Distribution Free Evolvability of Polynomial Functions over all Convex Loss Functions

Transcription

1 Disibuion Fee Evolvabiliy of Polynomial Funcions ove all Convex Loss Funcions Paul Valian UC Beeley Beeley, Califonia ABSTRACT We fomulae a noion of evolvabiliy fo funcions wih domain and ange ha ae eal-valued vecos, a compelling way of expessing many naual biological pocesses. We show ha linea and fixed-degee polynomial funcions ae evolvable in he following dually obus sense: Thee is a single evoluion algoihm ha fo all convex loss funcions conveges fo all disibuions. I is possible ha such dually obus esuls can be achieved by simple and moe naual evoluion algoihms. In he second pa of he pape we inoduce a simple and naual algoihm ha we call wide-scale andom noise and pove a coesponding esul fo he L 2 meic. We conjecue ha he algoihm wos fo moe geneal classes of meics. 1. INTRODUCTION Since he inoducion of he evolvabiliy model by L. Valian in [11], significan wo has been done o show boh he powe and he obusness o modeling vaiaions of his compuaional famewo fo invesigaing how complexiy can aise in a fixed envionmen [2, 3, 4, 7, 10]. In his wo we pesen wo complemenay consucions, which exend his body of wo in a new and vey naual diecion: while pevious papes sudied evolvabiliy of Boolean funcions (fom {0, 1} n { 1, 1}) we hee conside funcions fom R n R m. Many of he funcions ha evolve in biology, fo example, how should he concenaion of poein A in his cell vay in esponse o he concenaions of poeins B hough Z?, migh be much moe naually epesened as eal funcions as opposed o Boolean funcions. Of couse, eal funcions, when esiced o Boolean domain and ange, become Boolean funcions, so in some sense his model is moe geneal han he oiginal Boolean model of evolvabiliy. We fis define he noion of evolvabiliy ove he eals, and hen poceed in wo diecions. The fis diecion con- Reseach paially suppoed by NSF gan CC and by a Google Univesiy Reseach Awad Pemission o mae digial o had copies of all o pa of his wo fo pesonal o classoom use is ganed wihou fee povided ha copies ae no made o disibued fo pofi o commecial advanage and ha copies bea his noice and he full ciaion on he fis page. To copy ohewise, o epublish, o pos on seves o o edisibue o liss, equies pio specific pemission and/o a fee. ITCS 12, Januay 8 10, Cambidge, Massachuses, USA. Copyigh 2012 ACM $ siss of an adapaion of Feldman s esuls on evolvabiliy fom [4] o ou seing, which, because of he diffeen seing, immediaely yields esuls of a somewha diffeen naue: evolvabiliy can simulae abiay polynomialime opimizaion algoihms ha only equie appoximae access o he funcion being opimized. In he eminology of Lovasz [9], his is wea opimizaion. We show ha his consucion of polynomial ime wea convex opimizaion can be leveaged o yield evoluion algoihms fo linea funcions, and fuhe, fixed-degee polynomials, in a disibuion-independen sense, fo any convex loss funcion including a foioi he commonly consideed linea and quadaic (L 1 and L 2-squaed) loss funcions. Fo he oiginal Boolean famewo of evolvabiliy, Feldman showed ha, as long as he undelying disibuion is nown, hen he class SQ (saisical queies) defined by Keans in [8] exacly chaaceizes he classes of evolvable funcions [2]. SQ is boh a poweful and naual famewo, and seems o capue mos of he powe of PAC leaning. Howeve, he assumpion ha he evoluion algoihm mus now he undelying disibuion is decidedly unnaual, as one would hope fo evoluion o funcion acoss a boad ange of poenially quie inicae and vaying disibuions of condiions fo is ceaues. In subsequen wo, Feldman showed ha if one einepes he Boolean model by allowing hypoheses ha ae eal values (even hough he hypoheses ae hen compaed, via a loss funcion, o age values ha ae Boolean), hen if he pefomance meic is non-linea ha is, essenially anyhing excep he coelaion meic (he L 1 meic esiced o a small enough egion so ha i is a linea funcion, insead of he usual piecewise linea) one can ae advanage of a in in i o, in fac, evolve eveyhing in SQ (Theoem 4.3 of [4]). In a sense, by changing pa of he Boolean model o allow fo eal values, Feldman cicumvens an appaen limiaion of he puely Boolean evoluion seing. In his cuen wo we conside all elemens of he evolvabiliy model o be eal insead of Boolean. The esuls we deive ae of a ahe diffeen fom fom hose in he (paially) Boolean seings peviously consideed. We show ha hee exiss an algoihm ha evolves linea and fixed-degee polynomial funcions ove he eals, fo all disibuions and all convex meics (including boh he L 1 and L 2 meics). While hese esuls demonsae he powe of he eal evolvabiliy famewo, hey come a he expense of a ceain unnaualness of he undelying evoluion algoihm. We balance his ou in he second pa of his pape by consideing pehaps he simples and mos naual algoihm

