Lecture 6 Rutme aalyss RLS o OeMax trals of {,, },, l ( + ɛ) l ( ɛ)( ) l Algorthm Egeerg Group Hasso Platter Isttute, Uversty of Potsdam 9 May T, We wat to rgorously uderstad ths behavor 9 May / Rutme aalyss RLS o OeMax Let s suppose: durg the executo of RLS the curret strg x looks lke ths: Rutme aalyss RLS o OeMax x = p = 6 6 E(T ) = 6 6 p = 6 E(T ) = 6 p = 6 E(T ) = 6 exactly oe bts p = 6 E(T ) = 6 Let s look to p : probablty that RLS makes a mprovg move from x T : tme utl RLS makes a mprovg move from x p = 6 E(T ) = 6 p = 6 E(T ) = 6 9 May / 9 May /
Rutme aalyss RLS o OeMax Rutme aalyss RLS o OeMax Rutme T s the radom varable that couts the umber of steps (fucto evaluatos) take by RLS utl the optmum s geerated p = E(T ) = p = E(T ) = p = E(T ) = E(T ) = E(T ) + E(T ) + + E(T ) = /p + /p + + /p 6 6 = = p + = 6 = = = = 6 = 7 p = E(T ) = remag zero 9 May / 9 May / Coupo collector process Suppose there are dfferet kds of coupos We must collect all coupos durg a seres of trals I each tral, exactly oe of the coupos s draw, each oe equally lkely We must keep drawg each tral utl we have collected each coupo at least oce Startg wth zero coupos, what s the exact umber of trals eeded before we have all coupos? Theorem (Coupo collector theorem) Let T be the umber of trals utl all coupos are collected The E(T ) = = = p + = = = = H = (log + Θ()) = log + O() Coupo collector process: cocetrato bouds What s the probablty that T > l + O()? Theorem (Coupo collector upper boud) Let T be the umber of trals utl all coupos are collected The Pr(T ( + ɛ) l ) ɛ Proof Probablty of choosg a specfc coupo: / Probablty of ot choosg a specfc coupo: / Probablty of ot choosg a specfc coupo for t rouds: ( /) t Probablty that oe of the coupos s ot chose t rouds: ( /) t (uo boud) Let t = c l, Pr(T c l ) ( /) c l e c l = c = c+ 9 May 6 / 9 May 7 /
Coupo collector process: cocetrato bouds Theorem (Coupo collector lower boud) (Doerr, ) Let T be the umber of trals utl all coupos are collected The Pr(T < ( ɛ)( ) l ) e ɛ Corollary Let T be the tme for RLS to optmze OeMax The, Rutme aalyss RLS o OeMax T,,, l ( + ɛ) l ( ɛ)( ) l Pr(T ( + ɛ) l ) ɛ Pr(T < ( ɛ)( ) l ) e ɛ E(T ) = Θ( log ) What about (+) EA? Ca we use Coupo Collector? Why/why ot? 9 May 8 / 9 May 9 / Ftess levels Ftess levels Observato: ftess durg optmzato s always mootoe creasg Idea: partto the search space {, } to m sets A, A m such that j : A A j = m = A = {, } for all pots a A ad b A j, f(a) < f(b) f < j Law of total probablty: E(X) = F Pr(F )E(X F ) Pr((+) EA leaves A ) s A A A 7 A 6 A A ftess A We requre A m to cota oly optmal search pots Procedure: for each level A, boud the probablty of leavg a level A for a hgher level A j, j > 9 May / p(a ) be the probablty that a radom chose pot belogs to A s be the probablty to leave level A for level A j wth j > m ( E(T ) p(a ) + + ) ( + + ) = s s m s s m = Fgure adapted from D Sudholt, Tutoral m = s 9 May /
Rutme aalyss (+) EA o OeMax Rutme aalyss (+) EA o OeMax Theorem The expected rutme of the (+) EA o OeMax s O( log ) Proof We partto {, } to dsjot sets A, A,, A where x s A f ad oly f t has zeros ( oes) To escape A, t suffces to flp a sgle zero ad leave all other bts uchaged Ths gves oly a upper boud Maybe the (+) EA ca be much qucker For example t could be O() or eve somethg lke O( log log ) Thus, s We coclude ( ) e, ad s e E(T ) m = s = e = e H = O( log ) 9 May / 9 May / Rutme aalyss (+) EA o OeMax Theorem (Droste, Jase, Wegeer ) The expected rutme of the (+) EA o OeMax s Ω( log ) Lemma The probablty that the (+) EA eeds at least ( ) l steps s at least a costat c Rutme aalyss (+) EA o OeMax Proof of Lemma The tal soluto has at most / oe bts wth probablty at least / There s a costat probablty that ( ) l steps oe of the remag zero bts does ot flp: Probablty a partcular bt does t flp t steps: ( /) t Probablty t flps at least oce t steps: ( /) t Probablty / bts flp at least oce t steps: ( ( /) t ) / Probablty at least oe of the / bts does ot flp t steps: [ ( /) t ] / Set t = ( ) l The [ ( /) t ] / = [ ( /) ( ) l ] / [ (/e) l ] / = [ /] / = [ /] / ( (e)) / = c 9 May / 9 May /
