Game Theory Reveals the Future of Deep Learning
medium.com
medium.com
>Where do these regularizations come from and how can we select a good regularization? How do we handle impartial information? This is where a game theoretic viewpoint becomes important. Generalization is sometimes referred to as ‘structural risk minimization’.
Ok... and what useful insight do the game theorists have here? He doesn't elaborate at all. Regularization is a very well studied concept already. I don't see how this is connected to game theory at all. Regularization is all about priors and Occam's razor.
Nash Equilibrium from Wikipedia: "Amy is making the best decision she can, taking into account Phil's decision while Phil's decision remains unchanged, and Phil is making the best decision he can, taking into account Amy's decision while Amy's decision remains unchanged."
At the point this happens, a stable solution exists which is almost like saying that we have made the most rational decision given the information we had. Nash was novel in this theory to say that if the participants were rational they would converge on an equilibrium and not get lost in endless recursion. Endless recursion being something like participants endlessly guessing the other opponent move... If Amy knows Phil knows Amy knows Phil knows Amy knows Phil... Obviously it would be problematic for DL to plunge into infinite recursion.
I like to think of the Nash Equilibrium as a balanced teeter totter. When the teeter totter becomes off balanced rational systems will relearn to balance the odds again based the properties of risk (what Nash referred to as utility I believe). Once they do so, they once again converge on an equilibrium.
Intuitively, what can happen is that one of the problems is simpler to learn than another, and when one of the "players" becomes overwhelmingly good, then the other part of the network stops receiving useful feedback and is unable to find out a direction for improvement, the learning stops. A "sufficiently smart" method would be able to go to a Nash equilibrium even in this case, but current GAN methods can not, so you need to take steps to ensure that it doesn't happen - e.g. extra normalization or less training for the "simpler to learn" component.
It's not enough to train the "adversarial" networks to compete during each inference, on a meta-level you must ensure a form of cooperation to ensure that they are effective teachers for each other during the learning process. For a real life analogy, there's a difference between effective behavior during combat and effective behavior during sparring.
It seems like for AI a perfect equilibrium wouldn't be easy to reach, its often hard for the human brain to reach one after all. It'd be the fact that an equilbrium exists though and that it's trained to find it that generates the knowledge along the way. Kinda like a journey not the destination learning method. I'm just theorizing though.
My background is in more traditional statistical modeling. The adversarial networks paradigm is really fascinating to me because it's a new way to think about model fit problems. It may turn out to be the same as some traditional model fit statistics approaches, but they're usually not thought of as competitive, which seems to open up a new way of thinking of it.
At some point in the author's discussion of game theory, I began thinking that if DL does go that route, it will eventually rediscover Bayesian statistics, which can be derived from decision-theoretic principles. It seems like many things in DL have parallels in traditional statistical theory, but are expressed in a mechanistic way.
I guess my broader point is that I had the opposite reaction as a lot of people: rather than wondering how game theory might be relevant, to me it seemed inevitable, given the important role it plays in basic derivations of major inferential paradigms. Those derivations usually leave the mechanisms of decision theory a black box, and in DL the making the contents of the box is the focus.
The wiki entry of Nash equilibrium [1] and this recent preprint on improving GAN training [2] may help you understand the link between GAN and game theory. To me, this link seems plausible but very vague so far. Training GAN has not directly benefited from the math foundation of game theory. Correct me if I am wrong.
Now, since the competing networks can be formally characterized as a minimax two-player game, other theorems relating to these games may provide further insight in how to proceed.
(1) careful study, teaching, and applying the detailed mathematics of linear programming and, in particular, its approach to proving the basic von Neumann saddle point theorem of game theory,
(2) applications of game theory to US strategic systems analysis, e.g., the Blue side positioning submarines and targeting multi-warhead nuclear missiles and the Red side allocating submarine search resources so that if Red shoots first Blue gets maximum expected target value destroyed,
(4) study of game theory from
Guillermo Owen, Game Theory, W. B. Saunders.
(5) study of game theory, including Nash's result with a nice proof by Lemke and Sion's result, in
T. Parthasarathy and T. E. S. Raghavan, Some Topics in Two-Person Games, ISBN 0-444-00059-3, American Elsevier.
With this background, I read the OP and found no clear connections with anything in game theory.
Perhaps you mean [2], which does. However, the article seems to misinterpret this: the fact that that system approximates the Nash equilibrium should be expected. Any system (including humans) that performs well in those types of games will, almost by definition.
"We propose a new framework for estimating generative models via an adversarial process, in which we simultaneously train two models ... This framework corresponds to a minimax two-player game."