2 ha could be hoped o wo, and showing ha i in fac can epoduce a significan poion of hese esuls, albei less efficienly. In a geneaion of his algoihm, a paen poduces a polynomial numbe of neaby childen, each chosen in a unifomly and independenly andom diecion. Suvival of he fies uns his ino a ind of seepes descen saegy, which enables us o pove ha, fo quadaic loss funcions, consan pogess is made in each geneaion, which will apidly lead o he opimum. This algoihm, which we call wide-scale andom noise o emphasize is simple unsucued naue, has in fac been found in simulaions o convege apidly in many cases beyond ha of he quadaic loss funcion, hough how i achieves his convegence seems ahe diffeen han is povable behavio in he quadaic case. In paicula, while fo quadaic loss funcions, he algoihm povably consisenly poduces offsping which pefom bee han he paen, leading o a guaaneed impovemen, in conas fo he case of he L 1 loss funcion he abiliy o evolve o a descenden whose pefomance is wose han ha of he paen seems cucial fo efficien pogess. I would seem couneinuiive ha such bacacing would help in a convex landscape wih no spuious local minima. Howeve, his was exacly he effec found in a pape on simulaed annealing ha also consideed a vey simila L 1 opimizaion seing [6]. I would appea ha boh evolvabiliy and simulaed annealing seem effecive in unexpeced cases, and one migh hope ha new analysis of one migh shed ligh on he ohe DEFINITIONS We adap much of ou noaion fom [4]. We noe a concee example o moivae he following definiions: conside he as of ying o evolve linea funcions. Namely, hee is an unnown linea funcion f : R n R m efeed o as he age funcion, and an unnown disibuion D ove R n fom which naue daws es cases o evaluae he pefomance of a ceaue. Ceaues ae disinguished by he hypohesis encoded in hei genome, namely a funcion h : R n R m. The ceaue s life consiss of being pesened a se of samples fom he disibuion D; on each sample, i evaluaes h and is penalized by naue accoding o how is answes diffe fom he age funcion f. To mae his pecise, we mus inoduce he noion of a loss funcion. Definiion 1. Fo hypoheses having ange R m, a loss funcion is a nonnegaive funcion L : R m R m [0, ) such ha fo any x R m, L(x, x) = 0. Definiion 2. Given a loss funcion L and a age funcion f : R n R m, he pefomance of a hypohesis funcion h : R n R m elaive o a disibuion D (ove R n ) is defined as LPef f,d (h) = E x D [L(f(x), h(x))]. Given a posiive inege s, he s-sample empiical pefomance LPef s f,d(h) 1 Feldman has defined monoone evolvabiliy o be he esicion whee each geneaion s pefomance mus be a leas ha of he pevious geneaion, in pa inspied by a desie o conside evoluion algoihms ha seem moe naual. Monoone evolvabiliy has been shown in a disibuion-independen seing fo poin funcions unde he L 1 meic [4], conjuncions unde he L 2 meic [3], and vey ecenly, fo linea heshold funcions wih nonnegligible magin, fo L 2 and elaed meics [5]. is defined o be he andom vaiable esuling fom dawing s samples z 1,..., z s D and evaluaing 1 s s i=1 L(f(z i), h(z i )). In a manne which will be made pecise sholy, evoluion pics a hypohesis h which, empiically, is chosen o have small loss. In geneal, insead of consideing he class of linea funcions, we conside he evolvabiliy of a concep class C, consising of a subse of he funcions f : R n R m. And, as hee may be some pahological disibuions ha ba pogess, we may esic ouselves o a class D of disibuions ove R n. In paicula, woing ove he eal numbes inoduces poblems of scale ha ae no pesen in he Boolean case. Fo example, since he feedbac ha naue gives he evoluion algoihm in any geneaion is simply he choice of which, of a bounded (polynomial) numbe of poenial childen, suvives o he nex geneaion, hee is no way o evolve in bounded ime a good appoximaion o an unbounded eal numbe. I is hus impoan o wo wih concep classes C ha ae in some sense bounded. A elaed issue aises wih he disibuion class D. Suppose we ae woing wih he L 1 loss funcion, L(x, y) = x y, and suppose D is such ha, wih pobabiliy 1 τ, D samples he poin 0, and wih pobabiliy τ samples a poin moe han 1 τ -fa fom he oigin. If τ is supe-polynomially small, hen evoluion will liely neve see any samples ohe han 0, bu meanwhile, he expeced loss of hypoheses is unnown and poenially huge. Thus D (and L) mus also be easonably bounded. We mae pecise he inds of bounds we use in he paicula heoems. We now define he componens of evolvabiliy. Definiion 3 (Definiion 3.6 in [4]). Given paamee > 0, a