Rutme aalyss (+) EA o OeMax Theorem (Droste, Jase, Wegeer ) The expected rutme of the (+) EA o OeMax s Ω( log ) Proof Expected rutme: E(T ) = t Pr(T = t) ( ) l Pr(T ( ) l ) t= ( ) l c = Ω( log ) by prevous lemma Upper boud gve by ftess levels s tght Ftess levels There are several more advaced results that use the ftess levels techque: Expected rutme of the (+λ) EA o LeadgOes s O(λ + ) (Jase et al, ) Expected rutme of the (µ+) EA o LeadgOes s O(µ log + ) (Wtt, 6) Ftess levels for provg lower bouds (Sudholt, ) No-eltst populatos (Lehre, ) 9 May 6 / 9 May 7 / Drft Aalyss Cosder a process movg towards/away from a goal (possbly stochastcally) Model ths as a sequece of umbers X, X, where X t := dstace from the goal at tme t { f X t = (, 9, 8, 7, 6,,,,,, ) E(X t X t+ ) = otherwse (, 9, 8, 9, 8, 7, 6,,,,, )??? Defto The drft of a process at tme t s the expected decrease dstace from a goal: E(X t X t+ ) Drft aalyss allows us to relate the drft to the tme to reach the goal Drft Aalyss Determstc Process Cosder a process that moves as follows I each step, Wth probablty, move oe step toward the goal Startg at dstace, how may steps utl the goal s reached? Drft s E(X t X t+ ) = as log as X t > Expected tme to reach the goal: E(T ) = maxmum dstace drft = = 9 May 8 / 9 May 9 /
Drft Aalyss Stochastc Process Cosder a process that moves as follows: wth probablty /, move oe step toward the goal, wth probablty /, move oe step away from the goal Startg at dstace, how may steps utl the goal s reached? Drft s f X t =, X t X t+ = f X t, wth probablty /, f X t, wth probablty /, Expected tme to reach the goal: E(X t X t+ ) = + ( ) = E(T ) = maxmum dstace drft = / = = 9 May / Drft Aalyss Theorem (He ad Yao, ) Let {X t : t } be a Markov process over R + Let T := m{t : X t = } If there exsts δ > such that at ay tme step t ad at ay state X t >, the followg codto holds: the E(X t X t+ X t > ) δ, E(T X > ) X δ Example: (+) EA o OeMax: ad E(T ) E(X ) δ E(X t X t+ X t > ) ( ) e e = δ E(T X > ) E(X ) / δ /(e) = O( ) Obvously ot tght! 9 May / Drft Aalyss Drft Aalyss Observato: we do t have to use the dstace drectly! Idea: progress toward goal depeds o dstace from goal We ca use a potetal fucto Let X t = l( + ) where s the umber of zeros the btstrg Drft aalyss has may powerful varats: Multplcatve Drft (Doerr et al, ) Negatve Drft (Olveto ad Wtt, ) Drft Aalyss for Stochastc Populatos (Lehre, ) Varable Drft (Johase ) E(X t X t+ X t > ) l( + ) ( ) l( + ) l() e e = δ E(T X > ) X δ l( + ) l()/e = O( log ) Refemets allow for Upper ad lower bouds o expectato Tal equaltes 9 May / 9 May /
Further readg Petro Olveto ad X Yao A Getle Itroducto to the Tme Complexty Aalyss of Evolutoary Algorthms: http://wwwcsbhamacuk/~olvetps/mages/olvetotutoralpdf Frak Neuma ad Carste Wtt, Bospred Computato Combatoral Optmzato Algorthms ad Ther Computatoal Complexty Natural Computg Seres, Sprger, http://wwwbospredcomputatocom/ Ae Auger ad Bejam Doerr (edtors) Theory of Radomzed Search Heurstcs: Foudatos ad Recet Developmets World Scetfc, Thomas Jase, Aalyzg Evolutoary Algorthms The Computer Scece Perspectve Sprger, Lectures &6 are based part o these sldes (wth permsso) 9 May / Refereces Bejam Doerr, Aalyzg Radomzed Search Heurstcs: Tools from Probablty Theory Chapter of Theory of Radomzed Search Heurstcs: Foudatos ad Recet Developmets World Scetfc, Bejam Doerr, Dael Johase ad Carola Wze () Multplcatve drft aalyss I Proceedgs of the Twelfth Aual Coferece o Geetc ad Evolutoary Computato, pages 9 6 ACM Stefa Droste, Thomas Jase ad Igo Wegeer () O the aalyss of the (+) evolutoary algorthm Theoretcal Computer Scece, 76(-): 8 Thomas Jase, Ke A De Jog ad Igo Wegeer () O the choce of the offsprg populato sze evolutoary algorthms Evolutoary Computato, (): Dael Johase () Radom Combatoral Structures ad Radomzed Search Heurstcs PhD thess, Uverstät des Saarlades Per Krsta Lehre () Negatve drft populatos I Proceedgs of the Eleveth Iteratoal Coferece o Parallel Problem Solvg from Nature, pages Per Krsta Lehre () Ftess-levels for o-eltst populatos I Proceedgs of the Thrteeth Aual Coferece o Geetc ad Evolutoary Computato, pages 78 ACM Petro S Olveto ad Carste Wtt () Smplfed drft aalyss for provg lower bouds evolutoary computato Algorthmca, 9():69 86 Erratum: http://arxvorg/abs/78 Drk Sudholt () Geeral lower bouds for the rug tme of evolutoary algorthms I Proceedgs of the Eleveth Iteratoal Coferece o Parallel Problem Solvg from Nature, pages Sprger Wtt, C (6) Rutme aalyss of the (µ+) ea o smple pseudo-boolea fuctos evolutoary computato I Proceedgs of the Egth Aual Coferece o Geetc ad Evolutoary Computato, pages 668 ACM 9 May /