One Practical Algorithm for Both Stochastic and Adversarial Bandits


 Camilla Boyd
 2 years ago
 Views:
Transcription
1 One Prcicl Algorihm for Boh Sochsic nd Adversril Bndis Yevgeny Seldin Queenslnd Universiy of Technology, Brisbne, Ausrli Aleksndrs Slivkins Microsof Reserch, New York NY, USA Absrc We presen n lgorihm for mulirmed bndis h chieves lmos opiml performnce in boh sochsic nd dversril regimes wihou prior knowledge bou he nure of he environmen. Our lgorihm is bsed on ugmenion of he EXP lgorihm wih new conrol lever in he form of explorion prmeers h re ilored individully for ech rm. The lgorihm simulneously pplies he old conrol lever, he lerning re, o conrol he regre in he dversril regime nd he new conrol lever o deec nd exploi gps beween he rm losses. This secures problemdependen logrihmic regre when gps re presen wihou compromising on he worscse performnce gurnee in he dversril regime. We show h he lgorihm cn exploi boh he usul expeced gps beween he rm losses in he sochsic regime nd deerminisic gps beween he rm losses in he dversril regime. The lgorihm reins logrihmic regre gurnee in he sochsic regime even when some observions re conmined by n dversry, s long s on verge he conminion does no reduce he gp by more hn hlf. Our resuls for he sochsic regime re suppored by experimenl vlidion.. Inroducion Sochsic mulirmed bndis Thompson, 9; Robbins, 95; Li & Robbins, 985; Auer e l., nd dversril mulirmed bndis Auer e l., 995; b hve coexised in prllel for lmos wo decdes by now, in he sense h no lgorihm for sochsic mulirmed bndis is pplicble o dversril mulirmed bndis nd l Proceedings of he s Inernionl Conference on Mchine Lerning, Beijing, Chin, 4. JMLR: W&CP volume. Copyrigh 4 by he uhors. gorihms for dversril bndis re unble o exploi he simpler regime of sochsic bndis. The recen emp of Bubeck & Slivkins o bring hem ogeher did no mke i in he full sense of unificion, since he lgorihm of Bubeck nd Slivkins relies on he knowledge of ime horizon nd mkes oneime irreversible swich beween sochsic nd dversril operion modes if he beginning of he gme is esimed o exhibi dversril behvior. We presen n lgorihm h res boh sochsic nd dversril mulirmed bndi problems wihou disinguishing beween hem. Our lgorihm jus runs, s mos oher bndi lgorihms, wihou knowledge of ime horizon nd wihou mking ny hrd semens bou he nure of he environmen. We show h if he environmen hppens o be dversril he performnce of he lgorihm is jus fcor of worse hn he performnce of he EXP lgorihm wih he bes consns, s described in Bubeck & CesBinchi nd if he environmen hppens o be sochsic he performnce of our lgorihm is comprble o he performnce of UCB of Auer e l.. Thus, we cover he full rnge nd chieve lmos opiml performnce he exreme poins. Furhermore, we show h he new lgorihm cn exploi boh he usul expeced gps beween he rm losses in he sochsic regime nd deerminisic gps beween he rm losses in he dversril regime. We lso show h he lgorihm reins logrihmic regre gurnee in he sochsic regime even when some observions re dversrilly conmined, s long s on verge he conminion does no reduce he gp by more hn hlf. To he bes of our knowledge, no oher lgorihm hs been ye shown o be ble o exploi gps in he dversril or dversrilly conmined sochsic regimes. The conmined sochsic regime is very prcicl model, since in mny rellife siuions we re deling wih sochsic environmens wih occsionl disurbnces. Since he inroducion of Thompson s smpling Thompson, 9 which ws nlyzed only fer 8 yers Kufmnn e l., ; Agrwl & Goyl, vriey of l
2 One Prcicl Algorihm for Boh Sochsic nd Adversril Bndis gorihms were invened for he sochsic mulirmed bndi problem. The mos powerful for ody re KLUCB Cppé e l.,, EwS Millrd,, nd he foremenioned Thompson s smpling. I is esy o show h ny deerminisic lgorihm cn poenilly suffer liner regre in he dversril regime see he supplemenry meril for proof. Alhough nohing is known bou he performnce of rndomized lgorihms for sochsic bndis in he dversril regimes, empiriclly hey re exremely sensiive o deviions from he sochsic ssumpion. In he dversril world he mos powerful lgorihm for ody is INF Audiber & Bubeck, 9; Bubeck & Ces Binchi,. Neverheless, he EXP lgorihm of Auer e l. b sill reins n imporn plce, minly due o is simpliciy nd wide pplicbiliy, which covers combinoril bndis, pril monioring gmes, nd mny oher dversril problems. Since ny sochsic problem cn be seen s n insnce of n dversril problem, boh INF nd EXP hve he worscse roo regre gurnee in he sochsic regime, bu i is no known wheher hey cn do beer. Empiriclly in he sochsic regime EXP is inferior o ll oher known lgorihms for his seing, including he simples UCB lgorihm. I is ineresing o ke brief look ino he developmen of EXP. The lgorihm ws firs suggesed in Auer e l. 995 nd is prmerizion nd nlysis were improved in Auer e l. b. The EXP of Auer e. l. ws designed for he mulirmed bndi gme wih rewrds nd is plying sregy is bsed on mixing Gibbs disribuion lso known s exponenil weighs wih uniform explorion disribuion in proporion o he lerning re. The uniform explorion leves no hope for chieving logrihmic regre in he sochsic regime simulneously wih he roo regre in he dversril regime, since ech rm is plyed les Ω imes in rounds of he gme. By chnging he lerning re CesBinchi & Fischer 998 mnged o derive differen prmerizion of he lgorihm h ws shown o chieve logrihmic regre in he sochsic regime, bu i hd no regre gurnees in he dversril regime. Solz 5 hs observed h in he gme wih losses he roo regre gurnee in he dversril regime cn be chieved wihou mixing in he uniform