muaion algoihm A is defined by a pai (R, M) whee R is a epesenaion class of funcions R n R m. M is a andomized polynomial (in n, m, and 1/) ime Tuing machine ha, given R and 1/ as inpus oupus a epesenaion 1 R wih pobabiliy ha we denoe P A (, 1 ). The se of epesenaions ha can be oupu by M(, ) is efeed o as he neighbohood of fo and is denoed by Neigh A (, ). As fa as he epesenaion class, ecall ha he hypohesis funcions ae ulimaely soed as he genomes of ou ceaues, and hus ae epesened as sings ove a finie alphabe. Fo he esuls of Secion 3 we explicily epesen funcions as binay sings, hough he class of funcions epesened by he scheme of Secion 3 is somewha aificial. In Secion 4 we conside genomes ha can epesen he enie class of fixed-degee polynomial funcions, and implicily conside hese polynomials as being epesened by appoximaely epesening each eal-valued coefficien as a sho sing in he genome only limied pecision is equied. The muaion algoihm is he souce of poenial genomes fo he nex geneaion; which one suvives is deemined by he selecion ule, an efficienly-implemenable algoihm ha we imagine naue unning, defined as follows (fom Definiion 3.7 of [4]). Definiion 4. Fo a loss funcion L, oleance, candidae pool size p, and sample size s, he selecion ule SelNB[L,, p, s]

3 is an algoihm such ha fo any funcion f, disibuion D, muaion algoihm A = (R, M), epesenaion R and accuacy, SelNB[L,, p, s](f, D, A, ) oupus a andom vaiable ha aes a value 1 deemined as follows. Fis un M(, ) p imes and le Z be he se of epesenaions obained. Fo Z, le P Z ( ) be he elaive fequency wih which was geneaed among he p obseved epesenaions. Fo each Z {}, compue an empiical value of pefomance v( ) LPef s f,d( ). Le Bene(Z) denoe he se of empiically beneficial muaions, { Z v( ) v() } and Neu(Z) denoe he se of empiically neual muaions, { Z v( ) v() < }. Then (i) If Bene(Z) hen oupu a andom 1 Bene(Z) disibued wih elaive pobabiliies accoding o P Z. (ii) If Bene(Z) = and Neu(Z) hen oupu a andom 1 Neu(Z) disibued wih elaive pobabiliies accoding o P Z. (iii) If Neu(Z) Bene(Z) = hen oupu. The siuaion whee all childen pefom noiceably wose han he paen, in which case he selecion ule oupus is viewed as unnaual, and we view such a case as aboing. Ohewise, if fo a concep class (and class of disibuions) hee exiss a muaion algoihm ha, unde selecion ule SelNB, efficienly conveges o any age funcion in he class, hen we say ha he concep class is evolvable: Definiion 5 (See Definiion 3.3 of [4]). A concep class C, disibuion class D, and loss funcion L ae said o be evolvable if hee exiss a muaion algoihm A = (R, M), polynomials p(n, m, 1 ), s(n, m, 1 ), a poly-bounded oleance (, n, m, 1 ) and a polynomial numbe of geneaions g(n, m, 1 ) such ha fo all n, m, age funcions f C, disibuions D D, > 0, and any iniial genome 0 R, wih pobabiliy a leas 1 he andom sequence defined by i SelNB[L,, p, s](f, D, A, i 1 ) will have LPef f,d ( g ). (We noe ha he sign convenion mos naual fo he eal case is opposie ha used in pevious wo fo he Boolean case, and in paicula, a pefec oganism in ou seing has LPef = 0 while in [4] would have LPef = 1.) 3. EVOLVABILITY AS WEAK OPTIMIZA- TION The idea a he cene of his secion is ha evolvabiliy can epoduce any esul efficienly obainable fom appoximae oacle access o LPef. In his secion we demonsae his connecion, which les us hen leveage he enie field of opimizaion algoihms owads ou goal of evolvabiliy, yielding immediae fuis a he end of his secion. As noed in he inoducion, we pove his connecion via an adapaion of he analogous esul fom he Boolean case which appeas as Theoem 5.1 in [4]. The main hudle in boh cases is showing ha he selecion ule SelNB can efficienly simulae appoximae esponses o quesions of he fom: is LPef f,d (h) geae han a heshold θ? In paicula, his will be achieved in a single geneaion of evoluion. One diffeence beween he eal case and he Boolean case o, moe specifically, beween how LPef is defined hee vesus in [4] is ha in ou case we have no funcions whose pefomance we now a pioi, while in hei case, he Boolean funcion ha euns an independen unbiased coin flip is guaaneed o have pefomance 0. Wihou such a efeence poin, evolvabiliy has no hope of addessing such heshold queies. In lieu of an absolue benchma lie ha, we insead adop a elaive benchma, compaing pefomance always agains LPef(0). 