Christopher Olah wrote an incredibly insightful post on Deep Neural Nets (DNNs) titled “Deep Learning, NLP, and Representations“.  In his post, Chris looks at Deep Learning from a Natural Language Processing (NLP) point of view.  He discusses how many different deep neural nets designed for different NLP tasks learn the same things.   According to Chris and the many papers he cites, these DNNs will automatically learn to intelligently embed words into a vector space.  Words with related meanings will often be clustered together.  More surprisingly, analogies such as “France is to Paris as Italy is to Rome” or “Einstein is to scientist as Picasso is to Painter” are also learned by many DNNs when applied to NLP tasks.  Chris reproduced the chart of analogies below from “Efficient Estimation of Word Representations in Vector Space” by Mikolov, Chen, Corrado, and Dean (2013).

Relationship pairs in a word embedding. From Mikolov et al. (2013).

Additionally, the post details the implementation of recurrent deep neural nets for NLP.  Numerous papers are cited, but the writing is non-technical enough that anyone can gain insights into how DNNs work by reading Chris’s post.

So why don’t you just read it like NOW  — CLICK HERE.   :)

 

I was doing a little bit of research on Mulit-stage Markov Chains for a friend when I ran across this nice set of slides on Multi-stage Markov Chains for diffusion through porous media (i.e. oil or water flowing through the ground) by Pereira and Rahunanthan.

The authors first review the multi-scale equations for diffusion under pressure (see Darcy’s law) and mixed finite element analysis.  Then, interestingly they introduce a Bayesian Markov chain Monte Carlo method (MCMC) to get approximate solutions to the differential equation.  They use Multi-Stage Hastings-Metropolis and prefetching (to parallelize the computation) to improve the speed of MCMC.  The slides also display the results of their algorithm applied to aquifer contamination and oil recovery.

It’s really cool how methods used for probabilistic graphical models can be used to approximate the solutions to partial differential equations.

 

I’m hoping to get back to 2048 in a few weeks.  Turns out that it takes a long time to write code, make the code look nice, run the code, analyze the results, and then put together a blog post.  It’s much easier and quicker to read papers and summarize them or to try to explain things you already know.

 

Have a great weekend.  – Hein

 

 

So, I noticed Brian Rushton’s question on matheducators.stackexchange.com. He asked,

“What should be included in a freshman ‘Mathematics for computer programmers’ course?”

User dtldarek had a pretty cool reply broken up into three parts:

  • Math that is actually useful.
  • Math that you can run into, and is generally good to know.
  • Math that lets you build more awesome math.

His answer got me to think about and revise my “100 Most useful Theorems and Ideas in Mathematics” post.  I wondered which of the 100 I used every week. Rather than trying to think about every week, I just thought it would be interesting to identify which ideas and theorems I used this week excluding the class I am teaching.

The first several ideas on the top 100 list, you can’t seem to avoid:  counting, zero, decimal notation, addition, subtraction, fractions, and negative numbers. Everyone uses these ideas when they buy anything or check their change.  (I wonder how many people rarely count the change from the cashier.)

The next group was mostly about algebra:  equality, substitution, variables, equations, distributive property, polynomials, and associativity.  Every week I attend two math seminars (Tensor Networks and Approximation Theory), so I use these ideas every week, just to add to the discussion during the seminar.

Strangely, I can’t remember using scientific notation this week except while teaching.  During class, I briefly mentioned the dinosaur killing asteroid (see the Chicxulub crater).  It’s hard to calculate any quantity about the asteroid (energy = 4 ×1023 Joules, mass = 2 ×1015 kg ?,  velocity $\approx$ 20000 meters/sec) without using scientific notation.  But, other than during class, I don’t remember using scientific notation.

During the Approximation Theory Seminar, we discussed polynomial approximations to the absolute value function so we used polynomials, infinity, limits, irrational numbers (square roots), the idea of a function, inequalities, basic set theory, vector spaces esp. Hilbert/Banach Spaces, continuity, Karush–Kuhn–Tucker conditions, linear regression, the triangle inequality, linearity, and symmetry during our discussion.  That seems like a lot, but when you have been using these ideas for decades, they are just part of your vocabulary.

