I am building a 2048 AI, and it is leading to a rather peculiar observation (peculiar enough to me).
The optimizations are not up to the mark right now (coupled with the fact that the code is written in python), which is letting me reach till only a depth of 3 moves (plies).
As evident from the results, Expectimax is quite dominant over minimax (similar results can be seen without alpha-beta pruning in minimax) in terms of results produced. Both use the same evaluation function and do not proceed any further than 3 moves.
AFAIK, minimax should work optimally in such games, but that doesn't seem to be the case here. My question is, this observation is due to the fact that:
I am not going deep enough into the search tree?
2048 is a stochastic game, and that is hampering the performance of minimax (or boosting the performance of expectimax)?
The opponent (the 2048 game logic) is not playing optimally (90-10 % chance of putting a 2-4 tile, random adversary) (if yes, then why should this affect the performance of minimax)?
Anything else that is not apparent to me?
I have to create an AI which has to compete against other AIs.
Both AIs will run on the same hardware, have the same amount of processing time and memory. I know the opponent AI will be using the minimax algorithm with alpha beta pruning.
Now my question is - what are some approaches for beating such an opponent? If I use minimax myself - then both AI's perfectly predict each other's moves and the game resolves based on an inherent property of the game (first move wins etc).
The obvious solution is to somehow see further ahead into the possible moves which would allow for a better evaluation - since the processor time is the same I couldn't evaluate to a greater depth (assuming the opposing AI code is equally optimized). I could use a precomputed tree for an extra advantage but without a super computer I certainly couldn't "solve" any nontrivial game.
Is there some value in intentionally picking a non optimal node such as one that alpha beta would have pruned? This could potentially incur a CPU time penalty on the opponent as they'd have to go back and re-evaluate the tree. It would incur a penalty on me as well as I'd have to evaluate the minimax tree + alpha beta to see which nodes alpha beta would prune without reaping any direct benefits.
What are some other strategies for optimizing against such an opponent?
First, there isn't any value in choosing a non-optimal line of play. Assuming your opponent will play optimally (and that's a fundamental assumption of minimax search), your opponent will make a move that capitalizes on the mistake. A good game engine will have a hashed refutation table entry containing the countermove for your blunder, so you'll gain no time by making a wild move. Making bad moves allows a computer opponent to find good moves faster.
The key thing to realize with a game like Othello is that you can't be sure what the optimal move is until late in the game. That's because the search tree is almost always too large to be exhaustively searched for all won or lost positions, and so minimax can't tell you with certainty which moves will lead to victory or defeat. You can only heuristically decide where to stop searching, arbitrarily call those nodes "terminal", and then run an evaluation function that guesses the win/loss potential of a position.
The evaluation function's job is to assess the value of a position, typically using static metrics that can be computed without searching the game tree further. Piece counts, positional features, endgame tablebases, and even opponent psychology can play a role here. The more intelligence you put into your evaluation function, generally the better your engine will play. But the point of static evaluation is replace searches that would be too expensive. If your evaluation function does too much or does what it does too inefficiently, it can become slower than the game tree search needed to obtain the same information. Knowing what to put in an evaluation function and when to use static evaluation instead of search is a large part of the art of writing a good game engine.
There are a lot of ways to improve standard minimax with AB pruning. For example, the killer heuristic attempts to improve the order moves are looked at, since AB's efficiency is better with well-ordered moves.
A lot of information on different search enhancements and variations on AB can be found at chessprogramming.wikispaces.com.
So I'm reading on evolutionary algorithms and am confused.
What are the 'traditional' differences between evolutionary programming, evolutionary strategies and genetic algorithms as I believe in modern day they have basically converged to the same thing?
My understanding is that genetic algorithms vary 'genes' to produce results, evolutionary strategies vary parameters which somehow changes the individuals. What does it mean exactly numerical parameters as per (http://en.wikipedia.org/wiki/Evolutionary_algorithm)? Evolutionary programming then varies primarily by mutation on real numbers?
