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Moved FAIKR1 in year1
This commit is contained in:
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src/year1/fundamentals-of-ai-and-kr/module1/ainotes.cls
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src/year1/fundamentals-of-ai-and-kr/module1/ainotes.cls
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../../../ainotes.cls
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src/year1/fundamentals-of-ai-and-kr/module1/faikr1.tex
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src/year1/fundamentals-of-ai-and-kr/module1/faikr1.tex
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\documentclass[11pt]{ainotes}
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\title{Fundamentals of Artificial Intelligence and Knowledge Representation\\(Module 1)}
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\date{2023 -- 2024}
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\def\lastupdate{{PLACEHOLDER-LAST-UPDATE}}
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\begin{document}
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\makenotesfront
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\input{sections/_intro.tex}
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\input{sections/_search.tex}
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\input{sections/_local_search.tex}
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\input{sections/_swarm_intelligence.tex}
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\input{sections/_games.tex}
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\input{sections/_planning.tex}
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\input{sections/_constraints.tex}
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\eoc
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\end{document}
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\chapter{Constraints satisfaction}
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\begin{description}
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\item[Constraints satisfaction problem (CSP)] \marginnote{Constraints satisfaction problem (CSP)}
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Problem defined on a finite set of variables\\$(X_1, \dots, X_n)$,
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each belonging to a domain $(D_1, \dots, D_n)$
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(notation: $X_i :: [d_{i,1}, \dots, d_{i,m}]$ or $X_i \in [d_{i,1}, \dots, d_{i,m}]$).
|
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A constraint is a relationship between variables (e.g. $X < Y$).
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Formally, a constraint $c(X_{k_1}, \dots, X_{k_m})$ is a subset of the cartesian product
|
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$D_{k_1} \times \dots \times D_{k_m}$ of the admissible values of the variables.
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A solution to a CSP is an admissible assignment of all the variables.
|
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CSP can be solved as a search problem.
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\begin{descriptionlist}
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\item[A posteriori algorithms] \marginnote{A posteriori algorithms}
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Constraints are checked after the construction of the search tree.
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|
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\item[Propagation algorithms] \marginnote{Propagation algorithms}
|
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Once a variable has been assigned,
|
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constraints are propagated to the unassigned variables to prune part of the search space.
|
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|
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\item[Consistency techniques] \marginnote{Consistency techniques}
|
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Constraints are propagated to derive a simpler problem.
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\end{descriptionlist}
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\end{description}
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\section{A posteriori algorithms}
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\subsection{Generate and test}
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\marginnote{Generate and test}
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|
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The search tree is constructed depth-first.
|
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At level $i$, a value is assigned to the variable $X_i$.
|
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Constraints are only checked when a leaf has been reached.
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In other words, it computes all the permutations of the values of the variables.
|
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|
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|
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\subsection{Standard backtracking}
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\marginnote{Standard backtracking}
|
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|
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The search tree is constructed depth-first.
|
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At level $i$, a value is assigned to the variable $X_i$ and
|
||||
constraints involving $X_1, \dots, X_i$ are checked.
|
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In case of failure, the path is not further explored.
|
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A problem with this approach is that it requires to backtrack in case of failure
|
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and reassign all the variables in the worst case.
|
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|
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|
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|
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\section{Propagation algorithms}
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|
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\subsection{Forward checking}
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\marginnote{Forward checking}
|
||||
|
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The search tree is constructed depth-first.
|
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At level $i$, a value is assigned to the variable $X_i$ and
|
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constraints involving $X_i$ are propagated
|
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to restrict the domain of the remaining unassigned variables.
|
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If the domain of a variable becomes empty, the path is considered a failure and the algorithm backtracks.
|
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|
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\begin{example}
|
||||
Consider the variables:
|
||||
\[ X_1 :: [1, 2, 3] \hspace{1cm} X_2 :: [1, 2, 3] \hspace{1cm} X_3 :: [1, 2, 3] \]
|
||||
and the constraints $X_1 < X_2 < X_3$.
|
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|
||||
We assign the variables in lexicographic order, at each step we have that:
|
||||
\begin{enumerate}
|
||||
\item $X_1 = 1 \hspace{1cm} X_2 :: [\cancel{1}, 2, 3] \hspace{1cm} X_3 :: [\cancel{1}, 2, 3]$
|
||||
\item $X_1 = 1 \hspace{1cm} X_2 = 2 \hspace{1cm} X_3 :: [\cancel{1}, \cancel{2}, 3]$
|
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\item $X_1 = 1 \hspace{1cm} X_2 = 2 \hspace{1cm} X_3 = 3$
|
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\end{enumerate}
|
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|
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In another scenario, assume that we choose $X_1 = 2$ as first step:
|
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\begin{enumerate}
|
||||
\item $X_1 = 2 \hspace{1cm} X_2 :: [\cancel{1}, \cancel{2}, 3] \hspace{1cm} X_3 :: [\cancel{1}, \cancel{2}, 3]$
|
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\item $X_1 = 2 \hspace{1cm} X_2 = 3 \hspace{1cm} X_3 :: [\cancel{1}, \cancel{2}, \cancel{3}]$
|
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\end{enumerate}
|
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As the domain of $X_3$ is empty, a search on this branch fails and backtracking is required.
|
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\end{example}
|
||||
|
||||
|
||||
|
||||
\subsection{Look ahead}
|
||||
|
||||
\begin{description}
|
||||
\item[Partial look ahead (PLA)] \marginnote{Partial look ahead (PLA)}
|
||||
When a variable $X_i$ has been assigned, it is propagated to unassigned variables (as in forward checking).
|
||||
Then, for each unassigned variable $X_h$, we check its values against the ones of the variables $X_{h+1}, \dots, X_{n}$ in front of it
|
||||
(note that the check is not done in both directions).
|
||||
|
||||
When checking the values of an unassigned variable $X_p :: D_p$ against $X_q :: D_q$,
|
||||
we maintain a value $x \in D_p$ only if there is at least a corresponding value $y \in D_q$ such that
|
||||
an assignment $X_p=x$ and $X_q=y$ does not violate the constraints.
|
||||
$y$ is called support \marginnote{Support} of $x$.
|
||||
|
||||
\begin{example}
|
||||
Consider the variables and constraints:
|
||||
\[ X_1 :: [1, 2, 3] \hspace{0.5cm} X_2 :: [1, 2, 3] \hspace{0.5cm} X_3 :: [1, 2, 3] \hspace{1cm} X_1 < X_2 < X_3 \]
|
||||
|
||||
We assign the variables in lexicographic order. At each step, we have that:
|
||||
\begin{enumerate}
|
||||
\item $X_1 = 1 \hspace{1cm} X_2 :: [\cancel{1}, 2, \cancel{3}] \hspace{1cm} X_3 :: [\cancel{1}, 2, 3]$ \\
|
||||
Here, we assign $X_1=1$ and propagate to unassigned constraints.
|
||||
Then, we check the constraints of $(X_2, X_3)$: $X_2 = 3$ is not possible as it does not have a support in $X_3$.
|
||||
\item $X_1 = 1 \hspace{1cm} X_2 = 2 \hspace{1cm} X_3 :: [\cancel{1}, \cancel{2}, 3]$
|
||||
\item $X_1 = 1 \hspace{1cm} X_2 = 2 \hspace{1cm} X_3 = 3$
|
||||
\end{enumerate}
|
||||
\end{example}
|
||||
|
||||
\item[Full look ahead (FLA)] \marginnote{Full look ahead (FLA)}
|
||||
Same as partial look ahead, but the checks on unassigned variables are bidirectional.
|
||||
|
||||
\begin{example}
|
||||
Consider the variables and constraints:
|
||||
\[ X_1 :: [1, 2, 3] \hspace{0.5cm} X_2 :: [1, 2, 3] \hspace{0.5cm} X_3 :: [1, 2, 3] \hspace{1cm} X_1 < X_2 < X_3 \]
|
||||
|
||||
We assign the variables in lexicographic order. At each step, we have that:
|
||||
\begin{enumerate}
|
||||
\item $X_1 = 1 \hspace{1cm} X_2 :: [\cancel{1}, 2, \cancel{3}] \hspace{1cm} X_3 :: [\cancel{1}, \cancel{2}, 3]$ \\
|
||||
Here, we assign $X_1=1$ and propagate to unassigned constraints.
|
||||
Then, we check the constraints of:
|
||||
\begin{itemize}
|
||||
\item $(X_2, X_3)$: $X_2 = 3$ is not possible as it does not have a support in $X_3$.
|
||||
\item $(X_3, X_2)$: $X_3 = 2$ is not possible as it does not have a support in $X_2$.
|
||||
\end{itemize}
|
||||
Note that checking in a different order might result in different (but correct) domains.
|
||||
\item $X_1 = 1 \hspace{1cm} X_2 = 2 \hspace{1cm} X_3 = 3$
|
||||
\end{enumerate}
|
||||
\end{example}
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Search heuristics}
|
||||
When searching, we have two degrees of freedom:
|
||||
\begin{enumerate}
|
||||
\item Selection of the variable to assign.