2 Namely, ou evoluion algoihm will funcion as hough i had appoximae oacle access o LPef( ) LPef(0). We give an oveview of he inuiive idea fo he consucion o appoximaely answe, in a single geneaion, queies of he fom is LPef(h) LPef(0) > θ?. We assume genomes may epesen pobabilisic funcions, and, moeove, assume as a so of inducion hypohesis ha he paen s genome defines a funcion ha is he 0 funcion a lage facion of he ime. Leing q = q(n, m, 1 ) be a bound on he oal numbe of heshold queies we would eve need o esolve, we will guaanee going fowad ha he diffeence beween he paen s pobabiliy of expessing he 0 funcion and any child s is a mos 1. q Denoing he paen s genome by, is pefomance is LPef s f,d(), and fo a given oleance, he selecion ule SelNB eas childen vey diffeenly accoding o whehe hei obseved loss is wihin of his (neual muaions), moe han lowe han his (beneficial muaions), o moe han highe han his and doomed o be culled. Since ou goal is o mae he selecion ule have a shap heshold nea whee LPef(h) LPef(0) θ, and he selecion ule aleady has hese naual shap hesholds, he naual appoach, as in [4], is o mae use of hese hesholds fo ou puposes, having poduce wo ypes of childen, 0 ha oupus idenically o he paen, and 1 ha oupus he funcion 0 wih pobabiliy less han is paen and h wih θ pobabiliy moe han is paen. θ The deails of he poof follow he ideas in he appendix of [4] (specifically Theoem A.3) and ae given below. To sae he esul moe cleanly, we inoduce wea opimizaion eminology adaped fom [9]: Definiion 6. A -wea evaluaion oacle fo a funcion f : R R is an oacle ha on inpu x euns a numbe a such ha f(x) a <. Definiion 7. The ν-wea funcion minimizaion poblem fo a funcion f : R R is ha of finding an x such ha y R, f(y) > f(x) ν. Definiion 8. A class of funcions is wealy opimizable if hee exiss a andomized polynomial ime oacle algoihm A and a polynomial = (ν, 1 ) such ha fo evey ν > 0, and any funcion f : R R in he class, A solves he ν- wea funcion minimizaion poblem when given access o a (ν, 1 )-wea evaluaion oacle fo f. 2 We noe ha hee and fo he es of he pape, we use no special popeies of he 0 funcion, and indeed any abiay funcion fom he hypohesis class could be subsiued hee and houghou he pape. We use 0 simply o avoid inoducing fuhe noaion. A moe meiculous eade migh menally subsiue an abiaily chosen elemen of he hypohesis class fo 0 as i appeas in he esuls below, o handle he odd bu pefecly legiimae case ha 0 is no in he hypohesis class of Theoem 1.

4 Theoem 1. If L is a loss funcion, C is a concep class, and D is a disibuion class such ha hee is a polynomial b(n, m) ha bounds L(f 1 (x), f 2 (x)) fo any f 1, f 2 C and any x in he suppo of a disibuion in D, and such ha he class of funcions LPef f,d (h) LPef f,d (0) indexed by f C, D D and evaluaed on h C is wealy opimizable, hen (C, D, L) is evolvable. We will find i convenien o fis pove his esul in a esiced model efeed o as evolvabiliy wih iniializaion, whee Definiion 5 is modified so ha insead of assuming evoluion sas wih an abiay genome 0 R, we insead assume a fixed saing configuaion. (See Theoem A.1 of [4].) Lemma 1. Theoem 1 holds unde he esiced evolvabiliy wih iniializaion model whee Definiion 5 is changed by eplacing he phase any iniial genome 0 R by iniial genome 0 =. Poof. By assumpion, hee is a andomized polynomial ime algoihm and a polynomial = (ν, ) such ha fo evey ν > 0 and any f C and D D, he algoihm, when given -wea oacle access o LPef f,d ( ) LPef f,d (0), will eun a hypohesis h C ha is wihin ν of opimal. Denoing by T a (polynomial) bound on he unime of his algoihm, we noe ha we may equivalenly eexpess i as a deeminisic algoihm ha is given as auxiliay inpu a T -bi unifomly andom sing. Ou goal will be o show ha we can simulae he opeaion of his algoihm in he evolvabiliy famewo As a fis sep, we will eplace he wea evaluaion oacle wih a simple oacle, he wea compaison oacle. The -wea compaison oacle fo a funcion g will, on given an inpu x and a heshold θ, eun 1 if g(x) θ +, 0 if g(x) θ, and eihe 1 o 0 ohewise. We noe ha since by assumpion, b bounds he value of he funcion in quesion, ha is, LPef f,d ( ) LPef f,d (0), we have ha log b bounds he numbe of ounds of binay seach we need o -wealy appoximae he value of he funcion via wea compaison queies. Denoe his bound by β, which since b and ae polynomial, is hence polynomially bounded iself. We noe, as will be impoan lae, ha such a binay seach can be designed so ha none of he hesholds queied eve have magniude less han. We have hus ivially shown ha hee is a deeminisic algoihm ha, when given as an auxiliay inpu a T -bi unifomly andom sing, and given wea compaison oacle access o LPef f,d ( ) LPef f,d (0), will eun a ν-wea minimum wihin T β seps. We denoe his algoihm A, and fo he sae of conceeness, assume ha afe T β seps have passed, i hals and oupus a hypohesis, no mae wha. We now un o he as of expessing algoihm A in he evolvabiliy famewo. Recall ha by assumpion, he iniial genome is uniquely