population,
2. make them play against each other a certain number of games,
3. select the ones with more wins and discard the others,
4. replace the discarded players either by new random players, either by mutations and crossovers of the best players.
5. go back to step 2.

Still, with this simple algorithm, there are many possible parameters:

• the population size,
• how many individuals to keep? Keep the top x %? Or all the ones managing to have a certain score?
• how much mutation and crossover percentage should be applied?
• how many games to play to get a good confidence score on every player?

# The supporting UI

To evaluate the impacts of these parameters, I then built an UI on top of this to observe the evolution of the population and see how the best individuals evolve. The UI is made of a simple table, sorted by “performance” (that’s to say the percentage of wins). Every row shows one individual, its ranking, its age (since a single individual can survive for multiple generations), its original parent, and the number of total mutated generations it comes from.

Later, I also added at the bottom of the screen the progression of the score of the best individuals.

Here is a simple screenshot:

The green part represents the individuals that will be selected for the next generation. All individuals in red will be discarded. The gray ones are non NN implementations that can be used as “benchmarks” against the NN implementations. When the first population is created randomly, they are generally beaten very easily by those standard implementations, but we can see that after some iterations, the NNs selected one generation after the other end up beating those standard implementations. We’ll dig into that in the next post, along with the choice of the different parameters.

Next comes part 5.

## GANN-EWN / Part 3 / building a Neural Network from scratch

Welcome to the third part of the GANN-EWN (Genetic Algorithms for Neural Networks applied to the game “Einstein Würfelt Nicht”) series! If you haven’t read it yet, it’s probably time to read the first part and/or the second part.

In this part, we’ll start building a Neural Network from scratch.

I assume that you’re familiar with Neural Networks, which are basically a simplified stereotype of what people thought neurons in the brain were like in the 1950’s. 🙂 Since then, biological research has evolved a lot and we know now that real neurons and their interactions are much more complex than the general models of Computer Neural Networks that are around.

Anyway, there are many types of neural networks around, all used for different tasks. My attempt is to build a very generic neural network structure that can then be used in various environments, typically to play different games.

# Important structural decisions

Since the early days of neural network research, the transfer functions used in artificial neurons have been carefully chosen to enable backpropagation. In here, though, we want to use our neural network through Genetic Algorithms by selecting the best neural networks within a population, and we won’t necessarily be using backpropagation to make the network “learn”. So we can use any kinds of transfer functions we want.

At first, I also wondered if I would be using totally random graphs for my network structure, but it poses problems such as cycles which require more computational power to deal with so I stayed with a “simple” layer architecture. But because I wanted to code a network that would be as generic as possible, the only constraints I kept was to have a structure with layers: links between neurons of different layers can be as random as possible, with the only requirement that every neuron should have at least one input and one output (a neuron without any input or output would simply be useless and would waste resources unnecessarily). On the other hand, I didn’t put any restrictions on duplicate links from one neuron to another. After all, it could be useful to have the sinus of something and combine it with its raw value.

Another important decision was to limit to the strict minimum the manipulation of floating point values. There are two reasons for this, which are both linked to the projected use of GPUs for my research, and based on my own experience of using GPUs. The first one is that GPUs in general are not so great, performance-wise, with floating point values. The second is that the results of floating point operations may differ between CPU operations and GPU operations, due to rounding and approximating some functions on the GPU. Besides, most GPUs have limited support for double floating point operations, which makes portability some kind of a pain. We will actually discuss that in more detail in another article.

So I decided to take only integers as neuron and link input and output values, as well as the weights and shifts that could be applied on all operations. However, some operations will need some floating point numbers, such as sigmoid and tangent functions, but the results will be transformed back into integers as soon as possible so that floats are used as little as possible.

# Description of neurons and links

Besides, as I didn’t stick with “simple” linear or sigmoid functions for my neurons, the links themselves also have interesting properties. Here is a simple figure showing the basics of my neurons and links:

So the output of every link is of the form: out = unary_op ( (in + shift) x weight )

As for computing the result of a neuron, I used what is generally referred to as “genetic programming” where the genetic algorithm doesn’t mutates “parameters” for a program, but rather a tree of instructions (a program). So in my case, a neuron contains a tree of mathematical operations to be applied on the inputs.

