This document introduces Monte Carlo methods and their application to the game of Go. It discusses how Monte Carlo methods can be used to simulate games and approximate properties without a closed-form solution. It also describes the multi-armed bandit problem and the upper confidence bound algorithm for exploration-exploitation. Finally, it outlines how the UCT algorithm extends upper confidence bounds to tree search for move selection in Monte Carlo Go programs.

1. Introduction
Monte Carlo Go
My Work
Machine Learning applied to Go
Dmitry Kamenetsky
Supervisor: Nic Schraudolph
March 2007
Dmitry Kamenetsky Machine Learning applied to Go

2. Introduction
Monte Carlo Go
My Work
1 Introduction
2 Monte Carlo Go
3 My Work
Dmitry Kamenetsky Machine Learning applied to Go

3. Introduction
Monte Carlo Go
My Work
What is Go?
Two-player deterministic
board game
Originated in ancient China.
Today very popular in China,
Japan and Korea
19x19 grid, also 9x9 grid for
beginners
Simple rules, but very
complex strategy
Dmitry Kamenetsky Machine Learning applied to Go

4. Introduction
Monte Carlo Go
My Work
Why study Go?
If there are sentient beings on other planets, then they play Go
– Emanuel Lasker, former chess world champion
Go is one of the grand challenges of AI
– Ron Rivest, professor of Computer Science at MIT
Go is like life and life is like Go
- Chinese proverb
Dmitry Kamenetsky Machine Learning applied to Go

5. Introduction
Monte Carlo Go
My Work
Why study Go?
If there are sentient beings on other planets, then they play Go
– Emanuel Lasker, former chess world champion
Go is one of the grand challenges of AI
– Ron Rivest, professor of Computer Science at MIT
Go is like life and life is like Go
- Chinese proverb
Dmitry Kamenetsky Machine Learning applied to Go

6. Introduction
Monte Carlo Go
My Work
Why study Go?
If there are sentient beings on other planets, then they play Go
– Emanuel Lasker, former chess world champion
Go is one of the grand challenges of AI
– Ron Rivest, professor of Computer Science at MIT
Go is like life and life is like Go
- Chinese proverb
Dmitry Kamenetsky Machine Learning applied to Go

7. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Monte Carlo
Often used for problems that have no closed solution, e.g.
computational physics
Sample instances from some large population
Use samples to approximate some common property of the
population
In search, select the next state based on some fixed
distribution (usually uniform)
Recently, have been very successful in Go, causing a
mini-revolution
Dmitry Kamenetsky Machine Learning applied to Go

8. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Monte Carlo
Often used for problems that have no closed solution, e.g.
computational physics
Sample instances from some large population
Use samples to approximate some common property of the
population
In search, select the next state based on some fixed
distribution (usually uniform)
Recently, have been very successful in Go, causing a
mini-revolution
Dmitry Kamenetsky Machine Learning applied to Go

9. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Monte Carlo
Often used for problems that have no closed solution, e.g.
computational physics
Sample instances from some large population
Use samples to approximate some common property of the
population
In search, select the next state based on some fixed
distribution (usually uniform)
Recently, have been very successful in Go, causing a
mini-revolution
Dmitry Kamenetsky Machine Learning applied to Go

10. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Monte Carlo
Often used for problems that have no closed solution, e.g.
computational physics
Sample instances from some large population
Use samples to approximate some common property of the
population
In search, select the next state based on some fixed
distribution (usually uniform)
Recently, have been very successful in Go, causing a
mini-revolution
Dmitry Kamenetsky Machine Learning applied to Go

11. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Monte Carlo
Often used for problems that have no closed solution, e.g.
computational physics
Sample instances from some large population
Use samples to approximate some common property of the
population
In search, select the next state based on some fixed
distribution (usually uniform)
Recently, have been very successful in Go, causing a
mini-revolution
Dmitry Kamenetsky Machine Learning applied to Go

12. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
K-armed bandit problem
Slot machine with K arms. Each arm provides a reward
based on some unknown, but fixed distribution
Goal: to choose arms to play such that the total reward is
maximized
How should the gambler play at any given moment?
Choose arm with highest average reward seen so far
(exploitation)
Choose a sub-optimal arm in the hope that it it will lead to
a greater reward (exploration)
Neither - need a combination of exploitation and
exploration
Dmitry Kamenetsky Machine Learning applied to Go

13. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
K-armed bandit problem
Slot machine with K arms. Each arm provides a reward
based on some unknown, but fixed distribution
Goal: to choose arms to play such that the total reward is
maximized
How should the gambler play at any given moment?
Choose arm with highest average reward seen so far
(exploitation)
Choose a sub-optimal arm in the hope that it it will lead to
a greater reward (exploration)
Neither - need a combination of exploitation and
exploration
Dmitry Kamenetsky Machine Learning applied to Go

14. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
K-armed bandit problem
Slot machine with K arms. Each arm provides a reward
based on some unknown, but fixed distribution
Goal: to choose arms to play such that the total reward is
maximized
How should the gambler play at any given moment?
Choose arm with highest average reward seen so far
(exploitation)
Choose a sub-optimal arm in the hope that it it will lead to
a greater reward (exploration)
Neither - need a combination of exploitation and
exploration
Dmitry Kamenetsky Machine Learning applied to Go

15. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
K-armed bandit problem
Slot machine with K arms. Each arm provides a reward
based on some unknown, but fixed distribution
Goal: to choose arms to play such that the total reward is
maximized
How should the gambler play at any given moment?
Choose arm with highest average reward seen so far
(exploitation)
Choose a sub-optimal arm in the hope that it it will lead to
a greater reward (exploration)
Neither - need a combination of exploitation and
exploration
Dmitry Kamenetsky Machine Learning applied to Go

16. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
K-armed bandit problem
Slot machine with K arms. Each arm provides a reward
based on some unknown, but fixed distribution
Goal: to choose arms to play such that the total reward is
maximized
How should the gambler play at any given moment?
Choose arm with highest average reward seen so far
(exploitation)
Choose a sub-optimal arm in the hope that it it will lead to
a greater reward (exploration)
Neither - need a combination of exploitation and
exploration
Dmitry Kamenetsky Machine Learning applied to Go

17. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
K-armed bandit problem
Slot machine with K arms. Each arm provides a reward
based on some unknown, but fixed distribution
Goal: to choose arms to play such that the total reward is
maximized
How should the gambler play at any given moment?
Choose arm with highest average reward seen so far
(exploitation)
Choose a sub-optimal arm in the hope that it it will lead to
a greater reward (exploration)
Neither - need a combination of exploitation and
exploration
Dmitry Kamenetsky Machine Learning applied to Go

18. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Upper Confidence Bounds
Successive plays of arm i give rewards Xi,1 , Xi,2 , ... which
are i.i.d. with unknown E(Xi ) = µi
Let Ti (n) be the number of times arm i has been played
during the first n plays of machine
Upper Confidence Bounds - UCB (Auer et al. 2002)
Initialization: Play each arm once
2 log n
Loop: Play arm i that maximizes µi + Ti (n)
Proven to achieve optimal regret
Dmitry Kamenetsky Machine Learning applied to Go

19. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Upper Confidence Bounds
Successive plays of arm i give rewards Xi,1 , Xi,2 , ... which
are i.i.d. with unknown E(Xi ) = µi
Let Ti (n) be the number of times arm i has been played
during the first n plays of machine
Upper Confidence Bounds - UCB (Auer et al. 2002)
Initialization: Play each arm once
2 log n
Loop: Play arm i that maximizes µi + Ti (n)
Proven to achieve optimal regret
Dmitry Kamenetsky Machine Learning applied to Go

20. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Upper Confidence Bounds
Successive plays of arm i give rewards Xi,1 , Xi,2 , ... which
are i.i.d. with unknown E(Xi ) = µi
Let Ti (n) be the number of times arm i has been played
during the first n plays of machine
Upper Confidence Bounds - UCB (Auer et al. 2002)
Initialization: Play each arm once
2 log n
Loop: Play arm i that maximizes µi + Ti (n)
Proven to achieve optimal regret
Dmitry Kamenetsky Machine Learning applied to Go

21. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Upper Confidence Bounds
Successive plays of arm i give rewards Xi,1 , Xi,2 , ... which
are i.i.d. with unknown E(Xi ) = µi
Let Ti (n) be the number of times arm i has been played
during the first n plays of machine
Upper Confidence Bounds - UCB (Auer et al. 2002)
Initialization: Play each arm once
2 log n
Loop: Play arm i that maximizes µi + Ti (n)
Proven to achieve optimal regret
Dmitry Kamenetsky Machine Learning applied to Go

22. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Upper Confidence Bounds
Successive plays of arm i give rewards Xi,1 , Xi,2 , ... which
are i.i.d. with unknown E(Xi ) = µi
Let Ti (n) be the number of times arm i has been played
during the first n plays of machine
Upper Confidence Bounds - UCB (Auer et al. 2002)
Initialization: Play each arm once
2 log n
Loop: Play arm i that maximizes µi + Ti (n)
Proven to achieve optimal regret
Dmitry Kamenetsky Machine Learning applied to Go

23. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
Upper Confidence Bounds
Successive plays of arm i give rewards Xi,1 , Xi,2 , ... which
are i.i.d. with unknown E(Xi ) = µi
Let Ti (n) be the number of times arm i has been played
during the first n plays of machine
Upper Confidence Bounds - UCB (Auer et al. 2002)
Initialization: Play each arm once
2 log n
Loop: Play arm i that maximizes µi + Ti (n)
Proven to achieve optimal regret
Dmitry Kamenetsky Machine Learning applied to Go

24. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
UCT
UCB for minimax tree search (Kocsis and Szepesvari 2006)
Start at the current board position p
For i = 1 to 100,000 (number of simulations)
p ←p
until stopping criterion is reached (e.g. end of game)
p ← p + move given by UCB
create node p
Evaluate leaf: value ← winner of p
Update all the visited nodes with value
At p play move with highest winning percentage
Dmitry Kamenetsky Machine Learning applied to Go

25. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
UCT
UCB for minimax tree search (Kocsis and Szepesvari 2006)
Start at the current board position p
For i = 1 to 100,000 (number of simulations)
p ←p
until stopping criterion is reached (e.g. end of game)
p ← p + move given by UCB
create node p
Evaluate leaf: value ← winner of p
Update all the visited nodes with value
At p play move with highest winning percentage
Dmitry Kamenetsky Machine Learning applied to Go

26. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
UCT
UCB for minimax tree search (Kocsis and Szepesvari 2006)
Start at the current board position p
For i = 1 to 100,000 (number of simulations)
p ←p
until stopping criterion is reached (e.g. end of game)
p ← p + move given by UCB
create node p
Evaluate leaf: value ← winner of p
Update all the visited nodes with value
At p play move with highest winning percentage
Dmitry Kamenetsky Machine Learning applied to Go

27. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
UCT
UCB for minimax tree search (Kocsis and Szepesvari 2006)
Start at the current board position p
For i = 1 to 100,000 (number of simulations)
p ←p
until stopping criterion is reached (e.g. end of game)
p ← p + move given by UCB
create node p
Evaluate leaf: value ← winner of p
Update all the visited nodes with value
At p play move with highest winning percentage
Dmitry Kamenetsky Machine Learning applied to Go

28. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
UCT
UCB for minimax tree search (Kocsis and Szepesvari 2006)
Start at the current board position p
For i = 1 to 100,000 (number of simulations)
p ←p
until stopping criterion is reached (e.g. end of game)
p ← p + move given by UCB
create node p
Evaluate leaf: value ← winner of p
Update all the visited nodes with value
At p play move with highest winning percentage
Dmitry Kamenetsky Machine Learning applied to Go

29. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
UCT
UCB for minimax tree search (Kocsis and Szepesvari 2006)
Start at the current board position p
For i = 1 to 100,000 (number of simulations)
p ←p
until stopping criterion is reached (e.g. end of game)
p ← p + move given by UCB
create node p
Evaluate leaf: value ← winner of p
Update all the visited nodes with value
At p play move with highest winning percentage
Dmitry Kamenetsky Machine Learning applied to Go

30. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
UCT
UCB for minimax tree search (Kocsis and Szepesvari 2006)
Start at the current board position p
For i = 1 to 100,000 (number of simulations)
p ←p
until stopping criterion is reached (e.g. end of game)
p ← p + move given by UCB
create node p
Evaluate leaf: value ← winner of p
Update all the visited nodes with value
At p play move with highest winning percentage
Dmitry Kamenetsky Machine Learning applied to Go

31. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
UCT
UCB for minimax tree search (Kocsis and Szepesvari 2006)
Start at the current board position p
For i = 1 to 100,000 (number of simulations)
p ←p
until stopping criterion is reached (e.g. end of game)
p ← p + move given by UCB
create node p
Evaluate leaf: value ← winner of p
Update all the visited nodes with value
At p play move with highest winning percentage
Dmitry Kamenetsky Machine Learning applied to Go

32. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
UCT
UCB for minimax tree search (Kocsis and Szepesvari 2006)
Start at the current board position p
For i = 1 to 100,000 (number of simulations)
p ←p
until stopping criterion is reached (e.g. end of game)
p ← p + move given by UCB
create node p
Evaluate leaf: value ← winner of p
Update all the visited nodes with value
At p play move with highest winning percentage
Dmitry Kamenetsky Machine Learning applied to Go

33. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
UCT
UCB for minimax tree search (Kocsis and Szepesvari 2006)
Start at the current board position p
For i = 1 to 100,000 (number of simulations)
p ←p
until stopping criterion is reached (e.g. end of game)
p ← p + move given by UCB
create node p
Evaluate leaf: value ← winner of p
Update all the visited nodes with value
At p play move with highest winning percentage
Dmitry Kamenetsky Machine Learning applied to Go

34. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
MoGo
First Go program to use UCT (Gelly et. al 2006)
Store nodes and their statistics in a tree data structure
Stopping criterion is a node that is not yet in the tree
Leaf node evaluation:
Playout position randomly until no moves remain. Final
position is trivial to score
Enhanced through the use of patterns and playing near the
previous move
Pruning techniques, smart ordering of unexplored moves
Dmitry Kamenetsky Machine Learning applied to Go

35. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
MoGo
First Go program to use UCT (Gelly et. al 2006)
Store nodes and their statistics in a tree data structure
Stopping criterion is a node that is not yet in the tree
Leaf node evaluation:
Playout position randomly until no moves remain. Final
position is trivial to score
Enhanced through the use of patterns and playing near the
previous move
Pruning techniques, smart ordering of unexplored moves
Dmitry Kamenetsky Machine Learning applied to Go

36. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
MoGo
First Go program to use UCT (Gelly et. al 2006)
Store nodes and their statistics in a tree data structure
Stopping criterion is a node that is not yet in the tree
Leaf node evaluation:
Playout position randomly until no moves remain. Final
position is trivial to score
Enhanced through the use of patterns and playing near the
previous move
Pruning techniques, smart ordering of unexplored moves
Dmitry Kamenetsky Machine Learning applied to Go

37. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
MoGo
First Go program to use UCT (Gelly et. al 2006)
Store nodes and their statistics in a tree data structure
Stopping criterion is a node that is not yet in the tree
Leaf node evaluation:
Playout position randomly until no moves remain. Final
position is trivial to score
Enhanced through the use of patterns and playing near the
previous move
Pruning techniques, smart ordering of unexplored moves
Dmitry Kamenetsky Machine Learning applied to Go

38. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
MoGo
First Go program to use UCT (Gelly et. al 2006)
Store nodes and their statistics in a tree data structure
Stopping criterion is a node that is not yet in the tree
Leaf node evaluation:
Playout position randomly until no moves remain. Final
position is trivial to score
Enhanced through the use of patterns and playing near the
previous move
Pruning techniques, smart ordering of unexplored moves
Dmitry Kamenetsky Machine Learning applied to Go

39. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
MoGo
First Go program to use UCT (Gelly et. al 2006)
Store nodes and their statistics in a tree data structure
Stopping criterion is a node that is not yet in the tree
Leaf node evaluation:
Playout position randomly until no moves remain. Final
position is trivial to score
Enhanced through the use of patterns and playing near the
previous move
Pruning techniques, smart ordering of unexplored moves
Dmitry Kamenetsky Machine Learning applied to Go

40. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
MoGo
First Go program to use UCT (Gelly et. al 2006)
Store nodes and their statistics in a tree data structure
Stopping criterion is a node that is not yet in the tree
Leaf node evaluation:
Playout position randomly until no moves remain. Final
position is trivial to score
Enhanced through the use of patterns and playing near the
previous move
Pruning techniques, smart ordering of unexplored moves
Dmitry Kamenetsky Machine Learning applied to Go

41. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
MoGo’s success
Ranked first on 9x9 Computer Go Server since August
2006
Won two most recent tournaments on 9x9 and 13x13
Expected to reach the level of human professional on 9x9
board
Dmitry Kamenetsky Machine Learning applied to Go

42. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
MoGo’s success
Ranked first on 9x9 Computer Go Server since August
2006
Won two most recent tournaments on 9x9 and 13x13
Expected to reach the level of human professional on 9x9
board
Dmitry Kamenetsky Machine Learning applied to Go

43. Introduction Monte Carlo
Monte Carlo Go Bandit Problem
My Work MoGo
MoGo’s success
Ranked first on 9x9 Computer Go Server since August
2006
Won two most recent tournaments on 9x9 and 13x13
Expected to reach the level of human professional on 9x9
board
Dmitry Kamenetsky Machine Learning applied to Go

44. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Replacing UCB
UCT is ad-hoc. Lack of theoretical analysis, because
random variables (rewards Xi ) are not i.i.d.
Instead, use Beta distributions to model random variables
Beta distribution is a conjugate prior to binomial
distribution (game result)
Here α = wins from node, β = losses from node
Let p be parent’s winning percentage and 0 < a < 1
parameter
Pick a move that is most likely to have a winning
percentage greater than (1 − a)p + a
Dmitry Kamenetsky Machine Learning applied to Go

45. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Replacing UCB
UCT is ad-hoc. Lack of theoretical analysis, because
random variables (rewards Xi ) are not i.i.d.
Instead, use Beta distributions to model random variables
Beta distribution is a conjugate prior to binomial
distribution (game result)
Here α = wins from node, β = losses from node
Let p be parent’s winning percentage and 0 < a < 1
parameter
Pick a move that is most likely to have a winning
percentage greater than (1 − a)p + a
Dmitry Kamenetsky Machine Learning applied to Go

46. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Replacing UCB
UCT is ad-hoc. Lack of theoretical analysis, because
random variables (rewards Xi ) are not i.i.d.
Instead, use Beta distributions to model random variables
Beta distribution is a conjugate prior to binomial
distribution (game result)
Here α = wins from node, β = losses from node
Let p be parent’s winning percentage and 0 < a < 1
parameter
Pick a move that is most likely to have a winning
percentage greater than (1 − a)p + a
Dmitry Kamenetsky Machine Learning applied to Go

47. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Replacing UCB
UCT is ad-hoc. Lack of theoretical analysis, because
random variables (rewards Xi ) are not i.i.d.
Instead, use Beta distributions to model random variables
Beta distribution is a conjugate prior to binomial
distribution (game result)
Here α = wins from node, β = losses from node
Let p be parent’s winning percentage and 0 < a < 1
parameter
Pick a move that is most likely to have a winning
percentage greater than (1 − a)p + a
Dmitry Kamenetsky Machine Learning applied to Go

48. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Replacing UCB
UCT is ad-hoc. Lack of theoretical analysis, because
random variables (rewards Xi ) are not i.i.d.
Instead, use Beta distributions to model random variables
Beta distribution is a conjugate prior to binomial
distribution (game result)
Here α = wins from node, β = losses from node
Let p be parent’s winning percentage and 0 < a < 1
parameter
Pick a move that is most likely to have a winning
percentage greater than (1 − a)p + a
Dmitry Kamenetsky Machine Learning applied to Go

49. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Replacing UCB
UCT is ad-hoc. Lack of theoretical analysis, because
random variables (rewards Xi ) are not i.i.d.
Instead, use Beta distributions to model random variables
Beta distribution is a conjugate prior to binomial
distribution (game result)
Here α = wins from node, β = losses from node
Let p be parent’s winning percentage and 0 < a < 1
parameter
Pick a move that is most likely to have a winning
percentage greater than (1 − a)p + a
Dmitry Kamenetsky Machine Learning applied to Go

50. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving node evaluation
MoGo’s node evaluation is fast, but not so meaningful
Instead, use our cooperative scorer:
Initialization: Statically fill neutral territory with stones
Loop: players cooperate to make moves that do not affect
the score
Accurately predicts score: 96.3% on 9x9 and 89.2% on
19x19
Only 15 times slower than pure random
Dmitry Kamenetsky Machine Learning applied to Go

51. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving node evaluation
MoGo’s node evaluation is fast, but not so meaningful
Instead, use our cooperative scorer:
Initialization: Statically fill neutral territory with stones
Loop: players cooperate to make moves that do not affect
the score
Accurately predicts score: 96.3% on 9x9 and 89.2% on
19x19
Only 15 times slower than pure random
Dmitry Kamenetsky Machine Learning applied to Go

52. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving node evaluation
MoGo’s node evaluation is fast, but not so meaningful
Instead, use our cooperative scorer:
Initialization: Statically fill neutral territory with stones
Loop: players cooperate to make moves that do not affect
the score
Accurately predicts score: 96.3% on 9x9 and 89.2% on
19x19
Only 15 times slower than pure random
Dmitry Kamenetsky Machine Learning applied to Go

53. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving node evaluation
MoGo’s node evaluation is fast, but not so meaningful
Instead, use our cooperative scorer:
Initialization: Statically fill neutral territory with stones
Loop: players cooperate to make moves that do not affect
the score
Accurately predicts score: 96.3% on 9x9 and 89.2% on
19x19
Only 15 times slower than pure random
Dmitry Kamenetsky Machine Learning applied to Go

54. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving node evaluation
MoGo’s node evaluation is fast, but not so meaningful
Instead, use our cooperative scorer:
Initialization: Statically fill neutral territory with stones
Loop: players cooperate to make moves that do not affect
the score
Accurately predicts score: 96.3% on 9x9 and 89.2% on
19x19
Only 15 times slower than pure random
Dmitry Kamenetsky Machine Learning applied to Go

55. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving node evaluation
MoGo’s node evaluation is fast, but not so meaningful
Instead, use our cooperative scorer:
Initialization: Statically fill neutral territory with stones
Loop: players cooperate to make moves that do not affect
the score
Accurately predicts score: 96.3% on 9x9 and 89.2% on
19x19
Only 15 times slower than pure random
Dmitry Kamenetsky Machine Learning applied to Go

56. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Scorer Example
Dmitry Kamenetsky Machine Learning applied to Go

57. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Scorer Example
Dmitry Kamenetsky Machine Learning applied to Go

58. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Scorer Example
Dmitry Kamenetsky Machine Learning applied to Go

59. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving memory management
Tree data structure is memory-inefficient
Instead, use a hash table:
For each visited position p, key = ZobristHash(p)
Store statistics of p: hashTable[key] = (#wins, #runs, depth)
Collision handling
Use a small hash table with more information for
frequently visited nodes
Dmitry Kamenetsky Machine Learning applied to Go

60. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving memory management
Tree data structure is memory-inefficient
Instead, use a hash table:
For each visited position p, key = ZobristHash(p)
Store statistics of p: hashTable[key] = (#wins, #runs, depth)
Collision handling
Use a small hash table with more information for
frequently visited nodes
Dmitry Kamenetsky Machine Learning applied to Go

61. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving memory management
Tree data structure is memory-inefficient
Instead, use a hash table:
For each visited position p, key = ZobristHash(p)
Store statistics of p: hashTable[key] = (#wins, #runs, depth)
Collision handling
Use a small hash table with more information for
frequently visited nodes
Dmitry Kamenetsky Machine Learning applied to Go

62. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving memory management
Tree data structure is memory-inefficient
Instead, use a hash table:
For each visited position p, key = ZobristHash(p)
Store statistics of p: hashTable[key] = (#wins, #runs, depth)
Collision handling
Use a small hash table with more information for
frequently visited nodes
Dmitry Kamenetsky Machine Learning applied to Go

63. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving memory management
Tree data structure is memory-inefficient
Instead, use a hash table:
For each visited position p, key = ZobristHash(p)
Store statistics of p: hashTable[key] = (#wins, #runs, depth)
Collision handling
Use a small hash table with more information for
frequently visited nodes
Dmitry Kamenetsky Machine Learning applied to Go

64. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Improving memory management
Tree data structure is memory-inefficient
Instead, use a hash table:
For each visited position p, key = ZobristHash(p)
Store statistics of p: hashTable[key] = (#wins, #runs, depth)
Collision handling
Use a small hash table with more information for
frequently visited nodes
Dmitry Kamenetsky Machine Learning applied to Go

65. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Learning evaluation function
Convert the board position into a graph:
Collapse regions of the same colour into one node
Create edges between adjacent regions
Use Condition Random Fields (CRF) to learn from this
graph:
Learn final territory assignment
Predict the next move
Can use this with the scorer or for move generation
Dmitry Kamenetsky Machine Learning applied to Go

66. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Learning evaluation function
Convert the board position into a graph:
Collapse regions of the same colour into one node
Create edges between adjacent regions
Use Condition Random Fields (CRF) to learn from this
graph:
Learn final territory assignment
Predict the next move
Can use this with the scorer or for move generation
Dmitry Kamenetsky Machine Learning applied to Go

67. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Learning evaluation function
Convert the board position into a graph:
Collapse regions of the same colour into one node
Create edges between adjacent regions
Use Condition Random Fields (CRF) to learn from this
graph:
Learn final territory assignment
Predict the next move
Can use this with the scorer or for move generation
Dmitry Kamenetsky Machine Learning applied to Go

68. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Learning evaluation function
Convert the board position into a graph:
Collapse regions of the same colour into one node
Create edges between adjacent regions
Use Condition Random Fields (CRF) to learn from this
graph:
Learn final territory assignment
Predict the next move
Can use this with the scorer or for move generation
Dmitry Kamenetsky Machine Learning applied to Go

69. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Learning evaluation function
Convert the board position into a graph:
Collapse regions of the same colour into one node
Create edges between adjacent regions
Use Condition Random Fields (CRF) to learn from this
graph:
Learn final territory assignment
Predict the next move
Can use this with the scorer or for move generation
Dmitry Kamenetsky Machine Learning applied to Go

70. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Learning evaluation function
Convert the board position into a graph:
Collapse regions of the same colour into one node
Create edges between adjacent regions
Use Condition Random Fields (CRF) to learn from this
graph:
Learn final territory assignment
Predict the next move
Can use this with the scorer or for move generation
Dmitry Kamenetsky Machine Learning applied to Go

71. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Learning evaluation function
Convert the board position into a graph:
Collapse regions of the same colour into one node
Create edges between adjacent regions
Use Condition Random Fields (CRF) to learn from this
graph:
Learn final territory assignment
Predict the next move
Can use this with the scorer or for move generation
Dmitry Kamenetsky Machine Learning applied to Go

72. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Graph conversion example
Dmitry Kamenetsky Machine Learning applied to Go

73. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Graph conversion example
Dmitry Kamenetsky Machine Learning applied to Go

74. Beta Distributions
Introduction
Cooperative Scorer
Monte Carlo Go
Zobrist Hash
My Work
Conditional Random Fields
Questions?
You never ever know if you never ever GO!
Dmitry Kamenetsky Machine Learning applied to Go