|
||||
\item Selection of the value to assign.
|
||||
\item Strategy for propagation.
|
||||
\end{enumerate}
|
||||
|
||||
Heuristics can be used to make better selections (for the points 1. and 2.):
|
||||
\begin{descriptionlist}
|
||||
\item[Variable selection] \marginnote{Variable selection heuristics}
|
||||
\phantom{}
|
||||
\begin{description}
|
||||
\item[First-fail (minimum remaining values)] \marginnote{First-fail}
|
||||
Choose the variable with the smallest domain.
|
||||
|
||||
\item[Most-constrained principle] \marginnote{Most-constrained principle}
|
||||
Choose the variable that appears the most in the constraints.
|
||||
In this way, harder variables are ideally assigned first.
|
||||
\end{description}
|
||||
|
||||
\item[Value selection] \marginnote{Value selection heuristics}
|
||||
The choice of the value to assign to a variable does not have general rules.
|
||||
\end{descriptionlist}
|
||||
|
||||
Moreover, heuristics can be:
|
||||
\begin{descriptionlist}
|
||||
\item[Static] \marginnote{Static heuristics}
|
||||
Computed before the search.
|
||||
|
||||
\item[Dynamic] \marginnote{Dynamic heuristics}
|
||||
Can be computed at each assignment.
|
||||
\end{descriptionlist}
|
||||
|
||||
|
||||
|
||||
\section{Consistency techniques}
|
||||
|
||||
Consistency techniques reduce the original problem by removing domain values that are surely impossible.
|
||||
This class of methods can be applied statically before the search or after each assignment.
|
||||
|
||||
\begin{description}
|
||||
\item[Constraint graph] \marginnote{Constraint graph}
|
||||
Graph where nodes are variables and arcs are constraints.
|
||||
\begin{itemize}
|
||||
\item There is a directed arc from $X_i$ to $X_j$ if there is a binary constraint $X_i \circ X_j$.
|
||||
\item There is a self-loop to $X_i$ if there is a unary constraint on $X_i$.
|
||||
\end{itemize}
|
||||
|
||||
\item[Node consistency (level 1 consistency)] \marginnote{Node consistency}
|
||||
A node $X_i$ is consistent if all of its values in the domain satisfy the constraints.
|
||||
|
||||
A graph is node consistent if all its nodes are consistent.
|
||||
|
||||
\item[Arc consistency (level 2 consistency)] \marginnote{Arc consistency}
|
||||
An arc from $X_i :: D_i$ to $X_j :: D_j$ is consistent if each value of $D_i$ has a support in $D_j$ (as in look ahead).
|
||||
|
||||
A graph is arc consistent if all its arcs are consistent.
|
||||
|
||||
\begin{description}
|
||||
\item[AC-3 algorithm] \marginnote{AC-3 algorithm}
|
||||
A possible algorithm to achieve arc consistency.
|
||||
Uses a queue to store arcs to explore:
|
||||
\begin{enumerate}
|
||||
\item At the beginning, all arcs are in the queue.
|
||||
\item When evaluating an arc $(X_i, X_j)$,
|
||||
values in the domain of $X_i$ are removed if they don't have a support in the domain of $X_j$.
|
||||
\item If the domain of $X_i$ has been modified,
|
||||
for all the neighbors $X_k$ of $X_i$, the arc $(X_k, X_i)$ is added to the queue.
|
||||
\item Repeat until empty queue (quiescence).
|
||||
\end{enumerate}
|
||||
\end{description}
|
||||
|
||||
\begin{example}
|
||||
Note: when doing the exercises, it is sufficient to iterate until none of the nodes change, without using a queue.
|
||||
|
||||
Consider the variables and constraints:
|
||||
\[ X_1 :: [1, 2, 3] \hspace{0.5cm} X_2 :: [1, 2, 3] \hspace{0.5cm} X_3 :: [1, 2, 3] \hspace{1cm} X_1 < X_2 < X_3 \]
|
||||
The constraint graph and the iterations to achieve arc consistency are the following:
|
||||
|
||||
\begin{minipage}{0.7\textwidth}
|
||||
\begin{enumerate}
|
||||
\item We check the following pairs:
|
||||
\begin{itemize}
|
||||
\item $(X_1, X_2)$. Here, $X_1 :: [1, 2, \cancel{3}]$.
|
||||
\item $(X_2, X_1)$. Here, $X_2 :: [\cancel{1}, 2, 3]$.
|
||||
\item $(X_2, X_3)$. Here, $X_2 :: [\cancel{1}, 2, \cancel{3}]$.
|
||||
\item $(X_3, X_2)$. Here, $X_3 :: [\cancel{1}, \cancel{2}, 3]$.
|
||||
\item $(X_1, X_3)$. Here, $X_1 :: [1, 2, \cancel{3}]$ (no changes).
|
||||
\item $(X_3, X_1)$. Here, $X_3 :: [\cancel{1}, \cancel{2}, 3]$ (no changes).
|
||||
\end{itemize}
|
||||
The resulting domains are:
|
||||
\[ X_1 :: [1, 2, \cancel{3}] \hspace{0.5cm} X_2 :: [\cancel{1}, 2, \cancel{3}] \hspace{0.5cm} X_3 :: [\cancel{1}, \cancel{2}, 3] \]
|
||||
\item We check the following pairs:
|
||||
\begin{itemize}
|
||||
\item $(X_1, X_2)$. Here, $X_1 :: [1, \cancel{2}, \cancel{3}]$.
|
||||
\item Other pairs do nothing.
|
||||
\end{itemize}
|
||||
The resulting domains are:
|
||||
\[ X_1 :: [1, \cancel{2}, \cancel{3}] \hspace{0.5cm} X_2 :: [\cancel{1}, 2, \cancel{3}] \hspace{0.5cm} X_3 :: [\cancel{1}, \cancel{2}, 3] \]
|
||||
\end{enumerate}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.20\textwidth}
|
||||
\includegraphics[width=\linewidth]{img/_constraint_graph.pdf}
|
||||
\end{minipage}
|
||||
\end{example}
|
||||
|
||||
\item[Path consistency (level 3 consistency)] \marginnote{Path consistency}
|
||||
A path $(X_i, X_j, X_k)$ is path consistent if $X_i$ and $X_j$ are node consistent, $(X_i, X_j)$ is arc consistent and
|
||||
for every pair of values in the domains of $X_i$ and $X_j$, there is a support in the domain of $X_k$.
|
||||
|
||||
\item[K-consistency] \marginnote{K-consistency}
|
||||
Generalization of arc/path consistency.
|
||||
If a problem with $n$ variables is $n$-consistent, the solution can be found without search.
|
||||
|
||||
Usually, it is not applicable as it has exponential complexity.
|
||||
\end{description}
|
||||
146
src/year1/fundamentals-of-ai-and-kr/module1/sections/_games.tex
Normal file
146
src/year1/fundamentals-of-ai-and-kr/module1/sections/_games.tex
Normal file
@ -0,0 +1,146 @@
|
||||
\chapter{Games}
|
||||
|
||||
In this course, we are interested in two-player games with
|
||||
perfect knowledge (both players have the same information on the state of the game).
|
||||
|
||||
|
||||
\section{Minimax algorithm}
|
||||
\marginnote{Minimax algorithm}
|
||||
The minimax (min-max) algorithm allows to determine the optimal strategy for a player by
|
||||
building and propagating utility scores in a tree where each node represents a state of the game.
|
||||
It considers the player as the entity that maximizes (\textsc{Max}) its utility and
|
||||
the opponent as the entity that (optimally) minimizes (\textsc{Min}) the utility of the player.
|
||||
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.5\textwidth]{img/_minmax.pdf}
|
||||
\caption{Example of game tree with propagated scores}
|
||||
\end{figure}
|
||||
|
||||
In a game tree, each level represents the actions that a single player can do.
|
||||
In minimax, the levels where the player plays are the \textsc{Max} levels,
|
||||
while the levels of the opponent are the \textsc{Min} levels.
|
||||
|
||||
Given a node $n$ of a game tree,
|
||||
an iteration of the minimax algorithm can be described as follows:
|
||||
\begin{descriptionlist}
|
||||
\item[Expansion]
|
||||
Expansion of the sub-tree having $n$ as root by considering the possible moves starting from the state of $n$.
|
||||
|
||||
The expansion in height stops when a final state is reached or when some predetermined conditions are met
|
||||
(note that different branches may be expanded with different heights).
|
||||
|
||||
The expansion in width stops when all the possible moves have been considered or when some predetermined conditions are met.
|
||||
|
||||
\item[Evaluation]
|
||||
Each leaf is labeled with a score.
|
||||
For terminal states, a possible score assignment is:
|
||||
\[ \texttt{utility}(\texttt{state}) = \begin{cases}
|
||||
+1 & \text{Win} \\
|
||||
-1 & \text{Loss} \\
|
||||
0 & \text{Draw}
|
||||
\end{cases} \]
|
||||
For non-terminal states, heuristics are required.
