UCB bandit algorithms are quite simple to understand, and the generalisation of them by the UCT algorithm to tree search spaces seems like a pretty clean idea.
I had a go at using UCT to explore a search space earlier this year (not games, but selecting a good sequence of decisions inside a simulation).
The problem I ran into with this approach was that the algorithm has no way to generalise or lift experience from one explored state to other similar / unexplored states, and my search space was large. So having to evaluate each state at least once was prohibitive.
This isn't really a criticism of how well UCT can tackle the problem as framed, but more a criticism of how much better one might be able to do with a bit more structure -- e.g. perhaps with the addition of some kind of similarity metric between states, along with some bounds or assumptions on how the values of "similar" states should relate to each other.
It seems like combining some machine learning / statistical / approximation techniques to approximate the true state->value function with a UCT-like approach to guide exploration could be very effective for the right problem, but getting it to work well might be more of an art than a science.
If anyone has any pointers to reading about this kind of thing, I'd be most grateful! I have previously read about "approximate dynamic programming" techniques, e.g. [1]
Any sequence of one of more decisions with varying results is a game, but you probably know that.
Aside from estimating the value of a node from similar nodes, another option is to reduce the search space by putting them in the same information set. This means that to one player they appear indistinguishable, so the same strategy is played in all of them, while the other players might still be able to distinguish them. If your game has a huge state space it's likely that some of the state just hardly matters some of the time to some of the opponents. Other methods of fusing states have the disadvantage that all players have to care about the same information.
To implement MCTS with information sets, use the Information Set MCTS algorithm (ISMCTS) [1], which is a pretty obvious and intuitive extension of MCTS. (Has nobody really described ISMCTS before?) Whitehouse's PhD thesis [2] goes into more detail, and looks at using ISMCTS to reduce the search space in games ("ICARUS"), though I didn't read that chapter. (Earlier this year I mostly-implemented ISMCTS for Cordial Minuet, a game with simultaneous moves in addition to imperfect information. Information sets are unavoidable in simultaneous-move games.)
Also, you will find loads of papers specifically discussing speeding up MCTS for Go, many of which generalise.
Excellent. I did want to eventually write something showing how to modify this algorithm to handle games with randomness, and ISMCTS at a glance looks like it does just that. Thanks!
Actually, you don't need information sets to handle games with chance nodes. Standard MCTS works just fine for those. (Don't use a bandit algorithm at the chance nodes, just pick randomly.) Information sets are only needed if you have uncertainty, either because players make simultaneous moves or a player doesn't know (or forgets) the complete state of the game.
(That was before Google acquired DeepMind, and probably a large part of the reason.)
This uses a deep convolutional NN as an approximation function of the expected value. Interestingly there are only 3 hidden layers in this CNN, and it only gets fed the last 4 frames as state. Of course normally you would feed your approximator a direct encoding of the state. If you wanted to use a NN to evaluate nodes in gametree search you would have to invent some other way to train it. Would love to know whether there is research on that.
This algorithm (in it's regular form used often in games and examples) has one interesting "downside" I was exploring some time ago - selection is performed using the UCB formula. So basically it tries to maximize the player payout. But in the most games this is in fact impractical assumption, because we end up tending to expand branches that will be most likely not chosen by our opponent. As in the example (I assume gray moves are "our" moves) - we will much more likely choose to expand 5/6 branch instead of the 2/4, that will be in fact more likely chosen by our opponent.
In a game state where your opponent will be choosing the next move, you should select the next state to expand based on a UCB formula involving your opponent's expected payout, not your own.
But this requires storing and back propagating this info for the other players - something I really haven't seen in any examples (nor in this article). We cannot also assume that game is always zero-sum game and this information is not needed.
Actually, I think it should be 3/8, since it's supposed to be the opposing player's win count.
These diagrams were supposed to just be an update of the diagrams from the MCTS Wikipedia article, adding the correct backpropagation values, but I see there are more problems than that.
This is exactly the problem I've written about - AFAIK basic UCT algorithm does not model your opponent in the selection phase (only in simulation when some heavier than random logic is applied).
In your example script yes, but basic UCT does not do that - simply because UCT was not meant only for multi-player games in the beginning. And this is some assumption we make about our opponents (actually that they want to maximize their payout, not to e.g. win or minimize our score). Of course this is a very straightforward "application" of UCT or MCTS to the multiplayer games that was done in works of Sturtevant and Cazenave in 2008.
But it is not so easy to know about this, e.g. this is a very recent change to wikipedia page on the topic
https://en.wikipedia.org/w/index.php?title=Monte_Carlo_tree_... and very often people writing on the topic are not pointing this out at all, which I find very strange and misleading.
Also in the classic MCTS you should select move which has most visits, not the one with the highest percentage of wins.
In what appears to be the originating paper for UCT ("Bandit based Monte-Carlo Planning", L. Kocsis and C. Szepesvári, 2006),
* The choice of action to be returned is the one "with the highest average observed long-term reward"
* For simplicity, the payout value used in the paper is 1=win and 0=loss, which will result in the agents maximizing their wins. Presumably one could choose other payout values (i.e. points for that player in games that have that concept) to adjust the priority of the agents. The mathematics does not seem to forbid it.
* This paper uses as their first experiment a multi-player game. They state, "...for P-games UCT is modified to a negamax-style: In MIN nodes the negative of estimated action-values is used in the action selection procedures." It is straightforward and more generalizable to games with N > 2 players to simply record values from that player's standpoint in the first place, instead of manipulating it after the fact in this manner.
From the "An analysis of UCT in multi-player games", Nathan Sturtevant, 2008:
"Multi-player UCT is nearly identical to regular UCT. At the highest level of the algorithm, the tree is repeatedly sampled until it is time to make an action. The sampling process is illustrated in Figure 2. The only difference between this code and a two-player implementation is that in line 5 the average score for player p is used instead of a single average payoff for the state."
I think this is kind of a clear statement that original paper (and after it a lot of writing on the topic) may be lacking. Of course people used this simple generalization before and it is pretty straightforward, but it is not that obvious at a first glance.
And I've seen quite a lot of code examples, images explaining UCT for games and articles that were just not saying a word on this. Or even worse - just doing it wrong for multiplayer games.
Choice of action is a different topic, as I remember correctly there was also a paper proving that win rate and most robust branch are in the end performing the same ;)
Hope you will continue this series, because it is really good and code examples are really nice!
I had a go at using UCT to explore a search space earlier this year (not games, but selecting a good sequence of decisions inside a simulation).
The problem I ran into with this approach was that the algorithm has no way to generalise or lift experience from one explored state to other similar / unexplored states, and my search space was large. So having to evaluate each state at least once was prohibitive.
This isn't really a criticism of how well UCT can tackle the problem as framed, but more a criticism of how much better one might be able to do with a bit more structure -- e.g. perhaps with the addition of some kind of similarity metric between states, along with some bounds or assumptions on how the values of "similar" states should relate to each other.
It seems like combining some machine learning / statistical / approximation techniques to approximate the true state->value function with a UCT-like approach to guide exploration could be very effective for the right problem, but getting it to work well might be more of an art than a science.
If anyone has any pointers to reading about this kind of thing, I'd be most grateful! I have previously read about "approximate dynamic programming" techniques, e.g. [1]
[1] http://castlelab.princeton.edu/Papers/Powell-NRLWhat%20you%2...