Roshambo Part III – Representation Theory

In the last two posts, we’ve looked at using machine learning for playing iterated Roshambo.  Specifically, we saw how to use Bayes’ theorem to try to detect and exploit patterns, and then saw how Fourier transforms can give us a concrete measurement of the randomness (and non-randomness) in our opponent’s play.  Today’s post is about how we can use representation theory to improve our chances of finding interesting patterns.

Niels Henrik Abel, for whom 'Abelian groups' are named.
Niels Henrik Abel, for whom ‘Abelian groups’ are named.  These are groups where xy=yx for any x, y.

A Crash Course in Group Theory

I’m going to attempt to run through a few of the basic ideas of group and representation theory now without writing a full textbook.  There’s a huge amount of additional information out there; this is just enough to explain what I’m doing with Roshambo, and hopefully get you interested in learning more if you’re not already a mathematician!

Groups are mathematical structures that measure symmetry in a system.  They show up all over the place, really anywhere where there’s symmetry.  Some groups are finite, like the symmetries of a cube, and some groups are infinite, like the symmetries of a sphere.  There’s an abstract mathematical definition of a group which doesn’t really care so explicitly about the notion of symmetry, but the most important groups usually do arise as symmetries of important geometrical objects.  One of my favourite groups is the group of permutations, S_n consisting of all of the ways of permuting n things.  To see this as symmetries of an object, think of an equilateral triangle, with its three corners labelled.  Then any way of moving the triangle back onto itself is a symmetry which permutes the labels of the corners, giving an S_3 worth of symmetries.  Likewise, S_4 can be realized as symmetries of a regular tetrahedron, and S_n can be realized as the symmetries of a regular n-simplex.  The permutation group is also called the symmetric group; but as John Baez says, this is a bit redundant, since all groups measure symmetry and are very symmetric themselves.  A bit like calling it the ‘symmetric symmetries.’

Symmetries can be composed to get other symmetries.  Thinking of a equilateral triangle, rotation by 120 degress is a symmetry, as is flipping the triangle over.  If I rotate and then flip, I get another symmetry of the triangle.  Groups always have this kind of composition: Put together two elements of a group (ie, symmetries), and you get another element (symmetry) back.

Representation theory is, roughly speaking, the study of how mathematical groups show up in the world.  Technically, a representation is an action of a group on a vector space: an expression of symmetry of the vector space that encodes at least some of the symmetry of the group; representation theory is the study of such actions.  I identify vector spaces with ‘the world’ because, as scientists and mathematicians, we tend to linearize everything as quickly as possible, so that interesting group actions on a system usually translate into linear representations.

It turns out that for any finite group (like symmetries of a polygon or permutations or a finite cyclic group, like C_3), every representation breaks up into a collection of irreducible representations.  There are only finitely many irreducible representations, so if you can figure out what they are, you can break any representation into irreducible parts.  This process is analogous to taking a discrete Fourier transform for a cyclic group.

There are also analogues of the Fourier transform for certain infinite groups.  The ‘circle group’ corresponds to the usual continuous Fourier transform, but other important groups like the group of four-dimensional orthogonal matrices have interesting representation theory as well, and thus a notion of a Fourier transform.

In general, it’s very difficult to find all of the irreducible representations of a finite group.  But if the group is commutative it’s pretty easy to write down all of the irreducibles, and we also know how to write down the irreducible representations of the permutation group, as well as for many of the important infinite groups.

Group Theory In Probability and Statistics

Group theory becomes useful to us in statistics and probability when there are symmetries in our space of possible data.  A classic example is if you’re taking a political survey and ask respondents to give a ranked list of the presidential candidates.  The possible data is any permutation of the candidates; thus, there is a permutation group’s worth of symmetry available to us.  Taking the actual survey will give us a function on the permutation group: For each permutation, keep track of the number of people who gave that response.  Breaking up that function according to irreducible representations will then tell us quite a bit about our voting population, in much the same way that a Fourier transform tells us about a music sample.  (In fact, here’s a great paper on how to apply representation theory to voting theory; in less than twenty pages they re-prove many of Arrow’s famous theorems which were famously difficult to derive.)

A classic example of the use of group theory in statistics was Persi Diaconis’ analysis of card shuffling (and other random processes).  It was long known that shuffling eventually mixes up your cards well; Diaconis used representation theory of the permutation group to show that it takes about seven shuffles.  The methods he developed apply to a wide variety of other ‘mixing time’ problems, and ideas can be extended to other contexts in statistics quite easily.  (The idea in a nutshell: The Fourier transform on groups gives us a way to measure distance of a probability distribution from the uniform distribution.  Repeatedly applying a random process (like shuffling cards) corresponds to taking the convolution product of the function with itself; we can then measure the distance of repeated shuffles from the uniform distribution, and find a concrete place at which this distance is close to 0.)