Are evolutionary programming and genetic programming ways of finding a program to solve a problem while genetic algorithms and evolutionary strategies ways of finding a solution to a problem using candidates? The distinction is not visible to me and the only distinction I see in evolutionary strategies vs genetic algorithms is a list of parameters vs a chromosome and real numbers vs integers?
Thanks.
hoping this will clarify a bit things for you :
evolutionary algorithms: algorithms that try, inside a given population of candidates solutions, to find a "best fit" solution. The algorithm start with random candidates and try to evolve from there by reproduction, mutation, recombination and selection
From there, you have several "standard way" of calling several families of evolutionary algorithms. Theses names have been given by academic research ; I believe the subtle differences in the naming convention come both from subtle reasons and from lack of available vocabulary as time goes on :
genetic algorithm: The given population is defined as "a string of numbers" (traditionnaly a vector of 0 an 1s).
genetic programming: The given population is a set of structurally different computer programs. The question is which program is the "best fit" to solve a given problem. If you are familiar with Lisp processing, you will know that a whole program can be represented as a tree ; Imagine that your algorithm mutates or recombines these trees. You will end up with a lot of programs evolving out of your original candidates and some will be better than others at solving your problem.
evolutionary programming: Given a specific computer program that has a fixed structure but parameters, the studied population is the set of programs obtained by a variation of those parameters. For example, if you think your problem could be represented by a finite state machine, this technique can help you find the number of states, the authorized transitions between states, the probability of these transitions.
evolution strategy: the given population is a vector of real numbers + parameters like the mutation rate among studied candidates tries to reach an optimum. Imagine you start with a population of (c1,c2,c3) 3-dimensional vectors. At each generation you mutate them by adding a random value to c1, c2 and c3. The random value may be based on a Gaussian with a standard deviation S. It may be interesting to start with a big value of S, so that mutations will produce vectors all over the place and once you start finding interesting vectors, you start decreasing S in order to "focus" around these interesting vectors.
Remember that these names are just naming conventions. Some are good at describing what they mean, some are less optimized. Evolutionary naming in the field of AI is still a work in progress ;-)
From what I've read so far they seem very similar.
Differential evolution uses floating point numbers instead, and the solutions are called vectors? I'm not quite sure what that means.
If someone could provide an overview with a little bit about the advantages and disadvantages of both.
Well, both genetic algorithms and differential evolution are examples of evolutionary computation.
Genetic algorithms keep pretty closely to the metaphor of genetic reproduction. Even the language is mostly the same-- both talk of chromosomes, both talk of genes, the genes are distinct alphabets, both talk of crossover, and the crossover is fairly close to a low-level understanding of genetic reproduction, etc.
Differential evolution is in the same style, but the correspondences are not as exact. The first big change is that DE is using actual real numbers (in the strict mathematical sense-- they're implemented as floats, or doubles, or whatever, but in theory they're ranging over the field of reals.) As a result, the ideas of mutation and crossover are substantially different. The mutation operator is modified so far that it's hard for me to even see why it's called mutation, as such, except that it serves the same purpose of breaking things out of local minima.
On the plus side, there are a handful of results showing DEs are often more effective and/or more efficient than genetic algorithms. And when working in numerical optimization, it's nice to be able to represent things as actual real numbers instead of having to work your way around to a chromosomal kind of representation, first. (Note: I've read about them, but I've not messed extensively with them so I can't really comment from first hand knowledge.)
On the negative side, I don't think there's been any proof of convergence for DEs, yet.
Differential evolution is actually a specific subset of the broader space of genetic algorithms, with the following restrictions:
The genotype is some form of real-valued vector
The mutation / crossover operations make use of the difference between two or more vectors in the population to create a new vector (typically by adding some random proportion of the difference to one of the existing vectors, plus a small amount of random noise)
DE performs well for certain situations because the vectors can be considered to form a "cloud" that explores the high value areas of the solution solution space quite effectively. It's pretty closely related to particle swarm optimization in some senses.