|
||||
|
||||
\item[Propagation]
|
||||
Starting from the parents of the leaves, the scores are propagated upwards
|
||||
by labeling the parents based on the children's scores.
|
||||
|
||||
Given an unlabeled node $m$, if $m$ is at a \textsc{Max} level, its label is the maximum of its children's score.
|
||||
Otherwise (\textsc{Min} level), the label is the minimum of its children's score.
|
||||
\end{descriptionlist}
|
||||
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{10cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & Yes, if the game tree is finite \\
|
||||
\hline
|
||||
\textbf{Optimality} & Yes, assuming an optimal opponent (otherwise, it may need more moves) \\
|
||||
\hline
|
||||
\textbf{Time complexity}
|
||||
& $O(b^m)$, with breadth $m$ and branching factor $b$ \\
|
||||
\hline
|
||||
\textbf{Space complexity}
|
||||
& $O(b \cdot m)$, with breadth $m$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Minimax}
|
||||
\begin{lstlisting}[mathescape=true]
|
||||
def minimax(node, max_depth, who_is_next):
|
||||
if node.isLeaf() or max_depth == 0:
|
||||
eval = evaluate(node)
|
||||
elif who_is_next == ME:
|
||||
eval = -$\infty$
|
||||
for c in node.children:
|
||||
eval = max(eval, minimax(c, max_depth-1, OPPONENT))
|
||||
elif who_is_next == OPPONENT:
|
||||
eval = +$\infty$
|
||||
for c in node.children:
|
||||
eval = min(eval, minimax(c, max_depth-1, ME))
|
||||
return eval
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
\section{Alpha-beta cuts}
|
||||
\marginnote{Alpha-beta cuts}
|
||||
Alpha-beta pruning (cuts) allows to prune subtrees whose state will never be selected (when playing optimally).
|
||||
$\alpha$ represents the best choice found for \textsc{Max}.
|
||||
$\beta$ represents the best choice found for \textsc{Min}.
|
||||
|
||||
The best case for alpha-beta cuts is when the best nodes are evaluated first.
|
||||
In this scenario, the theoretical number of nodes to explore is decreased to $O(b^{d/2})$.
|
||||
In practice, the reduction is of order $O(\sqrt{b^d})$.
|
||||
In the average case of a random distribution, the reduction is of order $O(b^{3d/4})$.
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Minimax with alpha-beta cuts}
|
||||
\begin{lstlisting}[mathescape=true]
|
||||
def alphabeta(node, max_depth, who_is_next, $\alpha$=-$\infty$, $\beta$=+$\infty$):
|
||||
if node.isLeaf() or max_depth == 0:
|
||||
eval = evaluate(node)
|
||||
elif who_is_next == ME:
|
||||
eval = -$\infty$
|
||||
for c in node.children:
|
||||
eval = max(eval, alphabeta(c, max_depth-1, OPPONENT, $\alpha$, $\beta$))
|
||||
$\alpha$ = max(eval, $\alpha$)
|
||||
if eval >= $\beta$: break # cutoff
|
||||
elif who_is_next == OPPONENT:
|
||||
eval = +$\infty$
|
||||
for c in node.children:
|
||||
eval = min(eval, alphabeta(c, max_depth-1, ME, $\alpha$, $\beta$))
|
||||
$\beta$ = min(eval, $\beta$)
|
||||
if eval <= $\alpha$: break # cutoff
|
||||
return eval
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
\begin{figure}[h]
|
||||
\begin{subfigure}{.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{img/alphabeta_algo_example1.png}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{img/alphabeta_algo_example2.png}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{.4\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{img/alphabeta_algo_example3.png}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{.4\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{img/alphabeta_algo_example4.png}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{.4\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{img/alphabeta_algo_example5.png}
|
||||
\end{subfigure}
|
||||
\caption{Algorithmic (left values) and intuitive (right value) application of alpha-beta}
|
||||
\end{figure}
|
||||
215
src/year1/fundamentals-of-ai-and-kr/module1/sections/_intro.tex
Normal file
215
src/year1/fundamentals-of-ai-and-kr/module1/sections/_intro.tex
Normal file
@ -0,0 +1,215 @@
|
||||
\lohead{\color{gray} Not required for the exam}
|
||||
\lehead{\color{gray} Not required for the exam}
|
||||
\chapter{Introduction}
|
||||
|
||||
|
||||
\section{AI systems classification}
|
||||
|
||||
\subsection{Intelligence classification}
|
||||
Intelligence is defined as the ability to perceive or infer information and to retain the knowledge for future use.
|
||||
|
||||
\begin{description}
|
||||
\item[Weak AI] \marginnote{Weak AI}
|
||||
aims to build a system that acts as an intelligent system.
|
||||
|
||||
\item[Strong AI] \marginnote{Strong AI}
|
||||
aims to build a system that is actually intelligent.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Capability classification}
|
||||
\begin{description}
|
||||
\item[General AI] \marginnote{General AI}
|
||||
systems able to solve any generalized task.
|
||||
|
||||
\item[Narrow AI] \marginnote{Narrow AI}
|
||||
systems able to solve a particular task.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{AI approaches}
|
||||
\begin{description}
|
||||
\item[Symbolic AI (top-down)] \marginnote{Symbolic AI}
|
||||
Symbolic representation of knowledge, understandable by humans.
|
||||
|
||||
\item[Connectionist approach (bottom-up)] \marginnote{Connectionist approach}
|
||||
Neural networks. Knowledge is encoded and not understandable by humans.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Symbolic AI}
|
||||
\begin{description}
|
||||
\item[Deductive reasoning] \marginnote{Deductive reasoning}
|
||||
Conclude something given some premises (general to specific).
|
||||
It is unable to produce new knowledge.
|
||||
\begin{example}
|
||||
"All men are mortal" and "Socrates is a man" $\rightarrow$ "Socrates is mortal"
|
||||
\end{example}
|
||||
|
||||
\item[Inductive reasoning] \marginnote{Inductive reasoning}
|
||||
A conclusion is derived from an observation (specific to general).
|
||||
Produces new knowledge, but correctness is not guaranteed.
|
||||
\begin{example}
|
||||
"Several birds fly" $\rightarrow$ "All birds fly"
|
||||
\end{example}
|
||||
|
||||
\item[Abduction reasoning] \marginnote{Abduction reasoning}
|
||||
An explanation of the conclusion is found from known premises.
|
||||
Differently from inductive reasoning, it does not search for a general rule.
|
||||
Produces new knowledge, but correctness is not guaranteed.
|
||||
\begin{example}
|
||||
"Socrates is dead" (conclusion) and "All men are mortal" (knowledge) $\rightarrow$ "Socrates is a man"
|
||||
\end{example}
|
||||
|
||||
\item[Reasoning by analogy] \marginnote{Reasoning by analogy}
|
||||
Principle of similarity (e.g. k-nearest-neighbor algorithm).
|
||||
\begin{example}
|
||||
"Socrates loves philosophy" and Socrates resembles John $\rightarrow$ "John loves philosophy"
|
||||
\end{example}
|
||||
|
||||
\item[Constraint reasoning and optimization] \marginnote{Constraint reasoning}
|
||||
Constraints, probability, statistics.
|
||||
\end{description}
|
||||
|
||||
|
||||
\section{Machine learning}
|
||||
|
||||
\subsection{Training approach}
|
||||
\begin{description}
|
||||
\item[Supervised learning] \marginnote{Supervised learning}
|
||||
Trained on labeled data (ground truth is known).\\
|
||||
Suitable for classification and regression tasks.
|
||||
|
||||
\item[Unsupervised learning] \marginnote{Unsupervised learning}
|
||||
Trained on unlabeled data (the system makes its own discoveries).\\
|
||||
Suitable for clustering and data mining.
|
||||
|
||||
\item[Semi-supervised learning] \marginnote{Semi-supervised learning}
|
||||
The system is first trained to synthesize data in an unsupervised manner,
|
||||
followed by a supervised phase.
|
||||
|
||||
\item[Reinforcement learning] \marginnote{Reinforcement learning}
|
||||
An agent learns by simulating actions in an environment with rewards and punishments depending on its choices.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Tasks}
|
||||
\begin{description}
|
||||
\item[Classification] \marginnote{Classification}
|
||||
Supervised task that, given the input variables $X$ and the output (discrete) categories $Y$,
|
||||
aims to approximate a mapping function $f: X \rightarrow Y$.
|
||||
|
||||
\item[Regression] \marginnote{Regression}
|
||||
Supervised task that, given the input variables $X$ and the output (continuous) variables $Y$,
|
||||
aims to approximate a mapping function $f: X \rightarrow Y$.
|
||||
|
||||
\item[Clustering] \marginnote{Clustering}
|
||||
Unsupervised task that aims to organize objects into groups.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Neural networks}
|
||||
\marginnote{Perceptron}
|
||||
A neuron (\textbf{perceptron}) computes a weighted sum of its inputs and
|
||||
passes the result to an activation function to produce the output.