So the output of a neuron having n inputs is:

• the input itself if n==1
• the result of a tree of n-1 binary operations

For instance, the tree for the previous picture is the following:

The unary operations I have chosen are: id (identity), sinus, log, square, square root, tan, and negative.

The roughly described binary operations are:

• add (actually average, we’ll see why later),
• divide (leave numerator unchanged if denominator is 0),
• subtract,
• multiply (with an adjusting weight),
• max and min,
• an hyperbolic paraboloid and its reverse.

With this in mind, it is actually not difficult to switch this network to any type of network, including simple linear functions. Just limit the allowed operations to “+” and we’re pretty good to go. The same goes with links, just restrict them to use “identity” as their unary operations and set the weights as desired, just leave the offset (often called “bias” in neural network computing).

Besides, as the link structure is not constrained either, we can create any type of network we want by forcing the links to be in a certain way when we build the network.

# Integers and range

There is one remaining problem here: if you store everything as integers and never stop adding and multiplying numbers, you’ll overflow at some point.

So I kept all operations to be within a certain range and added weights and min/max to make sure that all numbers would be kept within that range. How to define the range? It has to be as big as possible to allow for enough precision (obviously if we keep only integer values from -10 to 10, we won’t be able to store Pi using a good precision, but if we go up to a million, then we can store 3.14159, which is already not that bad). On the other hand, the range should allow not to lose precision while doing operations such as multiplying integers. Integers in Java or OpenCl have a rough range of -2 billion to 2 billion so 10.000 should do it (the square is 100 million, perfectly in range).

So within the network, all values circulating should be within the -10.000 to 10.000 range, which means that some adjustments had to be made on the different binary and unary operations, in order to always keep results within that range. For instance, adding is converted to an average, to make sure the result is still within the desired range.

# Encoding and storing the networks

Because the ultimate goal is to run this on GPUs, the networks cannot be programmed as Objects. Instead, plain arrays of integers are used. One other advantage is that it is very simple to store an array of integers and it could be manipulated in any programming language (provided that we code the program that would interpret this array).

Here is the description of the array describing a network so far:

• [0]: the number of layers L, including the output which is considered as a layer, but excluding the input,
• [1 : L]: the number of neurons in each layer (including the output layer, but excluding the input),
• [L+1 : 2xL]: the number of links between each layer (at position L+1, we have the number of links between the input and the first hidden layer),
• [2xL+1:…]: all link parameters (we will go over them shortly),
• after the links: all binary operations for the links, for every layer connectivity there are as many as links minus one.

Every link contains 5 integers and is described as follows:

• the number of the source neuron in the source layer (this number is unique within the network, so the number of the first neuron of the first hidden layer is equal to the number of inputs),
• the number of the destination neuron in the destination layer,
• the applied offset on incoming values,
• the applied weight applied on the result of the offset,
• the code for the unary operation applied to that result.

Let’s calculate the size of an array needed to encode the following network: 20 connections between inputs and the first hidden layer, 10 neurons in the first hidden layer, 30 connections from the first hidden layer to the second hidden layer, 4 neurons in the second hidden layer, 10 connections from the second hidden layer to the 2 outputs.

1 + 2 x 2 + 5 x 50 + 19 + 29 = 303 elements

For now, the networks are stored within a simple local H2 database.

# Watching it in action

Finally, I built a small UI to watch how the network behaves when its inputs change. Later, it might come in handy to see how the network behaves during a real game:

The inputs are on the left, and can be entered either numerically or from sliders, and the outputs are read on the right (this one has only two outputs). The two hidden layers (containing 4 and 3 neurons) are clearly visible. Besides, every link has the offset represented as a bar (red means negative, green means positive, full gray is 0) and the weights are represented as filled circles.

In the next article, we’ll build the genetic algorithm part of the project.

The next post of the series is here.

## GANN-EWN / Part 2 / Building a Hand-Crafted Player For EWN

Welcome to this second part of the GANN-EWN (Genetic Algorithms for Neural Networks applied to the game “Einstein Würfelt Nicht”) series! If you haven’t read it yet, it’s probably time to read the first part.

The game is quite simple: at the beginning of the game every player has 6 pieces labeled from 1 to 6, and every player throws a die at every turn. Depending on the result of the throw, the player may move one or two of his own pieces. The goal is to reach the opposite side of the board (diagonally) or capture all opponent’s pieces.