So between the approximation theory seminar and daily restaurant mathematics, I used most of the first 40 items from the top 100 list.   Earlier in the week, I wrote up a simple category theory proof about isomorphisms and pull-backs.  Turns out, if you pullback an isomorphism and another arrow, then you get another isomorphism.  For example, in the commutative diagram below the top arrow and the bottom arrow are both isomorphisms (isometries even) and the entire diagram is a pullback.  If the bottom arrow is an isomorphism and the entire diagram is a pullback, then the top arrow is an isomorphism.  (The reverse is not true.)

 

FourCom

 

Working on the proof got me thinking about trig functions, modular arithmetic, injective/surjective functions, Fourier transforms, duality, eigenvalues, metric spaces, projections, the Reiz Representation Theorem, Plancherel’s theorem, and numerical integration.

Probability and information theory were largely missing from my week until I stopped by Carl’s house yesterday. He and I got into a discussion about category theory, information theory, and neural networks.  One thing we noticed was that 1-1 functions are the only functions that conserve the entropy of categorical variables. For example, if $X\in\{1,2,3,4,5,6\}$ is a die roll, then $Y=f(X)$ has the same entropy as $X$ only if $f$ is 1-1.  Turns out you can characterize many concepts from Category theory using entropy and information theory.  So we used random variables, probability distributions, histograms, the mean (entropy is the mean value of the log of the probability), standard deviations, the Pythagorean theorem, statistical independence, real numbers, correlation, the central limit theorem, Gaussian Distributions (the number e), integrals, chain rule, spheres, logarithms, matrices, conic sections (the log of the Gaussian is a paraboloid), Jacobians, Bayes Theorem, and compactness.

I almost avoided using the Pigeon Hole Principle, but then Carl argued that we could use the definition of average value to prove Pigeon Hole Principle.  I still don’t feel like I used it, but it did come up in a discussion.

Notably missing from my week were (surprisingly): derivatives, volume formulas (volume of a sphere or cube), Taylor’s theorem, the fundamental theorem of calculus, and about half of the ideas between #63 Boolean algebra and #99 uncountable infinity.

So, it looks like I typically use around 70 ideas from the list each week—more than I thought.

 

I will probably get back to 2048 stuff later this month and maybe a post on Fourier Transforms.

 

Have a great day!    - Hein

 

So far we have only considered blind strategies and a simple greedy strategy based on maximizing our score on every move.  The results were interesting because the blind cyclic strategy left-down-right-down did surprisingly well and we saw that a biased random strategy which strongly favored two orthogonal moves like left and down with a small amount of one other move like right outperformed all of the other biased random strategies.

If we want to do better, we need something more sophisticated than a blind strategy or pure, short-term, score greed.

 

The first approach used by game AI programmers in the olden days was to simply program in the ideas that the programmer had about the game into the game AI. If the programmer liked to put an X in the middle of a tic-tac-toe board, then he hard-wired that rule into the computer code.  That rule alone would not be enough to play tic-tac-toe, so the programmer would then just add more and more decision rules based on his own intuition until there were enough rules to make a decision in every game situation. For tic-tac-toe, the rules might have been

  • If you can win on your next move, then do it.
  • If your opponent has two in a row, then block his win.
  • Take the center square if you can.
  • Take a corner square if it is available.
  • Try to get two squares in a single row or column which is free of your opponent’s marks.
  • Move randomly if no other rule applies.

Though this approach can be rather successful, it tends to be hard to modify because it is brittle.  It is brittle in two ways: a new rule can totally change the behavior of the entire system, and it is very hard to make a tiny change in the rules.  These kind of rule based systems matured and incorporated reasoning in the nineties with the advent of “Expert Systems“.

Long before the nineties, the first game AI programmers came up with another idea that was much less brittle.  If we could just give every position a numerical rating, preferably a real number rather than an integer, then we can make small changes by adding small numbers to the rating for any new minor feature that we identify.

In tic-tac-toe, our numerical rating system might be as simple as:

  • Start with a rating of zero
  • Add 100 to the rating if I have won,
  • Subtract 100 if my opponent has won or will win on his next move,
  • Add 5 if I own the central square,
  • Add 2 points for each corner square that I own,
  • Add 1 point for every row, column, or diagonal in which I have two squares and my opponent has no squares.

So, for example, the position below would be scored as

2 (for the upper left corner x) + 2 (for the other corner) + 1 (for the top row) = 5.