It still has the usual GA problem of getting stuck in local minima however.
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I am trying to get a feel for the difference between the various classes of machine-learning algorithms.
I understand that the implementations of evolutionary algorithms are quite different from the implementations of neural networks.
However, they both seem to be geared at determining a correlation between inputs and outputs from a potentially noisy set of training/historical data.
From a qualitative perspective, are there problem domains that are better targets for neural networks as opposed to evolutionary algorithms?
I've skimmed some articles that suggest using them in a complementary fashion. Is there a decent example of a use case for that?
Here is the deal: in machine learning problems, you typically have two components:
a) The model (function class, etc)
b) Methods of fitting the model (optimizaiton algorithms)
Neural networks are a model: given a layout and a setting of weights, the neural net produces some output. There exist some canonical methods of fitting neural nets, such as backpropagation, contrastive divergence, etc. However, the big point of neural networks is that if someone gave you the 'right' weights, you'd do well on the problem.
Evolutionary algorithms address the second part -- fitting the model. Again, there are some canonical models that go with evolutionary algorithms: for example, evolutionary programming typically tries to optimize over all programs of a particular type. However, EAs are essentially a way of finding the right parameter values for a particular model. Usually, you write your model parameters in such a way that the crossover operation is a reasonable thing to do and turn the EA crank to get a reasonable setting of parameters out.
Now, you could, for example, use evolutionary algorithms to train a neural network and I'm sure it's been done. However, the critical bit that EA require to work is that the crossover operation must be a reasonable thing to do -- by taking part of the parameters from one reasonable setting and the rest from another reasonable setting, you'll often end up with an even better parameter setting. Most times EA is used, this is not the case and it ends up being something like simulated annealing, only more confusing and inefficient.
Problems that require "intuition" are better suited to ANNs, for example hand writing recognition. You train a neural network with a huge amount of input and rate it until you're done (this takes a long time), but afterwards you have a blackbox algorithm/system that can "guess" the hand writing, so you keep your little brain and use it as a module for many years or something. Because training a quality ANN for a complex problem can take months I'm worst case, and luck.
Most other evolutionary algorithms "calculate" an adhoc solution on the spot, in a sort of hill climbing pattern.
Also as pointed out in another answer, during runtime an ANN can "guess" faster than most other evolutionary algorithms can "calculate". However one must be careful, since the ANN is just "guessing" an it might be wrong.
Evolutionary, or more generically genetic algorithms, and neural networks can both be used for similar objectives, and other answers describe well the difference.
However, there is one specific case where evolutionary algorithms are more indicated than neural networks: when the solution space is non-differentiable.
Indeed, neural networks use gradient descent to learn from backpropagation (or similar algorithm). The calculation of a gradient relies on derivatives, which needs a continuous and derivative space, in other words that you can shift gradually and progressively from one solution to the next.
If your solution space is non-differentiable (ie, either you can choose solution A, or B, or C, but nothing in the middle like 0.5% A + 0.5% B, so that some solutions are impossible), then you are trying to fit a non-differentiable function, and then neural networks cannot work.
(Side note: discrete state space partially share the same issue and so are a common issue for most algorithms but there are usually some work done to workaround these issues, for example decision trees can work easily on categorical variables, while other models like svm have more difficulties and generally require encoding categorical variables into continuous values).
In this case, evolutionary and genetic algorithms are perfect, one could even say a god send, since they can "jump" from one solution to the next without any issue. They don't care that some solutions are impossible, nor that the gaps are big or small between subset of the possible state space, evolutionary algorithms can jump randomly far away or close by until they find appropriate solutions.
Also worth mentioning is that evolutionary algorithms are not subject to the curse of dimensionality as much as any other machine learning algorithm, including neural networks. This might seem a bit counter intuitive, since the convergence to a global maximum is not guaranteed, and the procedure might seem to be slow to evolve to a good solution, but in practice the selection procedure works fast and converges to a good local maximum.
