tex: bla bla ;)

[gostyle.git] / tex / gostyle.tex

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\hyphenation{op-tical net-works semi-conduc-tor}


\begin{document}
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% paper title
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\title{On Pattern Feature Trends in Large Go Game Corpus}

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\author{Petr~Baudis,~Josef~Moudrik% <-this % stops a space
\thanks{P. Baudis is student at the Faculty of Math and Physics, Charles University, Prague, CZ, and also does some of his Computer Go research as an employee of SUSE Labs Prague, Novell CZ.}% <-this % stops a space
\thanks{J. Moudrik is student at the Faculty of Math and Physics, Charles University, Prague, CZ.}}

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\markboth{Transactions on Computational Intelligence and AI in Games}%
{On Pattern Feature Trends in Large Go Game Corpus}
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\begin{abstract}
%\boldmath

We process a~large corpus of game records of the board game of Go and
propose a~way to extract per-player summary information on played moves.
We then apply several basic data-mining methods on the summary
information to identify the most differentiating features within the
summary information, and discuss their correspondence with traditional
Go knowledge. We show mappings of the features to player attributes
like playing strength or informally perceived "playing style" (such as
territoriality or aggressivity), and propose applications including
seeding real-work ranks of internet players, aiding in Go study, or
contribution to discussion within Go theory on the scope of "playing
style".

\end{abstract}
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\begin{IEEEkeywords}
board games, go, data mining, player strength, playing style
\end{IEEEkeywords}






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\section{Introduction}
% The very first letter is a 2 line initial drop letter followed
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\IEEEPARstart{T}{he} field of Computer Go usually focuses on the problem
of creating a~program to play the game, finding the best move from a~given
board position. We will make use of one method developed in the course
of such research and apply it to the analysis of existing game records
with the aim of helping humans to play the game better instead.

Go is a~two-player full-information board game played
on a~square grid (usually $19\times19$ lines) with black and white
stones; the goal of the game is to surround the most territory and
capture enemy stones. We assume basic familiarity with the game.

Many Go players are eager to play using computers (usually over
the internet) and review games played by others on computers as well.
This means that large amounts of game records are collected and digitally
stored, enabling easy processing of such collections. However, so far
only little has been done with the available data --- we are aware
only of uses for simple win/loss statistics (TODO: KGS Stats, KGS Analytics,
Pro Go Rating) and ''next move'' statistics on a~specific position (TODO:
Kombilo, Moyo Go Studio).

We present a~more in-depth approach --- from all played moves, we devise
a~compact evaluation of each player. We then explore correlations between
evaluations of various players in light of externally given information.
This way, we can discover similarity between moves characteristics of
players with the same playing strength, or discuss the meaning of the
"playing style" concept on the assumption that similar playing styles
should yield similar moves characteristics.


\section{Expert-based knowledge}
\label{style-vectors}
In order to provide a reference frame for our style analysis,
we have gathered some expert-based information about various
traditionally perceived style aspects.
Three high-level Go players (Alexander Dinerstein 3-pro, Motoki Noguchi
7-dan and Vit Brunner 4-dan) have judged style of several Go
professionals -- we call them \emph{reference playerse} -- chosen for both
being well-known within the community and having large number of played games in our collection.

This expert-based knowledge allows us to predict styles of unknown players based on
the similarity of their pattern vectors, as well as discover correlations between
styles and proportions of played patterns.

Experts were asked to assign each of player's style a number
on a scale from 1 to 10. These are interpreted
as shown in the table below.

\vspace{4mm}
\noindent
%\begin{table}
\begin{center}
%\caption{Styles}
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{3}{|c|}{Styles} \\ \hline
Style & 1 & 10\\ \hline
Territoriality & Moyo & Territorial \\
Orthodoxity & Classic & Novel \\
Aggressivity & Calm & Figting \\
Thickness & Safe & Shinogi \\ \hline
\end{tabular}
\end{center}
%\end{table}
\vspace{4mm}

Averaging this expert based evaluation yields
\emph{reference style vector} $\vec s_r$ (of dimension $4$) for each player $r$
from the set of \emph{reference players} $R$.

%-- each with a \emph{pattern vector} $\vec p_i$ and \emph{style vector} $\vec s_i$.

\section{Data Extraction}
\label{pattern-vectors}
In addition to the explicit expert knowledge, we use the data obtained by...

