From 4820aebed684fb9efd7e471614a4561b11313d81 Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Wed, 18 Oct 2023 20:56:56 +0200 Subject: [PATCH] Add local search --- .../module1/img/_genetic_crossover.pdf | Bin 0 -> 75302 bytes .../module1/img/_genetic_cycle.pdf | Bin 0 -> 80685 bytes .../module1/img/_genetic_mutation.pdf | Bin 0 -> 81241 bytes .../module1/img/_genetic_terminology.pdf | Bin 0 -> 75844 bytes .../module1/img/_local_search_neigh_graph.pdf | Bin 0 -> 112384 bytes .../img/_local_search_search_graph.pdf | Bin 0 -> 112388 bytes .../module1/img/tsp_2-exchange.png | Bin 0 -> 82547 bytes .../module1/img/tsp_3-exchange.png | Bin 0 -> 113178 bytes .../module1/main.tex | 1 + .../module1/sections/_local_search.tex | 295 ++++++++++++++++++ 10 files changed, 296 insertions(+) create mode 100644 src/fundamentals-of-ai-and-kr/module1/img/_genetic_crossover.pdf create mode 100644 src/fundamentals-of-ai-and-kr/module1/img/_genetic_cycle.pdf create mode 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b/src/fundamentals-of-ai-and-kr/module1/main.tex @@ -9,5 +9,6 @@ \input{sections/_intro.tex} \input{sections/_search.tex} + \input{sections/_local_search.tex} \end{document} \ No newline at end of file diff --git a/src/fundamentals-of-ai-and-kr/module1/sections/_local_search.tex b/src/fundamentals-of-ai-and-kr/module1/sections/_local_search.tex new file mode 100644 index 0000000..5ea2b5a --- /dev/null +++ b/src/fundamentals-of-ai-and-kr/module1/sections/_local_search.tex @@ -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 an Hamiltonian tour of minimum cost in an undirected graph. + + A possible neighborhood of a state applies the $k$-exchange that guarantees to maintain an 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 optima (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 in 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 finds 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 optima (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, 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 a individual to be chosen as parent of the next offspring. + Depends on the fitness. + \item[Generational replacement] + Create the new generation. Possibile 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} \ No newline at end of file

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