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.40\textwidth]{img/neuron.png}
|
||||
\caption{Representation of an artificial neuron}
|
||||
\end{figure}
|
||||
|
||||
\marginnote{Feed-forward neural network}
|
||||
A \textbf{feed-forward neural network} is composed of multiple layers of neurons, each connected to the next one.
|
||||
The first layer is the input layer, while the last is the output layer.
|
||||
Intermediate layers are hidden layers.
|
||||
|
||||
The expressivity of a neural network increases when more neurons are used:
|
||||
\begin{descriptionlist}
|
||||
\item[Single perceptron]
|
||||
Able to compute a linear separation.
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.25\textwidth]{img/1perceptron.png}
|
||||
\caption{Separation performed by one perceptron}
|
||||
\end{figure}
|
||||
\item[Three-layer network]
|
||||
Able to separate a convex region ($n_\text{edges} \leq n_\text{hidden neurons}$)
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.90\textwidth]{img/3layer.png}
|
||||
\caption{Separation performed by a three-layer network}
|
||||
\end{figure}
|
||||
\item[Four-layer network]
|
||||
Able to separate regions of arbitrary shape.
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.40\textwidth]{img/4layer.png}
|
||||
\caption{Separation performed by a four-layer network}
|
||||
\end{figure}
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{theorem}[Universal approximation theorem] \marginnote{Universal approximation theorem}
|
||||
A feed-forward network with one hidden layer and a finite number of neurons is
|
||||
able to approximate any continuous function with desired accuracy.
|
||||
\end{theorem}
|
||||
|
||||
\begin{description}
|
||||
\item[Deep learning] \marginnote{Deep learning}
|
||||
Neural network with a large number of layers and neurons.
|
||||
The learning process is hierarchical: the network exploits simple features in the first layers and
|
||||
synthesizes more complex concepts while advancing through the layers.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Automated planning}
|
||||
Given an initial state, a set of actions and a goal,
|
||||
\textbf{automated planning} aims to find a partially or totally ordered sequence of actions to achieve a goal. \marginnote{Automated planning}
|
||||
|
||||
An \textbf{automated planner} is an agent that operates in a given domain described by:
|
||||
\begin{itemize}
|
||||
\item Representation of the initial state
|
||||
\item Representation of a goal
|
||||
\item Formal description of the possible actions (preconditions and effects)
|
||||
\end{itemize}
|
||||
|
||||
|
||||
|
||||
\section{Swarm intelligence}
|
||||
\marginnote{Swarm intelligence}
|
||||
Decentralized and self-organized systems that result in emergent behaviors.
|
||||
|
||||
|
||||
|
||||
\section{Decision support systems}
|
||||
|
||||
\begin{description}
|
||||
\item[Knowledge based system] \marginnote{Knowledge based system}
|
||||
Use knowledge (and data) to support human decisions.
|
||||
Bottlenecked by knowledge acquisition.
|
||||
\end{description}
|
||||
|
||||
Different levels of decision support exist:
|
||||
\begin{descriptionlist}
|
||||
\item[Descriptive analytics] \marginnote{Descriptive analytics}
|
||||
Data are used to describe the system (e.g. dashboards, reports, \dots).
|
||||
Human intervention is required.
|
||||
|
||||
\item[Diagnostic analytics] \marginnote{Diagnostic analytics}
|
||||
Data are used to understand causes (e.g. fault diagnosis)
|
||||
Decisions are made by humans.
|
||||
|
||||
\item[Predictive analytics] \marginnote{Predictive analytics}
|
||||
Data are used to predict future evolutions of the system.
|
||||
Uses machine learning models or simulators (digital twins)
|
||||
|
||||
\item[Prescriptive analytics] \marginnote{Prescriptive analytics}
|
||||
Make decisions by finding the preferred scenario.
|
||||
Uses optimization systems, combinatorial solvers or logical solvers.
|
||||
\end{descriptionlist}
|
||||
|
||||
|
||||
\newpage
|
||||
\lohead{}
|
||||
\lehead{}
|
||||
@ -0,0 +1,295 @@
|
||||
\chapter{Local search}
|
||||
|
||||
\begin{description}
|
||||
\item[Local search] \marginnote{Local search}
|
||||
Starting from an initial state, iteratively improves it by making local moves in a neighborhood.
|
||||
|
||||
Useful when the path to reach the solution is not important (i.e. no optimality).
|
||||
|
||||
\item[Neighborhood] \marginnote{Neighborhood}
|
||||
Given a set of states $\mathcal{S}$, a neighborhood is a function:
|
||||
\[ \mathcal{N}: S \rightarrow 2^\mathcal{S} \]
|
||||
In other words, for each $s \in \mathcal{S}$, $\mathcal{N}(s) \subseteq \mathcal{S}$.
|
||||
|
||||
\begin{example}[Travelling salesman problem]
|
||||
Problem: find a Hamiltonian tour of minimum cost in an undirected graph.
|
||||
|
||||
A possible neighborhood of a state applies the $k$-exchange that guarantees to maintain a Hamiltonian tour.
|
||||
\begin{figure}[ht]
|
||||
\begin{subfigure}{.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=.70\linewidth]{img/tsp_2-exchange.png}
|
||||
\caption{2-exchange}
|
||||
\end{subfigure}%
|
||||
\begin{subfigure}{.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=.70\linewidth]{img/tsp_3-exchange.png}
|
||||
\caption{3-exchange}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
\end{example}
|
||||
|
||||
|
||||
\item[Local optima]
|
||||
Given an evaluation function $f$,
|
||||
a local optima (maximization case) is a state $s$ such that:
|
||||
\[ \forall s' \in \mathcal{N}(s): f(s) \geq f(s') \]
|
||||
|
||||
\item[Global optima]
|
||||
Given an evaluation function $f$,
|
||||
a global optimum (maximization case) is a state $s_\text{opt}$ such that:
|
||||
\[ \forall s \in \mathcal{S}: f(s_\text{opt}) \geq f(s) \]
|
||||
|
||||
Note: a larger neighborhood usually allows to obtain better solutions.
|
||||
|
||||
\item[Plateau]
|
||||
Flat area of the evaluation function.
|
||||
|
||||
\item[Ridges]
|
||||
Higher area of the evaluation function that is not directly reachable.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Iterative improvement (hill climbing)}
|
||||
\marginnote{Iterative improvement (hill climbing)}
|
||||
Algorithm that only performs moves that improve the current solution.
|
||||
|
||||
It does not keep track of the explored states (i.e. may return to a previously visited state) and
|
||||
stops after reaching a local optima.
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Iterative improvement}
|
||||
\begin{lstlisting}
|
||||
def iterativeImprovement(problem):
|
||||
s = problem.initial_state
|
||||
while noImprovement():
|
||||
s = bestOf(problem.neighborhood(s))
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
|
||||
\section{Meta heuristics}
|
||||
\marginnote{Meta heuristics}
|
||||
Methods that aim to improve the final solution.
|
||||
Can be seen as a search process over graphs:
|
||||
\begin{descriptionlist}
|
||||
\item[Neighborhood graph] The search space topology.
|
||||
\item[Search graph] The explored space.
|
||||
\end{descriptionlist}
|
||||
\begin{figure}[ht]
|
||||
\begin{subfigure}{.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=.55\linewidth]{img/_local_search_neigh_graph.pdf}
|
||||
\caption{Neighborhood graph}
|
||||
\end{subfigure}%
|
||||
\begin{subfigure}{.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=.60\linewidth]{img/_local_search_search_graph.pdf}
|
||||
\caption{Search graph. Edges are probabilities}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
|
||||
Meta heuristics aim to find a balance between:
|
||||
\begin{descriptionlist}
|
||||
\item[Intensification] \marginnote{Intensification}
|
||||
Look for moves near the neighborhood.
|
||||
\item[Diversification] \marginnote{Diversification}
|
||||
Look for moves somewhere else.
|
||||
\end{descriptionlist}
|
||||
|
||||
Different termination criteria can be used with meta heuristics:
|
||||
\begin{itemize}
|
||||
\item Time constraints.
|
||||
\item Iterations limit.
|
||||
\item Absence of improving moves (stagnation).
|
||||
\end{itemize}
|
||||
|
||||
|
||||
\subsection{Simulated annealing}
|
||||
\marginnote{Simulated annealing}
|
||||
Occasionally allow moves that worsen the current solution.
|
||||
The probability of this to happen is:
|
||||
\[
|
||||
\frac{e^{f(s) - f(s')}}{T}
|
||||
\]
|
||||
where $s$ is the current state, $s'$ is the next move and $T$ is the temperature.
|
||||
The temperature is updated at each iteration and can be:
|
||||
\begin{descriptionlist}
|
||||
\item[Logarithmic] $T_{k+1} = \Gamma / \log(k + k_0) $
|
||||
\item[Geometric] $T_{k+1} = \alpha T_k$, where $\alpha \in\, ]0, 1[$
|
||||
\item[Non-monotonic] $T$ is alternatively decreased (intensification) and increased (diversification).