fixed as. We hus as he muaion algoihm o, upon seeing he iniial genome, poduce childen whose genome encodes T bis unifomly geneaed a andom. In each subsequen sage of muaion, hese bis will be peseved in he genome; in his manne, fuue geneaions will have access o his andomly geneaed T -bi sing, as desied. All ha emains is o descibe how o simulae wea compaison queies. We will simulae one quey pe geneaion, wih he esul of he quey being soed in he genome fo he duaion. Thus a ime 0 he genome will consis of, a ime 1, of a T -bi andom sing, and a ime 1 + j we aim fo he genome o consis of he concaenaion of his sing wih a j-bi sing ha soes he esuls of he fis j wea compaison queies as specified by he algoihm A unde simulaion. Fo each such genome, we mus specify how he coesponding ceaue esponds o inpus. Fo he genome and any genome consising solely of a T -bi sing, we eun he 0 veco. Ohewise, le R be his T -bi andom sing, and le z be he emainde of he genome, whose lengh we denoe by j, and whose ih bi we denoe z i. Recall he algoihm A whose esuls we ae ying o epoduce. Ieaively simulae A saing wih sing R, and le (h 1, θ 1 ) be he fis quey sen o he wea compaison oacle; inepeing z 1 as he esul of his quey, le (h 2, θ 2 ) be he nex quey ased by A, and so on. We hus deive (h i, θ i) fo each i {0,..., j 1}, all compued in polynomial ime. We hus define he oupu behavio of ou genome on inpu x R n : fo each i {0,..., j 1} such θ i T β ha z i = 1, oupu h i(x) wih pobabiliy and ohewise oupu he veco 0. Since θ i is guaaneed o be a leas by consucion, he sum of he pobabiliies ove he (up o) T β geneaions involved will neve exceed 1. A complee specificaion of he scheme equies only ha we now specify he muaion pobabiliies. Namely, given he (andom) sing R of lengh T and a sing z of lengh j, whee we may deemine ha (h j+1, θ j+1 ) is he nex quey o be simulaed, we mus choose wih wha pobabiliy he muaion algoihm M should oupu T z0 as opposed o T z1. Vey simply, if θ j+1 < 0 hen oupu T z0 wih pobabiliy 1 and T z1 wih pobabiliy, ohewise oupu T z1 wih pobabiliy 1 and T z0 wih pobabiliy, whee = is chosen so ha in g (ou age numbe of 3g geneaions) ounds of coin flips, a -biased coin will neve land heads, excep wih pobabiliy somewha less han. We choose he oleance paamee, which specifies he T β. widh of he neual zone of pefomance, o equal We choose s, he numbe of samples aen o evaluae he empiical pefomance, o be lage enough so ha wih pobabiliy > 1 he empiical esimaes ae neve off by moe 3 han ove he enie couse of g geneaions. We analyze 2b he scheme in wo cases, noing ha, if we denoe he expeced pefomance of genome T z by ρ, hen he expeced pefomance of T z0 equals ρ while he expeced pefomance of T z1 equals ρ + [LPef θ j+1 T β f,d(h j+1 ) LPef f,d (0)], whee as jus defined, =. T β Case 1: θ j+1 < 0. If he wea compaison quey mus eun negaive, ha is, if he expeced value of LPef f,d (h j+1 ) LPef f,d (0) is a mos θ j+1, hen he expeced pefomance of T z1 is a mos ρ + (θ θ j+1 j+1 ) ρ. b Since by assumpion, excep wih pobabiliy <, he empiical pefomance will always appoximae he expeced 3 pefomance o wihin, we have ha T z1 will be found 2b o be beneficial, while T z0 will be found o be neual, and hus he nex genome will be T z1, coecly encoding he answe o he wea compaison quey. Convesely, if he wea compaison quey should eun posiive, hen by analogous agumen, he expeced pefomance of T z1 is a leas ρ +, and T z1 is hus eihe a neual o neg- b

5 aive muaion. Recall ha by consucion, in his case, an ovewhelming majoiy of he muaions in his geneaion wee consuced o be T z0 insead of T z1, and hus in eihe case, wih vey high pobabiliy (specifically, a leas = ) T z0 will hus be coecly chosen fo he nex 3g geneaion. Case 2: θ j+1 > 0. If he wea compaison quey mus eun posiive hen, fom he above agumen, he expeced loss of T z1 is a leas ρ + +, in which case T z1 is a b negaive muaion, and T z0 will be chosen fo he nex geneaion, as desied. Ohewise, he expeced loss of T z1 is a mos ρ +, which will be eihe neual o beneficial; b since he muaion algoihm will consuc T z1 insead of T z0 an ovewhelming facion of he ime (1 ), wih ovewhelming pobabiliy T z1 will hus by coecly chosen fo he nex geneaion. We conclude by sipulaing ha once he simulaion of A has compleed (which will occu wih pobabiliy a leas 1 ), he muaion algoihm will compue he esul ha A would have compued, and hus eun a saisfacoy hypohesis. We now pove Theoem 1, esuling fom Lemma 1 and a sho agumen ha iniializaion is no necessay fo he successful evoluion of ou algoihm. We ae a simple appoach han [4] hough a he expense of loose bounds. Poof Poof of Theoem 1. We noe ha he paamees in he poof