Tic-Tac-Toe

Now, in order to decide on a move, we could just look at every possible move, rate each of the resulting boards, and choose the board with the highest rating. If our tic-tac-toe position was the one below and we were the x player,

startTic

 

then we would have six possible moves.  Each move could be rated using our board rating system, and the highest rated board with x in the center and a rating of 8 would be chosen.

$\ $
toeChoice

 

This approach was first applied to computer game AI by the famous theoretician Claude Shannon in the paper “Programming a Computer for Playing Chess” (1949), but the idea without the computers actually predates Shannon by hundreds of years.  For instance, the chess theoretician Giambatista Lolli published “Osservazioni teorico-pratiche sopra il giuoco degli scacchi ossia il Giuoco degli Scacchi” in 1763.  He was the first to publish the idea of counting pawns as 1, bishops as 3, knights as 3, rooks as 5, and queens as 9 thereby giving a rating to every board position in chess (well, at least a rating for the pieces).  Shannon modified this rating by subtracting 0.5 for each “doubled” pawn, 0.5 for each “isolated” pawn, 0.5 for each “backward” pawn, and adding 0.1 for each possible move.  Thus, if the pieces were the same for both players, no captures were possible, and if the pawns were distributed over the board in a normal fashion, Shannon’s chess AI would choose the move which created the most new moves for itself and removed the most moves from his opponent.

Shannon significantly improved on the idea of an evaluation function by looking more than one move ahead.  In two players games, the main algorithm for looking a few moves ahead before applying the evaluation function is the alpha-beta search method which will be tested in a later post.

 

Let’s try to find and test a few evaluation functions for 2048.

Perhaps the simplest idea for evaluating a position is related to flying.  According the great philosopher Douglas Adams,

“There is an art to flying, or rather a knack. Its knack lies in learning to throw yourself at the ground and miss. … Clearly, it is this second part, the missing, that presents the difficulties.”  (HG2G).

Suffice it to say that in 2048, the longer you delay the end of the game (hitting the ground), the higher your score (the longer your flight).  The game ends when all of the squares are filled with numbers, so if we just count every empty square as +1, then the board rating must be zero at the end of the game.

I ran 1000 simulated games with the Empty Tile evaluation function and got an average ending tile total of 390.  The strategy ended the games with a maximum tile of  512 tile 25 times, a 256 tile 388 times, 128 491 times, 64 92 times, and 32 4 times.

Well, that didn’t seem like anything spectacular.  Let’s try to group like tiles—if there are two 128 tiles, let’s try to put them side-by-side.  Of course, the idea of putting like tiles together is obvious.  Slightly less obvious is the idea of trying to put nearly equal tiles together.  Over at stackoverflow.com, the user ovolve, the author of the first 2048 AI, suggested the criteria of “smoothness” in his great post about how his AI worked.  There are some interesting ways to measure smoothness with Fourier transforms, but just to keep things simple, let’s measure smoothness by totaling the differences between side-by-side tiles.  For example, the smoothness of the board below would be

2+2+4 (first row) + 30+32 (second row) + 14+ 48 + 64  (third row) + 12 + 16 (fourth) + 2+0+2 (first col) + 30+16 + 0 (second) + 4 + 64 + 64   + 128 + 128

which sums to 662.

randSlopy

Once again I ran a 1000 games using smoothness as an evaluation function.  The average ending tile total was 720, a new high.  The strategy had a maximum ending tile total of  1024 3 times, 512 263 times, 256 642 times, 128 87 times, and 64 5 times.  This looks very promising.

Let’s try ovolve’s last criteria — monotonicity.  The idea of monotonicity is that we prefer that each row or column is sorted.  On the board above, the top row

4 > 2 < 4 > 0 is not sorted.  The second 2 < 32 > 0 = 0 and forth 4 < 16 > 0 = 0 rows are sorted only if the spaces are ignored. The third row 2 < 16 < 64 < 128 is fully sorted.  Let’s measure the degree of monotonicity by counting how many times we switch from a less than sign to a greater than sign or vice-versa.  For the first row we get 2 changes because the greater than sign 4 > 2 changed to a less than sign 2 < 4 and then changed again in 4 > 0. We also get two changes for 2 < 32 > 0, no sign changes in the third row (perfectly monotonic), and two changes in the last row.

I ran 100 games using the monotonicity evaluation function.  The average ending tile total was 423.  The maximum tile was 512 16 times, 256 325 times, 128 505 times, 64 143 times, and 32 11 times.

