<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
(	O
MCTS	O
)	O
is	O
a	O
heuristic	B-Algorithm
search	B-Application
algorithm	I-Application
for	O
some	O
kinds	O
of	O
decision	O
processes	B-Operating_System
,	O
most	O
notably	O
those	O
employed	O
in	O
software	O
that	O
plays	O
board	O
games	O
.	O
</s>
<s>
MCTS	O
was	O
combined	O
with	O
neural	B-Architecture
networks	I-Architecture
in	O
2016	O
and	O
has	O
been	O
used	O
in	O
multiple	O
board	O
games	O
like	O
Chess	B-Application
,	O
Shogi	O
,	O
Checkers	O
,	O
Backgammon	B-Application
,	O
Contract	O
Bridge	O
,	O
Go	O
,	O
Scrabble	B-Application
,	O
and	O
Clobber	B-General_Concept
as	O
well	O
as	O
in	O
turn-based-strategy	O
video	O
games	O
(	O
such	O
as	O
Total	B-Application
War	I-Application
:	I-Application
Rome	I-Application
II	I-Application
's	O
implementation	O
in	O
the	O
high	O
level	O
campaign	O
AI	O
)	O
.	O
</s>
<s>
The	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
,	O
which	O
uses	O
random	B-Algorithm
sampling	I-Algorithm
for	O
deterministic	O
problems	O
which	O
are	O
difficult	O
or	O
impossible	O
to	O
solve	O
using	O
other	O
approaches	O
,	O
dates	O
back	O
to	O
the	O
1940s	O
.	O
</s>
<s>
In	O
his	O
1987	O
PhD	O
thesis	O
,	O
Bruce	O
Abramson	O
combined	O
minimax	B-Algorithm
search	I-Algorithm
with	O
an	O
expected-outcome	O
model	O
based	O
on	O
random	O
game	O
playouts	O
to	O
the	O
end	O
,	O
instead	O
of	O
the	O
usual	O
static	B-General_Concept
evaluation	I-General_Concept
function	I-General_Concept
.	O
</s>
<s>
He	O
experimented	O
in-depth	O
with	O
tic-tac-toe	O
and	O
then	O
with	O
machine-generated	O
evaluation	B-General_Concept
functions	I-General_Concept
for	O
Othello	O
and	O
chess	B-Application
.	O
</s>
<s>
Such	O
methods	O
were	O
then	O
explored	O
and	O
successfully	O
applied	O
to	O
heuristic	B-Algorithm
search	O
in	O
the	O
field	O
of	O
automated	B-Application
theorem	I-Application
proving	I-Application
by	O
W	O
.	O
Ertel	O
,	O
J	O
.	O
Schumann	O
and	O
C	O
.	O
Suttner	O
in	O
1989	O
,	O
thus	O
improving	O
the	O
exponential	O
search	O
times	O
of	O
uninformed	O
search	B-Application
algorithms	I-Application
such	O
as	O
e.g.	O
</s>
<s>
breadth-first	O
search	O
,	O
depth-first	O
search	O
or	O
iterative	B-Algorithm
deepening	I-Algorithm
.	O
</s>
<s>
In	O
1992	O
,	O
B	O
.	O
Brügmann	O
employed	O
it	O
for	O
the	O
first	O
time	O
in	O
a	O
Go-playing	B-Application
program	I-Application
.	O
</s>
<s>
proposed	O
the	O
idea	O
of	O
"	O
recursive	O
rolling	O
out	O
and	O
backtracking	O
"	O
with	O
"	O
adaptive	O
"	O
sampling	O
choices	O
in	O
their	O
Adaptive	O
Multi-stage	O
Sampling	O
(	O
AMS	O
)	O
algorithm	O
for	O
the	O
model	O
of	O
Markov	O
decision	O
processes	B-Operating_System
.	O
</s>
<s>
In	O
2006	O
,	O
inspired	O
by	O
these	O
predecessors	O
,	O
Rémi	O
Coulom	O
described	O
the	O
application	O
of	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
to	O
game-tree	O
search	O
and	O
coined	O
the	O
name	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
,	O
L	O
.	O
Kocsis	O
and	O
Cs	O
.	O
</s>
<s>
Google	B-Application
Deepmind	I-Application
developed	O
the	O
program	O
AlphaGo	B-Application
,	O
which	O
in	O
October	O
2015	O
became	O
the	O
first	O
Computer	B-Application
Go	I-Application
program	O
to	O
beat	O
a	O
professional	O
human	O
Go	O
player	O
without	O
handicaps	O
on	O
a	O
full-sized	O
19x19	O
board	O
.	O
</s>
<s>
In	O
March	O
2016	O
,	O
AlphaGo	B-Application
was	O
awarded	O
an	O
honorary	O
9-dan	O
(	O
master	O
)	O
level	O
in	O
19×19	O
Go	O
for	O
defeating	O
Lee	O
Sedol	O
in	O
a	B-Application
five-game	I-Application
match	I-Application
with	O
a	O
final	O
score	O
of	O
four	O
games	O
to	O
one	O
.	O
</s>
<s>
AlphaGo	B-Application
represents	O
a	O
significant	O