disribuion nd even led o beer consns. However, mixing in ny disribuion h elemenwise does no exceed he lerning re does no brek he worscse performnce of he lgorihm in he gme wih losses. We exploi his emerged freedom in order o derive modificion of he EXP lgorihm h chieves lmos opiml regre in boh dversril nd sochsic regimes wihou prior knowledge bou he nure of he environmen. Rewrds cn be rnsformed ino losses by king l = r.. Problem Seing We sudy he mulirmed bndi MAB gme wih losses. In ech round of he gme he lgorihm chooses one cion A mong K possible cions,.k.. rms, nd observes he corresponding loss l A. The losses of oher rms re no observed. There is lrge number of loss generion models, four of which re considered below. In his work we resric ourselves o loss sequences l }, h re genered independenly of he lgorihm s cions. Under his ssumpion we cn ssume h he loss sequences re wrien down before he gme srs bu no reveled o he lgorihm. We lso mke sndrd ssumpion h he losses re bounded in he [, inervl. The performnce of he lgorihm is qunified by regre, defined s he difference beween he expeced loss of he lgorihm up o round nd he expeced loss of he bes rm up o round : R = E [ l As s min E [ l s } The expecion is ken over he possible rndomness of he lgorihm nd loss generion model. The gol of he lgorihm is o minimize he regre. We consider wo sndrd loss generion models, he dversril regime nd he sochsic regime nd wo inermedie regimes, he conmined sochsic regime nd he dversril regime wih gp. Adversril regime. In his regime he loss sequences re genered by n unresriced dversry who is oblivious o he lgorihm s cions. This is he mos generl seing nd he oher hree regimes cn be seen s specil cses of he dversril regime. An rm rg min l s is known s bes rm in hindsigh for he firs rounds. Sochsic regime. In his regime he losses l re smpled independenly from n unknown disribuion h depends on, bu no on. We use µ = E [l o denoe he expeced loss of rm. Arm is clled bes rm if µ = min µ } nd subopiml oherwise; le denoe some bes rm. For ech rm, define he gp = µ µ. Le = min : > } denoe he miniml gp. Leing N be he number of imes rm ws plyed up o nd including round, he regre cn be rewrien s R = E [N. Conmined sochsic regime. In his regime he dversry picks some roundrm pirs, locions before he gme srs nd ssigns he loss vlues here in n.
3 One Prcicl Algorihm for Boh Sochsic nd Adversril Bndis rbirry wy. The remining losses re genered ccording o he sochsic regime. We cll conmined sochsic regime moderely conmined fer τ rounds if for ll τ he ol number of conmined locions of ech subopiml rm up o ime is mos /4 nd he number of conmined locions of ech bes rm is mos /4. By his definiion, for ll τ on verge over sochsiciy of he loss sequences he dversry cn reduce he gp of every rm by mos hlf. Adversril regime wih gp. An dversril regime is nmed by us n dversril regime wih gp if here exiss round τ nd n rm τ h persiss o be he bes rm in hindsigh for ll rounds τ. We nme such rm consisenly bes rm fer round τ. If no such rm exiss hen τ is undefined. Noe h if τ is defined for some τ hen τ is defined for ll τ > τ. We use λ = l s o denoe he cumulive loss of rm. Whenever τ is defined we define deerminisic gp of rm on round τ s: τ, = min τ λ λ τ If τ is undefined, τ, is defined s zero. }. Noion. We use E} o denoe he indicor funcion of even E nd = A=} o denoe he indicor funcion of he even h rm ws plyed on round.. Min Resuls Our min resuls include new lgorihm, which we nme EXP++, nd is nlysis in he four regimes defined in he previous secion. The EXP++ lgorihm, provided in Algorihm box, is generlizion of he EXP lgorihm wih losses. Algorihm Algorihm EXP++. Remrk: See ex for definiion of η nd ξ. : L =. for =,,... do β = ln K K. : ε = min K, β, ξ }. : ρ = e η L / e η L. : ρ = ε ρ + ε. Drw cion A ccording o ρ nd ply i. Observe nd suffer he loss l A. : l = la ρ. : L = L + l. end for The EXP++ lgorihm hs wo conrol levers: he lerning re η nd he explorion prmeers ξ. The EXP wih losses s described in Bubeck & CesBinchi is specil cse of he EXP++ wih η = β nd ξ =. The crucil innovion in EXP++ is he inroducion of explorion prmeers ξ, which re uned individully for ech rm depending on he ps observions. In he sequel we show h uning only he lerning re η suffices o conrol he regre of EXP++ in he dversril regime, irrespecive of he choice of he explorion prmeers ξ. Then we show h uning only he explorion prmeers ξ suffices o conrol he regre of EXP++ in he sochsic regime irrespecive of he choice of η, s long s η β. Applying he wo conrol levers simulneously we obin n lgorihm h chieves he opiml roo regre in he dversril regime up o logrihmic fcors nd lmos opiml logrihmic regre in he sochsic regime hough wih subopiml power in he logrihm. Then show h he new conrol lever is even more powerful nd llows o deec nd exploi he gp in even more chllenging siuions, including moderely conmined sochsic regime nd dversril regime wih gp. Adversril Regime Firs, we show uning η is sufficien o conrol he regre of EXP++ in he dversril regime. Theorem. For η = β nd ny ξ he regre of EXP++ for ny sisfies: R 4 K ln K. Noe h he regre bound in Theorem is jus fcor of worse hn he regre of EXP wih losses Bubeck & CesBinchi,. Sochsic Regime Now we show h for ny η β uning he explorion prmeers ξ suffices o conrol he regre of he lgorihm in he sochsic regime. By choosing η = β we obin lgorihms h hve boh he opiml roo regre scling in he dversril regime nd logrihmic regre scling in he sochsic regime. We consider number of differen wys of uning he explorion prmeers ξ, which led o differen prmerizions of EXP++. We sr wih n idelisic ssumpion h he gp is known, jus o give n ide of wh is he bes resul we cn hope for. Theorem. Assume h he gps re known. For ny choice of η β nd ny c 8, he regre of EXP++ wih ξ = c ln in he sochsic regime
