Caracterización de la base intelectual disciplinaria en el período 1995-2008:

Inthis paper,we studiedthe problemoflearning the CTRsof adsinsponsored search auctionswith truthful mecha-nisms.Thisproblemishighlychallengingsinceitrequiresthecombinationofonlinelearningtools(i.e.,regretminimization

18 FromthisexperimentisnotclearwhetherR RT= ˜O(q⁻_min¹),thusimplyingthatRT doesnotdependonqminatall,orR RT issublinearinqmin,which wouldcorrespondtoadependencyRT= ˜^O(q⁻_min^z)with0<z<1.

Fig. 14. Dependency of the relative regret R RT on N.

algorithms)andeconomictools(i.e.,truthfulmechanisms).Whilealmostalltheliteraturefocusedonsingle-slotscenarios, herewefocusedonmulti-slotscenarios.Withmultipleslotsitisnecessarytoadoptausermodeltocharacterizehowthe CTR ofanadvariesastheallocationofdisplayedadsvaries.Here,weadoptedthecascademodel,thatisthemostcommon modelusedintheliterature.Inthepaper,westudiedanumberofscenarios,eachwithaspecificinformationsettingof un-knownparameters.Foreachscenario,wedesignedatruthfullearningmechanism,studieditseconomicproperties,derived an upperbound overtheregret,and, forsome mechanisms,alsoalower bound.We consideredboth theregretover the auctioneer’srevenueandtheSW.

We showedthat forthe cascade modelwith only position-dependent externalities it is possible to design a truthful no-regret learning mechanismforthegeneralcasein whichall theparameters are unknown.Ourmechanismpresents a regret O˜(T²³) andit is DSIC in expectation w.r.t. therealization ofthe random component ofthe mechanism. However, it remains open whetheror not it is possible to obtain a regret O˜(T¹²). For specific cases, in which some parameters are known to the auctioneer, we obtained better results in terms of either incentive compatibility, obtaining dominant strategy truthfulness,orregret,obtaining a regretofzero.Weshowedthat forthecascademodelwiththeposition- and ad-dependentexternalitiesitispossibletodesignaDSIC aposteriori mechanismwitharegretO˜(T²³)whenonlythequality isunknown.Instead,evenwhenthecascademodelisonlywithad-dependentexternalitiesandnoparameterisknown,it isnot possibletoobtainano-regretDSIC aposteriori mechanism.Theproof ofthisresultwouldseemtosuggestthat the sameresultholdsalsowhena randommechanismisadoptedandthetruthfulness isinexpectationw.r.t.its realizations.

However, we didnotproduceanyproof forthat, leavingit forfutureworks.Finally,we empiricallyevaluatedthebounds weprovided,showingthatthedependencyoftheregretontheparametersismostlycorrectinaworst-casescenario.

Twomain questionsdeservefuture investigation.The firstquestion concerns thestudyofa lower bound forthecase in which thereare only position-dependentexternalities andtruthfulness isin expectationinexpectationw.r.t. only the realizationsoftherandomcomponentofthemechanismoralsow.r.t.theclickrealizations.Furthermore,itisopenwhether theseparationofexplorationandexploitationphasesisnecessaryand,inthenegativecase,whetheritispossibletoobtain aregret O˜(T¹²).Thesecondquestionconcernsasimilarstudyrelatedtothecasewithonlyad-dependentexternalities.

Appendix A. Vickrey–Clarke–Grovesmechanism

Consideragenericdirect-revelationmechanismM= (N ,V,,f,{^pi}i∈N)asdefinedinSection3.2.Differentlyfromthe SSA case,ingeneralthetypeofanagent,denotedbyviforconsistencywiththerestofthepaper,isavectorofparameters.

Wedefineafunctionval_i: ×V → R^{+,whichreturnsthevalueobtainedbyagenta}iwhenitstypeisv_iandtheallocation chosenbythemechanismisθ.

