The files in this directory constitute the tuning program that was used by
Deep Thought to adjust its evaluation function parameters based on a set
of some 868 grand-master games. I forgot where these games came from, but
we did not type them in. It was last used in the summer of 1988 and it is
believed that this program might be of historical interest to some chess
programmers. The Deep Thought hardware is probably no longer functional
and without the actual Deep Thought program, this code can only show how
DT evaluated chess positions, but it can not play any chess.
This program was written by me during the Spring of 1988 and included
suggestions and feedback from the entire Deep Thought team (Feng H. Hsu,
Thomas S. Anantharaman, Murray S. Campbell, Mike C. Browne and myself). It
includes the DT evaluation function, which was developed by Murray.
This file gives a brief description how this tuning program worked and how
to use it. Expect some errors and omissions because I'm writing this from
memory, 12 years after I last touched this code.
The basic method used the mathematical concept of least square fitting.
This was hardly new and it had been applied to chess evaluation functions
before. However, there are plenty of details on exactly how to go about
this, and our approach likely differed from earlier work.
Let's suppose that the evaluation function is a weighted sum of positional
features (later referred to as a feature vector):
E(P) = SUM Ai * Fi(P)
i
For a given chess position (= position of pieces on the board plus
castle and enpassant status), the evaluation is the sum of the
features recognized by Deep Thought times the weight given to each
feature . For example, a feature may be the number of white pawns minus
the number of black pawns. The corresponding weight would be the value for
one pawn. There were roughly 100 features that Deep Though used. Some were
implemented via a piece-placement table that could give a different weight
for each piece depending on where it is on the board. For example, a
gradient in the pawn value could be used to add a bonus for advanced
pawns. King centrality was implemented likewise. There were five other
tables in the hardware for more complex features, that could detect open
files, passed/doubled pawns etc. (four pawn structure tables of 8192
entries each, a rook evaluation table with 2048 entries and the 1024 entry
piece/placement table along with a few special bonus registers made up the
DT evaluation hardware. While these nearly 40,000 programmable parameters
of the DT hardware could be regarded as the components of DT's evaluation
function, they all were derived from the 89 to ~100 parameters mentioned
earlier). The basic DT move cycle consisted of computing these tables
before every search so that the weights of the evaluation function could
be adjusted according to the overall situation of the game (opening,
mid-game, endgame, etc.). This took some time, so DT was not good at fast
games (keep in mind that DT used '88 hardware and the VME interface from
the host, a Sun 3 or 4, was not a spead deamon).
Now suppose you had an oracle that could give the correct evaluation for a
position O(P). If we use this oracle on a sufficiently large set of
positions then we could minimize the squared evaluation error sum:
error = SUM (O(Pj) - E(Pj))^2
j
via partial differentiation of this expression for each parameter .
This leads to a linear equation system with one equation for each unknown
parameter of DT's evaluation function. If the positions are sufficiently
varied (they usually were), then this equation system can be solved and
out come the best values for our evaluation parameters.
The trouble was, we did not have such an oracle. So the next best thing we
had is the evaluation of DT. Murray made some initial guesses for each
parameter and we used that as a starting point. Obviously, if we use
our own as an oracle, we get the same parameters out of the least
square fit as we put in. So this is just a cumbersome way to compute the
identity: New(Ai) = Old(Ai) for all *. However, this was a great
debugging tool to see that we got the mathematics right.
In the tuning case, we did not just take the top-level evaluation, rather
we let DT search shallow 3ply trees with quiescence extensions. The
evaluation function is then computed symbolically: rather then plugging in
values, we propagated the feature vector of the best leaf node to the top.
The search itself was controlled by the current best guess of the
evaluation parameters. These were full min/max searches, rather than
alpha/beta searches. The tuning program cannot actually search these trees
because it does not know what a legal chess move is. Instead, the actual
DT was used to pre-search these trees and the results were stored in a
database (dbf_all). The tuning program merely traverses these trees.
