search space structure and sls performance · fundamental search space properties simple properties...

STOCHASTIC LOCAL SEARCHFOUNDATIONS AND APPLICATIONS

Search Space Structure

and SLS Performance

Holger H. Hoos & Thomas Stutzle

Outline

1. Fundamental Search Space Properties

2. Search Landscapes and Local Minima

3. Fitness-Distance Correlation

4. Ruggedness

5. Barriers and Basins

Stochastic Local Search: Foundations and Applications 2

Fundamental Search Space Properties

Simple properties of search space S :

I search space size #S

I number of (optimal) solutions #S ′, solution density #S ′/#S

I search space diameter diam(GN)(= maximal distance between any two candidate solutions)

I distribution of solutions within the neighbourhood graph


Example: Correlation between solution density and searchcost for GWSAT over set of hard Random-3-SAT instances:

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-log10(solution density)

sear

ch c

ost [

mea

n #

step

s]

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-log10(solution density)

P

cdfemp. pdf (�10)

logN[26.5, 1.26]


Search Landscapes

Given an SLS algorithm A and a problem instance π withassociated search space S(π), neighbourhood relation N(π) andevaluation function g(π) : S 7→ R, the search landscape of π, L(π),is defined as L(π) := (S(π),N(π), g(π)).

A landscape L := (S ,N, g) is . . .

I non-degenerate (or invertible), iff∀s, s ′ ∈ S : [g(s) = g(s ′) =⇒ s = s ′];

I locally invertible, iff∀r ∈ S : ∀s, s ′ ∈ N(r) ∪ {r} : [g(s) = g(s ′) =⇒ s = s ′];

I non-neutral, iff∀s ∈ S : ∀s ′ ∈ N(s) : [g(s) = g(s ′) =⇒ s = s ′].


Classification of search positions (according to evaluationfunction values of direct neighbours):

position type > = <

SLMIN (strict local min) + 0 0LMIN (local min) + + 0IPLAT (interior plateau) 0 + 0SLOPE + 0 +LEDGE + + +LMAX (local max) 0 + +SLMAX (strict local max) 0 0 +

“+” = present, “0” absent; table entries refer to neighbours with larger

(“>”) , equal (“=”), and smaller (“<”) evaluation function values


Example: Distribution of position types for hardRandom-3-SAT instances

instance avg sc SLMIN LMIN IPLATuf20-91/easy 13.05 0% 0.11% 0%uf20-91/medium 83.25 < 0.01% 0.13% 0%uf20-91/hard 563.94 < 0.01% 0.16% 0%

instance SLOPE LEDGE LMAX SLMAXuf20-91/easy 0.59% 99.27% 0.04% < 0.01%uf20-91/medium 0.31% 99.40% 0.06% < 0.01%uf20-91/hard 0.56% 99.23% 0.05% < 0.01%

(based on exhaustive enumaration of search space;sc refers to search cost for GWSAT)


Example: Distribution of position types for hardRandom-3-SAT instances

instance avg sc SLMIN LMIN IPLATuf50-218/medium 615.25 0% 47.29% 0%uf100-430/medium 3 410.45 0% 43.89% 0%uf150-645/medium 10 231.89 0% 41.95% 0%

instance SLOPE LEDGE LMAX SLMAXuf50-218/medium < 0.01% 52.71% 0% 0%uf100-430/medium 0% 56.11% 0% 0%uf150-645/medium 0% 58.05% 0% 0%

(based on sampling along GWSAT trajectories;sc refers to search cost for GWSAT)


Local Minima

Note: Local minima impede local search progress.

Simple measures related to local minima:

I number of local minima #lmin, local minima density#lmin/#S

I distribution of local minima within the neighbourhood graph

Problem: Determining these measures typically requiresexhaustive enumeration of search space

Solutions: Approximations based on sampling or estimation fromother measures (such as autocorrelation measures, see below)


Fitness-Distance Correlation (FDC)

Idea: Analyse (linear) correlation between solution quality (fitness)and distance to (closest) optimal solution.

Measure for FDC: empirical correlation coefficient

rfdc :=Cov(g , d)

σ(g) · σ(d),

where

Cov(g , d) :=1

m − 1

m∑i=1

(gi − g)(di − d),

σ(g) :=

√√√√ 1

m − 1

m∑i=1

(gi − g)2, σ(d) :=

√√√√ 1

m − 1

m∑i=1

(di − d)2

Note: rfdc depends on the given neighbourhood relation.Stochastic Local Search: Foundations and Applications 10

Fitness Distance Plots:

Graphical representation of fitness–distance correlation;distance from (closest) optimal solution vs relative solution quality.

Measuring FDC:

Sample locally optimal candidate solutions, as determinedby a (simple) SLS algorithm, e.g., iterative improvement.