TheVCG mechanismisobtainedcouplingthetwofollowingfunctions:

•^{theallocationfunction f whichreturnstheallocation}maximizing thesocialwelfare,i.e., f

(

v

ˆ ) =

^{arg max}

θ∈SW

(θ,

v

ˆ ) =

^{arg max}

θ∈

i∈N

val_i

(θ,

v

ˆ

_i

) ;

•^{thepaymentrule p}i,whichdefinesthepaymentrequiredfromagentai,i.e., pi

(ˆ

v

) =

^SW

(

f

(ˆ

v₋i

),

v

ˆ

₋i

) −

^SW−ⁱ

(

f

(

v

ˆ ),

v

ˆ )

=

j∈N ,^j=ⁱ

val_j

(

f

(ˆ

v_−i

),

v

ˆ

_j

) −

j∈N ,^j=ⁱ

val_j

(

f

(ˆ

v

),

v

ˆ

_j

),

wherewedenoteby f(ˆv₋i)theallocationreturnedby f whenagenti doesnotparticipatetotheauction.

Inthisquasi-linearenvironment,whentherearenointerdependenciesamongthetypesoftheagentsandthe no-single-agenteffect[3]holds,theVCG mechanismisAE, DSIC aposteriori,IR aposteriori,andWBB aposteriori.

Appendix B. MonotonicityandMyerson’spayments

Consideragenericdirect-revelationmechanismM= (N ,V,,f,{^pi}i∈N)asdefinedinSection3.2.Asingle-parameter linearenvironmentissuchthat

• ^{thetypeofeachagenta}iisascalar vi(single-parameterassumption),

• ^{theutility function ofagent a}i isu_i(ˆ^v)=^zi

Anallocationfunction f ismonotonic inasingle-parameterlinearenvironmentifforany^vˆ−i

zi possibletodesignaDSIC mechanismimposingthefollowingpayments[35]:

p_i

(

^v

ˆ ) =

^hi

(ˆ

^v₋i

) +

^zi Appendix C. ProofofrevenueregretinTheorem 2

WestartbyreportingtheproofofProposition 1.

ProofofProposition 1. Thederivation is a simpleapplicationof the Hoeffding’sbound.We first notice that each ofthe termsintheempiricalaverageq˜i(Eq.(11))isboundedin[⁰;¹/_π(i;θt)]^{.Thusweobtain}

Byreorderingthetermsinthepreviousexpressionwehave

η =

whichguaranteesthatalltheempiricalestimates^q˜iarewithin

η

ofq_i foralltheadswithprobability,atleast, 1− δ^. 2 Beforestatingthemainresultofthissection,weneedthefollowinglemma.

Lemma1.Foranyslots_mwithm∈K^,withprobability1− δ^,

Proof. TheproofisastraightforwardapplicationofProposition 1.Weconsidertheoptimalallocationθ^∗ definedinEq.(2) andtheoptimalallocation ˜θwhenestimatesq˜⁺areadopteddefinedinEq.(16).Wedenoteh=

α

(m;θ^∗)∈^{arg max}

i∈N(qivˆi;^m), i.e.,theindexoftheadallocatedinagenericslotinpositionm.Therearetwopossiblescenarios:

• ^If

π

(h;˜θ)<m (thead is displayed into ahigher slotin the approximatedallocation ˜θ), then ∃^j∈N ^s.t.

π

(j;θ^∗)<

m∧

π

(j;˜θ)≥^m.Thus

maxi∈N

(˜

q⁺_i v

ˆ

i

;

m

) ≥ ˜

q⁺_jv

ˆ

j

≥

qjv

ˆ

j

≥

qhv

ˆ

h

=

max

i∈N

(

qiv

ˆ

i

;

m

)

wherethesecondinequalityholdswithprobability1− δ^;

•^If

π

(h;˜θ)≥^{m (theadisdisplayedintoalowerorequalslotinthe}approximatedallocation ˜θ),then maxi∈N

(˜

q⁺_i v

ˆ

i

;

m

) ≥ ˜

q⁺_hv

ˆ

h

≥

qhvh

=

max

i∈N

(

qiv

ˆ

i

;

m

)

wherethesecondinequalityholdswithprobability 1− δ^. Inbothcases,thestatementfollows. 2

ProofofTheorem 2.