4 One Prcicl Algorihm for Boh Sochsic nd Adversril Bndis sisfies: R ln O + K Õ. The consns in his heorem re smll nd re provided explicily in he nlysis. We lso show h c cn be mde lmos s smll s. Nex we show h using he empiricl gp s n esime of he rue gp ˆ = min, L min L } we cn lso chieve polylogrihmic regre gurnee. We cll his lgorihm EXP++ AVG. Theorem. Le c 8 nd η β. Le be he miniml ineger h sisfies 4c K ln 4 nd le = mx, e / }. cln ˆ lnk The regre of EXP++ wih ξ = ermed EXP++ AVG in he sochsic regime sisfies: R ln O +. Alhough he ddiive consns in his heorem re very lrge, in he experimenl secion we show h minor modificion of his lgorihm performs comprbly o UCB in he sochsic regime nd hs he dversril regre gurnee in ddiion. In he following heorem we show h if we ssume known ime horizon T, hen we cn elimine he ddiive erm e / in he regre bound. The lgorihm in Theorem 4 replces he empiricl gp esime in he definiion of ξ wih lower confidence bound on he gp nd slighly djuss oher erms. We nme his lgorihm EXP++ LCBT. Theorem 4. Consider he sochsic regime wih known ime horizon T. The EXP++ LCBT lgorihm wih ny η β nd ppropriely defined ξ chieves regre RT Olog T. The precise definiion of EXP++ LCBT nd he proof of Theorem 4 re provided in he supplemenry meril. I seems h simulneous eliminion of he ssumpion on he known ime horizon nd he exponenilly lrge ddiive erm is very chllenging problem nd we defer i for fuure work. Conmined Sochsic Regime Nex we show h EXP++ AVG cn susin modere conminion in he sochsic regime wihou significn deeriorion in performnce. Theorem 5. Under he prmerizion given in Theorem, for = mx, e 4/ }, where is defined s before, he regre of EXP++ AVG in he sochsic regime h is moderely conmined fer τ rounds sisfies: R ln O + mx, τ}. The price h is pid for modere conminion fer τ rounds is he scling of by fcor of / nd he ddiive fcor of τ. The scling of ffecs he definiion of nd he consn in O ln. As before, he regre gurnee of Theorem 5 comes in ddiion o he gurnee of Theorem. Adversril Regime wih Gp Finlly, we show h EXP++ AVG cn lso ke dvnge of deerminisic gp in he dversril regime. Theorem 6. Under he prmerizion given in Theorem, he regre of EXP++ AVG in he dversril regime sisfies: R min mx, τ, e / τ,} } ln + O. τ τ, We remind he reder h in he bsence of consisenly bes rm τ, is defined s zero nd he regre bound is vcuous bu he regre bound of Theorem sill holds. We lso noe h τ, is nondecresing funcion of τ. Therefore, here is rdeoff: incresing τ increses τ,, bu loses he regre gurnee on he rounds before τ for simpliciy, we ssume h we hve no gurnees before τ. Theorem 6 llows o pick τ h minimizes his rdeoff. An imporn implicion of he heorem is h if he deerminisic gp is growing wih ime he regre gurnee improves oo. 4. Proofs We prove he heorems from he previous secion in he order hey were presened. The Adversril Regime The proof of Theorem relies on he following lemm, which is n inermedie sep in he nlysis of EXP by Bubeck see lso Bubeck & CesBinchi. Lemm 7. For ny K sequences of nonnegive numbers X, X,... indexed by,..., K} nd ny nonincresing posiive sequence η, η,..., for ρ =
5 One Prcicl Algorihm for Boh Sochsic nd Adversril Bndis exp η X s h exp η X s exponen is zero we hve: ssuming for = he sum in he T = ρ T X min = X T η ρ X + ln K. η T = More precisely, we re using he following corollry, which follows by llowing X s o be rndom vribles nd king expecions of he wo sides of nd using he fc h E [min[ min [E [. We decompose expecions of incremenl sums ino incremenl sums of condiionl expecions nd use E [ o denoe expecions condiioned on relizion of ll rndom vribles up o round. Corollry 8. Le X, X,... for,..., K} be nonnegive rndom vribles nd le η nd ρ s defined in Lemm 7. Then: [ T [ [ T E E ρ X min E E [X = = [ T [ η E E ρ X + ln K. η T = Proof of Theorem. We ssocie X in wih l in he EXP++ lgorihm. We hve E [ l = l nd since ρ = ε ρ ε ρ ε nd l [, we lso hve: E [ ρ l [ E ρ ε l E [l A ε. As well, we hve: [ E ρ l = E ρ [ ρ E ρ = = l A ρ ρ ρ ρ ε ρ + ε K, where he ls inequliy follows by he fc h ε by he definiion of ε. Subsiuion of he bove clculions ino Corollry 8 yields: [ T [ T R = E l A min E l K T = = η + ln K η T + = ε K T = η + ln K η T. The resul of he heorem follows by he choice of η. The Sochsic Regime Our proofs re bsed on he following form of Bernsein s inequliy, which is minor improvemen over CesBinchi & Lugosi 6, Lemm A.8 bsed on he ides from Boucheron e l., Theorem.. Theorem 9 Bernsein s inequliy for mringles. Le X,..., X n be mringle difference sequence wih respec o filrion F = F i in nd le S i = i j= X j be he ssocied mringle. Assume h here exis posiive numbers ν nd c, such h X j c for ll j wih probbiliy nd [ n i= E X i Fi ν wih probbiliy. Then for ll b > : P [ S n > νb + cb e b. We re lso using he following echnicl lemm, which is proved in he supplemenry meril. Lemm. For ny c > : = e c = O c. The proof of Theorems nd is bsed on he following lemm. Lemm. Le ε } = be nonincresing deerminisic sequences, such h ε ε wih probbiliy nd ε ε for ll nd. Define ν = ε s nd define he even E L L ν + ν b +.5b ε. E Then for ny posiive sequence b, b,... nd ny he number of imes rm is plyed by EXP++ up o round is bounded s: E [N + e bs + ε s E } s= s= + e ηsgs, s= where g = b ε + ε.5b ε.