The rules with the dice are simple as well. If the die corresponds exactly to one of his pieces, he has to play that piece in any direction forward (the three squares: up, up left, or left), like in here where the die shows “5”:

If no piece with the same number are on the board, then one of the two pieces that are the closest to the die can be played. For instance, if the die was “2”, then the “1” and “3” can be played here:

Again, if the die is “3” then on this board the “2” or the “5” can be played:

Note that if there was no “2” on this board, then only the “5” could be played since there is no lower number than “3” on the board.

The first thing to do was to code a simple board implementing the rules of the game, which is quite straightforward. I chose Java, which is probably the programming language I’m most comfortable with, but also for another reason, which I will develop later.

To have something to compare my neural networks to, and also to test my implementation of the EWN Board and game logic, my first first step was to implement a simple (and I can call it quite “naive”) implementation of a player, based on my very limited experience of the game. Remember that I just discovered EWN, so after having a look at some games from the best players, I realized that one common strategy is to “eat up” some of your own pieces early in the game, to give momentum to your remaining pieces. Although this is a “classic” strategy, it does have one drawback: you opponent may be able to take all your remaining pieces and you lose. Nevertheless, I decided to implement this rule, along with a few others. Here is the list of rules that are hard-coded in this first player:

• if one move is a winning move (capture the last opponent’s piece or reach the opposite side of the board), then play it,
• if an opponent piece is close to winning (basically at a certain distance from the goal), always take it,
• always take your own pieces if they are within a certain number (typically from 2 to 5) if you have more than a certain amount of pieces (you don’t want to take your pieces if you have too few of them, that could be suicide in some situations),
• move forward always the most advanced piece (rushing towards the goal),
• preferably move forward-up rather than simply forward or up.

And that’s it!

So yes, this is a very simple player, but one funny fact is that this player beats me consistently! That’s how weak I am at EWN as a player.

But as those rules can have parameters (how far must an adversary piece so that we capture it? etc.), I ran a simple computer simulation using a range of possible parameters (including totally disabling a rule). And those simulation showed me that, by far, the best parameters are:

• capture pieces that are at most at a distance of 2 from the goal, but actually this rule is not very important,
• take your own pieces from 2 to 5 until you have only 2 pieces left,
• move always forward-up,

Of course, every single rule taken on its own may have some exceptions, but my “simple player” doesn’t go that far.

I also created a “random” player which picks moves at random, again to have something to compare to that doesn’t vary much in its crappiness. 🙂 And indeed my simple player was definitely better than the random one, at least.

I published this first player on the littlegolem site, with a different user than my regular user.

This is when I discovered that the site was actually crawling with bots playing EWN! Nice, the challenge was getting interesting! That’s when I met with a German researcher who also wrote a bot to play EWN on littlegolem. He even made part of his PhD on this project, so needless to say that my own simple player was quite ridiculous compared to his, and got a very bad beating!

This is when I realized that EWN is actually a statistics-driven game, since you make a choice depending on the probabilities of the next dice throws. You have a “2” ahead and want to give it momentum to reach the opposite side of the board faster? Capture your own “1”, “3” and possibly “4” and “5” pieces. The enemy has one piece slightly ahead but that piece has a 1/6 chance to be picked by the dice? Just ignore it. But you will capture that piece if it has a 1/2 chance of being played. As the Wikipedia page points out, one of the first approach is to calculate probability tables for each piece to reach the goal. All the dynamics of that game are based on probabilities. Well, almost. So that makes it a very good candidate for a Monte-Carlo type approach.

As a reminder, the general principle behind the Monte-Carlo approach is: “at a given turn, for each possible move play as many random games as possible and play the one that leads to victory more often than the others.”

Brutal. But very effective and the easiest “AI” to program. I guess. At least on a CPU.

So I implemented a simple Monte-Carlo (that I will abbreviate as “MC” now on) algorithm and launched it against my simple player. And sure enough, the MC player won, even if it was just playing 1000 games with random moves to decide which move to pick. And I’m not speaking about a “little better” here. The MC player won almost 70 games out of 100.

Then I thought: picking random moves is not really a likely outcome for a normal game, so the results for the MC player are somehow skewed. What if I use my simple player instead of a random player to play the games?

