I'm working on an AI to play a fairly simple game, using minimax and genetic algorithms to find weights to score board states with.
The game resembles 4x4 tictactoe, but a turn can be spent to move a piece to an adjacent space, pieces come in different sizes, and larger pieces can cover up smaller pieces.
I want to score the board by looking at a variety of factors, such as how close they are to completing a 4 in a row, and how many adjacent enemy pieces could potentially be moved onto, but I have no idea what these factors should specifically be.
My ideas:
For each line, making a scoring expression based on number of friendly pieces, number of empty spaces, and number of enemy pieces, but I can't think of a simple expression to score that with weights, since the value probably won't be a linear function.
For each line, making a piecewise scoring expression split based on the number of enemy pieces in the row, and an expression based on the number of allies. Hence having 1 piece in an empty row might be worth more than having 1 piece in a row full of enemies, thus blocking them off, and the inverse would be true for having 3 in a row in a row that is already blocked.
Some complications I've noticed:
Having 3 pieces in a row, but then one of the enemy large pieces also in the row, is virtually worthless for anything except preventing their piece's movement.
Having 3 pieces in a row, with a small enemy piece in that row is almost a win, if you can place a large piece adjacent to their small piece to move onto it. This seems particularly hard to detect. It's also possible if this is worked into a factor, the above "number of adjacent enemies that can be moved onto" won't be necessary.
Thanks for any help. I have no clue how to proceed.
Related
I'm refering mostly to this paper here: http://clgiles.ist.psu.edu/papers/UMD-CS-TR-3617.what.size.neural.net.to.use.pdf
Current Setup:
I'm currently trying to port the neural-genetic AI solution that I have laying around to get into a multi-purpose multi-agent tool. So, for example, it should work as an AI in a game engine for moving around entities and let 'em shoot and destroy the enemy (so e.g. 4 inputs like distance x,y and angle x,y and 2 outputs like accelerate left,right).
The state so far is that I'm using the same amount of genomes as there are agents to determine the fittest agents. 20% of the fittest agents are combined with each other (zz, zw genomes selected) and create 2 babies for the new population each. The rest of the new population per-new-generation is selected randomly across the old population, including the fittest with-an-unfit-genome.
That works pretty well to prime the AI, after generation 50-100 it is pretty much human-unbeatable in a Breakout clone and a little Tank game where you can shoot and move around.
As I had the idea to use on evolution population for each "type of Agent" the question is now if it is possible to determine the amount of hidden layers and the amount of neurons in the hidden layers generically.
My setup for the tank game is 4 inputs, 3 outputs and 1 hidden layer with 12 neurons that worked the best (around 50 generations to be really strong).
My setup for a breakout game is 6 inputs, 2 outputs and 2 hidden layers with 12 neurons that seems to work best.
Done Research:
So, back to the paper: On page 32 you can see that it seems that more neurons per hidden layer need of course more time for priming, but the more neurons are in between, the more are the chances to get into the function without noise.
I currently prime my AI only using the fitness increase on successfully being better than the last try.
So in a tank game it means he successfully shot the other tank (wounded him 4 times is better, then enemy is dead) and won the round.
In the breakout game it's similar as I have a paddle that the AI can move around and it can collect points. "Getting shot" or negative treatment here is that it forgot to catch the ball. So potential noise input would be 2 output values (move-left, move-right) that depend on 4 input values (ball x, y, degx, degy).
Questions:
So, what kind of calculation for the amount of hidden layers and amount of neurons do you think can be a good tradeoff to have no noise that kills the genome evolution?
What is the minimum amount of agents until you can say that "it evolves further"? My current training setup is always around having 50 agents that learn in parallel (so they basically simulate 50 games in parallel "behind the scenes").
In sum, for most problems, one could probably get decent performance (even without a second optimization step) by setting the hidden layer configuration using just two rules: (i) number of hidden layers equals one; and (ii) the number of neurons in that layer is the mean of the neurons in the input and output layers.
-doug
In short. It's an ongoing area of research. Most (All that I know of) ANN using numerous neurons and H-Layers don't set a static number of either, instead they use algorithms to continuously modify these values. Usually constructing and destroying when outputs converge/diverge.
Since it sounds like you're already using some evolutionary computing, consider looking into Andrew Turner's work on CGPANN, I remember it getting pretty decent improvements on benchmarks similar to your work.