I have summarized the results of these three simple evaluation functions in the table below.

 

Evaluation Func AETT Max Tile
Empty 390 512
Smoothness 720 1024
Monotonicity 420 512
Cyclic LDRD 680 512
Pure Random 250 256

 

Well, clearly Smoothness was the best.

There are three great advantages to numerical evaluation functions over if-then based rules.  One, you can easily combine completely different evaluation functions together using things like weighted averages.  Two, you can plot the values of evaluation functions as the game proceeds, or even plot the probability of winning against one or two evaluation functions.  Three, in 2048, there is the uncertainty where the next random tile will fall.  With numerical evaluation functions, we can average together the values of all the possible boards resulting from the random tile to get the “expected value”  (or average value) of the evaluation function after the random tile drop.

Well, that’s it for my introduction to evaluation functions.  Next time we will focus on more sophisticated evaluation functions and the powerful tree search algorithms that look more than one move ahead.

Have a great day.  - Hein

 

 

Looks like KUKA Robotics decided to demonstrate the versatility, agility and speed of their Agilus robot by programming it to play ping-pong against a the world-class ping-pong player Timo Boll.  Check out the overly dramatic but cool video on Youtube.  A few more details can be found at Robohub.

 

 

I will be continuing to post articles on AI for the game 2048.  The first two articles were:

 

PROGRAMMING

In this article, I will make some comments on the cyclic blind strategies and report on random blind strategies.  The cyclic results can be generated from the Mathematica source code that I posted earlier.  The Biased Random results below can be obtained by just modifying the strategy function in the Mathematica source code.

 

CYCLIC BLIND

In the Cyclic Blind Article, we saw that left-down-right-down had the highest performance of any length 4 or less cyclic strategy with an average end tile total (AETT) of about 680.  The other cyclic strategies fell into the following seven groupings:

 

  1. All four directions alternating vertical and horizontal (dlur, drul)  420 AETT.
  2. All four directions without alternating vertical and horizontal (durl, dulr, dlru, …)  270 AETT.
  3. Three directions with one direction repeated twice in a row where the directions that were not repeated were opposite (durr, dllu, dull, drru, …) 390 AETT
  4. Three directions with one direction repeated twice in a row where the directions that were not repeated were perpendicular (drrl, duur, ddlu, …) 330 AETT
  5. Three directions with one direction repeated, but not twice in a row, and the directions that were not repeated were perpendicular (dudr, duru, dldu …) 295 AETT
  6. Three directions with no repetition (dlu, dur, …)  360 AETT
  7. One Horizontal Direction + One vertical direction with or without repetition 60 AETT.

 

BLIND RANDOM BIASED STRATEGIES

The other blind strategy to consider is independent random biased selection of directions.  For example, someone might try rolling a six sided die and going left on 1, right on a 2 and down for rolls of 3, 4, 5, or 6.  I ran 1000 games each for every possible biased random strategy where the strategy was of the form

left with probability L/T, right with probability R/T,  up with probability U/T, down with probability D/T,

where T was the total of L, R, U, and D, and L, R, U and D between 0 and 4.