6 One Prcicl Algorihm for Boh Sochsic nd Adversril Bndis Proof. Noe h elemens of } he mringle difference sequence l l re upper bounded by = ε +. Since ε ε /K /4 we cn simplify he upper bound by using ε +.5 ε. Furher noe h = [ E s l s [ E s l s [ E s ls l s l s [ + E s l s p s + p s ε s + ε s ε s + ε s = ν + ν wih probbiliy. Le E denoe he complemen of even E. Then by Bernsein s inequliy P [ E b. The number of imes rm is plyed up o round is bounded s: E [N = = P [A s = P [ A s = [ E s P E s + P [ A s = [ Es P E s P [ A s = E s E s } + P [ Es P [ A s = E s E s } + e bs. For he erms of he sum bove we hve: P [ A = E E s } = ρ E s } ρ + ε E s } L = ε + e η L e η E s } ε + e η L L E s } ε E s } + e ηg, Where in he ls inequliy we used he fcs h even E holds nd h since ε is nonincresing sequence ν ε. Subsiuion of his resul bck ino he compuion of E [N complees he proof. Proof of Theorem. The proof is bsed on Lemm. Le b = ln nd ε = ε. For ny c 8 nd ny, where is he miniml ineger for which 4c K ln 4 lnk, we hve: g = b ε + ε b ε.5b ε =.5 c c..5b ε The choice of ensures h for ll subopiml cions we hve ε = ξ, which slighly simplifies he clculions. Also noe h since ε = min K, β }, sympoiclly /ε erm in g domines /ε erm nd wih bi more creful bounding c cn be mde lmos s smll s. By subsiuion of he lower bound on g ino Lemm we hve: E [N + ln + c ln + c ln e 4 s lnk K + ln K + O +, where we used Lemm o bound he sum of he exponens. Noe h is of order Õ K 4. Proof of Theorem. Noe h since by our definiion ˆ } he sequence ε = ε = min K, β, c ln sisfies he condiion of Lemm. Also noe h for lrge enough, so h 4c K ln 4 ln K, we hve ε = c ln. Le b = ln nd le be lrge enough, so h for ll we hve 4c K ln 4 ln K nd e. We re going o bound he hree erms in he bound on E [N in Lemm. Bounding s= e bs is esy. For bounding s= ε s E s } we noe h when E holds nd c 8 we hve: ˆ L min L L L g = b.5b 4 ε ε =.5 c ln c ln.5 c c,
7 One Prcicl Algorihm for Boh Sochsic nd Adversril Bndis where in 4 we used he fc h E holds nd in he ls line we used he fc h for we hve ln /. Thus ε E s } cln ˆ 4c ln nd s= ε s E s } = O ln. Finlly, for he ls erm in Lemm we hve lredy shown s n inermedie sep in he clculion of he bound on ˆ h for we hve g. Therefore, he ls erm K is of order O. By king ll hese clculions ogeher we obin he resul of he heorem. Noe h he resul holds for ny η β. The Conmined Sochsic Regime Proof of Theorem 5. The key elemen of he previous proof ws highprobbiliy lower bound on L L. We show h we cn obin similr lower bound in he conmined seing oo. Le, denoe he indicor funcion of conminion in locion,, kes vlue if conminion occurred nd oherwise. Le m =,l +, µ, in oher words, if eiher ws conmined on round hen m is he dversrilly ssigned vlue of he loss of rm on round nd oherwise i is he expeced loss. Le M = m s hen M M L L is mringle. By definiion of moderely conmined fer τ rounds process, for τ nd ny subopiml cion he ol number of rounds up o where eiher iself or were conmined is mos /. Therefore, M M / / /. Define even B : L L ν b +.5b, B ε where ε is defined in he proof of Theorem nd ν = ε. Then by Bernsein s inequliy P [ B s b. The reminder of he proof is idenicl o he proof of Theorem wih replced by /. The Adversril Regime wih Gp The proof of Theorem 6 is bsed on he following lemm, which is n nlogue of Theorems nd 5. Lemm. Under he prmerizion given in Theorem, he number of imes subopiml rm is plyed by EXP++ AVG in n dversril regime wih gp sisfies: E [N mx, τ, e / τ,} ln + O τ,. Proof. Agin, he only modificion we need is highprobbiliy lower bound on L L τ. We noe h λ λ τ L L τ is mringle nd h by definiion for τ we hve λ λ τ τ,. Define he evens W : τ, L L τ ν b +.5b, W ε where ε nd ν re s in he proof of Theorem 5. By Bernsein s inequliy P [ W b. The reminder of he proof is idenicl o he proof of Theorem. Proof of Theorem 6. Noe h by definiion τ, is nondecresing sequence of τ. Since Lemm is deerminisic resul i holds for ll τ simulneously nd we re free o choose he one h minimizes he bound. 5. Empiricl Evluion: Sochsic Regime We consider he sochsic mulirmed bndi problem wih Bernoulli rewrds. For ll he subopiml rms he rewrds re Bernoulli wih bis.5 nd for he single bes rm he rewrd is Bernoulli wih bis.5 +. We run he experimens wih K =, K =, nd K =, nd =. nd =. in ol, six combinions of K nd. We run ech gme for 7 rounds nd mke en repeiions of ech experimen. The solid lines in he grphs in Figure represen he men performnce over he experimens nd he dshed lines represen he men plus one sndrd deviion sd over he en repeiions of he corresponding experimen. In he experimens EXP++ is prmerized by ξ = ln ˆ ˆ, where ˆ is he empiricl esime of defined in. In order o demonsre h in he sochsic regime he explorion prmeers re in full conrol of he performnce we run he EXP++ lgorihm wih wo differen lerning res. EXP++ EMP corresponds o η = β nd EXP++ ACC corresponds o η =. Noe h only he EXP++ EMP hs performnce gurnee in he dversril regime. We compre EXP++ lgorihm wih he EXP lgorihm s described in Bubeck & CesBinchi, he UCB lgorihm of Auer e l., nd Thompson s smpling. Since i ws demonsred empiriclly in Seldin e l. h in he bove experimens he performnce of Thompson smpling is comprble or superior o he performnce of EwS nd KLUCB, he ler wo lgorihms re excluded from he comprison. For he EXP++ nd he EXP lgorihms we rnsform he rewrds ino losses vi l = r rnsformion, oher lgorihms opere direcly on he rewrds.
