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Grégoire Grzeczkowicz
sho
Commits
31c9fa5a
Commit
31c9fa5a
authored
Jan 06, 2020
by
Johann Dreo
Browse files
fix latex in LESSON
parent
045d4b9a
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LESSON.md
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31c9fa5a
Metaheuristics (IA308)
=======================
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
\i
n 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:

monoobjective,

multimodal,

multiobjectives.
...
...
@@ 77,11 +82,13 @@ Performance evaluation
### What is performance
Main performances axis:

time,

quality,

probability.
Additional performance axis:

robustness,

stability.
...
...
@@ 95,6 +102,7 @@ Proofreality 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 pvalue.
How many runs?

not always "as many as possible",

maybe "as many as needed",

generally: 15 (min for nonparametric tests)  20 (min for parametricgaussian 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: ERTECDF.
```
$ERTECDF(\{X_0,\dots,X_i,\dots,X_r\}, \delta, f, t) := \#\{x_t \in X_t  f(x_t^*)>=\delta \}$
$\delta \in [0, max_{x \in \mathcal{X}}(f(x))]$
$X_i := \{\{ x_0^0, \dots, x_i^j, \dots, x_p^u  p\in[1,\infty[ \}  u \in [0,\infty[ \} \in \mathcal{X}$
```
with $p$ the sample size, $r$ the number of runs, $u$ the nubmer of iterations, $t$ the number of calls to the objective
$$ERTECDF(
\{
X_0,
\d
ots,X_i,
\d
ots,X_r
\}
,
\d
elta, f, t) :=
\#\{
x_t
\i
n X_t  f(x_t^
*
)>=
\d
elta
\}
$$
$$
\d
elta
\i
n
\l
eft[0,
\m
ax_{x
\i
n
\m
athcal{X}}(f(x))
\r
ight]$$
$$X_i :=
\l
eft
\{\l
eft
\{
x_0^0,
\d
ots, x_i^j,
\d
ots, x_p^u  p
\i
n[1,
\i
nfty[
\r
ight
\}
 u
\i
n [0,
\i
nfty[
\r
ight
\}
\i
n
\m
athcal{X}$$
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 ERTECDF can be easily computed for quality (EQTECDF).
### 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,

quasiergodicity.
...
...
@@ 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: grammarbased algorithm selection
Example: modular CMAES.
Parameter setting tools:

ParamILS,

SPO,

irace.
Design tools:

ParadisEO.

ParadisEO,

jMetal,

Jenetics,

ECJ,

DEAP,

HeuristicLab.
### Landscapeaware algorithms
...
...
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