Example: FDC plot for TSPLIB instance rat783, based on2500 local optima obtained from a 3-opt algorithm

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distance to global optimum

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distance to best known solution


Implications of FDC for SLS behaviour:

I High FDC (close to one):

I ‘Big valley’ structure of landscape provides guidance forlocal search;

I high-quality local minima provide good starting points;

I search diversification: perturbation is better than restart;

I search initialisation: high quality starting points help;

I typical for TSP.

I FDC close to zero:

I global structure of landscape does not provide guidance forlocal search;

I indicative of harder problems, such as certain instance types ofQAP (Quadratic Assignment Problem)


Ruggedness

Idea: Rugged landscapes, i.e., landscapes with with many localminima, are hard to seach.

Measures for landscape ruggedness:

I autocorrelation function [Weinberger, 1990; Stadler, 1995]

I correlation length [Stadler, 1995]

I autocorrelation coefficient [Angel & Zissimopoulos, 1997]


Empirical autocorrelation function r(i):

r(i) :=1/(m − i) ·

∑m−ik=1(gk − g) · (gk+i − g)

1/m ·∑m

k=1(gk − g)2

Empirical autocorrelation coefficient (ACC) ξ:

ξ = 1/(1− r(1))

Note: r(i) and ξ depend on the given neighbourhood relation.


Implications of ACC on SLS behaviour:

I High ACC (close to one):

I “smooth” landscape;

I evaluation function values for neighbouring candidate solutionsare close on average;

I low local minima density;

I problem typically relative easy for local search.

I Low ACC (close to zero):

I very rugged landscape;

I evaluation function values for neighbouring candidate solutionsare almost uncorrelated;

I high local minima density;

I problem typically relatively hard for local search.


Measuring ACC:

I measure series g = (g1, . . . , gm) of evaluation function valuesalong uninformed random walk;

I estimate ACC based on autocorrelation function on g,where distance is measured in search steps.

computationally cheap compared to, e.g., FDC analysis.

Note: (Bounds on) ACC can be theoretically derived in manycases, including TSP with 2-exchange neighbourhood.


Plateaus

Intuition: Plateaus, i.e., ‘flat’ regions in the search landscape,can impede search progress due to lack of guidance by theevaluation function.

Definition (1)

I region: connected subgraph of GN .

I border of region R: set of s ∈ S with direct neighboursthat are not contained in R (border positions).


Definition (continued)

I plateau region: region in which all positions havethe same level, i.e., evaluation function value, l .

I plateau: maximally extended plateau region,i.e., plateau region in which no border position has anydirect neighbours at the plateau level l .

I exit of plateau region R: direct neighbour s of a borderposition of R with lower level than plateau level l .

I open / closed plateau: plateau with / without exits.


Measures of plateau structure:

I plateau diameter = diameter of corresponding subgraph of GN

I plateau width = maximal distance of any plateau position tothe respective closest border position

I plateau branching factor = fraction of neighbours of a plateauposition that are also on the plateau.

I number of exits, exit density

I distribution of exits within a plateau, exit distance distribution(in particular: avg./max distance to closest exit)


Some plateau structure results for SAT:

I Plateaus typically don’t have an interiour, i.e., almost everyposition is on the border.

I The diameter of plateaus, particularly at higher levels, iscomparable to the diameter of search space. (In particular:plateaus tend to span large parts of the search space, but arequite well connected internall.)

I For open plateaus, exits tend to be clustered, but the averageexit distance is typically relatively small.


Barriers and Basins

I positions s, s ′ are mutually accessible at level liff there is a path connecting s ′ and s in the neighbourhoodgraph that visits only positions t with g(t) ≤ l

I The barrier level between positions s, s ′

is the lowest level l at which s ′ and s ′ are mutually accessible.

I Basin below position s = set of search positions s ′ at levelg(s ′) < g(s) such that s and s ′ are mutually accessible atlevel g(s).


I A gradient walk from position s to s ′ is a possible trajectory ofiterative best improvement (= gradient descent) from s to s ′.

I The gradient basin of position s is the sets of all positions s ′

such that there is a gradient walk from s ′ to s.


Barries trees and plateau connection graphs

I Barrier trees and plateau connection graphs are based oncollapsing positions on the same plateau or in the same basininto ‘macro positions’ and illustrate connections betweenthese regions.

I This type of search space analysis can give much deeperinsights into SLS behaviour and problem hardness than globalmeasures, such as FDC or ACC.

I This type of analysis is computationally expensive and requiresenumeration of large parts of the search space.


Example: Search space structure (plateau connection graph)of easy Random 3-SAT instance

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Example: Search space structure (plateau connection graph)of hard Random 3-SAT instance

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search space structure and sls performance · fundamental search space properties simple properties...

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