My question is about this topic I've been reading about a bit. Basically my understanding is that in higher dimensions all points end up being very close to each other.
The doubt I have is whether this means that calculating distances the usual way (euclidean for instance) is valid or not. If it were still valid, this would mean that when comparing vectors in high dimensions, the two most similar wouldn't differ much from a third one even when this third one could be completely unrelated.
Is this correct? Then in this case, how would you be able to tell whether you have a match or not?
Basically the distance measurement is still correct, however, it becomes meaningless when you have "real world" data, which is noisy.
The effect we talk about here is that a high distance between two points in one dimension gets quickly overshadowed by small distances in all the other dimensions. That's why in the end, all points somewhat end up with the same distance. There exists a good illustration for this:
Say we want to classify data based on their value in each dimension. We just say we divide each dimension once (which has a range of 0..1). Values in [0, 0.5) are positive, values in [0.5, 1] are negative. With this rule, in 3 dimensions, 12.5% of the space are covered. In 5 dimensions, it is only 3.1%. In 10 dimensions, it is less than 0.1%.
So in each dimension we still allow half of the overall value range! Which is quite much. But all of it ends up in 0.1% of the total space -- the differences between these data points are huge in each dimension, but negligible over the whole space.
You can go further and say in each dimension you cut only 10% of the range. So you allow values in [0, 0.9). You still end up with less than 35% of the whole space covered in 10 dimensions. In 50 dimensions, it is 0.5%. So you see, wide ranges of data in each dimension are crammed into a very small portion of your search space.
That's why you need dimensionality reduction, where you basically disregard differences on less informative axes.
Here is a simple explanation in layman terms.
I tried to illustrate this with a simple illustration shown below.
Suppose you have some data features x1 and x2 (you can assume they are blood pressure and blood sugar levels) and you want to perform K-nearest neighbor classification. If we plot the data in 2D, we can easily see that the data nicely group together, each point has some close neighbors that we can use for our calculations.
Now let's say we decide to consider a new third feature x3 (say age) for our analysis.
Case (b) shows a situation where all of our previous data comes from people the same age. You can see that they are all located at the same level along the age (x3) axis.
Now we can quickly see that if we want to consider age for our classification, there is a lot of empty space along the age(x3) axis.
The data that we currently have only over a single level for the age. What happens if we want to make a prediction for someone that has a different age(red dot)?
As you can see there are not enough data points close this point to calculate the distance and find some neighbors. So, If we want to have good predictions with this new third feature, we have to go and gather more data from people of different ages to fill the empty space along the age axis.
(C) It is essentially showing the same concept. Here assume our initial data, were gathered from people of different ages. (i.e we did not care about the age in our previous 2 feature classification task and might have assumed that this feature does not have an effect on our classification).
In this case , assume our 2D data come from people of different ages ( third feature). Now, what happens to our relatively closely located 2d data, if we plot them in 3D? If we plot them in 3D, we can see that now they are more distant from each other,(more sparse) in our new higher dimension space(3D). As a result, finding the neighbors becomes harder since we don't have enough data for different values along our new third feature.
You can imagine that as we add more dimensions the data become more and more apart. (In other words, we need more and more data if you want to avoid having sparsity in our data)
'Proximity' is a strategy game of territorial domination similar to Othello, Go and Risk.
Two players, uses a 10x12 hex grid. Game invented by Brian Cable in 2007.
Seems to be a worthy game for discussing a) optimal algorithm then b) how to build an AI.
Strategies are going to be probabilistic or heuristic-based, due to the randomness factor, and the insane branching factor (20^120).
So it will be kind of hard to compare objectively.
A compute time limit of 5 seconds max per turn seems reasonable => this rules out all brute-force attempts. (Play the game's AI on Expert level to get a feel - it does a very good job based on some simple heuristic)
Game: Flash version here, iPhone version iProximity here and many copies elsewhere on the web
Rules: here
Object: to have control of the most armies after all tiles have been placed. You start with an empty hexboard. Each turn you receive a randomly numbered tile (value between 1 and 20 armies) to place on any vacant board space. If this tile is adjacent to any ALLY tiles, it will strengthen each of those tile's defenses +1 (up to a max value of 20). If it is adjacent to any ENEMY tiles, it will take control over them IF its number is higher than the number on the enemy tile.