8 One Prcicl Algorihm for Boh Sochsic nd Adversril Bndis 7 K =. =. 5 K =. =..5 x 4 K =. =. Cumulive Regre Cumulive Regre 4 Cumulive Regre x 6 K =, = x 6 b K =, = x 6 c K =, =. Cumulive Regre K =. =. Cumulive Regre 4 x K =. =. Cumulive Regre x UCB Thom EXP EXP++ EMP EXP++ ACC K =. = x 6 d K =, = x 6 e K =, = x 6 f K =, =. Figure. Comprison of UCB, Thompson smpling Thom, EXP, nd EXP++ lgorihms in he sochsic regime. The legend in figure f corresponds o ll he figures. EXP++ EMP is he Empiricl EXP++ lgorihm nd EXP++ ACC is n Accelered Empiricl EXP++, where we ke η =. Solid lines correspond o mens over repeiions of he corresponding experimens nd dshed lines correspond o he mens plus one sndrd deviion. The resuls re presened in Figure. We see h in ll he experimens he performnce of EXP++ EMP is lmos idenicl o he performnce of UCB. However, unlike UCB nd Thompson s smpling, EXP++ EMP is secured gins he possibiliy h he gme is conrolled by n dversry. In he supplemenry meril we show h ny deerminisic lgorihm is vulnerble gins n dversry. The EXP++ ACC lgorihm cn be seen s eser for fuure work. I performs beer hn EXP++ EMP, bu i does no hve he dversril regime performnce gurnee. However, we do no exclude he possibiliy h by some more sophisiced simulneous conrol of η nd ε s i my be possible o design n lgorihm h will hve boh beer performnce in he sochsic regime nd regre gurnee in he dversril regime. An exmple of such sophisiced conrol of he lerning re in he full informion gmes cn be found in de Rooij e l Discussion We presened generlizion of he EXP lgorihm, he EXP++ lgorihm, which ugmens he EXP lgorihm wih new conrol lever in he form explorion prmeers ε h re uned individully for ech rm. We hve shown h he new conrol lever is exremely useful in deecing nd exploiing he gp in wide rnge of regimes, while he old conrol lever lwys keeps he worscse performnce of he lgorihm under conrol. Due o he cenrl role of he EXP lgorihm in he dversril nlysis h sreches fr beyond he dversril bndis nd due o he simpliciy of our generlizion we believe h our resul will led o muliude of new lgorihms for oher problems h exploi he gps wihou compromising on he worscse performnce gurnees. There is lso room for furher improvemen of he presened echnique h we pln o pursue in fuure work. Acknowledgmens The uhors would like o hnk Sébsien Bubeck nd Wouer Koolen for useful discussions nd Csb Szepesvári for bringing up he reference o CesBinchi & Fischer 998. This reserch ws suppored by n Ausrlin Reserch Council Ausrlin Luree Fellowship FL8.
9 One Prcicl Algorihm for Boh Sochsic nd Adversril Bndis References Agrwl, Shipr nd Goyl, Nvin. Furher opiml regre bounds for Thompson smpling. In AISTATS,. Audiber, JenYves nd Bubeck, Sébsien. Minimx policies for dversril nd sochsic bndis. In Proceedings of he Inernionl Conference on Compuionl Lerning Theory COLT, 9. Auer, Peer, CesBinchi, Nicolò, Freund, Yov, nd Schpire, Rober E. Gmbling in rigged csino: The dversril mulirmed bndi problem. In Proceedings of he Annul Symposium on Foundions of Compuer Science, 995. Seldin, Yevgeny, Szepesvári, Csb, Auer, Peer, nd Abbsi Ydkori, Ysin. Evluion nd nlysis of he performnce of he EXP lgorihm in sochsic environmens. In JMLR Workshop nd Conference Proceedings, volume 4 EWRL,. Solz, Gilles. Incomplee Informion nd Inernl Regre in Predicion of Individul Sequences. PhD hesis, Universié Pris Sud, 5. Thompson, Willim R. On he likelihood h one unknown probbiliy exceeds noher in view of he evidence of wo smples. Biomerik, 5, 9. Auer, Peer, CesBinchi, Nicolò, nd Fischer, Pul. Finieime nlysis of he mulirmed bndi problem. Mchine Lerning, 47,. Auer, Peer, CesBinchi, Nicolò, Freund, Yov, nd Schpire, Rober E. The nonsochsic mulirmed bndi problem. SIAM Journl of Compuing,, b. Boucheron, Séphne, Lugosi, Gábor, nd Mssr, Pscl. Concenrion Inequliies A Nonsympoic Theory of Independence. Oxford Universiy Press,. Bubeck, Sébsien. Bndis Gmes nd Clusering Foundions. PhD hesis, Universié Lille,. Bubeck, Sébsien nd CesBinchi, Nicolò. Regre nlysis of sochsic nd nonsochsic mulirmed bndi problems. Foundions nd Trends in Mchine Lerning, 5,. Bubeck, Sébsien nd Slivkins, Aleksndrs. The bes of boh worlds: sochsic nd dversril bndis. In Proceedings of he Inernionl Conference on Compuionl Lerning Theory COLT,. Cppé, Olivier, Grivier, Aurélien, Millrd, OdlricAmbrym, Munos, Rémi, nd Solz, Gilles. KullbckLeibler upper confidence bounds for opiml sequenil llocion. Annls of Sisics, 4,. CesBinchi, Nicolò nd Fischer, Pul. Finieime regre bounds for he mulirmed bndi problem. In Proceedings of he Inernionl Conference on Mchine Lerning ICML, 998. CesBinchi, Nicolò nd Lugosi, Gábor. Predicion, Lerning, nd Gmes. Cmbridge Universiy Press, 6. de Rooij, Seven, vn Erven, Tim, Grünwld, Peer D., nd Koolen, Wouer M. Follow he leder if you cn, hedge if you mus. Journl of Mchine Lerning Reserch, 4. Kufmnn, Emilie, Kord, Nhniel, nd Munos, Rémi. Thompson smpling: An opiml finie ime nlysis. In Proceedings of he Inernionl Conference on Algorihmic Lerning Theory ALT,. Li, Tze Leung nd Robbins, Herber. Asympoiclly efficien dpive llocion rules. Advnces in Applied Mhemics, 6, 985. Millrd, OdlricAmbrym. Apprenissge Séqueniel: Bndis, Sisique e Renforcemen. PhD hesis, INRIA Lille,. Robbins, Herber. Some specs of he sequenil design of experimens. Bullein of he Americn Mhemicl Sociey, 95.