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Meta heuristics -- Simulated annealing}
|
||||
\begin{lstlisting}[mathescape=true]
|
||||
def simulatedAnnealing(problem, T0):
|
||||
s = problem.initial_state
|
||||
T, k = T0, 0
|
||||
while not terminationConditions():
|
||||
s_next = randomOf(problem.neighborhood(s))
|
||||
if (problem.f(s_next) > problem.f(s) or
|
||||
downHillWithProbability($e^{\texttt{problem.f(\texttt{s}) - problem.f(s\_next)}} \texttt{/ T}$)):
|
||||
s = s_next
|
||||
k += 1
|
||||
update(T, k)
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
\subsection{Tabu search}
|
||||
\marginnote{Tabu search}
|
||||
Keep track of the last $n$ explored solutions in a tabu list and forbid them.
|
||||
Allows to escape from local optima and cycles.
|
||||
|
||||
Since keeping track of the visited solutions is inefficient,
|
||||
moves can be stored instead but, with this approach, some still not visited solutions may be cut off.
|
||||
\textbf{Aspiration criteria} can be used to allow forbidden moves in the tabu list to be evaluated.
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Meta heuristics -- Tabu search}
|
||||
\begin{lstlisting}[mathescape=true]
|
||||
def tabuSearch(problem, T0):
|
||||
s = problem.initial_state
|
||||
tabu_list = [] # limited to n elements
|
||||
T, k = T0, 0
|
||||
while not terminationConditions():
|
||||
allowed_s = $\{ s' \in \mathcal{N}(s): s' \notin \texttt{tabu\_list} \texttt{ or } \texttt{aspiration condition satisfied} \}$
|
||||
s = bestOf(allowed_s)
|
||||
updateTabuListAndAspirationConditions()
|
||||
k += 1
|
||||
update(T, k)
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
\subsection{Iterated local search}
|
||||
\marginnote{Iterated local search}
|
||||
Based on two steps:
|
||||
\begin{descriptionlist}
|
||||
\item[Subsidiary local search steps] Efficiently reach a local optimum (intensification).
|
||||
\item[Perturbation steps] Escape from a local optima (diversification).
|
||||
\end{descriptionlist}
|
||||
In addition, an acceptance criterion controls the two steps.
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Meta heuristics -- Iterated local search}
|
||||
\begin{lstlisting}[mathescape=true]
|
||||
def tabuSearch(problem):
|
||||
s = localSearch(problem.initial_state)
|
||||
while not terminationConditions():
|
||||
s_perturbation = perturbation(s, history)
|
||||
s_local = localSearch(s_perturbation)
|
||||
s = acceptanceCriterion(s, s_local, history)
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
\subsection{Population based (genetic algorithm)}
|
||||
\marginnote{Population based (genetic algorithm)}
|
||||
|
||||
Population based meta heuristics are built on the following concepts:
|
||||
\begin{descriptionlist}
|
||||
\item[Adaptation] Organisms are suited to their environment.
|
||||
\item[Inheritance] Offspring resemble their parents.
|
||||
\item[Natural selection] Fit organisms have many offspring while others become extinct.
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{table}[ht]
|
||||
\centering
|
||||
\begin{tabular}{c | c}
|
||||
\textbf{Biology} & \textbf{Artificial intelligence} \\
|
||||
\hline
|
||||
Individuals & Possible solution \\
|
||||
Fitness & Quality \\
|
||||
Environment & Problem \\
|
||||
\end{tabular}
|
||||
\caption{Biological evolution metaphors}
|
||||
\end{table}
|
||||
|
||||
% The evolutionary cycle of a genetic algorithm has the following steps:
|
||||
% \begin{descriptionlist}
|
||||
% \item[Recombination] Combine the genetic material of the parents.
|
||||
% \item[Mutation] Introduce variability.
|
||||
% \item[Selection] Selection of the parents.
|
||||
% \item[Replacement/Insertion] Define the new population.
|
||||
% \end{descriptionlist}
|
||||
The following terminology will be used:
|
||||
\begin{descriptionlist}
|
||||
\item[Population] Set of individuals (solutions).
|
||||
\item[Genotypes] Individuals of a population.
|
||||
\item[Genes] Units of chromosomes.
|
||||
\item[Alleles] Domain of values of a gene.
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width=0.5\textwidth]{img/_genetic_terminology.pdf}
|
||||
\caption{}
|
||||
\end{figure}
|
||||
|
||||
|
||||
Genetic operators are:
|
||||
\begin{descriptionlist}
|
||||
\item[Recombination/Crossover]
|
||||
Cross-combination of two chromosomes.
|
||||
\begin{center}
|
||||
\includegraphics[width=0.5\textwidth]{img/_genetic_crossover.pdf}
|
||||
\end{center}
|
||||
\item[Mutation]
|
||||
Random modification of genes.
|
||||
\begin{center}
|
||||
\includegraphics[width=0.2\textwidth]{img/_genetic_mutation.pdf}
|
||||
\end{center}
|
||||
\item[Proportional selection]
|
||||
Probability of an individual to be chosen as parent of the next offspring.
|
||||
Depends on the fitness.
|
||||
\item[Generational replacement]
|
||||
Create the new generation. Possible approaches are:
|
||||
\begin{itemize}
|
||||
\item Completely replace the old generation with the new one.
|
||||
\item Keep the best $n$ individual from the new and old population.
|
||||
\end{itemize}
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{example}[Real-valued genetic operators]
|
||||
Solution $x \in [a, b]$ with $a, b \in \mathbb{R}$.
|
||||
\begin{descriptionlist}
|
||||
\item[Mutation] Random perturbation: $x \rightarrow x \pm \delta$, as long as $x \pm \delta \in [a, b]$.
|
||||
\item[Crossover] Linear combination: $x = \lambda_1 y_1 + \lambda_2 y_2$, as long as $x \in [a, b]$.
|
||||
\end{descriptionlist}
|
||||
\end{example}
|
||||
|
||||
\begin{example}[Permutation genetic operators]
|
||||
Solution $x = (x_1, \dots, x_n)$ is a permutation of $(1, \dots, n)$.
|
||||
\begin{descriptionlist}
|
||||
\item[Mutation] Random exchange of two elements at index $i$ and $j$, with $i \neq j$.
|
||||
\item[Crossover] Crossover avoiding repetitions.
|
||||
\end{descriptionlist}
|
||||
\end{example}
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width=0.4\textwidth]{img/_genetic_cycle.pdf}
|
||||
\caption{Evolutionary cycle}
|
||||
\end{figure}
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Meta heuristics -- Genetic algorithm}
|
||||
\begin{lstlisting}[mathescape=true]
|
||||
def geneticAlgorithm(problem):
|
||||
population = problem.initPopulation()
|
||||
evaluate(population)
|
||||
while not terminationConditions():
|
||||
offspring = []
|
||||
while not offspringComplete(offspring):
|
||||
p1, p2 = selectParents(population)
|
||||
new_individual = crossover(p1, p2)
|
||||
new_individual = mutation(new_individual)
|
||||
offspring.append(new_individual)
|
||||
population = offspring
|
||||
evaluate(population)
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
1039
src/year1/fundamentals-of-ai-and-kr/module1/sections/_planning.tex
Normal file
1039
src/year1/fundamentals-of-ai-and-kr/module1/sections/_planning.tex
Normal file
File diff suppressed because it is too large
Load Diff
409
src/year1/fundamentals-of-ai-and-kr/module1/sections/_search.tex
Normal file
409
src/year1/fundamentals-of-ai-and-kr/module1/sections/_search.tex
Normal file
@ -0,0 +1,409 @@
|
||||
\chapter{Search problems}
|
||||
|
||||
|
||||
|
||||
\section{Search strategies}
|
||||
\begin{description}
|
||||
\item[Solution space] \marginnote{Solution space}
|
||||
Set of all the possible sequences of actions an agent may apply.
|
||||
Some of these lead to a solution.
|
||||
|
||||
\item[Search algorithm] \marginnote{Search algorithm}
|
||||
Takes a problem as input and returns a sequence of actions that solves the problem (if exists).