The results are tabulated below

AETT    MAX   
        TILE  L/T   R/T   U/T   D/T
11.138   16   1.    0.    0.    0.
55.416   128  0.57  0.    0.43  0.
56.33    128  0.67  0.    0.33  0.
57.17    64   0.5   0.    0.5   0.
57.732   128  0.6   0.    0.4   0.
70.93    16   0.6   0.4   0.    0.
71.246   32   0.57  0.43  0.    0.
71.294   32   0.5   0.5   0.    0.
71.752   32   0.67  0.33  0.    0.
234.42   256  0.4   0.4   0.1   0.1
238.182  256  0.38  0.38  0.12  0.12
241.686  256  0.44  0.33  0.11  0.11
245.49   256  0.44  0.44  0.11  0.
246.148  256  0.4   0.3   0.2   0.1
247.046  256  0.5   0.25  0.12  0.12
247.664  256  0.36  0.36  0.18  0.09
250.372  256  0.43  0.29  0.14  0.14
253.09   256  0.44  0.22  0.22  0.11
254.002  512  0.5   0.38  0.12  0.
255.178  256  0.33  0.33  0.22  0.11
255.242  256  0.33  0.33  0.17  0.17
255.752  256  0.33  0.33  0.25  0.08
257.468  512  0.43  0.43  0.14  0.
258.662  512  0.36  0.27  0.27  0.09
258.832  256  0.57  0.29  0.14  0.
258.9    256  0.36  0.27  0.18  0.18
260.052  256  0.3   0.3   0.2   0.2
261.676  256  0.38  0.25  0.25  0.12
261.902  256  0.31  0.31  0.23  0.15
262.414  256  0.27  0.27  0.27  0.18
263.702  256  0.27  0.27  0.27  0.2
264.034  256  0.31  0.23  0.23  0.23
264.17   512  0.57  0.14  0.14  0.14
264.2    256  0.29  0.29  0.29  0.14
264.226  256  0.31  0.23  0.31  0.15
264.232  256  0.29  0.29  0.21  0.21
264.33   256  0.5   0.17  0.17  0.17
265.328  256  0.33  0.22  0.22  0.22
265.512  256  0.4   0.2   0.3   0.1
265.724  256  0.4   0.2   0.2   0.2
265.918  512  0.33  0.25  0.25  0.17
265.986  256  0.3   0.3   0.3   0.1
266.61   256  0.25  0.25  0.25  0.25
266.796  256  0.29  0.21  0.29  0.21
266.812  256  0.31  0.31  0.31  0.08
267.096  256  0.33  0.17  0.33  0.17
267.5    512  0.33  0.25  0.33  0.08
267.804  256  0.36  0.18  0.27  0.18
267.99   256  0.3   0.2   0.3   0.2
268.414  512  0.43  0.14  0.29  0.14
268.634  256  0.44  0.11  0.33  0.11
271.202  256  0.38  0.12  0.38  0.12
271.646  256  0.33  0.22  0.33  0.11
271.676  512  0.5   0.33  0.17  0.
272.128  256  0.36  0.18  0.36  0.09
273.632  256  0.5   0.12  0.25  0.12
279.82   256  0.4   0.1   0.4   0.1
281.958  256  0.4   0.4   0.2   0.
291.702  512  0.44  0.33  0.22  0.
306.404  256  0.38  0.38  0.25  0.
306.508  512  0.5   0.25  0.25  0.
310.588  512  0.43  0.29  0.29  0.
310.596  512  0.36  0.36  0.27  0.
321.016  512  0.4   0.3   0.3   0.
333.788  512  0.33  0.33  0.33  0.
340.574  512  0.5   0.17  0.33  0.
341.238  512  0.57  0.14  0.29  0.
342.318  512  0.44  0.22  0.33  0.
347.778  512  0.36  0.27  0.36  0.
355.194  512  0.38  0.25  0.38  0.
366.72   512  0.5   0.12  0.38  0.
369.35   512  0.43  0.14  0.43  0.
371.63   512  0.4   0.2   0.4   0.
384.962  512  0.44  0.11  0.44  0.

There are a few things you might notice.

  • The  best strategy was 44% Left, 11% Right, and 44% down with a AETT of 385 which was much better than pure random (25% in each direction), but not nearly as good as the cyclic strategy left-down-right-down (AETT 686).
  • The AETT varies smoothly with small changes in the strategy.  This is not surprising.  You would think that 55% up, 45% down would have nearly the same result as 56% up and 44% down.  The Extreme value theorem then tells us that there must be a best possible random biased strategy.
  • The top 17 strategies in the table prohibit one direction with AETTs of 280 to 385.
  • If the strategy uses just two directions, then it does very poorly.

 

Well that’s it for blind strategies.  Over the next few posts, we will introduce evaluation functions, null moves, quiescent search, alpha-beta prunning, and expectimax search.

 

 

A guy named Phil asked me to post my code, so here it is.  It’s in Mathematica, so I imagine that most people will have trouble reading it.  I will probably rewrite the code in a more common, faster language later.

