Suppose we choose our feature vector to be

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x$=[$n

white

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$,$n

black

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$],$ the number of white and black pieces on the board, respectively and we design our $1-$layer neural network

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;

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f$($x;$\backslash$theta $)=\backslash$sigma $($z$),$ where

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$0$

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$+$

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$1$

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$+$

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z$=$w

$0$

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n

white

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$+$w

$1$

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n

black

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$+$b$,$

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$0$

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$1$

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$\backslash$theta $=[$w

$0$

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$,$w

$1$

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$,$b$]$ and

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$\backslash$sigma is the sigmoid function.

Screenshot$\_2023-04-17\_$at$\_7\mathrm{.}09\mathrm{.}47\_$PM$.$png

We preprocess the data and organize it by features. The organized data comes in the form of

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k tuples:

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$($x

i

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$,$n

i

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$,$m

i

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$)$ for

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$1$

$,$

$...$

$,$

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i$=1,...,$k$,$ where

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x

i

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is the feature vector,

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n

i

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is the number of games played starting from states with those features, and

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m

i

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is the number of games won of those

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n

i

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games.

We decide that a reasonable utility function should be the probability of success and the output of our neural network will be the the probability

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p of winning each game starting from the given state.

Our goal is to learn the parameters

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$\backslash$theta of

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$($

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;

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p$=$f$($x;$\backslash$theta $)$ which outputs the most likely

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p for a given feature vector

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x$,$ given our training data.

What is the log likelihood function that we are trying to maximize? Keep your answer in terms of

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;

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n

i

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$,$m

i

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$,$x

i

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$,$f$($x

i

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;$\backslash$theta $).$ Hint: You may want to use rules for logs to expand out the log$-$likelihood to make the next part easier.