of Lemma 1 wee chosen so ha he pobabilisic pocedue descibed will deviae fom is expeced behavio wih pobabiliy a mos ove g geneaions, whee, significanly, g is a paamee ha we ae sill fee o specify. Inuiively, evoluion will follow he pocedue se up in he poof of Lemma 1, which aes 1+T β geneaions, excep ha a evey geneaion, hee is pobabiliy ρ o be defined sholy of einiializing, ha is, aemping o sa evoluion fom scach again. We will exhibi a einiializaion pocedue ha aes 2b geneaions. Thus one ound of complee einiializaion and evoluion will ae 1+T β+ 2b geneaions, while in expecaion his will happen only once evey 1 geneaions. Le ρ = 1 / (1 + T β + 2b ). ρ 2 Since afe log geneaions, his pocedue will have ρ occued a leas once wih pobabiliy a leas 1, we 2 have ha fo any momen in ime afe his, he pobabiliy ha evoluion is a a wea opimum is a leas 1. Thus leing g = log yields he heoem wih pobabiliy ρ of success a leas 1 2. We hus epaameeize 2. We now illusae he vey simple einiializaion pocedue, which will ae 2b geneaions, a numbe we denoe hee as c. Fo each genome epesenaion G in he scheme of Lemma 1, wih he excepion of, we add copies labeled by ineges i {0,... c 1}, which we denoe as G i wih he inepeaion ha G i is G afe i ou of c seps owads einiializaion. The muao descibed in Lemma 1 we modify so ha evey ime i migh oupu a ceain epesenaion G, now wih pobabiliy ρ i will insead oupu G 0. The muaion ule fo G i is even simple: if i c 1 hen oupu G i+1, and if i = c 1 hen oupu, ha is, einiialize. We now define how elemens G i evaluae an inpu x: wih pobabiliy c i oupu whaeve G would oupu; wih pobabiliy i oupu he 0 veco. We noe ha since he pe- c c fomance diffeence beween 0 and any ohe hypohesis is a mos b, ha he expeced change in pefomance ove any geneaion of einiializaion is hus a mos b c = 2, namely, hese ae all neual muaions, and, by he paamee choice of Lemma 1 will be ecognized as such, which guaanees ha his pocedue will opeae as claimed. While i is faily immediae ha ou noion of evolvabiliy iself is indeed a wea opimizaion pocedue, he supising consequence of his heoem is he convese, ha any opimizaion echnique ha is noise-olean o in ou noaion wea may be leveaged by evoluion. We may hus immediaely leap o wha is pehaps he mos poweful and obus famewo fo opimizaion: he ellipsoid mehod. The ellipsoid mehod is famously nown o solve any (easonably bounded) convex opimizaion poblem, and in paicula, is wea fomulaions [9]. (Specifically, boh he domain and ange of he funcions should be bounded.) We hus have ha, as long as we can aange fo LPef o be convex and bounded, he associaed iple (C, D, L) is evolvable. As an immediae and impoan consequence, conside a degee d polynomial p : R n R m, wih D a disibuion of bounded suppo. Then fo a hypohesis h, pefomance is evaluaed by aing a sample x D and evaluaing L(p(x), h(x)). We noe ha if h is a degee d polynomial, consideed as a veco of is m (n+d ) n coefficiens, hen h(x) is a linea funcion of his coefficien veco (hough no linea in x!). Thus, if L is a convex funcion of is agumens, L will be a convex funcion of he coefficiens of h. In sho, finding he coefficiens of h is a convex opimizaion poblem when L is convex: Theoem 2. Fo any consan posiive inege d and posiive numbe, and an abiay convex loss funcion L bounded on he adius ball, he class of degee d polynomials fom R n R m wih coefficiens bounded by is evolvable wih espec o all disibuions ove he adius ball. We noe ha he case whee m > 1, hough ivial fo us o incopoae hee, is in fac quie poweful fo geneal choices of loss funcion L. Fo example, i migh seem naual and sufficien o decompose a funcion p : R n R m ino a veco of m sepaae funcions R n R and opimize he pefomance of each sepaaely, applying he loss funcion o he veco whee each of he ohe funcions is assumed o ae some defaul value, pehaps 0. Howeve, his appoach is in some sense analogous o ying o evolve waling by opimizing each leg sepaaely, assuming each ohe leg wee fixed immobile. Evoluion seems an inheenly high-dimensional poblem, in many senses, hus why we emphasize he m > 1 case hee. We noe ha we insis on consan and d in Theoem 2 because he definiion of evolvabiliy (Definiion 5) insiss ha each paamee of pefomance mus be bounded by a (polynomial) funcion of only n, m, and 1. Howeve, each of he paamees of evoluion in fac depends as mildly as migh be expeced on and d, depending polynomially on he numbe of coefficiens needed o descibe he hypohesis class of degee-d polynomials, = m (n+d ) n, and on d, which capues he gowh of he oupu of degee d polynomials on inpus of magniude up o. This same will hold ue fo he main esul of he nex secion, Theoem 3.