(* Mathematica code for simulating 2048
   - by Hein Hundal (Public Domain)

   Visit  http://gabrielecirulli.github.io/2048/

   for more detials.
*)

(* collapse[v] takes a list of values and returns a 
   collapsed list of values where two consecutive equal 
   values are summed into one value. *)

collapse[v_List] := PadRight[
   collapseAux[Cases[v, _Integer]]  , 4, "."];

collapseAux[{}] = {};
collapseAux[{x_}] = {x};
collapseAux[v_List] := If[ v[[1]] == v[[2]],
   Prepend[ collapseAux[Drop[v, 2]], v[[1]]*2],
   Prepend[ collapseAux[Drop[v, 1]], v[[1]]]];

vGlobalMoves = Characters["lrud"];
mGlobalEmptyBoard = Table[".", {4}, {4}];

move[mBoard_, sMove_String] := Switch[sMove,
   "l", collapse /@ mBoard,
   "r", Reverse /@ collapse /@ Reverse /@ mBoard,
   "u", Transpose[ collapse /@ Transpose[ mBoard ]],
   "d", Reverse[ Transpose[ 
        collapse /@ Transpose[ Reverse[ mBoard]]]],
   _, Throw[{"move::illeagal move", sMove}]];

(* game1Turn[ mStart_List, randFunc_, moveStrat_]
      Performs one turn of the game.
      - mStart is a 4 x4 game matrix where every elemet 
        is either a number 2, 4, 8, ... or the string ".".
      - randFunc is any function that take a positive 
        integer n as input and outputs a positive integer 
        between 1 and n.
      - moveStrat is any function that takes a game board as 
        an input and gives as an output one of the four 
        characters u, d, l, r.
      - The output of game1Turn is a new board state.  *)

game1Turn[ mStart_List, randFunc_, moveStrat_] :=
  Module[{sMove, mBoard, mEmpty, iSpot, iVal},
   sMove = moveStrat[mStart];
   mBoard = move[mStart, sMove];

   (* only add a new piece if the board changed *)
   If[ mBoard =!= mStart,
      mEmpty = Position[mBoard, "."];
      iSpot = randFunc[Length[mEmpty]];

      (* the new board tile will either be a 4 or a 2 *)
      iVal = If[ randFunc[10] == 1, 4, 2];
      mBoard = ReplacePart[mBoard, mEmpty[[iSpot]] -> iVal]
    ];
   mBoard];
(*  gameManyTurns  - executes iDo turns of the game  *)
gameManyTurns[mStart_List, randFunc_, moveStrat_, iDo_Integer] := 
   NestList[game1Turn[#, randFunc, moveStrat] &, mStart, iDo];

(******************* Display Results of Multiple Runs **********)

periodTo0[m_List]  :=  m /. "." -> 0;
maxTile[m_List]    := Max[Flatten[periodTo0[m]]]
totalTiles[m_List] := Total[Flatten[periodTo0[ m ]]];

rand1[i_Integer] := 1 + RandomInteger[i - 1];

(* rand2[m]   replaces a random entry on the board m with a 2 *)
rand2[m_List] := ReplacePart[ m, 
   (RandomInteger[3, {2}] + 1) -> 2];

runSeveralGames[ randFunc_, moveStrat_, iDo_Integer]  := 
   Module[{},
   Table[
    (* run a single game for 100 turns *)
    ten1 = gameManyTurns[rand2[mGlobalEmptyBoard], 
               randFunc, moveStrat, 100];

    (* keep going until there is not change for 50 moves *)
    While[ ten1[[-50]] =!= ten1[[-1]]  && Length[ten1] < 10000,
       ten1 =  Join[ten1, gameManyTurns[ten1[[-1]], 
                    randFunc, moveStrat, 100]]
    ];

    ten2 = TakeWhile[ ten1, # =!= ten1[[-1]] &];

    (* output a list {# turns of the game, tile Total,
    maximum tile} for each game *)
    {Length[ten2], totalTiles[Last[ten1]], maxTile[ten1[[-1]]]},
    {iDo}]];

stats[mRes_List] := Module[{mRN = N@mRes},
   {Mean[mRN], StandardDeviation[mRN], Max[mRes[[All, 3]]],
    Tally[mRes[[All, 3]]] // Sort}];

(******** Blind Cyclic Strategy ****************************)

(* createCyclicStrategy - creates a cyclic strategy function 
   from the string s.  If s = "uddl", then the strategy function 
   will repeat the sequence move up, move down, move down, and 
   move left indefinitely. *)

createCyclicStrategy[sMoves_String] := Module[
   {exHeld, iCount = 1},
   exHeld = Hold[
     Function[ m, chars[[ Mod[iCount++, iStringLength] + 1]]]];
   ReleaseHold[
    exHeld /. {chars -> Characters[sMoves] ,
      iStringLength -> StringLength[sMoves]}]];

testOneStrategy[] := Module[{},
   stratDRDL = createCyclicStrategy["drdl"];
   mRes = runSeveralGames[rand1, stratDRDL, 100];
   stats[mRes]];