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Search tree}
|
||||
\begin{description}
|
||||
\item[Expansion] \marginnote{Expansion}
|
||||
Starting from a state, apply a successor function and generate a new state.
|
||||
|
||||
\item[Search strategy] \marginnote{Search strategy}
|
||||
Choose which state to expand.
|
||||
Usually is implemented using a fringe that decides which is the next node to expand.
|
||||
|
||||
\item[Search tree] \marginnote{Search tree}
|
||||
Tree structure to represent the expansion of all states starting from a root
|
||||
(i.e. the representation of the solution space).
|
||||
|
||||
Nodes are states and branches are actions.
|
||||
A leaf can be a state to expand, a solution or a dead-end.
|
||||
\Cref{alg:search_tree_search} describes a generic tree search algorithm.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.25\textwidth]{img/_search_tree.pdf}
|
||||
\caption{Search tree}
|
||||
\end{figure}
|
||||
|
||||
Each node contains:
|
||||
\begin{itemize}
|
||||
\item The state
|
||||
\item The parent node
|
||||
\item The action that led to this node
|
||||
\item The depth of the node
|
||||
\item The cost of the path from the root to this node
|
||||
\end{itemize}
|
||||
\end{description}
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Tree search} \label{alg:search_tree_search}
|
||||
\begin{lstlisting}
|
||||
def treeSearch(problem, fringe):
|
||||
fringe.push(problem.initial_state)
|
||||
# Get a node in the fringe and expand it if it is not a solution
|
||||
while fringe.notEmpty():
|
||||
node = fringe.pop()
|
||||
if problem.isGoal(node.state):
|
||||
return node.solution
|
||||
fringe.pushAll(expand(node, problem))
|
||||
return FAILURE
|
||||
|
||||
def expand(node, problem):
|
||||
successors = set()
|
||||
# List all neighboring nodes
|
||||
for action, result in problem.successor(node.state):
|
||||
s = new Node(
|
||||
parent=node, action=action, state=result, depth=node.dept+1,
|
||||
cost=node.cost + problem.pathCost(node, s, action)
|
||||
)
|
||||
successors.add(s)
|
||||
return successors
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
\subsection{Strategies}
|
||||
\begin{description}
|
||||
\item[Non-informed strategy] \marginnote{Non-informed strategy}
|
||||
Domain knowledge is not available. Usually does an exhaustive search.
|
||||
|
||||
\item[Informed strategy] \marginnote{Informed strategy}
|
||||
Use domain knowledge by using heuristics.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Evaluation}
|
||||
\begin{description}
|
||||
\item[Completeness] \marginnote{Completeness}
|
||||
if the strategy is guaranteed to find a solution (when exists).
|
||||
|
||||
\item[Time complexity] \marginnote{Time complexity}
|
||||
time needed to complete the search.
|
||||
|
||||
\item[Space complexity] \marginnote{Space complexity}
|
||||
memory needed to complete the search.
|
||||
|
||||
\item[Optimality] \marginnote{Optimality}
|
||||
if the strategy finds the best solution (when more solutions are possible).
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Non-informed search}
|
||||
|
||||
\subsection{Breadth-first search (BFS)}
|
||||
\marginnote{Breadth-first search}
|
||||
Always expands the least deep node. The fringe is implemented as a queue (FIFO).
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{10cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & Yes \\
|
||||
\hline
|
||||
\textbf{Optimality} & Only with uniform cost (i.e. all edges have the same cost) \\
|
||||
\hline
|
||||
\textbf{\makecell{Time and space\\complexity}}
|
||||
& $O(b^d)$, where the solution depth is $d$ and the branching factor is $b$ (i.e. each non-leaf node has $b$ children) \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
The exponential space complexity makes BFS impractical for large problems.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.30\textwidth]{img/_bfs.pdf}
|
||||
\caption{BFS visit order}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{Uniform-cost search}
|
||||
\marginnote{Uniform-cost search}
|
||||
Same as BFS, but always expands the node with the lowest cumulative cost.
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{10cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & Yes \\
|
||||
\hline
|
||||
\textbf{Optimality} & Yes \\
|
||||
\hline
|
||||
\textbf{\makecell{Time and space\\complexity}}
|
||||
& $O(b^d)$, with solution depth $d$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.50\textwidth]{img/_ucs.pdf}
|
||||
\caption{Uniform-cost search visit order. $(n)$ is the cumulative cost}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{Depth-first search (DFS)}
|
||||
\marginnote{Depth-first search}
|
||||
Always expands the deepest node. The fringe is implemented as a stack (LIFO).
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{10cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & No (loops) \\
|
||||
\hline
|
||||
\textbf{Optimality} & No \\
|
||||
\hline
|
||||
\textbf{Time complexity}
|
||||
& $O(b^m)$, with maximum depth $m$ and branching factor $b$ \\
|
||||
\hline
|
||||
\textbf{Space complexity}
|
||||
& $O(b \cdot m)$, with maximum depth $m$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.30\textwidth]{img/_dfs.pdf}
|
||||
\caption{DFS visit order}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{Depth-limited search}
|
||||
\marginnote{Depth-limited search}
|
||||
Same as DFS, but introduces a maximum depth.
|
||||
A node at the maximum depth will not be explored further.
|
||||
|
||||
This allows to avoid infinite branches (i.e. loops).
|
||||
|
||||
|
||||
|
||||
\subsection{Iterative deepening}
|
||||
\marginnote{Iterative deepening}
|
||||
Runs a depth-limited search by trying all possible depth limits.
|
||||
It is important to note that each iteration is executed from scratch (i.e. a new execution of depth-limited search).
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Iterative deepening}
|
||||
\begin{lstlisting}
|
||||
def iterativeDeepening(G):
|
||||
for c in range(G.max_depth):
|
||||
sol = depthLimitedSearch(G, c)
|
||||
if sol is not FAILURE:
|
||||
return sol
|
||||
return FAILURE
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
Both advantages of DFS and BFS are combined.
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{10cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & Yes \\
|
||||
\hline
|
||||
\textbf{Optimality} & Only with uniform cost \\
|
||||
\hline
|
||||
\textbf{Time complexity}
|
||||
& $O(b^d)$, with solution depth $d$ and branching factor $b$ \\
|
||||
\hline
|
||||
\textbf{Space complexity}
|
||||
& $O(b \cdot d)$, with solution depth $d$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
|
||||
|
||||
\section{Informed search}
|
||||
\marginnote{Informed search}
|
||||
Informed search uses evaluation functions (heuristics) to reduce the search space and
|
||||
estimate the effort needed to reach the final goal.
|
||||
|
||||
|
||||
\subsection{Best-first search}
|
||||
\marginnote{Best-first seacrh}
|
||||
Uses heuristics to compute the desirability of the nodes (i.e. how close they are to the goal).
|
||||
The fringe is ordered according to the estimated scores.
|
||||
|
||||
|
||||
\begin{description}
|
||||
\item[Greedy search / Hill climbing]
|
||||
\marginnote{Greedy search / Hill climbing}
|
||||
The heuristic only evaluates nodes individually and does not consider the path to the root
|
||||
(i.e. expands the node that currently seems closer to the goal).
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{9cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & No (loops) \\
|
||||
\hline
|
||||
\textbf{Optimality} & No \\
|
||||
\hline
|
||||
\textbf{\makecell{Time and space\\complexity}}
|
||||
& $O(b^d)$, with solution depth $d$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
% The complexity can be reduced depending on the heuristic.
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width=0.65\textwidth]{img/_greedy_best_first_example.pdf}
|
||||
\caption{Hill climbing visit order}
|
||||
\end{figure}
|
||||
|
||||
\item[A$^\textbf{*}$]
|
||||
\marginnote{A$^*$}
|
||||
The heuristic also considers the cumulative cost needed to reach a node from the root.
|
||||
The score associated to a node $n$ is:
|
||||
\[ f(n) = g(n) + h'(n) \]
|
||||
where $g$ is the depth of the node and $h'$ is the heuristic that computes the distance to the goal.
|
||||
|
||||
\begin{description}
|
||||
\item[Optimistic/Feasible heuristic]
|
||||
\marginnote{Optimistic/Feasible heuristic}
|
||||
Given $t(n)$ that computes the true distance of a node $n$ to the goal.
|
||||
An heuristic $h'(n)$ is optimistic (i.e. feasible) if:
|
||||
\[ h'(n) \leq t(n) \]
|
||||
In other words, $h'$ is optimistic if it always underestimates the distance to the goal.
|
||||
\end{description}
|
||||
|
||||
\begin{theorem}
|
||||
If the heuristic used by A${^*}$ is optimistic $\Rightarrow$ A${^*}$ is optimal
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
Consider a scenario where the queue contains:
|
||||
\begin{itemize}
|
||||
\item A node $n$ whose child is the optimal solution
|
||||
\item A sub-optimal solution $G_2$
|
||||
\end{itemize}
|
||||
\begin{center}
|
||||
\includegraphics[width=0.5\textwidth]{img/_a_start_optimality.pdf}
|
||||
\end{center}
|
||||
We want to prove that A$^*$ will always expand $n$.
|
||||
|
||||
Given an optimistic heuristic $f(n) = g(n) + h'(n)$ and
|
||||
the true distance of a node $n$ to the goal $t(n)$,
|
||||
we have that:
|
||||
\[
|
||||
\begin{split}
|
||||
f(G_2) &= g(G_2) + h'(G_2) = g(G_2) \text{, as } G_2 \text{ is a solution: } h'(G_2)=0 \\
|
||||
f(G) &= g(G) + h'(G) = g(G) \text{, as } G \text{ is a solution: } h'(G)=0
|
||||
\end{split}
|
||||
\]
|
||||
Moreover, $g(G_2) > g(G)$ as $G_2$ is suboptimal.