6 4. A DIRECT APPROACH In his secion we consuc wha is pehaps he simples conceivable andom muao ha could hope o do evoluionay hill-climbing (echnically, in ou case, he less glamoous sounding valley descen ) and show ha i is in fac supisingly adep. In paicula, i is capable of efficienly evolving he same class of geneal mulivaiae polynomials as we consideed a he end of he las secion. While we only deive esuls fo he case of a quadaic loss funcion, ha is, L(x, y) = x y 2 2, we conjecue ha simila esuls hold fo a much wide ange of loss funcions, including, pehaps, any loss funcion L(x, y) = x y c fo c > 0 including specifically hose funcions fo c < 1 which ae no convex. Definiion 9. The -dimensional wide-scale andom noise paameeized by a lowe and uppe bound (l, u) is he esul of he following pocess: choose a unifomly andom numbe ρ fom he ineval [ln l, ln u]; eun e ρ imes a andomly chosen elemen of he -dimensional uni ball. Ou muaion algoihm consiss simply of poducing seveal offsping each chosen by adding o he paen an independen sample of wide-scale andom noise. 3 Specifically: Definiion 10. Given a concep class of degee d polynomials fom R n R m wih bounded ) coefficiens, conside hei coefficiens as = m (n+d n -dimensional vecos in R. The muaion algoihm A = (R, M) fo wide-scale andom noise is defined as: R is he epesenaion of vecos in R, and M consiss of geneaing wide-scale andom noise and adding i o he paen. Theoem 3. Given any posiive inege consan d hen, fo any eal numbe hee exis bounds l, u and an inege c = poly(n, m, ) such ha he wide-scale andom noise muao wih scale in [l, u] and c childen pe geneaion evolves he class of degee d polynomials fom R n R m wih coefficiens a mos ove he class of disibuions on he n-dimensional adius ball, wih espec o he quadaic loss funcion. To pove his heoem, we fis will show ha pogess can always be made if we choose he igh adius, and hen will obseve ha, because of he exponenial way in which he adius is chosen, i is vey liely o choose a adius ha is almos exacly igh. Fo he fis pa, we noe ha he expecaion (ove any disibuion) of he quadaic loss funcion beween a polynomial and an abiay funcion, is a posiive semidefinie quadaic funcion of he polynomial s paamees. This is simply because, fo any elemen in he suppo of he disibuion, x R n, he value of he polynomial is a linea funcion of is coefficiens; he value of he ohe funcion is fixed; and hence he squae of he discepancy beween hese wo 3 Vialy Feldman has poined ou [pesonal communicaion] ha one can deandomize his pocedue by insead of choosing andom elemens of he -dimensional uni ball, ahe aing each of he sandad uni basis vecos, and hei negaions. I may be, howeve, ha in evoluion andomizaion is he moe naual. is posiive semidefinie. Inegaing hese posiive semidefinie funcions ove he disibuion will hus yield a posiive semidefinie funcion. Conside a oaion and anslaion of his posiive semidefinie funcion so ha i has he fom i=1 c i x 2 i, fo nonnegaive c i, whee {x i } ae a oaed and anslaed fom of he polynomial s = m (n+d ) n paamees. Viewing he expeced loss of a genome evolving in he conex of Theoem 3 in his fom, we show he following lemma, implying ha pogess can always be made: Lemma 2. Given > 0 and a veco of non-negaive coefficiens, (c 1,..., c ), wih σ = i=1 c i, hen he quadaic funcion p : R R defined as p(x) = i=1 c i x 2 i has he popey ha fo any veco x of lengh a mos 1, if p(x) > hen wih pobabiliy a leas 1, a andomly chosen veco y in he ball of adius abou x will 4 have p(y) < p(x) 2. 12σ 6σ The esicion ha x has lengh a mos 1 is fo he sae of convenience of he poof; when we apply he lemma in he conex of Theoem 3, we will scale he inpus so ha he adius ball becomes a diamee 1 ball. Poof Poof of Lemma 2. To aid wih he poof, we fis noe he following elemenay fac: (see fo example Chape 1 of [1]) Fac: A -dimensional ball of uni adius has a leas 1 of is volume in he egion whee is fis 4 coodinae exceeds 1 3. Conside p esiced o he line connecing x o he oigin. Since p has value 0 and deivaive 0 a he oigin, and is quadaic, i mus have deivaive (along his line) of 2p(x) x a x; since by assumpion p(x) and x 1, his is a leas 2. Since his is jus he deivaive in one diecion, he gadien a x mus have magniude a leas his. We fuhe noe ha he second deivaive in any diecion is a mos 2σ, fom he definiion of σ. Conside he value of p in he ball of adius 6σ aound x, and specifically, in he poion ha is a leas 3 in he diecion of he (downwad) gadien fom x. By he above fac, his poion compises a leas a quae of he ball. Consideing he second-degee Taylo expansion of p abou x which is exac, since p is quadaic we noe ha he linea conibuion is a decease of a leas 2 (ou lowe bound on he magniude of he gadien) imes he disance aveled in he diecion of he gadien, namely 3 =, 18σ yielding 2. The quadaic conibuion is bounded by 1 9σ 2 imes he diecional second deivaive in ou diecion imes he squae of he disance, which is bounded by in ou ball, yielding σ 2 = 2. Subacing yields he desied 36σ bound. We now assemble he pieces ino a poof of Theoem 3. Poof Poof of Theoem 3. Le = m (n+d ) n be he dimension of ou degee d polynomials when viewed as a veco space, and le b be a bound on he loss of any pai of hypoheses funcions evaluaed a any poin in he n- dimensional adius ball. These ae boh bounded by polynomials fo consan d.