Thoughts on strategy: Here are some initial thoughts; setting the computer AI to Expert will probably teach a lot:
minimizing your perimeter seems to be a good strategy, to prevent flips and minimize worst-case damage
like in Go, leaving holes inside your formation is lethal, only more so with the hex grid because you can lose armies on up to 6 squares in one move
low-numbered tiles are a liability, so place them away from your main territory, near the board edges and scattered. You can also use low-numbered tiles to plug holes in your formation, or make small gains along the perimeter which the opponent will not tend to bother attacking.
a triangle formation of three pieces is strong since they mutually reinforce, and also reduce the perimeter
Each tile can be flipped at most 6 times, i.e. when its neighbor tiles are occupied. Control of a formation can flow back and forth. Sometimes you lose part of a formation and plug any holes to render that part of the board 'dead' and lock in your territory/ prevent further losses.
Low-numbered tiles are obvious-but-low-valued liabilities, but high-numbered tiles can be bigger liabilities if they get flipped (which is harder). One lucky play with a 20-army tile can cause a swing of 200 (from +100 to -100 armies). So tile placement will have both offensive and defensive considerations.
Comment 1,2,4 seem to resemble a minimax strategy where we minimize the maximum expected possible loss (modified by some probabilistic consideration of the value ß the opponent can get from 1..20 i.e. a structure which can only be flipped by a ß=20 tile is 'nearly impregnable'.)
I'm not clear what the implications of comments 3,5,6 are for optimal strategy.
Interested in comments from Go, Chess or Othello players.
(The sequel ProximityHD for XBox Live, allows 4-player -cooperative or -competitive local multiplayer increases the branching factor since you now have 5 tiles in your hand at any given time, of which you can only play one. Reinforcement of ally tiles is increased to +2 per ally.)
A former member of the U of A GAMES group here.
That branching factor is insane. Far worse than Go.
Basically, you're hooped.
The problem with this game is that it is not deterministic due to the selection of a random tile. This actually adds another layer of nodes between each existing layer of nodes in the tree. You'll be interested in my publications on *-Minimax to learn about techniques for searching in stochastic domains.
In order to complete one-ply searches before the end of this century, you're going to need some very aggressive forward pruning techniques. Throw provably best move out the window early and concentrate on building good move ordering.
For general algorithms, I would suggest you to check the research done by the Alberta University AI Games group: http://games.cs.ualberta.ca Many of the algorithms there guarantee to find optimal policies. However, I doubt you're really interested in finding the optimal, aim for the "good enough" unless you want to sell that game in Korea :D
From your description, I have understood the game to be a two-player with full-observability i.e. no hidden units and such and fully deterministic i.e. player's actions outcomes do not require rolling, then you should take a look at the real-time bounded-search minimax derivatives proposed by the U Alberta guys. However, being able to do bound as well the depth of the backups of the value function would perhaps be a nice way to add a "difficulty level" to your game. They have been doing some work - a bit fishy imo - on sampling the search space for improving value function estimates.
About the "strategy" section you describe: in the framework I am mentioning, you will have to encode that knowledge as an evaluation function. Look at the work of Michael Büro and others - also in the U Alberta group - for examples of such knowledge engineering.
Another possibility would be to pose the problem as a Reinforcement Learning problem, where adversary moves are compiled as "afterstates". Look that up on the Barto & Sutton book: http://webdocs.cs.ualberta.ca/~sutton/book/the-book.html However the value function for a RL problem resulting from such a compilation might prove a bit difficult to solve optimally - the number of states will blow up like an H-Bomb. However, if you see how to use a factored representation, things can be much easier. And your "strategy" could perhaps be encoded as some shaping function, which would be speeding up the learning process considerably.
EDIT: Damn English prepositions
I once wrote a Tetris AI that played Tetris quite well. The algorithm I used (described in this paper) is a two-step process.
In the first step, the programmer decides to track inputs that are "interesting" to the problem. In Tetris we might be interested in tracking how many gaps there are in a row because minimizing gaps could help place future pieces more easily. Another might be the average column height because it may be a bad idea to take risks if you're about to lose.
The second step is determining weights associated with each input. This is the part where I used a genetic algorithm. Any learning algorithm will do here, as long as the weights are adjusted over time based on the results. The idea is to let the computer decide how the input relates to the solution.