|
||||
Therefore, $\bm{f(G_2) > f(G)}$.
|
||||
|
||||
Furthermore, as $h'$ is feasible, we have that:
|
||||
\[
|
||||
\begin{split}
|
||||
h'(n) \leq t(n) &\iff g(n) + h'(n) \leq g(n) + t(n) = g(G)=f(G) \\
|
||||
&\iff \bm{f(n) \leq f(G)}
|
||||
\end{split}
|
||||
\]
|
||||
In the end, we have that $f(G_2) > f(G) \geq f(n)$.
|
||||
So we can conclude that A$^*$ will never expand $G_2$ as:
|
||||
\[ f(G_2) > f(n) \]
|
||||
\end{proof}
|
||||
|
||||
\begin{center}
|
||||
\def\arraystretch{1.2}
|
||||
\begin{tabular}{c | m{9cm}}
|
||||
\hline
|
||||
\textbf{Completeness} & Yes \\
|
||||
\hline
|
||||
\textbf{Optimality} & Only if the heuristic is optimistic \\
|
||||
\hline
|
||||
\textbf{\makecell{Time and space\\complexity}}
|
||||
& $O(b^d)$, with solution depth $d$ and branching factor $b$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
In general, it is better to use heuristics with large values (i.e. heuristics that don't underestimate too much).
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width=0.65\textwidth]{img/_a_start_example.pdf}
|
||||
\caption{A$^*$ visit order}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
|
||||
\section{Graph search}
|
||||
\marginnote{Graph search}
|
||||
Differently from a tree search, searching in a graph requires to keep track of the explored nodes.
|
||||
\begin{algorithm}
|
||||
\caption{Graph search} \label{alg:search_graph_search}
|
||||
\begin{lstlisting}
|
||||
def graphSearch(problem, fringe):
|
||||
closed = set()
|
||||
fringe.push(problem.initial_state)
|
||||
# Get a node in the fringe and
|
||||
# expand it if it is not a solution and is not closed
|
||||
while fringe.notEmpty():
|
||||
node = fringe.pop()
|
||||
if problem.isGoal(node.state):
|
||||
return node.solution
|
||||
if node.state not in closed:
|
||||
closed.add(node.state)
|
||||
fringe.pushAll(expand(node, problem))
|
||||
return FAILURE
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
\subsection{A$^\textbf{*}$ with graphs}
|
||||
\marginnote{A$^*$ with graphs}
|
||||
The algorithm keeps track of closed and open nodes.
|
||||
The heuristic $g(n)$ evaluates the minimum distance from the root to the node $n$.
|
||||
|
||||
\begin{description}
|
||||
\item[Consistent heuristic (monotone)] \marginnote{Consistent heuristic (monotone)}
|
||||
An heuristic is consistent if for each $n$, for any successor $n'$ of $n$ (i.e. nodes reachable from $n$ by making an action)
|
||||
holds that:
|
||||
\[
|
||||
\begin{cases}
|
||||
h(n) = 0 & \text{if the corresponding status is the goal} \\
|
||||
h(n) \leq c(n, a, n') + h(n') & \text{otherwise}
|
||||
\end{cases}
|
||||
\]
|
||||
where $c(n, a, n')$ is the cost to reach $n'$ from $n$ by taking the action $a$.
|
||||
|
||||
In other words, $f$ never decreases along a path.
|
||||
In fact:\\
|
||||
\begin{minipage}{.48\linewidth}
|
||||
\[
|
||||
\begin{split}
|
||||
f(n') &= g(n') + h(n') \\
|
||||
&= g(n) + c(n, a, n') + h(n') \\
|
||||
&\geq g(n) + h(n) \\
|
||||
&= f(n)
|
||||
\end{split}
|
||||
\]
|
||||
\end{minipage}
|
||||
\begin{minipage}{.48\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=0.3\textwidth]{img/monotone_heuristic.png}
|
||||
\end{minipage}
|
||||
|
||||
|
||||
\begin{theorem}
|
||||
If $h$ is a consistent heuristic, A$^*$ on graphs is optimal.
|
||||
\end{theorem}
|
||||
\end{description}
|
||||
@ -0,0 +1,255 @@
|
||||
\chapter{Swarm intelligence}
|
||||
|
||||
\begin{description}
|
||||
\item[Swarm intelligence] \marginnote{Swarm intelligence}
|
||||
Group of locally interacting agents that
|
||||
shows an emergent behavior without a centralized control system.
|
||||
|
||||
A swarm intelligent system has the following features:
|
||||
\begin{itemize}
|
||||
\item Individuals are simple and have limited capabilities.
|
||||
\item Individuals are not aware of the global view.
|
||||
\item Individuals have local direct or indirect communication patterns.
|
||||
\item The computation is distributed and not centralized.
|
||||
\item The system works even if some individuals "break" (robustness).
|
||||
\item The system adapts to changes.
|
||||
\end{itemize}
|
||||
|
||||
Agents interact with each other and obtain positive and negative feedbacks.
|
||||
|
||||
\item[Stigmergy] \marginnote{Stigmergy}
|
||||
Form of indirect communication where an agent modifies the environment and the others react to it.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Ant colony optimization (ACO)}
|
||||
|
||||
|
||||
Ants release pheromones while walking from the nest to the food.
|
||||
They also tend to prefer paths marked with the highest pheromone concentration.
|
||||
|
||||
\begin{description}
|
||||
\item[Ant colony optimization] \marginnote{Ant colony optimization (ACO)}
|
||||
Probabilistic parametrized model that builds the solution incrementally.
|
||||
A problem is solved by making stochastic steps in a fully connected graph (construction graph) $G=(C, L)$ where:
|
||||
\begin{itemize}
|
||||
\item The vertexes $C$ are the solution components.
|
||||
\item The edges $L$ are connections.
|
||||
\item The paths on $G$ are states.
|
||||
\end{itemize}
|
||||
Additional constraints may be added if needed.
|
||||
|
||||
\begin{example}[Travelling salesman]
|
||||
The construction graph can be defined as:
|
||||
\begin{itemize}
|
||||
\item Nodes are cities.
|
||||
\item Edges are connections between cities.
|
||||
\item A solution is a Hamiltonian path in the graph.
|
||||
\item Constraints to avoid sub-cycles (i.e. avoid visiting a city multiple times).
|
||||
\end{itemize}
|
||||
\end{example}
|
||||
|
||||
\item[Pheromone] \marginnote{Pheromone}
|
||||
Value associated to each node and each edge to estimate the quality of the solution.
|
||||
|
||||
\item[Heuristic values] \marginnote{Heuristic values}
|
||||
Value associated to each node and each edge to represent the prior background knowledge.
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{ACO system}
|
||||
|
||||
\begin{description}
|
||||
\item[Transition rule] \marginnote{ACO system}
|
||||
Ants build a path on the construction graph based on a transition rule that uses pheromones and heuristics.
|
||||
The probability of choosing the node $j$ starting from $i$
|
||||
is parametrized on $\alpha$ (pheromones) and $\beta$ (heuristics), and is defined as:
|
||||
\[ p_{\alpha, \beta}(i, j) = \begin{cases}
|
||||
\frac{(\tau_{ij})^\alpha \cdot (\eta_{ij})^\beta}{\sum_{k \in \texttt{feasible\_nodes}} (\tau_{ik})^\alpha \cdot (\eta_{ik})^\beta} & \text{if } $j$ \text{ consistent} \\
|
||||
0 & \text{otherwise}
|
||||
\end{cases} \]
|
||||
where $\tau_{ij}$ is the pheromone trail from $i$ to $j$ and $\eta_{ij} = \frac{1}{d_{ij}}$ is the heuristic ($d_{ij}$ is the distance).
|
||||
|
||||
\item[Pheromone update]
|
||||
After each step, the pheromone trail is updated depending on an evaporation factor $\rho \in [0, 1]$:
|
||||
\[ \tau_{ij} = (1 - \rho) \tau_{ij} + \sum_{k=1}^{n_\text{ants}} \Delta \tau_{ij}^{(k)} \]
|
||||
$\Delta \tau_{ij}^{(k)}$ of the $k$-th ant is defined as:
|
||||
\[ \Delta \tau_{ij}^{(k)} = \begin{cases}
|
||||
\frac{1}{L_k} & \text{if ant } k \text{ used the arch } (i, j) \\
|
||||
0 & \text{otherwise}
|
||||
\end{cases} \]
|
||||
where $L_k$ is the length of the path followed by the $k$-th ant.
|
||||
|
||||
For the best ant, the update also affects all the components on the path crossed by the ant.
|
||||
|
||||
\item[Daemon actions]
|
||||
Centralized actions performed on each solution built by the ants.