Machine learning is essentially weaponized applied statistics: Given silly large piles of data, how can we work the existing tools of statistical modelling into computer algorithms that can operate at scale?  Numerous algorithms have been developed for machine learning, and approaching a problem involves both identifying an algorithm that applies well to your use-case, but also cleaning up and choosing data to make it as easy as possible for the algorithms to succeed.  In fact, improving data quality often has as much impact on the outcome as choice of algorithm.  One of the main lessons learned from group theory in statistics is that recognizing symmetries in the space of data improves our ability to analyze a problem, and thus, to learn.

Distance from Uniformity

There are a number of precise ways that recognizing group structure in our data can help us out.

As mentioned in the last post, we can evaluate easily whether the data is uniformly distributed.  This means we need to measure the distance between the observed probability and the uniform distribution.  Diaconis uses the variation distance, given by max_{A \subset G} |P(A)-Q(A)| for two distributions P and Q on a group G.  This looks for the biggest difference in probability for any subset of G picked according to P and Q.  It’s a theorem that this distribution is estimated by taking the sums of the magnitudes of the non-trivial Fourier coefficients.  (See Diaconis’ notes above for the precise statement.)  The inequality is:

d(P, U)^2 \leq \frac{1}{4}( -1+\sum \hat{P}\hat{P}* )

In our RPS game, this can give us an easy way to evaluate whether we’re observing a pattern, and also opens us up to evaluating whether the pattern we’re observing is significant, relative to the amount of data we’ve collected.  I’ve mainly been using the former method in writing my RPS algorithm: Given the games we’ve played so far, look for non-uniform distributions and then use them to choose plays.

Here’s an example.  SwitchBot plays RPS by choosing uniformly at random a move that is different from its last move.  So if its last move was rock, it will play paper or scissors at random.  After playing 1000 rounds, the Bayesian estimate for the distribution of moves might look like this:

{'p': [173, 1, 167], 'r': [1, 169, 165], 's': [161, 170, 1]}

This says that, for example, if ‘p’ (or paper) is played, we’ve observed 172 instances of rock and 166 instances of scissors as the next move (remembering that we initialize the algorithm with one instance each of every possible combination of prior move and next move).  The probability of rock given that the last move is paper is then estimated as \frac{173}{341}, or 0.507; very close to the correct chance of 50%, and greater than the 33% we would expect if the opponent were playing at random.

If we take a Fourier transform of P(r|p), we get three complex coefficients.  The trivial component is 1 (as it always is for a probability), and the other two components are -0.495+0.030j and -0.495-0.030j, where j stands for the square root of -1.  Each of these two components has magnitude around 0.25, giving a total distance from the uniform distribution of (at most) \sqrt{0.125} \sim 0.35, which is actually pretty significant.

For the sake of comparison, let’s see what happens with the player who always throws rock.  In this case, we get this dictionary after a thousand plays:

{'r': [999, 1, 1]}

The Fourier transform for P(r|r) has three coefficients, of magnitudes 1, 0.99, and 0.99, giving a total distance from the uniform distribution of (at most) \sqrt{2}\sim 1.41.  This is about as un-random as you can get.  Meanwhile, we assume that the other possibilities, ‘p’ and ‘s’ give uniform distributions when we see them, since we’ve never seen them.  Then these probabilities P(r|s), for example, have distance 0 from the uniform distribution.

Extending to Bigger Groups

Here are two questions we can ask which will lead to looking at representations of bigger groups:

  1. How evenly distributed is the data?
  2. What if we have to declare throws (say) three at a time, instead of playing one round at a time?

Suppose we’re looking at combinations of three moves at a time to try to predict the next move.  There are 27 possible data states to consider; how uniformly distributed is the data we’ve seen so far?  To answer this, we can find the distribution P(D) for each possible combination of data, and consider this as a function on C_3^3, which is a product group.  We can write down a character table for this group, and use that character table to compute a Fourier coefficients for the distribution of data.

Another way to extend to bigger groups is to play a variant in which we need to choose three moves ahead of actually playing them.  In this case, we need to predict the probability of a given triple of plays given past data: The same kind of Fourier transform that we used to measure the distribution of data in the simple RPS game will now be of use for making predictions in this game.  The algorithm works exactly the same, but now we have a larger group to work with.

We can also consider another extension of the game: Suppose we play Rock-Paper-Scissors-Lizard-Spock (an extension to C_5, where you throw one sign with each hand, and lose if you throw the same sign on both hands.  In this game, our output is a partial permutation consisting of two of the five elements of C_5.  It can be considered as a homogeneous space for the permutation group S_5.  We can still do a similar analysis for this game, which I’ll come back to consider soon.