This makes evolutionary algorithms a very versatile and generic tool to approach naively any problem, and one of the very few tools to deal with either non-differentiable functions, discrete functions, or with astronomically high dimensional datasets.
Look at Neuro Evolution. (NE)
The current best methods is NEAT and HyperNEAT by Kenneth Stanley.
Genetic Algorithms only find a genome of some sort; It's great to create the genome of a neural network, because you get the reactive nature of the neural network, rather than just a bunch of static genes.
There's not many limits to what it can learn. But it takes time of course. Neural topology have to be evolved through the usual mutation and crossover, as well as weights updated. There can be no back propagation.
Also you can train it with a fitness function, which is thus superior to back propagation when you do not know what the output should be. Perfect for learning complex behaviour for systems that you do not know any optimal strategies for. Only problem is that it'll learn behaviour you didn't anticipate. Often that behaviour can be very alien, although it does exactly what you rewarded it for in the fitness function. Thus you'll be using as much time deriving fitness functions as you would have creating output sets for backpropagation :P
Evolutionary algorithms (EAs) are slow because they rely on unsupervised learning: EAs are told that some solutions are better than others, but not how to improve them. Neural networks are generally faster, being an instance of supervised learning: they know how to make a solution better by using gradient descent within a function space over certain parameters; this allows them to reach a valid solution faster. Neural networks are often used when there isn't enough knowledge about the problem for other methods to work.
In terms of problem domains, I compare artificial neural networks trained by backpropagation to an evolutionary algorithm.
An evolutionary algorithm deploys a randomized beamsearch, that means your evolutionary operators develop candidates to be tested and compared by their fitness. Those operators are usually non deterministic and you can design them so they can both find candidates in close proximity and candidates that are further away in the parameter space to overcome the problem of getting stuck in local optima.
However the success of a EA approach greatly depends on the model you develop, which is a tradeoff between high expression potential (you might overfit) and generality (the model might not be able to express the target function).
Because neural networks usually are multilayered the parameter space is not convex and contains local optima, the gradient descent algorithms might get stuck in. The gradient descent is a deterministic algorithm, that searches through close proximity. That's why neural networks usually are randomly initialised and why you should train many more than one model.
Moreover you know each hidden node in a neural network defines a hyperplane you can design a neural network so it fits your problem well. There are some techniques to prevent neural networks from overfitting.
All in all, neural networks might be trained fast and get reasonable results with few efford (just try some parameters). In theory a neural network that is large enough is able to approximate every target function, which on the other side makes it prone to overfitting. Evolutionary algorithms require you to make a lot of design choices to get good results, the hardest probably being which model to optimise. But EA are able to search through very complex problem spaces (in a manner you define) and get good results quickly. AEs even can stay successful when the problem (the target function) is changing over time.
Tom Mitchell's Machine Learning Book:
http://www.cs.cmu.edu/~tom/mlbook.html
Evolutionary algorithms (EA) represent a manner of training a model, where as neuronal nets (NN) ARE a model. Most commonly throughout the literature, you will find that NNs are trained using the backpropagation algorithm. This method is very attractive to mathematicians BUT it requires that you can express the error rate of the model using a mathematical formula. This is the case for situations in which you know lots of input and output values for the function that you are trying to approximate. This problem can be modeled mathematically, as the minimization of a loss function, which can be achieved thanks to calculus (and that is why mathematicians love it).
But neuronal nets are also useful for modeling systems which try to maximize or minimize some outcome, the formula of which is very difficult to model mathematically. For instance, a neuronal net could control the muscles of a cyborg to achieve running. At each different time frame, the model would have to establish how much tension should be present in each muscle of the cyborg's body, based on the input from various sensors. It is impossible to provide such training data. EAs allow training by only providing a manner of evaluation of the model. For our example, we would punish falling and reward the traveled distance across a surface (in a fixed timeframe). EA would just select the models which do their best in this sense. First generations suck but, surprisingly, after a few hundred generations, such individuals achieve very "natural" movements and manage to run without falling off. Such models may also be capable of dealing with obstacles and external physical forces.