In my last post, I described how to play 2048 and reported the results of some simple simulations.  Namely, I reported the results of two very simple strategies:

  • Random play will achieve an average ending tile total (AETT) of 252 and usually the highest tile will be a 64 tile or a 128 tile (about an 85% chance of ending with one of those).
  • A pure greedy strategy where the player just tries to make the largest new tile possible results in an average ending tile total (AETT) of 399 and usually the highest tile will be a 128 tile or a 256 tile (about an 87% chance of ending with one of those).

 

Blind Strategies

Perhaps surprisingly, there are “blind” strategies that will do even better than the pure greedy strategy.  Blind strategies are strategies that ignore the current state of the board.  I tried two types of simple blind strategies “biased random” and “repeating sequences”.  I will summaries the results of the biased random trials in my next post.

The simplest repeating strategy would be just hitting the same key every time—down, down, down, ….   Of course that does not do well.  Nor would down, up, down, up, ….  So the simplest repeating strategy worth trying would be down, right, down, right or some other combination of horizontal and vertical moves.  I ran 100 games for every possible repeated sequence of two to four moves and found that the best of these sequences was down, right, down, left  or a version of that sequence in disguise.  The results are captured in the table below.

Notice that restricting your moves to only two out of the four possibilities does poorly.  The Down, Left, Down, Left… strategy had an average tile total of only 44.38 at the end of the game and that strategy never created a tile above 64.

The best strategies were down-right-down-left or that same strategy in disguise (dlul, dldr, and drur are really the same strategy).  Notice that this strategy does better than pure greedy which had an AETT of 399.

move sequence AETT Largest Tile
dl 44.38 64
ddll 46.98 64
drrd 47.98 64
dld 51.18 64
ddrr 52.68 64
drdd 53.32 64
dll 53.46 64
dlld 54.32 64
dr 55.76 64
drd 56. 64
ddrd 57.48 64
dldd 57.66 128
dlll 58. 128
ddl 58.5 64
ddr 58.58 64
drr 59.9 64
drrr 60.56 64
ddld 61.62 64
dddr 66.78 64
dddl 68.46 64
dulr 262.44 256
durl 266.8 256
drlu 267.38 256
dlru 273.4 256
drlr 290.96 256
drdu 295.4 256
dlrl 298.5 256
duld 298.58 256
dldu 307.06 256
duru 308.8 256
dllr 309.32 512
dlrr 312.56 256
dudr 313.38 512
druu 314.58 256
duul 315.74 512
dudl 315.82 512
dulu 317.16 256
dluu 322.48 512
ddul 323.92 256
dlud 325.92 256
ddur 326.6 512
ddru 328.5 256
durd 329.12 512
drud 333.7 256
drll 337.92 512
ddlu 338.6 512
duur 345.62 512
dlu 345.88 256
dru 348.1 512
drrl 348.26 512
dur 352.72 256
dul 354.64 512
ddrl 357.3 512
dlr 361.1 512
ddlr 372.24 512
drl 373.32 256
drru 375.3 512
dull 377.38 512
dllu 379. 512
durr 385.38 512
dlrd 404.16 512
drld 404.6 512
drul 435.72 512
dlur 440.06 512
drur 614.9 512
dldr 620.36 512
dlul 627.06 512
drdl 686.28 512

 

So I played the game 2048 and thought it would be fun to create an AI for the game.  I wrote my code in Mathematica because it is a fun, terse language with easy access to numerical libraries since I was thinking some simple linear or logit regression ought to give me some easy to code strategies.

The game itself is pretty easy to code in Mathematica.  The game is played on a 4×4 grid of tiles.  Each tile in the grid can contain either a blank (I will hereafter use a _ to denote a blank), or one of the numbers 2,4,8,16,32,64,128, 256, 512, 1024, or 2048.  On every turn, you can shift the board right, left, up, or down. When you shift the board, any adjacent matching numbers will combine.

The object of the game is to get one 2048 tile which usually requires around a thousand turns.