TODO rozvest uvod, nemuze se zacinat jenom As the input...

As the input, we assume a~collection of game records\footnote{We
use the SGF format (TODO) in our implementation.} organized by player names.
We use two collections; the first one is GoGoD Winter 2009 (TODO) containing 42000 (TODO)
professional games, dating from the early Go history 1500 years ago to the present.
We use this collection for style analysis and detailed correlation analysis
of well-known Go professionals.
The other source is Go Teaching Ladder reviews (TODO). These include 7600 games
of players spanning over all strength levels; we use this collection
for finding correlations between moves of players of the same strength rank.

In order to generate the required compact description of most frequently played moves,
we construct a set of $n$ most occuring patterns (\emph{top patterns})
across all players and games from the database\footnote{We use $n=500$ in our analysis.}.
For each player, we then count how many times was each of those $n$ patterns played
during all his games and finally assign him a~{\em pattern vector} $\vec p$ of dimension $n$, with each
dimension corresponding to the relative number of occurences of a given pattern
(with respect to player's most played \emph{top pattern}). Using relative numbers of occurences ensures that
each dimension of player's \emph{pattern vector} is scaled to range $[0,1]$ and
therefore even players with different number of games in the database have comparable \emph{pattern vectors}.

\subsection{Pattern Features}
TODO sladit aby to navazovalo na predchozi odstavec

Of course a big question is how to compose the pattern descriptions.
There are some tradeoffs in play - overly general descriptions carry too few
information to discern various player attributes; too specific descriptions
gather too few specimen over the games and the differences in vectors are
not statistically significant.

We have chosen an intuitive and simple approach inspired by pattern features
used when computing ELO ratings for candidate patterns in Computer Go play.
\cite{ELO} Each pattern is a~combination of several {\em pattern features}
matched at the position of the played move. We use these features:

\begin{itemize}
\item board edge distance
\item capture move flag
\item atari move flag
\item atari escape flag
\item spatial pattern --- a configuration of stones around the played move,
with differentiated color to play; configurations with diameter between
2 and 9 are matched\footnote{The diameter determines maximal distance from
center in the {\em gridcular} metric: $d(x,y) = |\delta x| + |\delta y| + \max(|\delta x|, |\delta y|)$}
\end{itemize}

\subsection{Implementation}

We have implemented the data extraction by making use of the pattern
features matching implementation within the Pachi go-playing program
(TODO). We extract information on players by converting the SGF game
records to GTP (TODO) stream that feeds Pachi's {\tt patternscan}
engine which outputs a~single patternspec per move. We can then gather
all encountered patternspecs belonging to a~given player and summarize
them; the $\vec p$ vector then consists of normalized counts of
the given $n$ most frequent patternspecs.


\section{Data Mining}
\label{data-mining}

To assess the properties of gathered \emph{pattern vectors} and their influence on playing styles,
we have analysed the data by a~few basic data minining techniques.

First two methods rely purely on data gathered from the GoGoD database. Principal component
analysis finds orthogonal vector components that have biggest variance. Reversing the process then
indicates which patterns correlate with each style. Additionally, PCA can be used as a vector-preprocessing
for methods that are negatively sensitive to \emph{pattern vector} component correlations.

A~second method -- Kohonen's networks -- is based on the theory of self-organizing maps of neurons that
compete against each other for representation of the input space. Because neurons in the network are
organized in a two-dimensional plane, the trained network virtually spreads vectors to the 2D plane,
allowing for simple visualization.

In addition to methods operating just upon the set of \emph{pattern vectors}, we have used and compared two methods
that approximate an unknown \emph{style vector} $\vec S$ of a player with known \emph{pattern vector} $\vec P$.

First of them is called $k$-Nearest Neighbor (kNN) classification.
Simpy said kNN approximates $\vec S$ by the ``average'' of \emph{style vectors} of $k$ nearest \emph{pattern vectors}.
The last method is based on the theory of Neural Networks -- networks of artificial neurons, that are used
for their generalization abilities. Neural network can learn correlations between input and output vectors and
generalize the ``knowledge'' to unknown vectors sufficiently good.

TODO rozdelit na algo/results?? 

\subsection{Principal Component Analysis}
\label{data-mining}
Principal Component Analysis \emph{PCA} \cite{Jolliffe1986} is a~method we use to reduce the dimensions of
\emph{pattern vectors} while preserving as much information as possible.