7 Fo any -dimensional uni veco v, egaded as a degee d polynomial, and any poin x in he n-dimensional adius ball, he loss of an abiay muliple of v, αv, elaive o he zeo polynomial, evaluaed on he poin x mus be a quadaic funcion cα 2, fo c b. Thus fo an abiay disibuion in he n-dimensional 2 adius ball, and abiay age funcion, he expeced loss will be a posiive semidefinie quadaic fom ha can be oaed and anslaed ino he fom i=1 cix2 i wih each c i 0, and if we fuhe scale he inpu by 1 so ha he adius ball in 2 dimensions maps ino a egion of diamee 1, hen we have σ i=1 c i 4b. We hus conside he applicaion of Lemma 2 o his ansfomed expeced loss funcion. If hee exiss a genome in he ball wih expeced loss geae han, hen Lemma 2 6σ guaanees ha hee exiss his magic adius = such ha moving he genome by a veco andomly chosen in he -dimensional ball of adius will, wih pobabiliy a leas 1 4, impove he expeced loss by a leas 2 12σ. Since we have polynomial uppe bounds on σ, Lemma 2 hus povides fo invese-polynomial pogess, in exacly hose cases whee we ae no aleady wihin of opimal. We noe ha we have aleady bounded 24 ; o uppe bound, we noe ha i=1 c ix 2 i is bounded by σ since x 1, and hence by assumpion, σ >, yielding he bound < 1 6 a faco of 1. Recalling ha we scaled he coodinaes by o apply Lemma 2, we have ha in he oiginal 2 coodinaes and poblem seup: eihe evoluion is wihin (plus sampling eo) of opimal, o hee is a magic adius beween 12 and 3 such ha a andomly chosen muaion in he -dimensional ball of his adius will yield significan impovemen. We hus declae he lowe and uppe bounds of ou wide-scale andom noise o be 12 and 3 especively. We noe in geneal ha a pai of -dimensional balls whose adii, have logaihms ae wihin 1 of each ohe will shae a consan facion of hei volume. Thus wih a leas invese-polynomial pobabiliy, choosing a adius ha is ( e o he ) powe ( of a ) numbe unifomly chosen beween ln 12 and ln 3 will yield wih consan pobabiliy a muaion ha impoves he expeced loss by a leas 2. The candidae pool size is chosen so ha wih ovewhelming pobabiliy, say, a leas 1, such a muaion 12σ 2 will be pesen in each geneaion. Thus we may choose s he numbe of samples wih which o evaluae he empiical pefomance high enough ha wih pobabiliy a leas 1, ove he enie couse of he 2 algoihm all esimaes will be accuae o wihin a hid of he minimum impovemen, 2. Fuhe, we choose 36σ, he heshold fo declaing a muaion beneficial, o be equal o, so ha, assuming each empiical esimae is 2 18σ in fac accuae o wihin 2, hen each of he beneficial 36σ muaions guaaneed by Lemma 2 will be ecognized and declaed o be beneficial. Thus wih pobabiliy a leas 1, he pefomance of evey geneaion will be a leas bee han ha of he pevious, unless we ae aleady 2 36σ wihin + 2 of opimum, yielding he desied esul. 36σ 5. REFERENCES [1] K. Ball. An Elemenay Inoducion o Moden Convex Geomey. MSRI Publicaions, Volume 31, [2] V. Feldman. Evolvabiliy fom Leaning Algoihms. STOC [3] V. Feldman. A Complee Chaaceizaion of Saisical Quey Leaning wih Applicaions o Evolvabiliy. FOCS, [4] V. Feldman. Robusness of Evolvabiliy. COLT, [5] V. Feldman. Disibuion-Independen Evolvabiliy of Linea Theshold Funcions. Manuscip, Febuay [6] A. Kalai and S. Vempala. Simulaed Annealing fo Convex Opimizaion. Mahemaics of Opeaions Reseach, 31(2), 2006, pp [7] V. Kanade, L. Valian, and J. Vaughan. Evoluion wih Difing Tages. COLT, [8] M. Keans. Efficien noise-olean leaning fom saisical queies. Jounal of he ACM, 25(6): , [9] L. Lovász. An Algoihmic Theoy of Numbes, Gaphs, and Convexiy, Chape 2. SIAM, [10] L. Michael. Evolvabiliy via he Fouie Tanfom. Theoeical Compue Science, o appea. [11] L. Valian. Evolvabiliy. Jounal of he ACM, 56(1), 2009.