Here’s an example.  Suppose you have the board below.

start

 

If you shift right, the top row will become _, _, 4, 16, because the two twos will combine.  Nothing else will combine.

movEr

 

Shifting left gives the same result as shifting right except all of the tiles are on the left side instead of the right.

Shift Left

If we look back at the original image (below), we can see that there are two sixteen tiles stacked vertically.

original

So the original board (above) shifted down combines the two sixteens into a thirty-two (below) setting some possible follow-up moves like combining the twos on the bottom row or combining the two thirty-twos into a sixty-four.

Shifted down

There are a few other game features that make things more difficult.  After every move, if the board changes, then a two or four tile randomly fills an empty space. So, the sum of the tiles increases by two or four every move.

 

Random Play

The first strategy I tried was random play.  A typical random play board will look something like this after 50 moves.  rand50

 

If you play for a while, you will think that this position is in a little difficult because the high numbers are not clumped, so they might be hard to combine.  A typical random game ends in a position like this one.

randEnd

In the board above, the random player is in deep trouble.  His large tiles are near the center, the 128 is not beside the 64, there are two 32 tiles separated by the 64, and the board is almost filled.  The player can only move up or down.  In this game the random player incorrectly moved up combining the two twos in the lower right and the new two tile appeared in the lower right corner square ending the game.

I ran a 1000 random games and found that the random player ends the game with an average tile total of 252.  Here is a histogram of the resulting tile totals at the end of all 1000 games.

histRand

(The random player’s tile total was between 200 and 220 one hundred and twenty seven times.  He was between 400 and 420 thirteen times.)

The random player’s largest tile was 256 5% of the time, 128 44% of the time, 64 42% of the time, 32 8% of the time, and 16 0.5% of the time.  It seems almost impossible to have a max of 16 at the end.  Here is how the random player achieved this ignominious result.

randEnd16

The Greedy Strategy

So what’s a simple improvement?  About the simplest possible improvement is to try to score the most points every turn—the greedy strategy.  You get the most points by making the largest tile possible.  In the board below, the player has the options of going left or right combining the 32 tiles, or moving up or down combining the two 2 tiles in the lower left.

combine2048combine2048Opt

The greedy strategy would randomly choose between left or right combining the two 32 tiles.  The greedy action avoids splitting the 32 tiles.

I ran 1000 greedy player games games.  The average tile total at the end of a greedy player simulation was 399—almost a 60% improvement over random play. Here is a histogram of the greedy player’s results.

GreedyResults

If you look closely, you will see that is a tiny bump just above 1000.  In all 1000 games, the greedy player did manage to get a total above 1000 once, but never got the elusive 1024 tile.  He got the 512 tile 2% of the time, the 256 tile 43% of the time, the 128 44%, the 64 11%, and in three games, his maximum tile was 32.

I ran a number of other simple strategies before moving onto the complex look ahead strategies like UCT.  More on that in my next post.

 

  1. Enjoying John Baez’s blog Azimuth.  Especially the posts on good research practices and an older post on levels of mathematical understanding.
  2. García-Pérez, Serrano, and Boguñá wrote a cool paper on primes, probability, and integers as a bipartite network.
  3. Loved the idea behind the game theoretical book “Survival of the Nicest”  (see Yes Magazine for a two page introduction).
  4. Scott Young is learning Chinese quickly.
  5. Cyber warriors to the rescue.
  6. Mao, Fluxx, and Douglas Hofstadter‘s Nomic are fun games.
  7. Healy and Caudell are applying category theory to semantic and neural networks.
  8. Some MOOCs for data science and machine learning.
  9. Here an old but good free online course on the Computational Complexity of Machine Learning.
  10. Great TeX graphics.
  11. Watch this Ted Video to learn anything in 20 hours (YMMV).
  12. Where are all the Steeler fans?  Cowboy fans?  ….
  13. Productivity Hints.
  14. Copper + Magnets = Fun
  15. Stray dogs on the subway.
  16. Deep learning on NPR.
  17. Happy 40th birthday D&D
  18. Google is applying deep learning to images of house numbers
  19. Deep learning in your browser.
  20. How to write a great research paper.
  21. Do Deep Nets Really Need to be Deep?
  22. A variation on neural net dropout.
  23. Provable algorithms for Machine Learning
  24. 100 Numpy Exercises
  25. Learn and Practice Applied Machine Learning | Machine Learning Mastery

 

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