|
||||
These actions cannot be performed by single ants and are useful to improve the solution using global information.
|
||||
In practice, a local search can be applied to push the result towards a better solution.
|
||||
\end{description}
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{ACO system}
|
||||
\begin{lstlisting}[mathescape=true]
|
||||
def acoSystem(problem, $\alpha$, $\beta$):
|
||||
initPheromones()
|
||||
while not terminationConditions():
|
||||
antBasedSolutionConstruction($\alpha$, $\beta$)
|
||||
pheromonesUpdate()
|
||||
daemonActions() # Optional
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
|
||||
\section{Artificial bee colony algorithm (ABC)}
|
||||
\marginnote{Artificial bee colony algorithm (ABC)}
|
||||
|
||||
System where the position of nectar sources represents the solutions and
|
||||
the quantity of nectar sources represents the fitness of the solution.
|
||||
Artificial bees can be:
|
||||
\begin{descriptionlist}
|
||||
\item[Employed]
|
||||
Bee associated to a specific nectar source (intensification) (i.e. it represents a solution).
|
||||
\item[Onlooker]
|
||||
Bee that is observing the employed bees and is choosing its nectar source.
|
||||
\item[Scout]
|
||||
Bee that discovers new food sources (diversification).
|
||||
\end{descriptionlist}
|
||||
|
||||
The algorithm has the following phases:
|
||||
\begin{descriptionlist}
|
||||
\item[Initialization]
|
||||
The initial nectar source of each bee is determined randomly.
|
||||
Each solution (nectar source) is a vector $\vec{x}_m \in \mathbb{R}^n$ and
|
||||
each of its components is initialized constrained to a lower ($l_i$) and upper ($u_i$) bound:
|
||||
\[ \vec{x}_m\texttt{[}i\texttt{]} = l_i + \texttt{rand}(0, 1) \cdot (u_i - l_i) \]
|
||||
|
||||
\item[Employed bees]
|
||||
Starting from their assigned nectar source, employed bees look in their neighborhood for a new food source with more fitness (more nectar).
|
||||
The fitness (for minimization problems) of a food source $\vec{x}_m$ is determined as:
|
||||
\[ \texttt{fit}(\vec{x}_m) = \begin{cases}
|
||||
\frac{1}{1 + \texttt{obj}(\vec{x}_m)} & \text{if } \vec{x}_m \geq 0 \\
|
||||
1 + \vert \texttt{obj}(\vec{x}_m) \vert & \text{if } \vec{x}_m < 0 \\
|
||||
\end{cases} \]
|
||||
where \texttt{obj} is the objective function.
|
||||
|
||||
\item[Onlooker bees]
|
||||
Onlooker bees stochastically choose their food source.
|
||||
Each food source $\vec{x}_m$ has a probability associated to it defined as:
|
||||
\[ p_m = \frac{\texttt{fit}(\vec{x}_m)}{\sum_{i=1}^{n_\text{bees}} \texttt{fit}(\vec{x}_i)} \]
|
||||
This provides a positive feedback as more promising solutions have a higher probability of being chosen.
|
||||
|
||||
\item[Scout bees]
|
||||
Scout bees choose a nectar source randomly.
|
||||
|
||||
An employed bee that cannot improve its solution after a given number of attempts {\tiny(gets fired and)} becomes a scout (negative feedback).
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{ABC}
|
||||
\begin{lstlisting}[mathescape=true]
|
||||
def abcAlgorithm(problem):
|
||||
initPhase()
|
||||
sol = None
|
||||
while not terminationConditions():
|
||||
employedBeesPhase()
|
||||
onlookerBeesPhase()
|
||||
scoutBeesPhase()
|
||||
sol = getCurrentBest()
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
|
||||
\section{Particle swarm optimization (PSO)}
|
||||
\marginnote{Particle swarm optimization (PSO)}
|
||||
|
||||
In a bird flock, the movement of the individuals tends to:
|
||||
\begin{itemize}
|
||||
\item Follow the neighbors.
|
||||
\item Stay in the flock.
|
||||
\item Avoid collisions.
|
||||
\end{itemize}
|
||||
However, a model based on these rules does not have a common objective.
|
||||
|
||||
PSO introduces as a common objective the search for food.
|
||||
Each individual who finds food can:
|
||||
\begin{itemize}
|
||||
\item Move away from the flock and reach the food.
|
||||
\item Stay in the flock.
|
||||
\end{itemize}
|
||||
Following the movement rules, the entire flock will gradually move towards promising areas.
|
||||
|
||||
Applied to optimization problems, the bird flock metaphor can be interpreted as:
|
||||
\begin{descriptionlist}
|
||||
\item[Bird]
|
||||
Agent that represents a possible solution that is progressively improved (exploration).
|
||||
|
||||
\item[Social interaction]
|
||||
Exploiting the knowledge of other agents to move towards a global solution (exploitation).
|
||||
|
||||
\item[Neighborhood]
|
||||
Individuals are affected by the actions of others close to them and are part of one or more sub-groups.
|
||||
|
||||
Note that sub-groups are not necessarily defined by physical proximity.
|
||||
\end{descriptionlist}
|
||||
|
||||
Given a cost function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ to minimize (gradient is not known),
|
||||
PSO initializes a swarm of particles (agents) whose movement is guided by the best-known position.
|
||||
Each particle is described by:
|
||||
\begin{itemize}
|
||||
\item Its position $\vec{x}_i \in \mathbb{R}^n$ in the search space.
|
||||
\item A velocity $\vec{v}_i \in \mathbb{R}^n$ that controls the movement of the particle.
|
||||
\item The best solution $\vec{p}_i$ it has found so far.
|
||||
\end{itemize}
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{PSO}
|
||||
\begin{lstlisting}[mathescape=true]
|
||||
def pso(f, n_particles, $\vec{l}$, $\vec{u}$, $\omega$, $\varphi_p$, $\varphi_g$):
|
||||
particles = [Particle()] * n_particles
|
||||
gb = None # Global best
|
||||
for particle in particles:
|
||||
particle.value = randomUniform($\vec{l}$, $\vec{u}$) # Search space bounds
|
||||
particle.vel = randomUniform($-\vert \vec{u}-\vec{l} \vert$, $\vert \vec{u}-\vec{l} \vert$)
|
||||
particle.best = particle.value
|
||||
if f(particles.best) < f(gb): gb = particles.best
|
||||
|
||||
while not terminationConditions():
|
||||
for particle in particles:
|
||||
$r_p$, $r_g$ = randomUniform(0, 1), randomUniform(0, 1)
|
||||
$\vec{x}_i$, $\vec{p}_i$, $\vec{v}_i$ = particle.value, particle.best, particle.vel
|
||||
particle.vel = $\omega$*$\vec{v}_i$ + $\varphi_p$*$r_p$*($\vec{p}_i$-$\vec{x}_i$) + $\varphi_g$*$r_g$*(gb-$\vec{x}_i$)
|
||||
particle.value = particle.value + particle.vel
|
||||
if f(particle.value) < f(particle.best):
|
||||
particle.best = particle.value
|
||||
if f(particle.best) < f(gb): gb = particle.best
|
||||
\end{lstlisting}
|
||||
\end{algorithm}
|
||||
|
||||
% \begin{algorithm}
|
||||
% \caption{PSO algorithm}
|
||||
% \begin{lstlisting}[mathescape=true]
|
||||
% def pco(f, $\omega$, $\varphi_p$, $\varphi_g$):
|
||||
% particles = initParticles($\vec{l}$, $\vec{u}$)
|
||||
% particles_best = [None] * len(particles)
|
||||
% velocities = [None] * len(particles)
|
||||
% global_best = None
|
||||
% for i in range(len(particles)):
|
||||
% velocities[i] = randomUniform($-\vert \vec{u}-\vec{l} \vert$, $\vert \vec{u}-\vec{l} \vert$)
|
||||
% particles_best[i] = particles[i]
|
||||
% if f(particles_best[i]) < f(g): g = particles_best[i]
|
||||
% while not terminationConditions():
|
||||
% for i in range(len(particles)):
|
||||
% $r_p$, $r_g$ = randomUniform(0, 1), randomUniform(0, 1)
|
||||
% $\vec{x}_i$, $\vec{p}_i$, $\vec{v}_i$ = particles[i], particles_best[i], velocities[i]
|
||||
% $\vec{g}$ = global_best
|
||||
% velocities[i] = $\omega$*$\vec{v}_i$ + $\varphi_p$*$r_p$*($\vec{p}_i$-$\vec{x}_i$) + $\varphi_g$*$r_g$*($\vec{g}$-$\vec{x}_i$)
|
||||
% particles[i] = particles[i] + velocities[i]
|
||||
% if f(particles[i]) < f(particles_best[i]):
|
||||
% particles_best[i] = particles[i]
|
||||
% if f(particles_best[i]) < f(g): g = particles_best[i]
|
||||
% \end{lstlisting}
|
||||
% \end{algorithm}
|
||||
Reference in New Issue
Block a user