Shortly, PCA is an eigenvalue decomposition of a~covariance matrix of centered \emph{pattern vectors}.
It can be thought of as a~mapping $o$ from $n$-dimensional vector space to a~reduced $m$-dimensional vector space.
The base of this reduced vector space comprises $m$ eigenvectors of original vectors' covariance matrix.
We choose them to be the eigenvectors with biggest eigenvalues. 
Ordered by decreasing eigenvalues, the eigenvectors form rows of the transformation matrix $W$.

Finally, we represent reduced \emph{pattern vectors} as a vector of coeficients of this eigenvector-base.
For each original \emph{pattern vector} $\vec p_i$, we obtain its new representation $\vec r_i$ as shown
in the following equation:
\begin{equation}
\vec r_i = W * \vec p_i
\end{equation}

The whole process is described in the Algorithm \ref{alg:pca}.

\begin{algorithm}
\caption{PCA -- Principal Component analysis}
\begin{algorithmic}
\label{alg:pca}
\REQUIRE{$m > 0$, set of players $R$ with \emph{pattern vectors} $p_r$}
\STATE $\vec \mu \leftarrow 1/|R| * \sum_{r \in R}{\vec p_r}$
\FOR{ $r \in R$}
\STATE $\vec p_r \leftarrow \vec p_r - \vec \mu$
\ENDFOR
\FOR{ $(i,j) \in \{1,... ,n\} \times \{1,... ,n\}$}
\STATE $\mathit{Cov}[i,j] \leftarrow 1/|R| * \sum_{r \in R}{\vec p_{ri} * \vec p_{rj}}$
\ENDFOR
\STATE Compute Eigenvalue Decomposition of $\mathit{Cov}$ matrix
\STATE Get $m$ biggest eigenvalues 
\STATE According eigenvectors ordered by decreasing eigenvalues form rows of matrix $W$ 
\FOR{ $r \in R$}
\STATE $\vec r_r\leftarrow W \vec p_r$
\ENDFOR
\end{algorithmic}
\end{algorithm}

\subsection{Kohonen Maps}
\label{koh}
Kohonen map is a self-organizing network with neurons organized in a two-dimensional plane.
Neurons in the map compete for representation of input vector space. Each neuron $\vec n$ represents a vector
and the network is trained so that the neurons that are topologically close tend to represent vectors that
are close as well. That is realised as follows. Initially random network is sequentially trained;
in each iterations we choose a random training vector $\vec t$ and find neuron $\vec w$ that is closest
to $\vec t$ in Euclidean metric (we call $\vec w$ a \emph{winner neuron}).

We then adapt neurons from the neighbourhood of $\vec w$ employing an equation:
\begin{equation}
\vec n = \vec n + \alpha * \mathit{Influence}(\vec w, \vec n) * (\vec t - \vec n) 
\end{equation}
where $\alpha$ is a learning parameter, usually decreasing in time. $Influence()$ is a function that forces neurons
to spread. Such function is usually realised using a mexican hat function or a difference-of-gaussians (see \cite{TODO}
for details). A state of the network can be valued by calculating mean square difference between each $\vec t \in T$
and its corresponding \emph{winner neuron} $\vec w_t$:
\begin{equation}
\mathit{Error}(N,T) = \sum_{\vec t \in T}{|\vec w_t - \vec t|}
\end{equation}


\begin{algorithm}
\caption{Kohonen maps -- training}
\begin{algorithmic}
\label{alg:koh}
\REQUIRE{Set of training vectors $T$, input dimension $D$}
\REQUIRE{max number of iterations $M$, desired error $E$}
\STATE $N \leftarrow \{\vec n | \vec n$ random, $\mathit{dim}(\vec n) = D\}$
\REPEAT
\STATE $\mathit{It} \leftarrow \mathit{It} + 1$
\STATE $\vec t \leftarrow \mathit{PickRandom}(T)$
\FORALL{$\vec n \in N$}
\STATE $D[\vec n] \leftarrow \mathit{EuclideanDistance}(\vec n, \vec t)$
\ENDFOR
\STATE Find $ \vec w \in N$ so that $D[\vec w] <= D[\vec m], \forall \vec m \in N$
\FORALL{$\vec n \in \mathit{TopologicalNeigbors}(N, \vec w)$}
\STATE $\vec n \leftarrow \vec n + \alpha(It) * \mathit{Influence}(\vec w, \vec n) * ( \vec t - \vec n ) $
\ENDFOR
\UNTIL{$\mathit{Error}(N, T) < E$ or $ \mathit{It} > M$}
\end{algorithmic}
\end{algorithm}


