## Deep Learning for … Chess

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#### by Erik Bernhardsson | February 2, 2017

About Erik: Dad and CTO (Chief Troll Officer) at a fintech startup in NYC. Ex-Spotify, co-organizing NYC ML meetup, open supply typically (Luigi, Annoy), blogs random stuff

### Deep studying for… chess

I’ve been that means to study Theano for some time and I’ve additionally wished to construct a chess AI in some unspecified time in the future. So why not mix the 2? That’s what I assumed, and I ended up spending means an excessive amount of time on it.

### What’s the idea?

Chess is a sport with a finite variety of states, that means in the event you had infinite computing capability, you possibly can truly clear up chess. Every place in chess is both a win for white, a win for black, or a pressured draw for each gamers. We can denote this by the operate
f(place) . If we had an infinitely quick machine we might compute this by

1. Assign all the ultimate positions the worth −1,0,1 relying on who wins.
2. Use the recursive rule

\$\$f(p) = max_{p rightarrow p’} -f(p’)\$\$

the place \$p rightarrow p’\$ denotes all of the authorized strikes from place p. The minus signal is as a result of the gamers alternate between positions, so if place
p is white’s flip, then place p′ is black turns (and vice versa). This is identical factor as minimax.

There’s roughly 10^43 positions, so there’s no means we are able to compute this. We must resort to approximations to f(p).

### What’s the purpose of utilizing machine learning for this?

What machine learning actually boils all the way down to is approximating capabilities given information. So assuming we are able to get a variety of information to study this from, we are able to study this operate
f(p). Once we’ve got a mannequin, an goal, and coaching information, we are able to go knock ourselves out.

I downloaded 100M video games from FICS Games Database and started coaching a machine learning mannequin. My operate
f(p) is discovered from information by utilizing two ideas

1. Players will select an optimum or near-optimal transfer. This implies that for two place in succession
p→q noticed within the sport, we may have f(p)=−f(q).
2. For the identical cause above, going from q , however to a random place p→r , we will need to have
f(r)>f(q) as a result of the random place is best for the subsequent participant and worse for the participant that made the transfer.

### The mannequin

We assemble f(p) as a three layer deep 2048 items extensive synthetic neural community, with rectified
linear items in every layer. The enter is a 8 * 8 * 12 = 768 extensive layer which
signifies whether or not every bit (there are 12 varieties) is current in every sq.
(there are 8 * Eight squares). After three matrix multiplications (every adopted by
a nonlinearity), there’s a last dot product with a 2048-wide vector to condense
it all the way down to a single worth.

In complete there’s roughly 10M unknown parameters within the community.

To practice the community, I current it with
(p,q,r) triplets. I feed it by means of the community. Denoting by
\$S(x) = 1 / (1 + exp(-x))\$, the sigmoid operate, the full goal is:

\$\$sum_{(p, q, r)} log S(f(q) – f(r)) + kappa log (f(p) + f(q)) + kappa log (-f(q) – f(p))\$\$

This is the log chance of the “soft” inequalities f(r)>f(q), f(p)>−f(q), and
f(p)<−f(q). The final two are only a means of expressing a “soft” equality
f(p)=−f(q). I additionally use \$kappa\$ to place extra emphasis on getting the equality proper.
I set it to 10.0. I don’t assume the answer is tremendous delicate to the worth of
\$kappa\$.

Notice that the operate we study has no concept concerning the guidelines of chess.
We’re not even educating it how every bit transfer. We ensure the mannequin has
the expressiveness to work out authorized strikes, however we don’t encode any data
concerning the sport itself. The mannequin learns this data by observing tons of
chess video games.

Note that I’m additionally not making an attempt to study something from who gained the sport.
The cause is that the coaching information is filled with video games performed by amateurs.
If a grandmaster got here into the center of a sport, s/he might most likely
fully flip it round. This means the ultimate rating is a fairly weak label.
Still, even an beginner participant most likely makes near-optimal strikes for most time.

### Training the mannequin

I rented a GPU occasion from AWS and skilled it on 100M video games for about 4 days
utilizing stochastic gradient descent with Nesterov momentum. I put all (p, q, r)
triplets right into a HDF5 information file. I used to be messing round with studying charges for a
whereas however after some time I spotted I simply wished one thing that will give me
good leads to a number of days. So I ended utilizing a barely unorthodox studying fee scheme:
\$0.03 cdot exp(-mbox{time in days})\$. Since I had a lot coaching information, regularization
wasn’t needed, so I wasn’t utilizing both dropout or L2 regularization.