Improper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b].
Improper Inegrls Dr. Philippe B. lvl Kennesw Se Universiy Sepember 9, 25 Absrc Noes on improper inegrls. Improper Inegrls. Inroducion In Clculus II, sudens defined he inegrl f (x) over finie inervl [,
More informationSolution of a Class of Riccati Equations
Proceedings of he 8h WSEAS Inernionl Conference on APPLIED MATHEMATICS, Tenerife, Spin, December 68, 005 (pp334338) Soluion of Clss of Ricci Equions K BUSAWON, P JOHNSON School of Compuing, Engineering
More informationDetecting Network Intrusions via Sampling : A Game Theoretic Approach
Deecing Nework Inrusions vi Smpling : A Gme Theoreic Approch Murli Kodilm T. V. Lkshmn Bell Lborories Lucen Technologies 101 Crwfords Corner Rod Holmdel, NJ 07733, USA {murlik, lkshmn}@belllbs.com Absrc
More informationOptimal Contracts in a ContinuousTime Delegated Portfolio Management Problem
Opiml Conrcs in Coninuousime Deleged Porfolio Mngemen Problem Hui OuYng Duke Universiy nd Universiy of Norh Crolin his ricle sudies he conrcing problem beween n individul invesor nd professionl porfolio
More informationDynamic Magnification Factor of SDOF Oscillators under. Harmonic Loading
Dynmic Mgnificion Fcor of SDOF Oscillors under Hrmonic Loding Luis Mrí GilMrín, Jun Frncisco CronellMárquez, Enrique HernándezMones 3, Mrk Aschheim 4 nd M. PsdsFernández 5 Asrc The mgnificion fcor
More informationTermbased composition of security protocols
Termsed composiion of securiy proocols B Genge P Hller R Ovidiu I Ign Peru ior Universiy of Trgu ures Romni genge@upmro phller@upmro oroi@engineeringupmro Technicl Universiy of Cluj poc Romni IosifIgn@csuclujro
More informationExample What is the minimum bandwidth for transmitting data at a rate of 33.6 kbps without ISI?
Emple Wh is he minimum ndwidh for rnsmiing d re of 33.6 kps wihou ISI? Answer: he minimum ndwidh is equl o he yquis ndwidh. herefore, BW min W R / 33.6/ 6.8 khz oe: If % rolloff chrcerisic is used, ndwidh
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationPhysics 1402: Lecture 21 Today s Agenda
ecure 4 Physics 142: ecure 21 Tody s Agend Announcemens: nducion, R circuis Homework 6: due nex Mondy nducion / A curren Frdy's w ds N S B v B B S N B v 1 ecure 4 nducion Selfnducnce, R ircuis X X X X
More informationA MODEL OF FIRM BEHAVIOUR WITH EQUITY CONSTRAINTS AND BANKRUPTCY COSTS.
WORKING PAPERS Invesigção  Trblhos em curso  nº 134, Novembro de 003 A Model of Firm Behviour wih Euiy Consrins nd Bnrupcy Coss Pedro Mzed Gil FACULDADE DE ECONOMIA UNIVERSIDADE DO PORTO www.fep.up.p
More informationIn the last decades, due to the increasing progress in genome
Se Esimion for Geneic Regulory Neworks wih TimeVrying Delys nd RecionDiffusion Terms Yunyun Hn, Xin Zhng, Member, IEEE, Ligng Wu, Senior Member, IEEE nd Yno Wng rxiv:59.888v [mh.oc] Sep 5 Absrc This
More informationInformation Technology Investment and Adoption: A Rational Expectations Perspective
Informion Technology Invesmen nd Adopion: A Rionl Expecions Perspecive Yoris A. Au Rober J. Kuffmn Docorl Progrm, Informion nd Decision CoDirecor, MIS Reserch Cener nd Sciences, Crlson School of Mngemen,
More informationA Dynamic Model of Health Insurance Choices and Health Care Consumption 1. Jian Ni Johns Hopkins University Email: jni@jhu.edu
A Dynmic Model of Helh Insurnce Choices nd Helh Cre Consumpion Jin Ni Johns Hopins Universiy Emil: jni@jhu.edu Niin Meh Universiy of Torono Emil: nmeh@romn.uorono.c Knnn Srinivsn Crnegie Mellon Universiy
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationReuseBased Test Traceability: Automatic Linking of Test Cases and Requirements
Inernionl Journl on Advnces in Sofwre, vol 7 no 3&4, yer 2014, hp://www.irijournls.org/sofwre/ ReuseBsed Tes Trcebiliy: Auomic Linking of Tes Cses nd Requiremens 469 Thoms Nock, Thoms Krbe Technische
More informationInfluence of Network Load on the Performance of Opportunistic Scanning
Influence of Nework Lod on he Performnce of Opporunisic Scnning Mrc Emmelmnn, Sven Wiehöler, nd HyungTek Lim Technicl Universiy Berlin Telecommunicion Neworks Group TKN Berlin, Germny Emil: emmelmnn@ieee.org,
More informationPhys222 W12 Quiz 2: Chapters 23, 24. Name: = 80 nc, and q = 30 nc in the figure, what is the magnitude of the total electric force on q?