\subsection{k-nearest Neighbors Classifier}
\label{knn}
K-nearest neigbors is an essential classification technique.
We use it to approximate player's \emph{style vector} $\vec S$, assuming that his \emph{pattern vector} $\vec P$ is known.
To achieve this, we utilize \emph{reference style vectors} (see section \ref{style-vectors}).

The idea is based on a assumption that similarities in players' \emph{pattern vectors}
correlate with similarities in players' \emph{style vectors}. We try to approximate $\vec S$
as a weighted average of \emph{style vectors}
$\vec s_i$ of $k$ players with \emph{pattern vectors} $\vec p_i$ closest to $\vec P$.
This is illustrated in the Algorithm \ref{alg:knn}.
Note that the weight is a function of distance and it is not explicitly defined in Algorithm \ref{alg:knn}.
During our research, exponentialy decreasing weight has proven to be sufficient.

\begin{algorithm}
\caption{k-Nearest Neighbors}
\begin{algorithmic}
\label{alg:knn}
\REQUIRE{pattern vector $\vec P$, $k > 0$, set of reference players $R$}
\FORALL{$r \in R$ }
\STATE $D[r] \leftarrow \mathit{EuclideanDistance}(\vec p_r, \vec P)$
\ENDFOR
\STATE $N \leftarrow \mathit{SelectSmallest}(k, R, D)$
\STATE $\vec S \leftarrow \vec 0$
\FORALL{$r \in N $}
\STATE $\vec S \leftarrow \vec S + \mathit{Weight}(D[r]) * \vec s_r $
\ENDFOR
\end{algorithmic}
\end{algorithm}

\subsection{Neural Network Classifier}
\label{neural-net}

As an alternative to the k-Nearest Neigbors algorithm (section \ref{knn}), we have used 
a classificator based on feed-forward artificial neural networks \cite{TODO}.
Neural networks (NN) are known for their ability to generalize and find correlations and patterns between
input and output data. Neural network is an adaptive system and it 
must undergo a certain training before it can be reasonably used. Basically, we use
information for \emph{reference players} (for which either \emph{pattern vectors} and
\emph{style vectors} are known) as training data.

\subsubsection{Computation and activation of the NN}
Technically, neural network is a network of interconnected computational units called neurons.
A feedforward neural network has a layered topology; it usually has one \emph{input layer}, one \emph{output
layer} and an arbitrary number of \emph{hidden layers} between.

Each neuron $i$ is connected to all neurons in the previous layer and each connection has its weight $w_{ij}$

The computation proceeds in a discrete time steps.
In the first step, \emph{activation} of neurons in the \emph{input layer} is set according to the \emph{input vector}.
Then, we iteratively compute output of each neuron in next layer until the output layer is reached. The activity of
output layer is then presented as a result.

The activation $y_i$ of neuron $i$ from the layer $I$ is computed using the following equation:
\begin{equation}
y_i = f(\sum_{j \in J}{w_{ij} y_j})
\end{equation}
where $J$ is a previous layer, while $y_j$ is the activation for neurons from $J$ layer. Function $f()$ is
called \emph{activation function} and its purpose is to bound outputs of neurons. A typical example of an activation
function is a sigmoid function.\footnote{The sigmoid function is a special case of the logistic function; it is defined by the formula
$\sigma(x)=\frac{1}{1+e^{-(rx+k)}}$, parameters control the growth rate ($r$) and the x-position ($k$).}

\subsubsection{Training}
The training of the feedforward neural network usually involves some
modification of supervised Backpropagation learning algorithm. \cite{TODO}
We use first-order optimization algorithm called RPROP \cite{Riedmiller1993}.

Because the \emph{reference set} is not usually very large, we have devised a simple method for its extension.
This enhancement is based upon adding random linear combinations of \emph{style and pattern vectors} to the training set.