Using these inputs and their weights we can determine the value of taking any action. For example, if putting the straight line shape all the way in the right column will eliminate the gaps of 4 different rows, then this action could get a very high score if its weight is high. Likewise, laying it flat on top might actually cause gaps and so that action gets a low score.
I've always wondered if there's a way to apply a learning algorithm to the first step, where we find "interesting" potential inputs. It seems possible to write an algorithm where the computer first learns what inputs might be useful, then applies learning to weigh those inputs. Has anything been done like this before? Is it already being used in any AI applications?
In neural networks, you can select 'interesting' potential inputs by finding the ones that have the strongest correlation, positive or negative, with the classifications you're training for. I imagine you can do similarly in other contexts.
I think I might approach the problem you're describing by feeding more primitive data to a learning algorithm. For instance, a tetris game state may be described by the list of occupied cells. A string of bits describing this information would be a suitable input to that stage of the learning algorithm. actually training on that is still challenging; how do you know whether those are useful results. I suppose you could roll the whole algorithm into a single blob, where the algorithm is fed with the successive states of play and the output would just be the block placements, with higher scoring algorithms selected for future generations.
Another choice might be to use a large corpus of plays from other sources; such as recorded plays from human players or a hand-crafted ai, and select the algorithms who's outputs bear a strong correlation to some interesting fact or another from the future play, such as the score earned over the next 10 moves.
Yes, there is a way.
If you choose M selected features there are 2^M subsets, so there is a lot to look at.
I would to the following:
For each subset S
run your code to optimize the weights W
save S and the corresponding W
Then for each pair S-W, you can run G games for each pair and save the score L for each one. Now you have a table like this:
feature1 feature2 feature3 featureM subset_code game_number scoreL
1 0 1 1 S1 1 10500
1 0 1 1 S1 2 6230
...
0 1 1 0 S2 G + 1 30120
0 1 1 0 S2 G + 2 25900
Now you can run some component selection algorithm (PCA for example) and decide which features are worth to explain scoreL.
A tip: When running the code to optimize W, seed the random number generator, so that each different 'evolving brain' is tested against the same piece sequence.
I hope it helps in something!
I want to program a chess engine which learns to make good moves and win against other players. I've already coded a representation of the chess board and a function which outputs all possible moves. So I only need an evaluation function which says how good a given situation of the board is. Therefore, I would like to use an artificial neural network which should then evaluate a given position. The output should be a numerical value. The higher the value is, the better is the position for the white player.
My approach is to build a network of 385 neurons: There are six unique chess pieces and 64 fields on the board. So for every field we take 6 neurons (1 for every piece). If there is a white piece, the input value is 1. If there is a black piece, the value is -1. And if there is no piece of that sort on that field, the value is 0. In addition to that there should be 1 neuron for the player to move. If it is White's turn, the input value is 1 and if it's Black's turn, the value is -1.
I think that configuration of the neural network is quite good. But the main part is missing: How can I implement this neural network into a coding language (e.g. Delphi)? I think the weights for each neuron should be the same in the beginning. Depending on the result of a match, the weights should then be adjusted. But how? I think I should let 2 computer players (both using my engine) play against each other. If White wins, Black gets the feedback that its weights aren't good.
So it would be great if you could help me implementing the neural network into a coding language (best would be Delphi, otherwise pseudo-code). Thanks in advance!
In case somebody randomly finds this page. Given what we know now, what the OP proposes is almost certainly possible. In fact we managed to do it for a game with much larger state space - Go ( https://deepmind.com/research/case-studies/alphago-the-story-so-far ).
I don't see why you can't have a neural net for a static evaluator if you also do some classic mini-max lookahead with alpha-beta pruning. Lots of Chess engines use minimax with a braindead static evaluator that just adds up the pieces or something; it doesn't matter so much if you have enough levels of minimax. I don't know how much of an improvement the net would make but there's little to lose. Training it would be tricky though. I'd suggest using an engine that looks ahead many moves (and takes loads of CPU etc) to train the evaluator for an engine that looks ahead fewer moves. That way you end up with an engine that doesn't take as much CPU (hopefully).
Edit: I wrote the above in 2010, and now in 2020 Stockfish NNUE has done it. "The network is optimized and trained on the [classical Stockfish] evaluations of millions of positions at moderate search depth" and then used as a static evaluator, and in their initial tests they got an 80-elo improvement when using this static evaluator instead of their previous one (or, equivalently, the same elo with a little less CPU time). So yes it does work, and you don't even have to train the network at high search depth as I originally suggested: moderate search depth is enough, but the key is to use many millions of positions.