Now we are still lacking an oracle. Instead we assumed that in our
collection of grand-master games, the winner of each games should serve as
a substitute oracle for DT. Discarding the moves of the losing side and
the first few opening moves, we considered all available moves for each
position. Each position was searched and evaluated. Suppose that there
were 20 legal moves in one position then each of these moves lead to
a new position ... , for which we get the evaluations
... . Let's assume that the winning GM played the move leading to
from . Then there are two cases:
1. DT's evaluation concurs, that is: E(P0) > E(P1...19)
2. DT evaluated some other move as best.
So the objective of our tuning procedure was to maximize the first case
and minimize the second case.
The GM games tell us which moves are better, relative to other moves. So
rather than having an absolute oracle that assigns an absolute value to a
position, we have a kind of relative oracle: The move from to is
expected to be better than the moves from to . Hence ought to be larger than . These differences of
evaluations were used for the tuning process, but this did not change much:
we still end up with linear combinations of elements of the feature vector
that can be dealt with via least square fitting. In the program and on the
display, the evaluations , and refer to the evaluation
of the root position, the position of the winning GM move and the evaluation
of move .
Naturally, not all changes to the evaluation of the board are positional
in nature and/or are captured by the evaluation function. The fact that
the pre-computed search trees include the results after quiescence search
should minimize differences due to exchanges, but this is not always the
case. Therefore in order to avoid interference from elements that DT
discovers via search and not via its evaluation function, the tuning
program includes a threshold: if the evaluation of the root position and
the position after a move differ by too much (more than 64 = half of a
pawn value), then it is assumed that there is something going on that the
evaluation function cannot grasp and this data-point is ignored
(out-of-bound).
The first idea was to apply a force function. Instead of using as
an oracle, we used as an oracle. A simple force function
would be used to add a constant force value if the position is the
result of the GM move and 0 otherwise.
This doesn't really work as it leads to a value inflation. So instead of
adding no bonus for the wrong positions, we add <- force/n>, where is
the number of wrong moves (positions).
This is in fact what an early version of the tuning code did. Add a small
force as described above, compute a new parameter set via least square
approximation and iterate until the number of correctly evaluated
positions does not increase anymore.
This works, but it did not really yield better play of DT. It became clear
that maximizing the agreement between DT's evaluation and the GM move
choices and better play were not really the same thing. Also, DT searched
9+ plys and we could tune only on 3ply searches due to compute time
limitations.
We observed that deeper searches in the tuning code lead to better
results, even though it could be argued that evaluations should be orthogonal
to searching. Perhaps an explanation for this effect is that deeper
searches lead to more differences in the positions that are being related
because they are more moves apart. Therefore, the tuning process collects
more information on how individual components of the evaluation relate to
each other. For example, the tuning process did result in a better
understanding of the piece-values with respect to each others and as
a function of the amount of material left on the board (this was used
to control the transitions from opening, mid-game to end-game).
The next refinement was to make the force dependent on the amount of
miss-evaluation. If DT is just a little bit off, use a small force, if DT
misses the position in a big way, add more force. This relationship was
subject to much debate and it is unclear which monotonic function is best
suited. Hence, the tuning program gives a number of options from linear,
quadratic, square-root, logarithmic, to reciprocal (the idea behind a
reciprocal force function was this: if DT is just a little off then there
is hope that it can learn how to evaluate this position correctly. If it
is off by a lot, don't bother to try - it will only screw up things
elsewhere because this is likely due to a concept that is missing from
DT's evaluation function). Which force-function to use eventually depended
on which parameters were subjected to tuning and became more of a trial
and error procedure.
A somewhat different idea for a force function was to count how many moves
were evaluated ahead of the GM-move. If this is 0, DT's evaluation
function is on target. Numbers greater than 0 are undesirable and lead to
an increased correcting force. This number was also used to compute a
histogram to see how DT's evaluation function scores against GM moves (see
the *.stat files after a multi-iteration tuning run). This is to be taken
with a large grain of salt, because the 3ply searches are clearly not good
enough to isolate positional evaluation from tactics.
