Commit beee6b3c authored by Michael Keller's avatar Michael Keller
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Gradient frist draft

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\chapter{Gradient Descent}
\chapter{Gradient Ascent}
Gradient Descent is a popular method
Gradient Ascent is a method
for searching through large search spaces.
When given a point
in the solution space, we find the locally
most beneficial direction to move in
in order to increase the score. In a
second step we move slightly in this direction.
That process is repeated until we converge
to a solution.
As from the helper theorems \ref{makeInt1} and
\ref{makeInt2} we can from fractional fields
build almost entirely integer fields,
we can actually use Gradient Descent
we can use Gradient Descent
for the solution space consisting of all
fractional solutions.
fractional solutions. This is natural
as it allows for adding and removing
small amounts of crop from a pixel, an important
feature of the small step size gradient descent has.
There are now two remaining issues we need
to tackle. The first is computing the gradient.
The second is that after following the gradient
we might no longer be within the valid solution
space. An example would
be the gradient suggesting we plant more than
one unit of crop per pixel to achieve a higher score.
While this would be beneficial to the score,
it no longer represents a valid fractional solution
for our problem. Thus may need to find the closest
valid solution. In the following sections
we tackle these issues.
\section{Finding the closest valid solution}
Again the Birkhoff Polytope offers a solution.
There are methods for finding points within
the Birkhoff Polytope that have a minimal
Euclidian distance to any arbitrary point TODO: Cite.
Hence if we could represent all valid fractional
solutions within a Birkhoff Polytope we could use these
The core idea is that we expand the fractional
solution representation slightly in order to get
a Birkhoff polytope. Usually for every pixel we
have $C$ variables, which would lead to a $(X \cdot Y) \times C$
matrix. However, for the birkhoff polytope we need
a square matrix where both all rows and all columns sum to $1$.
To achieve this we ``clone'' crops. That is, if crop $c$ should
be represented $n$ times within the field, then we introduce
$n$ crops, that all behave identically to $c$,
which all should be represented once within the field.
As the total amount of crops to be planted
is the same as the number of pixels, we must get
a square $(X \cdot Y) \times (X \cdot Y)$ matrix.
This process should be familiar from the section
on the linear programming method.
Every matrix of that form within the the corresponding
Birkhoff Polytope is a valid
fractional solution, as every pixel contains exactly
a total amount of $1$ planted crops (rows
sum to $1$) and we have one of each crop (columns sum to $1$).
As we have one of each crop and some crops behave exactly
the same, we can group these together to obtain
the original fractional solution space we started out with.
Hence every element of our Birkhoff Polytope maps
to a valid fractional solution. This means we can
project any $(X \cdot Y) \times (X \cdot Y)$ matrix
to the closest valid fractional solution.
\section{Computing the gradient}
We assume we are given a valid fractional
solution as a matrix $F$ in the Birkhoff Polytope
discussed above. Thus we have a
$(X \cdot Y) \times (X \cdot Y)$ matrix. Computing
the gradient is now straightforward. For every
pixel $p$ contains crop $c$ variable $p_c$
we take the partial derivative of the score
\frac{\partial}{\partial p_c} \sum_{n \in N(p)} \sum_{c_i \in \Cps} R(c, c_i) \cdot p_{c} \cdot n_{c_j}
= \sum_{n \in N(p)} \sum_{c_i \in \Cps} R(c, c_i) \cdot n_{c_j}
Using these elements we now build the following algorithm:
\caption{Random Gradient Ascent Algorithm}
\State $x_0 \gets \text{Random initial valid field}$
\While{$x_0$ not converged}
\State $G \gets gradient(x_0)$
\State $x_0 \gets x_0 + \text{stepsize} \cdot G$
\State $x_0 \gets project(x_0)$
\Return $integerize(x_0)$
\ No newline at end of file
......@@ -191,7 +191,7 @@ on large fields and requires no conditions of $R$.
\node[main node] (2) [right of=1] {V};
\path[every node/.style={font=\sffamily\small}]
(1) edge[bend left] node [above] {$\lambda \cdot c_1$} (2)
(1) edge[bend left] node [above] {$\lambda \cdot c_1$} (2);
(2) edge[bend left] node [below] {$\lambda \cdot c_2$} (1);
......@@ -300,8 +300,8 @@ statement with a condition on $R$ might be helpful:
\node[main node, fill=black!20,] (11) [below of=10] {N};
\node[main node, fill=black!20,] (12) [below of=11] {N};
\path[every node/.style={font=\sffamily\small}]
(1) edge[bend left] node [above] {$c_\alpha$} (2)
\path[every node/.style={font=\sffamily\small}];
(1) edge[bend left] node [above] {$c_\alpha$} (2);
(2) edge[bend left] node [below] {$c_\beta$} (1);
......@@ -33,9 +33,9 @@ their preferred crop:
\node[main node] (2) [right of=hidden] {V};
\node[main node] (3) [above of=hidden] {W};
\path[every node/.style={font=\sffamily\small}]
(1) edge[left] node [above] {$b$} (2)
(2) edge[left] node [above] {$c$} (3)
\path[every node/.style={font=\sffamily\small}];
(1) edge[left] node [above] {$b$} (2);
(2) edge[left] node [above] {$c$} (3);
(3) edge[right] node [above] {$a$} (1);
......@@ -20,7 +20,10 @@
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\BOOKMARK [0][-]{appendix.A}{\376\377\000C\000a\000l\000c\000u\000l\000a\000t\000i\000o\000n\000s\000\040\000A\000p\000p\000e\000n\000d\000i\000x}{}% 28
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......@@ -24,7 +24,10 @@
\contentsline {chapter}{\chapternumberline {8}The Linear Programming Method}{25}{chapter.8}%
\contentsline {section}{\numberline {8.1}Problem Setup}{25}{section.8.1}%
\contentsline {section}{\numberline {8.2}Method}{28}{section.8.2}%
\contentsline {chapter}{\chapternumberline {9}Gradient Descent}{31}{chapter.9}%
\contentsline {chapter}{\chapternumberline {9}Gradient Ascent}{31}{chapter.9}%
\contentsline {section}{\numberline {9.1}Finding the closest valid solution}{31}{section.9.1}%
\contentsline {section}{\numberline {9.2}Computing the gradient}{32}{section.9.2}%
\contentsline {section}{\numberline {9.3}Method}{32}{section.9.3}%
\contentsline {chapter}{\chapternumberline {10}Conclusion}{33}{chapter.10}%
\contentsline {appendix}{\chapternumberline {A}Calculations Appendix}{35}{appendix.A}%
\contentsline {chapter}{Bibliography}{37}{appendix*.4}%
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