Been there, done that. Since there is no continuity in your problem (the value of a position is not closely related to an other position with only 1 change in the value of one input), there is very little chance a NN would work. And it never did in my experiments.
I would rather see a simulated annealing system with an ad-hoc heuristic (of which there are plenty out there) to evaluate the value of the position...
However, if you are set on using a NN, is is relatively easy to represent. A general NN is simply a graph, with each node being a neuron. Each neuron has a current activation value, and a transition formula to compute the next activation value, based on input values, i.e. activation values of all the nodes that have a link to it.
A more classical NN, that is with an input layer, an output layer, identical neurons for each layer, and no time-dependency, can thus be represented by an array of input nodes, an array of output nodes, and a linked graph of nodes connecting those. Each node possesses a current activation value, and a list of nodes it forwards to. Computing the output value is simply setting the activations of the input neurons to the input values, and iterating through each subsequent layer in turn, computing the activation values from the previous layer using the transition formula. When you have reached the last (output) layer, you have your result.
It is possible, but not trivial by any means.
https://erikbern.com/2014/11/29/deep-learning-for-chess/
To train his evaluation function, he utilized a lot of computing power to do so.
To summarize generally, you could go about it as follows. Your evaluation function is a feedforward NN. Let the matrix computations lead to a scalar output valuing how good the move is. The input vector for the network is the board state represented by all the pieces on the board so say white pawn is 1, white knight is 2... and empty space is 0. An example board state input vector is simply a sequence of 0-12's. This evaluation can be trained using grandmaster games (available at a fics database for example) for many games, minimizing loss between what the current parameters say is the highest valuation and what move the grandmasters made (which should have the highest valuation). This of course assumes that the grandmaster moves are correct and optimal.
What you need to train a ANN is either something like backpropagation learning or some form of a genetic algorithm. But chess is such an complex game that it is unlikly that a simple ANN will learn to play it - even more if the learning process is unsupervised.
Further, your question does not say anything about the number of layers. You want to use 385 input neurons to encode the current situation. But how do you want to decide what to do? On neuron per field? Highest excitation wins? But there is often more than one possible move.
Further you will need several hidden layers - the functions that can be represented with an input and an output layer without hidden layer are really limited.
So I do not want to prevent you from trying it, but chances for a successful implemenation and training within say one year or so a practically zero.
I tried to build and train an ANN to play Tic-tac-toe when I was 16 years or so ... and I failed. I would suggest to try such an simple game first.
The main problem I see here is one of training. You say you want your ANN to take the current board position and evaluate how good it is for a player. (I assume you will take every possible move for a player, apply it to the current board state, evaluate via the ANN and then take the one with the highest output - ie: hill climbing)
Your options as I see them are:
Develop some heuristic function to evaluate the board state and train the network off that. But that begs the question of why use an ANN at all, when you could just use your heuristic.
Use some statistical measure such as "How many games were won by white or black from this board configuration?", which would give you a fitness value between white or black. The difficulty with that is the amount of training data required for the size of your problem space.
With the second option you could always feed it board sequences from grandmaster games and hope there is enough coverage for the ANN to develop a solution.
Due to the complexity of the problem I'd want to throw the largest network (ie: lots of internal nodes) at it as I could without slowing down the training too much.
Your input algorithm is sound - all positions, all pieces, and both players are accounted for. You may need an input layer for every past state of the gameboard, so that past events are used as input again.
The output layer should (in some form) give the piece to move, and the location to move to.
Write a genetic algorithm using a connectome which contains all neuron weights and synapse strengths, and begin multiple separated gene pools with a large number of connectomes in each.
Make them play one another, keep the best handful, crossover and mutate the best connectomes to repopulate the pool.
Read blondie24 : http://www.amazon.co.uk/Blondie24-Playing-Kaufmann-Artificial-Intelligence/dp/1558607838.
It deals with checkers instead of chess but the principles are the same.
Came here to say what Silas said. Using a minimax algorithm, you can expect to be able to look ahead N moves. Using Alpha-beta pruning, you can expand that to theoretically 2*N moves, but more realistically 3*N/4 moves. Neural networks are really appropriate here.
Perhaps though a genetic algorithm could be used.