At this point things started to improve somewhat (mostly measured via
self-play tournaments starting from various seed positions). But also
funny things happened to DT's evaluation function: the range of values
that it would produce during a search became smaller. The evaluation
function became less discriminative: the values for the good and bad moves
were progressively moving closer together. It did match more GM moves and
played slightly better, but it also became more erratic. Interestingly,
this reduction in value range was not due to simply reducing the weight
for the positional components of the evaluation function, rather it
involved balancing the various components against each other.
So a compensating force was added to the correction force so that it would
also encourage to keep a certain variance of evaluations. You see this in
the code commented as variance compensation.
As we learned how to tune more effectively, it became clear that tuning
all parameters at once was not necessarily a good idea. Certain parameters
were used very infrequently in our set of GM games, so allowing these to
be tuned can pick up the wrong idea for lack of sufficient data points.
868 games were not really enough to tune this many parameters.
It also became clear that the best parameter sets were usually obtained
after 10-15 iterations, and before the maximum match to the GM games was
reached, which typically required 20 to 30 iterations.
Finally, the last improvement of the tuning came after we added the
ability to tune tables. In the DT evaluation function are a number of
tables, for example the pawn advancement gradient, or the King centrality
table. Instead of just making up a linear gradient, we allowed the tuning
code to change the values of these table and allow more complex gradients.
This resulted in some strange 3D curves that upon inspection by Murray
were found to contain some known chess heuristics, which were not
originally part of DT's evaluation function (for example: the pawn
advancement gradient became more pronounced near the center and tapered
off towards the promotion squares). However, the overall impact of table
fitting was minor and was never fully exploited because the code became
stable only near the end of DT's life and we did not have enough compute
cycles to experiment a lot with it. The few experiments we did were great
fun because we extracted some general rules out of the game database in an
unbiased, neutral and fully automated way. This was quite important
because we had to avoid the slightest hint of suspicion that we took
anything from the competing Hitech effort, which was the officially funded
chess project at CMU at that time. Because of our automated tuning
process, we had a demonstratably independent and effective way to
incorporate chess knowledge into DT.
I just compiled the tuning code on an Compaq/Alpha Unix workstation and
it survived 12 years of bit-rot. So you may get it to run as well. Here are
a few hints on how to play with it:
1. The database is a binary file and the moves are stored in 2 byte fields.
These may need to be swapped depending on the byte-order of your computer.
Either enable or disable the "SUN_ENDIANESS" #define.
2. When it runs, it needs a very wide terminal window to show the board and
all Eval-parameters (142 columns).
3. After starting the tuning program via 'texam dbf_all' you should see a
small chess board and the set of parameters, their weights and
contributions to the current position. At this point there are several
single character commands available:
' ' (space bar) steps through all legal moves
n steps through the move played by the GM
N goes to the beginning of the next game
g goes to a specific game, the number of which is prompted
f does a parameter fit
S/R saves and restores a parameter set to/from a file (*.par)
F does a multi-iteration fit (a *.stat file will be generated
along with parameter file after each iteration. The *.stat file
summarizes the fit, shows tends and gives the error-distributions)
q exits the program
For the other commands (there are several more, you would need to look
into the source code. In particular, there are ways to edit parameters
manually and to control if they should participate in a tuning run or not)
4. All of this assumes a Unix system with the curses library.
5. There are 5 parameter files included in this directory. Q*.par were
certain starting points during development and the other ones appeared
to be the most recent ones. I don't know which parameter set was used
in what DT game: I have hundreds of them and no idea which one was
used for what. The USO.par file is probably the set that was used
by DT in the 1988 USopen tournament.
I would like to thank Feng H. Hsu and Murray Campbell for their permission
to release this code. Also let me add a note of appreciation for the
Computer Science Department at the Carnegie Mellon University, its open
and free-spirited environment made the Deep Thought project possible. If
you would like to know more about the roots of Chiptest and Deep Thought,
look out for Feng Hsu's upcoming book on his Deep Blue experience.
Share and enjoy,
-- A. Nowatzyk
January 2000
agn@acm.org
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