improvement	O
over	O
previous	O
Go	B-Application
programs	I-Application
as	O
well	O
as	O
a	O
milestone	O
in	O
machine	O
learning	O
as	O
it	O
uses	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
with	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
(	O
a	O
deep	B-Algorithm
learning	I-Algorithm
method	O
)	O
for	O
policy	O
(	O
move	O
selection	O
)	O
and	O
value	O
,	O
giving	O
it	O
efficiency	O
far	O
surpassing	O
previous	O
programs	O
.	O
</s>
<s>
MCTS	O
algorithm	O
has	O
also	O
been	O
used	O
in	O
programs	O
that	O
play	O
other	O
board	O
games	O
(	O
for	O
example	O
Hex	O
,	O
Havannah	O
,	O
Game	O
of	O
the	O
Amazons	O
,	O
and	O
Arimaa	O
)	O
,	O
real-time	O
video	O
games	O
(	O
for	O
instance	O
Ms.	B-Application
Pac-Man	I-Application
and	O
Fable	B-Device
Legends	I-Device
)	O
,	O
and	O
nondeterministic	O
games	O
(	O
such	O
as	O
skat	O
,	O
poker	O
,	O
Magic	B-Application
:	I-Application
The	I-Application
Gathering	I-Application
,	O
or	O
Settlers	O
of	O
Catan	O
)	O
.	O
</s>
<s>
The	O
focus	O
of	O
MCTS	O
is	O
on	O
the	O
analysis	O
of	O
the	O
most	O
promising	O
moves	O
,	O
expanding	O
the	O
search	B-Data_Structure
tree	I-Data_Structure
based	O
on	O
random	B-Algorithm
sampling	I-Algorithm
of	O
the	O
search	O
space	O
.	O
</s>
<s>
The	O
application	O
of	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
in	O
games	O
is	O
based	O
on	O
many	O
playouts	O
,	O
also	O
called	O
roll-outs	O
.	O
</s>
<s>
Each	O
round	O
of	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
consists	O
of	O
four	O
steps	O
:	O
</s>
<s>
The	O
section	O
below	O
says	O
more	O
about	O
a	O
way	O
of	O
biasing	O
choice	O
of	O
child	O
nodes	O
that	O
lets	O
the	O
game	O
tree	O
expand	O
towards	O
the	O
most	O
promising	O
moves	O
,	O
which	O
is	O
the	O
essence	O
of	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
.	O
</s>
<s>
A	O
playout	O
may	O
be	O
as	O
simple	O
as	O
choosing	O
uniform	O
random	O
moves	O
until	O
the	O
game	O
is	O
decided	O
(	O
for	O
example	O
in	O
chess	B-Application
,	O
the	O
game	O
is	O
won	O
,	O
lost	O
,	O
or	O
drawn	O
)	O
.	O
</s>
<s>
DeepMind	B-Application
's	O
AlphaZero	B-Application
replaces	O
the	O
simulation	O
step	O
with	O
an	O
evaluation	O
based	O
on	O
a	O
neural	B-Architecture
network	I-Architecture
.	O
</s>
<s>
UCT	O
is	O
based	O
on	O
the	O
UCB1	O
formula	O
derived	O
by	O
Auer	O
,	O
Cesa-Bianchi	O
,	O
and	O
Fischer	O
and	O
the	O
probably	O
convergent	O
AMS	O
(	O
Adaptive	O
Multi-stage	O
Sampling	O
)	O
algorithm	O
first	O
applied	O
to	O
multi-stage	O
decision-making	O
models	O
(	O
specifically	O
,	O
Markov	O
Decision	O
Processes	B-Operating_System
)	O
by	O
Chang	O
,	O
Fu	O
,	O
Hu	O
,	O
and	O
Marcus	O
.	O
</s>
<s>
Most	O
contemporary	O
implementations	O
of	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
are	O
based	O
on	O
some	O
variant	O
of	O
UCT	O
that	O
traces	O
its	O
roots	O
back	O
to	O
the	O
AMS	O
simulation	O
optimization	O
algorithm	O
for	O
estimating	O
the	O
value	O
function	O
in	O
finite-horizon	O
Markov	O
Decision	O
Processes	B-Operating_System
(	O
MDPs	O
)	O
introduced	O
by	O
Chang	O
et	O
al	O
.	O
</s>
<s>
Although	O
it	O
has	O
been	O
proven	O
that	O
the	O
evaluation	O
of	O
moves	O
in	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
converges	O
to	O
minimax	B-Algorithm
,	O
the	O
basic	O
version	O
of	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
converges	O
only	O
in	O
so	O
called	O
"	O
Monte	O
Carlo	O
Perfect	O
"	O
games	O
.	O
</s>
<s>
However	O
,	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
does	O
offer	O
significant	O
advantages	O
over	O
alpha	B-Algorithm
–	I-Algorithm
beta	I-Algorithm
pruning	I-Algorithm
and	O
similar	O
algorithms	O
that	O
minimize	O
the	O
search	O
space	O
.	O
</s>