No surprise, using the simple player within the MC player improved the results against the simple player by quite a bit. But against the random MC player, the difference was not that big, it just improved it slightly, but not really significantly, only by 2.5 to 3% which is not bad but not as much as I would have expected. Note that running this simulation on a CPU is costing a lot of cycles. 1000 games with each roughly 15 to 20 moves. And at each turn, play 1000 games for every possible move (there are 1 to 6 possible moves). That’s a lot of moves to run! I’ve accelerated that to run on a GPU, but that will come later.

And still, the MC player lost to the German PhD’s carefully hand-crafted player. Well if a simple MC approach would do it, that’s what he would have picked in the first place!

This is it for this part. Next, we’ll start building a Neural Network with its associated Genetic Algorithm.

## GANN-EWN / Part 1 / GANNs (Genetic Algorithms applied to Neural Networks)

Combining Genetic Algorithms and Neural Networks is an idea that has been troubling my mind for the last 20 years. Unfortunately, life went in the way and I didn’t have a real chance of putting this idea into practice. Neural networks are very often used to do some “classifying” jobs, and they are very good at it. At least they got far beyond what we humans could program ourselves. However, as I have ideas for developing very interesting games, I also had in mind to have an AI play those games, and an AI that would actually learn how to play them, rather than programing it myself. This is exactly what the guys at DeepMind have been doing in the recent years. From Atari games to Go and Chess (and Shogi), they have amazed us all.

As I’m going back to my original interest in AI, I have taken upon myself to build an AI from scratch that would do exactly what I had in mind 20 years ago: learn how to play games.

In the meantime, things in the AI field have evolved. Graphics cards have opened new horizons for training bigger and faster Neural Networks. I have had some graphics cards that I used at some point to mine some cryptocurrencies for fun. Those cards were AMD cards, and they were getting hot very easily (the 7970 was going up to 100 degrees Celsius, I suspect that the fitting of the radiators were poorly made). I set up a watercooling system for the whole miner. Although it was a very interesting experience, it came to an end rather quickly. since ASICs kicked in and rendered graphics cards useless.

At the time, I started a project for handwriting character recognition (more on this project later…), and I used one of those cards to apply some graphics transformations (Hough transform and such). But they also consume a lot of electricity. So, sorry AMD, but I switched to Nvidia (I don’t own any stocks of either of those companies… or any other company, by the way, but I do believe that AMD is seriously losing ground here), which also has some great support with tools to build Neural Networks like TensorFlow.

So I wanted to start with a simple game and a simple goal:

• create a neural network framework from scratch, with NNs as generic as possible, including non linear functions, spikes, max and many operations, the ones that will be best will be selected by the genetic algorithm,
• a genetic algorithm to make those NNs evolve,
• all this should be able to run both on CPUs and GPUs (at least the most power-consuming parts), and I originally planed to learn some Cuda although I had prior OpenCl experience.

But everything had to start somewhere. And I didn’t want to tackle Chess or Go as my first GANN project. Anyway, the DeepMind guys have done that already!

So I had to aim for something simple for a start, and at the same time some game that I didn’t know well so that I wouldn’t be tempted to direct the algorithm and make “Stockfish for game XXX”.  On the same note, another strong criteria was also that it shouldn’t be a game that had been solved by mainstream software, like Chess, Checkers or Reversi/Othello. At the website littlegolem on which I play board games, there are some games that I don’t know well (yet) and also many games for which mainstream software is not really available. One of them is called “Einstein Würfelt Nicht” (which I will call EWN in the future) and is actually a dice game, played on a 5×5 board. When I started this project, I had never played that game but the rules seemed quite simple and I decided to have a go at it. This was of course the first attempt, the first prototype, basically some starting point, but certainly not the end point.

There are also two variants of the game on the site, which would be also nice to tackle. My initial goal there was to defeat human players and become a top player on the site. Note that this particular site is also very flexible with its players, and doesn’t ban anyone. If you don’t like playing against cheaters or robots, then that site is definitely not for you. But personally I learn a lot by playing with any kind of opponent, it doesn’t matter if I win or if I lose.

I will develop that story in the next articles, and I will also post some of the code soon on my gitlab account. However, I will not post the full code, don’t expect to be able to have a running bot that would play for you at EWN from there. But the generic code will be there and will be adaptable to other games as well.

The next part is here.