A trick I did was to encode the boards as 64 bytes after which remodel the board
right into a 768 items extensive float vector on the GPU. This gave a fairly substantial
efficiency enhance since there’s rather a lot much less I/O.

### How does a chess AI work?

Every chess AI begins with some operate f(p) that approximates the worth of
the place. This is called analysis operate.

This operate can also be mixed with a deep search of many hundreds of thousands of
positions down the sport tree. It seems that an approximation of
f(p) is only a small a part of the enjoying chess effectively. All chess AI’s
deal with good search algorithms, however the variety of positions explode
exponentially down the search tree, so in follow you’ll be able to’t go deeper
than say 5-10 positions forward. What you do is you employ some approximation
to guage leaf nodes after which use some number of negamax to guage
a sport tree of a bunch of potential subsequent strikes.

By making use of some good looking out algorithm, we are able to take just about any approximation
and make it higher. Chess AI’s sometimes begin with some easy analysis operate
like: each pawn is value 1 level, each knight is value three factors, and so forth.

We’re going to take the operate we discovered and use it to guage leaves
within the sport tree. Then attempt to search deep. So we’re first going to study the operate
f(p) from information, then we’re going to plug it right into a search algorithm.

### Does it work?

I coined my chess engine Deep Pink as an homage to Deep Blue.
As it seems, the operate we study can undoubtedly play chess. It beats me, each time. But I’m a horrible chess participant.

Does Deep Pink beat current chess AI’s? Sometimes.

I pit it in opposition to one other chess engine: Sunfish by Thomas Dybdahl Ahle. Sunfish
is written solely in Python. The cause I selected to stay to the identical language
was that I didn’t need this to be an countless train of creating quick transfer
era. Deep Pink additionally depends closely on fast transfer era, and I
didn’t need to spend weeks understanding edge circumstances with bitmaps in C++ to be
capable of compete with the cutting-edge engines. That would simply be an
arms race. So to have the ability to set up one thing helpful, I picked a pure
Python engine.

The apparent factor in hindsight is: the principle factor you need out of any analysis operate
f(p) isn’t accuracy, it’s accuracy per time unit. It doesn’t matter that one
analysis operate is barely higher than one other if it’s ten occasions slower,
as a result of you’ll be able to take the quick (however barely worse) analysis operate and search
extra nodes within the sport tree. So you actually need to take note of the time spent
by the engine. Without additional ado, right here’s some outcomes of enjoying in opposition to the engine many occasions:

Notice the log-scale. The x-axis and y-axis aren’t tremendous related right here,
the principle factor is the gap to the diagonal, as a result of that tells us which engine
spent extra CPU time. Every sport I randomized the parameters for every engine:
the max depth for Deep Pink, and the max variety of nodes for Sunfish. (I didn’t
embody attracts as a result of each engines wrestle with it).

Not surprisingly, the extra time benefit both facet has, the higher it performs.
Overall, Sunfish is best, profitable the vast majority of the video games, however Deep Pink
most likely wins 1/three of the time.
I’m truly fairly inspired by this. I feel
with some optimizations, Deep Pink might truly play considerably higher:

• Better search algorithm. I’m at the moment utilizing Negamax with alpha-beta pruning, whereas Sunfish makes use of MTD-f
• Better analysis operate. Deep Pink performs fairly aggressively, however makes a variety of dumb errors. By producing “harder” coaching examples (ideally fed from errors it made) it ought to study a greater mannequin
• Faster analysis operate: It could be potential to coach a smaller (however perhaps deeper) model of the identical neural community
• Faster analysis operate: I didn’t use the GPU for enjoying, solely for coaching.

### Summary

I’m inspired by this. I feel it’s actually cool that

1. It’s potential to study an analysis operate straight from uncooked information, with no preprocessing
2. A reasonably sluggish analysis operate (a number of orders of magnitude slower) can nonetheless play effectively if it’s extra correct

I’m fairly curious to see if this might fare effectively for Go or
different video games the place AI’s nonetheless don’t play effectively. Either means, the conclusions above include one million caveats.
The greatest one is clearly that I haven’t challenged a “real” chess engine. I’m undecided if I’ve the
time to begin hacking on chess engines, but when anybody is , I’ve put all of the supply code up on Github.

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