As insinuated above, the training set consist of pairs of input vectors (\emph{pattern vectors}) and
desired output vectors (\emph{style vectors}). The training set $T$ is then enlarged by adding linear combinations:
\begin{equation}
T_\mathit{base} = \{(\vec p_r, \vec s_r) | r \in R\}\\
\end{equation}
\begin{equation}
T_\mathit{ext} = \{(\vec p, \vec s) | \exists D \subseteq R : \vec p = \sum_{d \in D}{g_d \vec p_d}, \vec s = \sum_{d \in D}{g_d \vec s_d}\}
\end{equation}
TODO zabudovat $g_d$ dovnitr?
where $g_d, d \in D$ are random coeficients, so that $\sum_{d \in D}{g_d} = 1$. The training set
is then constructed as:
\begin{equation}
T = T_\mathit{base} \cup \mathit{SomeFiniteSubset}(T_\mathit{ext})
\end{equation}

The network is trained as shown in Algorithm \ref{alg:tnn}.

\begin{algorithm}
\caption{Training Neural Network}
\begin{algorithmic}
\label{alg:tnn}
\REQUIRE{Train set $T$, desired error $e$, max iterations $M$}
\STATE $N \leftarrow \mathit{RandomlyInitializedNetwork}()$
\STATE $\mathit{It} \leftarrow 0$
\REPEAT
\STATE $\mathit{It} \leftarrow \mathit{It} + 1$
\STATE $\Delta \vec w \leftarrow \vec 0$
\STATE $\mathit{TotalError} \leftarrow 0$
%\FORALL{$(\overrightarrow{Input}, \overrightarrow{DesiredOutput}) \in T$}
%\STATE $\overrightarrow{Output} \leftarrow Result(N, \overrightarrow{Input})$
%\STATE $E \leftarrow |\overrightarrow{DesiredOutput} - \overrightarrow{Output}|$
\FORALL{$(\mathit{Input}, \mathit{DesiredOutput}) \in T$}
\STATE $\mathit{Output} \leftarrow \mathit{Result}(N, \mathit{Input})$
\STATE $\mathit{Error} \leftarrow |\mathit{DesiredOutput} - \mathit{Output}|$
\STATE $\Delta \vec w \leftarrow \Delta \vec w + \mathit{WeightUpdate}(N,\mathit{Error})$
\STATE $\mathit{TotalError} \leftarrow \mathit{TotalError} + \mathit{Error}$
\ENDFOR
\STATE $N \leftarrow \mathit{ModifyWeights}(N, \Delta \vec w)$
\UNTIL{$\mathit{TotalError} < e$ or $ \mathit{It} > M$}
\end{algorithmic}
\end{algorithm}


\subsubsection{Architecture details}
TODO num layers, num neurons, ..


\subsection{Implementation}

We have implemented the data mining methods as an open-source project ``gostyle'' \cite{TODO},
licensed under GNU GPL.
PCA: In our implementation, we use a~library called MDP \cite{MDP}.
TODO libfann


\section{Strength Estimation Analysis}

PCA analysis yielded X, chi-square test blabla...

We then tried to apply the NN classifier with linear output function on the dataset
and that yielded Y (see fig. Z), with MSE abcd.


\section{Style Components Analysis}

PCA analysis yielded X, chi-square test...

We then tried to apply the NN classifier with linear output function on the dataset
and that yielded Y (see fig. Z), with MSE abcd.


\section{Proposed Applications}

.


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\section{Conclusion}
The conclusion goes here.
We have shown brm and proposed brm.

Since we are not aware of any previous research on this topic and we
are limited by space and time constraints, plenty of research remains
to be done. There is plenty of room for further research in all parts
of our analysis --- different methods of generating the $\vec p$ vectors
can be explored; other data mining methods could be tried.
It can be argued that many players adjust their style by game conditions
(Go development era, handicap, komi and color, time limits, opponent)
or styles might express differently in various game stages.
Finally, more professional players could be consulted on the findings
and for style scales calibration.

TODO: Future research --- Sparse PCA




% if have a single appendix:
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\section*{Acknowledgment}
\label{acknowledgement}


We would like to thank X for reviewing our paper.
We appreciate helpful comments on our general methodology
by John Fairbairn, T. M. Hall, Robert Jasiek
and several GoDiscussions.com users. \cite{GoDiscThread}
Finally, we are very grateful for ranking of go styles of selected professionals
by Alexander Dinerstein 3-pro, Motoki Noguchi 7-dan and Vit Brunner 4-dan.


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