Edit: Regarding the paper you linked to. The fact that a Nash equilibrium is expected is an important part of why it's interesting to structure things this way: it's an optimization process guaranteed to stabilize (whereas it's problematic that other approaches will diverge).
"When applied to Leduc poker, Neural Fictitious Self-Play (NFSP) approached a Nash equilibrium, whereas common reinforcement learning methods diverged"
The articles claims (based on the paper I linked, and others) that:
What we see in these 3 players are 3 different ways game theory plays in Deep Learning. (1) As a means of describing and analyzing new DL architectures. (2) As a way to construct a learning strategy and (3) A way to predict behavior of human participants. The last application can make your skin crawl!
Of these claims, I think all are wrong.
One can certainly make the case that it is interesting to measure the performance of a system compared to the Nash equilibrium (when possible), but the author seems to think that the designers of the system are somehow using game theory to design the system.
There's a lot to game theory, e.g., one I omitted, differential game theory as in a missile chasing an airplane.
But the likely most relevant, first cut, simple view is that each of the two players, Red and Blue, has a list of moves they can make. Then there is one (von Neumann case) or two (Nash case) payoff matrices. Then each of Red and Blue picks a move or a probability distribution of moves.
The broad idea is to have and use a saddle point, maybe available only by making moves probabilistically in which case each of Red and Blue finds a probability distribution over the moves they can make and then picks a move according to their distribution and independent of everything else relevant.
So, just as in the classic children's game paper-scissors-rock, each player picks a move in secret, and then both players show their moves at the same time. Each player seeks to maximize their long run expected gain. As can be shown easily from linear programming and its duality theory, for paper-scissors-rock the saddle point probability distributions are that each player picks each of the three moves paper, scissors, rock with probability 1/3rd. Of course, Red and Blue pick their moves independently and in secret. The matrix for paper-scissors-rock is 3 x 3. But the theory and the linear programming solution applies the same for positive integers m and n for m moves for Red, n moves for Blue, and a matrix m x n. All the components of the matrix are >= 0 and represent, say, a gain for Red and a loss for Blue.
The classic von Neumann result shows how to get a saddle point from a payoff matrix with one row for each move for Red and one column for each move for Blue. Again, we can get this result from linear programming duality theory.
The Nash result is a little more general.
Sion's result is really just a case of max-min in mathematical programming, i.e., optimization.
Yes, convexity can play an important role, but we would need to be clear on just what is convex. Convexity is central to Sion's result. Yes, the theory gets involved with some classic fixed point results.
So, for machine learning, for a start, is the game even roughly in the sense of classic von Neumann and paper-scissors-rock? If so, what are the moves and, then, the payoffs? Getting that far, are we looking for two probability distributions? Then with the distributions, how do we apply those to the machine learning calculations, iterations, optimizations, improvements, etc.?
Little questions like those.
So, to start to apply game theory, let's start with a problem and formulate it as a problem in some appropriate part of game theory.
Also let alone so many other problem which come under umbrella of deep learning such as recognition / detection/ Seq to Sequence prediction where there is no "Game theory" component.
To be fair, the article does a kind of weird thing in discussing three different ways game theory has shown up in the context of machine learning recently—but they're all separate ideas really.
Edit: a more precise description in the abstract here: https://arxiv.org/abs/1406.2661
Ah, but there is! GANS can be applied to all those problems.
Conversely, machine learning seldom provides a story (at least in a human readable form) but it is predictive by nature.
1. There wasn't a nuclear war today
2. Every telco in the world gives large chunks of money to governements due to auctions
3. Ebay sellers get higher prices than they should
Also, I can read association rules, decision trees and lisp programs ok... and yet my machine learning programs (ok ok nowadays the ones I get out of various apis and packages) have written them...
There is an argument that game theory helped lead to 1, I suppose. But, in general, I think that is giving them more credit than they would even ask for. Mutually assured destruction is far from their central theorem.
2 could just as easily be explained by standard supply/demand scenarios. There is a very limited supply of that is auctioned off to telcos and they have a very vested interest in buying as much as they can. Even if they are not going to utilize it.
1) You can allocate credit how you like, but there's a dominant strategy for that too.
2)Ok for the UK case see
http://www.nuff.ox.ac.uk/users/klemperer/biggestpaper.pdf
And look them up.
Interestingly your position makes me wonder if you are actually an exec working in a senior finance position in a major telco?
I gladly admit I am using a layperson view that the spectrum auctioned off to telcos is a) likely to just get bought up by entrenched actors, and b) is unlikely to lead to any innovations in the marketplace.
Evidence that I am wrong on either of those points would be fun to read. My second claim, in particular, is one that I suspect is ill-founded.