Nme: 1. A pricle (m = 5 g, = 5. µc) is relesed from res when i is 5 cm from second pricle (Q = µc). Deermine he mgniude of he iniil ccelerion of he 5g pricle.. 54 m/s b. 9 m/s c. 7 m/s d. 65 m/s e. 36
More informationE0 370 Statistical Learning Theory Lecture 20 (Nov 17, 2011)
E0 370 Saisical Learning Theory Lecure 0 (ov 7, 0 Online Learning from Expers: Weighed Majoriy and Hedge Lecurer: Shivani Agarwal Scribe: Saradha R Inroducion In his lecure, we will look a he problem of
More informationINTERFEROMETRIC TECHNIQUES FOR TERRASARX DATA. Holger Nies, Otmar Loffeld, Baki Dönmez, Amina Ben Hammadi, Robert Wang, Ulrich Gebhardt
INTERFEROMETRIC TECHNIQUES FOR TERRASARX DATA Holger Nies, Omr Loffeld, Bki Dönmez, Amin Ben Hmmdi, Rober Wng, Ulrich Gebhrd Cener for Sensorsysems (ZESS), Universiy of Siegen PulBonzSr. 9, D5768
More informationAge Biased Technical and Organisational Change, Training and Employment Prospects of Older Workers
DISCUSSION PAPER SERIES IZA DP No. 5544 Age Bised Technicl nd Orgnisionl Chnge, Trining nd Employmen Prospecs of Older Workers Luc Behghel Eve Croli Muriel Roger Mrch 2011 Forschungsinsiu zur Zukunf der
More informationHuman Body Tracking with Auxiliary Measurements
IEEE Inernionl Workshop on Anlysis nd Modeling of Fces nd Gesures, 003. Humn Body Trcking wih Auxiliry Mesuremens Mun Wi Lee, Isc Cohen Insiue for Roboics nd Inelligen Sysems Inegred Medi Sysems Cener
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationMost contracts, whether between voters and politicians or between house owners and contractors, are
Americn Poliicl Science Review Vol. 95, No. 1 Mrch 2001 More Order wih Less Lw: On Conrc Enforcemen, Trus, nd Crowding IRIS BOHNET Hrvrd Universiy BRUNO S. FREY Universiy of Zürich STEFFEN HUCK Universiy
More information2. The econometric model
Age Bised Technicl nd Orgnisionl Chnge, Trining nd Employmen Prospecs of Older Workers * Luc BEHAGHEL (Pris School of Economics (INRA) nd CREST) Eve CAROLI (Universiy Pris Duphine, LEDLEGOS, Pris School
More informationCULTURAL TRANSMISSION AND THE EVOLUTION OF TRUST AND RECIPROCITY IN THE LABOUR MARKET. Gonzalo Olcina and Vicente Calabuig
Preliminr version o be published s Working Pper b BBVA Foundion CULTURAL TRANSMISSION AND THE EVOLUTION OF TRUST AND RECIPROCITY IN THE LABOUR MARKET Gonzlo Olcin nd Vicene Clbuig Universi of Vlenci nd
More informationMULTIPLE LIFE INSURANCE PENSION CALCULATION *
ULIPLE LIFE INSURANCE PENSION CALCULAION * SANISŁA HEILPERN Universi of Economics Dermen of Sisics Komndors 82 54345 rocł Polnd emil: snislheilern@uerocl Absrc he conribuion is devoed o he deenden mulile
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationMr. Kepple. Motion at Constant Acceleration 1D Kinematics HW#5. Name: Date: Period: (b) Distance traveled. (a) Acceleration.
Moion Consn Accelerion 1D Kinemics HW#5 Mr. Kepple Nme: De: Period: 1. A cr cceleres from 1 m/s o 1 m/s in 6.0 s. () Wh ws is ccelerion? (b) How fr did i rel in his ime? Assume consn ccelerion. () Accelerion
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationUSE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES
USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationAn Online Learningbased Framework for Tracking
An Online Learningbased Framework for Tracking Kamalika Chaudhuri Compuer Science and Engineering Universiy of California, San Diego La Jolla, CA 9293 Yoav Freund Compuer Science and Engineering Universiy
More informationBusato, Francesco; Chiarini, Bruno; Marzano, Elisabetta. Working Paper Moonlighting production, tax rates and capital subsidies
econsor www.econsor.eu Der OpenAccessPublikionsserver der ZBW LeibnizInformionszenrum Wirschf The Open Access Publicion Server of he ZBW Leibniz Informion Cenre for Economics Buso, Frncesco; Chirini,
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationTSGRAN Working Group 1 (Radio Layer 1) meeting #3 Nynashamn, Sweden 22 nd 26 th March 1999
TSGRAN Working Group 1 (Radio Layer 1) meeing #3 Nynashamn, Sweden 22 nd 26 h March 1999 RAN TSGW1#3(99)196 Agenda Iem: 9.1 Source: Tile: Documen for: Moorola Macrodiversiy for he PRACH Discussion/Decision
More informationHORIZONTAL POSITION OPTIMAL SOLUTION DETERMINATION FOR THE SATELLITE LASER RANGING SLOPE MODEL
HOIZONAL POSIION OPIMAL SOLUION DEEMINAION FO HE SAELLIE LASE ANGING SLOPE MODEL Yu Wng,* Yu Ai b Yu Hu b enli Wng b Xi n Surveying nd Mpping Insiue, No. 1 Middle Yn od, Xi n, Chin, 710054640677@qq.com
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationResource allocation in multiserver dynamic PERT networks using multiobjective programming and Markov process. Email: bagherpour@iust.ac.
IJST () A: 7 Irnin Journl of Science & Technology hp://www.shirzu.c.ir/en Resource llocion in uliserver dynic PERT neworks using uliobjecive progring nd Mrkov process S. Yghoubi, S. Noori nd M. Bgherpour
More informationACCOUNTING, ECONOMICS AND FINANCE. School Working Papers Series 2004 SWP 2004/08
FACULTY OF BUSINESS AND LAW School of ACCOUNTING, ECONOMICS AND FINANCE School Workin Ppers Series 4 SWP 4/8 STRUCTURAL EFFECTS AND SPILLOVERS IN HSIF, HSI AND S&P5 VOLATILITY Gerrd Gnnon* Deprmen of Accounin,
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationThis work is licensed under a Licença Creative Commons Attribution 3.0.
3.0. Ese rblho esá licencido sob um Licenç Creive Commons Aribuion This work is licensed under Licenç Creive Commons Aribuion 3.0. Fone: hp:///rigos.sp?sesso=redy&cod_rigo=255. Acesso em: 11 nov. 2013.
