### fix latex in LESSON

parent 045d4b9a
 Metaheuristics (IA-308) ======================= Compile as PDF: pandoc -f markdown --toc -o LESSON.pdf LESSON.md. Introduction ------------ Metaheuristics are mathematical optimization algorithms solving $\argmin_{x \in X} f(x)$ (or argmax). Metaheuristics are mathematical optimization algorithms solving $argmin_{x \in X} f(x)$ (or argmax). Synonyms: - search heuristics, - evolutionary algorithms, - stochastic local search. ... ... @@ -53,11 +56,13 @@ Way to model a solution: encoding. ### Main models Encoding: - continuous (s. numeric), - discrete metric (integers), - combinatorial (graph, permutation). Fitness: - mono-objective, - multi-modal, - multi-objectives. ... ... @@ -77,11 +82,13 @@ Performance evaluation ### What is performance Main performances axis: - time, - quality, - probability. Additional performance axis: - robustness, - stability. ... ... @@ -95,6 +102,7 @@ Proof-reality gap is huge, thus empirical performance evaluation is gold standar Empirical evaluation = scientific method. Basic rules of thumb: - randomized algorithms => repetition of runs, - sensitivity to parameters => design of experiments, - use statistical tools, ... ... @@ -103,11 +111,13 @@ Basic rules of thumb: ### Useful statistical tools Statistical tests. Statistical tests: - classical null hypothesis: test equality of distributions. - beware of p-value. How many runs? - not always "as many as possible", - maybe "as many as needed", - generally: 15 (min for non-parametric tests) -- 20 (min for parametric-gaussian tests). ... ... @@ -118,12 +128,14 @@ Use robust estimators: median instead of mean, Inter Quartile Range instead of s ### Expected Empirical Cumulative Distribution Functions On Run Time: ERT-ECDF.  $ERTECDF(\{X_0,\dots,X_i,\dots,X_r\}, \delta, f, t) := \#\{x_t \in X_t | f(x_t^*)>=\delta \}$ $\delta \in [0, max_{x \in \mathcal{X}}(f(x))]$ $X_i := \{\{ x_0^0, \dots, x_i^j, \dots, x_p^u | p\in[1,\infty[ \} | u \in [0,\infty[ \} \in \mathcal{X}$  with $p$ the sample size, $r$ the number of runs, $u$ the nubmer of iterations, $t$ the number of calls to the objective $$ERTECDF(\{X_0,\dots,X_i,\dots,X_r\}, \delta, f, t) := \#\{x_t \in X_t | f(x_t^*)>=\delta \}$$ $$\delta \in \left[0, \max_{x \in \mathcal{X}}(f(x))\right]$$ $$X_i := \left\{\left\{ x_0^0, \dots, x_i^j, \dots, x_p^u | p\in[1,\infty[ \right\} | u \in [0,\infty[ \right\} \in \mathcal{X}$$ with $p$ the sample size, $r$ the number of runs, $u$ the number of iterations, $t$ the number of calls to the objective function. The number of calls to the objective function is a good estimator of time because it dominates all other times. ... ... @@ -136,6 +148,7 @@ The dual of the ERT-ECDF can be easily computed for quality (EQT-ECDF). ### Other tools Convergence curves: do not forget the golden rule and show distributions: - quantile boxes, - violin plots, - histograms. ... ... @@ -146,11 +159,13 @@ Algorithm Design ### Neighborhood Convergence definition(s). Convergence definition(s): - strong, - weak. Neighborhood: subset of solutions atteinable after an atomic transformation: - ergodicity, - quasi-ergodicity. ... ... @@ -158,17 +173,20 @@ Neighborhood: subset of solutions atteinable after an atomic transformation: ### Structure of problem/algorithms Structure of problems to exploit: - locality (basin of attraction), - separability, - gradient, - funnels. Structure with which to capture those structures: - implicit, - explicit, - direct. Silver rule: choose the algorithmic template that adhere the most to the problem model. - taking constraints into account, - iterate between problem/algorithm models. ... ... @@ -188,12 +206,19 @@ Most generic way of thinking about algorithms: grammar-based algorithm selection Example: modular CMA-ES. Parameter setting tools: - ParamILS, - SPO, - i-race. Design tools: - ParadisEO. - ParadisEO, - jMetal, - Jenetics, - ECJ, - DEAP, - HeuristicLab. ### Landscape-aware algorithms ... ...
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