The cynic in me sees this as yet another way that economists have found to increase the wealth generation capabilities of valuable things, while at the same time losing sight of the utility that was there.
https://www.google.co.uk/url?sa=t&source=web&rct=j&url=http:...
This is not true. What makes you think this way?
Game theory studies games with stochastic outcomes, adversaries, and partial observability. Bayesian Nash equilibrium, trembling hand equilibrium and correlated equilibrium are generalizations of Nash equilibrium into these domains. Game theory also extends to continuous environments and sequential games that have concept of past and future.
All the generalizations you cite are fancy economics methods which loose applicability at the first contact with complex real world scenarios.
Game Theory is like asking question such as "Is chess a win for white? [1]" Sure answering this question might be important theoretically but its of no use when designing an agent that plays and learns.
loose applicability at the first contact
One example of the power of simplicity is the habit of theoretical computer scientists to use simplistic devices such as Turing machines as models of computing and to conflate "effective computability" with the complexity class P and polynomial reducibility (in LOGSPACE). We know that this 'looses applicability at the first contact with reality', but it's currently our only approach to have a meaningful theory of computational complexity and has lead to many deep insights already such as interactive proof systems and zero-knowledge proofs.
If Game Theory was being used as an "explanation" to show why deep learning outperforms other methods then maybe it would have been fine. But then we already have Statistical Learning Theory and PAC learning which try to explain/prove bounds on Machine Learning models.
There is nothing wrong with having a theory but these days Deep Learning has become a bandwagon that any/everything gets related to. And I just don't see how Game Theory of all mathematical tools is applicable to Deep Learning.
Finally the misguided theory driven approach was exactly the reason why Deep Learning was so controversial initially. Turns out making a theory like SLT and deriving ML models like SVMs is a really bad idea. Deep Learning succeeded because the ML and Vision community adopted empiricism over the fanciest/longest proof with covex loss. So when someone goes around claiming Game Theory as a savior/future of Deep Learning I find it perplexing.
Quote: What should be noted here is that the format of his intended talks follows exactly the development of game theory up to 1928, beginning with Zermelo on chess, and culminating with von Neumann groping for a theory for three and more players: game theory was still the mathematics of parlor games
Although "distance from Nash equilibrium" can be a useful metric to measure the performance of some (game playing) systems.
Untrue; game theory has plenty to say about such environments, though that goes well beyond the kind of trivial level of exposure lots of people who are aware of game theory have of it.
The post is not talking about making a new paradigm primarily based on game theory concepts.
Imagine Robert Axelrod's iterated artificial intelligence agents tournaments [0], reloaded; let's pit GANs with LSTM cells, deep-q-networks, style-transferring networks, random forest algos, MCMC algorithms, word2vec architectures, etc. against one another, if only to see possible new failure modes of tit-for-tat.
The advancements that are interesting in ML mostly come from deep statistics and information theory and very sophisticated analysis, not combining the most popular black boxes that you read a medium post about.
The article is kind of confusing because it's discussing three different ways game theory has shown up in the context of machine learning recently—but they're all totally separate ideas.
A lot of machine learning is about predicting human participants. Thing like "winning" at ImageNet have devolved into predicting the exact labels humans will put on a subset of the ImageNet dataset (as the "easy" labels are solved and it now comes down to working out what the most humans labelled confusing images as).
If that makes the authors skin crawl, one would have to wonder what they think ML actually is?
Unsupervised learning - is learning based on predicting future, present or past data from current data.
Both use immediate feedback.
Reinforcement learning is based on learning from dynamic datasets as opposed from static datasets.
Games are like dynamic datasets. Of course game theory has a lot to do with deep learning. If the signal you are learning is dynamic, it's related to game theory.
Reinforcement learning is learning from sequences of states/actions and a reward signal. The sequences could be totally deterministic and similar across different runs of the environment, and hence what I'm guessing you're calling "static" ("static" really has no meaning in the ML field). Or not. Go is totally deterministic, for example, but RL is still applied there.
Unsupervised learning is kind of vague, but it's supervised learning except the "desired output" (label) is generated from the input, or some desired statistic is inferred from the input. Inferring past/present or predicting future data is only a subset of the first case.
I'm not really familiar with the literature in this area, so sorry if this sounds dumb. But what you're saying here seems to make sense and sounds almost profound - like there's a vast amount of research (game theory) that's immediately related in some way. Is there somewhere I can read more about the relationship?
Now, it can teach you something about your data, but it is then up to you to turn that into something that can let you predict about your data. (So, typically cluster analysis.)
Right?
The idea of making Artificial Neural Networks compete with each other based on a selection function has been around for many years. I remember learning about it during my undergrad studies 6 years ago.
This just sounds like a more refined version of that. Using RNNs instead of ANNs.