Step1:expectedpayments. The proof follows steps similar to those in theproofs in [20]. We first recall that since the mechanismisDSIC in expectationw.r.t.the clicks,then wecan directlyfocusontheregretwhen theactualvalues v are bid.Foranyadaisuchthat

π

(i;θ^∗)≤^{K ,theexpectedpaymentsofthe}VCG mechanisminthiscasereducetoEq.(9):

while,giventhedefinitionofA-VCG1 reportedinSection4.1,theexpectedpaymentsforatt-thiterationoftheauctionare

˜

Step2:per-stepexplorationregret. Sinceforany1≤^t≤

τ

,A-VCG1setsallthepaymentsto0,theper-stepregretis r_t

=

Step3:per-stepexploitationregret. Now we focus on the expected (w.r.t. click realizations) per-step regret during the exploitation phase.According to thedefinition ofpayments, ateach stept∈ {

τ

+¹,. . . ,T} ^ofthe exploitation phase we boundtheper-stepregretr as

rt

=

Bydefinitionofthemaxoperator,sincel+1>m,itfollowsthat max

withprobabilityatleast1− δ^{.Noticethat,bydefinitionof}l,K

l=ml= m− K+¹= m.Furthermore,fromthe defini-tionof^q˜⁺_i ^{andusingEq.(14)wehavethatforanyada}i:

˜

q⁺_i

−

^qi

= ˜

^qi

−

^qi

+ η _≤

2

η ,

withprobabilityatleast1− δ.Thus,thedifferencebetweenthepaymentsbecomes rt

≤

^2vmax

Step4:cumulativeregret. Wefirstconsiderthe(low-probability)eventinwhichtheboundonq˜⁺_i derivedinProposition 1.

Inthiscase,we cannotguaranteeanythingaboutthebehaviorofthemechanism, sincethepaymentsare veryinaccurate estimatesoftheCTRs, andthusthelargestpossibleregretissuffered.Inparticular,weconsidertheworstcaselossofvmax foreachslotforeach step,leadingto atotalregretof v_maxK

m=¹m



T withprobability δ.Bysummingup theregrets reportedinEq.(C.3)duringtheexplorationphaseandEq.(C.6)duringtheexploitationphaseandbyconsideringthatthese boundsholdwithprobabilityatleast1− δ(upper-boundedby1inthefollowing),weobtainanexpectedregret

RT

≤

^vmax

where R_ei istheupperboundonthe regretsufferedduring theexploitationphase (whichholdswithprobability atleast 1− δ^{), R}er istheupperboundontheregretsufferedduring theexploitationphase(which holdswithprobabilityatleast 1− δ^)andRδ istheupperboundontheregretwhentheboundsdonothold(withprobabilityatmostδ).Thisboundcan befurthersimplified,giventhat_K

m=¹m≤^{K ,as}

Step5:parametersoptimization. BesidedescribingtheperformanceofA-VCG1,thepreviousboundalsoprovidesguidance fortheoptimizationoftheparameters

τ

andδ.WefirstsimplifytheboundinEq.(C.7)as

R_T

≤

^vmaxK deriva-tiveofthepreviousboundw.r.t.

τ

,setittozeroandobtain

v_maxK

Substitutingthisvalueof

τ

intoEq.(C.8)leadstotheoptimizedbound

RT

≤

^vmaxK

19 Noticethatinthelogarithmictermthefactorof2wehaveinProposition 1disappearssinceinthisproofweonlyneedtheone-sidedversionofthe bound.

Wearenowleftwiththechoiceoftheconfidenceparameterδ∈ (⁰,1),whichcanbeeasilysettooptimizetheasymptotic rate(i.e.,ignoringconstantsandlogarithmicfactors)as

δ =

^{K−13T−13N13}

Wethusobtainthefinalbound

R_T

≤

^4vmax



⁻

2 3

minK²³T²³N¹³



log K¹³T¹³N²³



¹

3

.

We havetoimposetheconstraintsthat T> ^N_K (givenby δ <1)andthat T>

τ

,i.e., T> ^N

K²_minlog^N_δ.The twoconstraints imply:

T

>

^N

In document Ciencia de la Información y Paradigma Social: (página 108-118)