More informationEfficient Onetime Signature Schemes for Stream Authentication *
JOURNAL OF INFORMATION SCIENCE AND ENGINEERING, 61164 (006) Efficien Oneime Signaure Schemes for Sream Auhenicaion * YONGSU PARK AND YOOKUN CHO + College of Informaion and Communicaions Hanyang Universiy
More informationIdentifying Merger Unilateral Effects: HHI or Simulation?
Idenifying Merger Unilerl Effecs: HHI or Simulion? Jerome FONCEL Universiy of Lille, Frnce erome.foncel@univlille3.fr Mrc IVALDI Toulouse School of Economics, nd CEPR, Frnce ivldi@cic.fr Jrissy MOTIS
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationSTRATEGIC PLANNING COMMITTEE Wednesday, February 17, 2010
em: STATEGC PLANNNG COMMTTEE Wednesdy, Februry 17, 2010 SUBJECT: EQUEST FO APPOVAL TO NAME THE WALKWAY FOM DADE AVENUE TO PAKNG GAAGE 2 DVESTY WAY ON THE BOCA ATON CAMPUS. POPOSED COMMTTEE ACTON Provide
More informationRESTORING FISCAL SUSTAINABILITY IN THE EURO AREA: RAISE TAXES OR CURB SPENDING? Boris Cournède and Frédéric Gonand *
RESTORING FISCAL SUSTAINABILITY IN THE EURO AREA: RAISE TAXES OR CURB SPENDING? Boris Cournède nd Frédéric Gonnd Wih populion geing fiscl consolidion hs become of prmoun impornce for euro re counries.
More informationForecasting and Information Sharing in Supply Chains Under QuasiARMA Demand
Forecasing and Informaion Sharing in Supply Chains Under QuasiARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in
More informationTime Series Analysis Using SAS R Part I The Augmented DickeyFuller (ADF) Test
ABSTRACT Time Series Analysis Using SAS R Par I The Augmened DickeyFuller (ADF) Tes By Ismail E. Mohamed The purpose of his series of aricles is o discuss SAS programming echniques specifically designed
More informationERSİTES ABSTRACTT. The. optimal. bulunmasına yönelik. Warsaw Tel: 482 256 486 17,
ANADOLU ÜNİVE ERSİTES Bilim ve Tenoloji Dergisi BTeori Bilimler Cil: 2 Syı: 2 203 Syf:2742 ARAŞTIRMAA MAKALESİ / RESEARCH ARTICLE Ann DECEWİCZ MARKOV MODELS IN CALCULATING ABSTRACTT The er resens mehod
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationPermutations and Combinations
Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes  ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he kvalue for he middle erm, divide
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationUNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE
UNVERSTY OF NOTTNGHAM Discussion Ppers in Economics Discussion Pper No. 04/15 STRATEGC SECOND SOURCNG N A VERTCAL STRUCTURE By Arijit Mukherjee September 004 DP 04/15 SSN 10438 UNVERSTY OF NOTTNGHAM Discussion
More informationNetwork Effects, Pricing Strategies, and Optimal Upgrade Time in Software Provision.
Nework Effecs, Pricing Sraegies, and Opimal Upgrade Time in Sofware Provision. YiNung Yang* Deparmen of Economics Uah Sae Universiy Logan, UT 84322353 April 3, 995 (curren version Feb, 996) JEL codes:
More informationA Multiagent Trading Platform for Electricity Contract Market
1 A Muligen Trding Plform for Elecriciy Conrc Mrke Yun Jihi, Yu Shunkun nd Hu Zhogung Absrc An genbsed negoiion plform for power genering nd power consuming (purchsing) compnies in conrc elecriciy
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationDDoS Attacks Detection Model and its Application
DDoS Aacks Deecion Model and is Applicaion 1, MUHAI LI, 1 MING LI, XIUYING JIANG 1 School of Informaion Science & Technology Eas China Normal Universiy No. 500, DongChuan Road, Shanghai 0041, PR. China
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationA NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
More informationGoRA. For more information on genetics and on Rheumatoid Arthritis: Genetics of Rheumatoid Arthritis. Published work referred to in the results:
For more informaion on geneics and on Rheumaoid Arhriis: Published work referred o in he resuls: The geneics revoluion and he assaul on rheumaoid arhriis. A review by Michael Seldin, Crisopher Amos, Ryk
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationMultiprocessor SystemsonChips
Par of: Muliprocessor SysemsonChips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationHow to calculate effect sizes from published research: A simplified methodology
WORKLEARNING RESEARCH How o alulae effe sizes from published researh: A simplified mehodology Will Thalheimer Samanha Cook A Publiaion Copyrigh 2002 by Will Thalheimer All righs are reserved wih one exepion.
More informationR&D Costs and Accounting Profits
R&D Coss nd Accouning Profis by Doron Nissim nd Jcob homs Columbi Business School New York, NY 10027 April 27, 2000 R&D Coss nd Accouning Profis Absrc Opponens of SFAS2, which required he immedie expensing
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More information4.2 Trigonometric Functions; The Unit Circle
4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationThe Kinetics of the Stock Markets
Asia Pacific Managemen Review (00) 7(1), 14 The Kineics of he Sock Markes Hsinan Hsu * and BinJuin Lin ** (received July 001; revision received Ocober 001;acceped November 001) This paper applies he
More informationKeldysh Formalism: Nonequilibrium Green s Function
Keldysh Formalism: Nonequilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Nonequilibrium
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationSampling TimeBased Sliding Windows in Bounded Space
Sampling TimeBased Sliding Windows in Bounded Space Rainer Gemulla Technische Universiä Dresden 01062 Dresden, Germany gemulla@inf.udresden.de Wolfgang Lehner Technische Universiä Dresden 01062 Dresden,
More informationOnline Learning with Sample Path Constraints
Journal of Machine Learning Research 0 (2009) 569590 Submied 7/08; Revised /09; Published 3/09 Online Learning wih Sample Pah Consrains Shie Mannor Deparmen of Elecrical and Compuer Engineering McGill
More informationTowards IncentiveCompatible Reputation Management
Towards InceniveCompaible Repuaion Managemen Radu Jurca, Boi Falings Arificial Inelligence Laboraory Swiss Federal Insiue of Technology (EPFL) INEcublens, 115 Lausanne, Swizerland radu.jurca@epfl.ch,
More information