<s>
In	O
particular	O
,	O
pure	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
does	O
not	O
need	O
an	O
explicit	O
evaluation	B-General_Concept
function	I-General_Concept
.	O
</s>
<s>
As	O
such	O
,	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
can	O
be	O
employed	O
in	O
games	O
without	O
a	O
developed	O
theory	O
or	O
in	O
general	B-Algorithm
game	I-Algorithm
playing	I-Algorithm
.	O
</s>
<s>
The	O
game	O
tree	O
in	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
grows	O
asymmetrically	O
as	O
the	O
method	O
concentrates	O
on	O
the	O
more	O
promising	O
subtrees	O
.	O
</s>
<s>
Thus	O
,	O
it	O
achieves	O
better	O
results	O
than	O
classical	O
algorithms	O
in	O
games	O
with	O
a	O
high	O
branching	B-Data_Structure
factor	I-Data_Structure
.	O
</s>
<s>
It	O
is	O
believed	O
that	O
this	O
may	O
have	O
been	O
part	O
of	O
the	O
reason	O
for	O
AlphaGo	B-Application
's	I-Application
loss	I-Application
in	I-Application
its	I-Application
fourth	I-Application
game	I-Application
against	I-Application
Lee	I-Application
Sedol	I-Application
.	O
</s>
<s>
Various	O
modifications	O
of	O
the	O
basic	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
method	O
have	O
been	O
proposed	O
to	O
shorten	O
the	O
search	O
time	O
.	O
</s>
<s>
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
can	O
use	O
either	O
light	O
or	O
heavy	O
playouts	O
.	O
</s>
<s>
Light	O
playouts	O
consist	O
of	O
random	O
moves	O
while	O
heavy	O
playouts	O
apply	O
various	O
heuristics	B-Algorithm
to	O
influence	O
the	O
choice	O
of	O
moves	O
.	O
</s>
<s>
These	O
heuristics	B-Algorithm
may	O
employ	O
the	O
results	O
of	O
previous	O
playouts	O
(	O
e.g.	O
</s>
<s>
the	O
Last	O
Good	O
Reply	O
heuristic	B-Algorithm
)	O
or	O
expert	O
knowledge	O
of	O
a	O
given	O
game	O
.	O
</s>
<s>
Paradoxically	O
,	O
playing	O
suboptimally	O
in	O
simulations	O
sometimes	O
makes	O
a	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
program	O
play	O
stronger	O
overall	O
.	O
</s>
<s>
A	O
related	O
method	O
,	O
called	O
progressive	O
bias	O
,	O
consists	O
in	O
adding	O
to	O
the	O
UCB1	O
formula	O
a	O
element	O
,	O
where	O
is	O
a	O
heuristic	B-Algorithm
score	O
of	O
the	O
-th	O
move	O
.	O
</s>
<s>
The	O
basic	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
collects	O
enough	O
information	O
to	O
find	O
the	O
most	O
promising	O
moves	O
only	O
after	O
many	O
rounds	O
;	O
until	O
then	O
its	O
moves	O
are	O
essentially	O
random	O
.	O
</s>
<s>
Heuristics	B-Algorithm
used	O
in	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
often	O
require	O
many	O
parameters	O
.	O
</s>
<s>
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
can	O
be	O
concurrently	O
executed	O
by	O
many	O
threads	B-Operating_System
or	O
processes	B-Operating_System
.	O
</s>
<s>
There	O
are	O
several	O
fundamentally	O
different	O
methods	O
of	O
its	O
parallel	B-Operating_System
execution	O
:	O
</s>
<s>
Leaf	O
parallelization	B-Operating_System
,	O
i.e.	O
</s>
<s>
parallel	B-Operating_System
execution	O
of	O
many	O
playouts	O
from	O
one	O
leaf	O
of	O
the	O
game	O
tree	O
.	O
</s>
<s>
Root	O
parallelization	B-Operating_System
,	O
i.e.	O
</s>
<s>
building	O
independent	O
game	O
trees	O
in	O
parallel	B-Operating_System
and	O
making	O
the	O
move	O
basing	O
on	O
the	O
root-level	O
branches	O
of	O
all	O
these	O
trees	O
.	O
</s>
<s>
Tree	O
parallelization	B-Operating_System
,	O
i.e.	O
</s>
<s>
parallel	B-Operating_System
building	O
of	O
the	O
same	O
game	O
tree	O
,	O
protecting	O
data	O
from	O
simultaneous	O
writes	O
either	O
with	O
one	O
,	O
global	O
mutex	B-Operating_System
,	O
with	O
more	O
mutexes	B-Operating_System
,	O
or	O
with	O
non-blocking	B-Operating_System
synchronization	I-Operating_System
.	O
</s>
