Follow the Leader If You Can, Hedge If You Must


 Brent Clifton Fleming
 3 years ago
 Views:
Transcription
1 Journal of Machine Learning Research 15 (2014) Submied 1/13; Revised 1/14; Published 4/14 Follow he Leader If You Can, Hedge If You Mus Seven de Rooij VU Universiy and Universiy of Amserdam Science Park 904, P.O. Box 94323, 1090 GH Amserdam, he Neherlands Tim van Erven Déparemen de Mahémaiques Universié ParisSud, Orsay Cedex, France Peer D. Grünwald Wouer M. Koolen Leiden Universiy (Grünwald) and Cenrum Wiskunde & Informaica (Grünwald and Koolen) Science Park 123, P.O. Box 94079, 1090 GB Amserdam, he Neherlands Edior: Nicolò CesaBianchi Absrac FollowheLeader (FTL) is an inuiive sequenial predicion sraegy ha guaranees consan regre in he sochasic seing, bu has poor performance for worscase daa. Oher hedging sraegies have beer worscase guaranees bu may perform much worse han FTL if he daa are no maximally adversarial. We inroduce he FlipFlop algorihm, which is he firs mehod ha provably combines he bes of boh worlds. As a sepping sone for our analysis, we develop AdaHedge, which is a new way of dynamically uning he learning rae in Hedge wihou using he doubling rick. AdaHedge refines a mehod by CesaBianchi, Mansour, and Solz (2007), yielding improved worscase guaranees. By inerleaving AdaHedge and FTL, FlipFlop achieves regre wihin a consan facor of he FTL regre, wihou sacrificing AdaHedge s worscase guaranees. AdaHedge and FlipFlop do no need o know he range of he losses in advance; moreover, unlike earlier mehods, boh have he inuiive propery ha he issued weighs are invarian under rescaling and ranslaion of he losses. The losses are also allowed o be negaive, in which case hey may be inerpreed as gains. Keywords: advice Hedge, learning rae, mixabiliy, online learning, predicion wih exper 1. Inroducion We consider sequenial predicion in he general framework of Decision Theoreic Online Learning (DTOL) or he Hedge seing (Freund and Schapire, 1997), which is a varian of predicion wih exper advice (Lilesone and Warmuh, 1994; Vovk, 1998; CesaBianchi and Lugosi, 2006). Our goal is o develop a sequenial predicion algorihm ha performs well no only on adversarial daa, which is he scenario mos sudies worry abou, bu also when he daa are easy, as is ofen he case in pracice. Specifically, wih adversarial daa, he worscase regre (defined below) for any algorihm is Ω( T ), where T is he number of predicions o be made. Algorihms such as Hedge, which have been designed o achieve his lower bound, ypically coninue o suffer regre of order T, even for easy daa, where c 2014 Seven de Rooij, Tim van Erven, Peer D. Grünwald and Wouer M. Koolen.
2 De Rooij, Van Erven, Grünwald and Koolen he regre of he more inuiive bu less robus FollowheLeader (FTL) algorihm (also defined below) is bounded. Here, we presen he firs algorihm which, up o consan facors, provably achieves boh he regre lower bound in he wors case, and a regre no exceeding ha of FTL. Below, we firs describe he Hedge seing. Then we inroduce FTL, discuss sophisicaed versions of Hedge from he lieraure, and give an overview of he resuls and conens of his paper. 1.1 Overview In he Hedge seing, predicion proceeds in rounds. A he sar of each round = 1, 2,..., a learner has o decide on a weigh vecor w = (w,1,..., w,k ) R K over K expers. Each weigh w,k is required o be nonnegaive, and he sum of he weighs should be 1. Naure hen reveals a Kdimensional vecor conaining he losses of he expers l = (l,1,..., l,k ) R K. Learner s loss is he do produc h = w l, which can be inerpreed as he expeced loss if Learner uses a mixed sraegy and chooses exper k wih probabiliy w,k. We denoe aggregaes of perrial quaniies by heir capial leer, and vecors are in bold face. Thus, L,k = l 1,k l,k denoes he cumulaive loss of exper k afer rounds, and H = h h is Learner s cumulaive loss (he Hedge loss). Learner s performance is evaluaed in erms of her regre, which is he difference beween her cumulaive loss and he cumulaive loss of he bes exper: R = H L, where L = min k L,k. We will always analyse he regre afer an arbirary number of rounds T. We will omi he subscrip T for aggregae quaniies such as L T or R T wherever his does no cause confusion. A simple and inuiive sraegy for he Hedge seing is FollowheLeader (FTL), which pus all weigh on he exper(s) wih he smalles loss so far. More precisely, we will define he weighs w for FTL o be uniform on he se of leaders {k L 1,k = L 1 }, which is ofen jus a singleon. FTL works very well in many circumsances, for example in sochasic scenarios where he losses are independen and idenically disribued (i.i.d.). In paricular, he regre for FollowheLeader is bounded by he number of imes he leader is overaken by anoher exper (Lemma 10), which in he i.i.d. case almos surely happens only a finie number of imes (by he uniform law of large numbers), provided he mean loss of he bes exper is sricly smaller han he mean loss of he oher expers. As demonsraed by he experimens in Secion 5, many more sophisicaed algorihms can perform significanly worse han FTL. The problem wih FTL is ha i breaks down badly when he daa are anagonisic. For example, if one ou of wo expers incurs losses 1 2, 0, 1, 0,... while he oher incurs opposie losses 0, 1, 0, 1,..., he regre for FTL a ime T is abou T/2 (his scenario is furher discussed in Secion 5.1). This has promped he developmen of a muliude of alernaive algorihms ha provide beer worscase regre guaranees. The seminal sraegy for he learner is called Hedge (Freund and Schapire, 1997, 1999). Is performance crucially depends on a parameer η called he learning rae. Hedge can be inerpreed as a generalisaion of FTL, which is recovered in he limi for η. In many analyses, he learning rae is changed from infiniy o a lower value ha opimizes 1282
3 Follow he Leader If You Can, Hedge If You Mus some upper bound on he regre. Doing so requires precogniion of he number of rounds of he game, or of some propery of he daa such as he evenual loss of he bes exper L. Provided ha he relevan saisic is monoonically nondecreasing in (such as L ), a simple way o address his issue is he socalled doubling rick: seing a budge on he saisic, and resaring he algorihm wih a double budge when he budge is depleed (CesaBianchi and Lugosi, 2006; CesaBianchi e al., 1997; Hazan and Kale, 2008); η can hen be opimised for each individual block in erms of he budge. Beer bounds, bu harder analyses, are ypically obained if he learning rae is adjused each round based on previous observaions, see e.g. (CesaBianchi and Lugosi, 2006; Auer e al., 2002). The Hedge sraegy presened by CesaBianchi, Mansour, and Solz (2007) is a sophisicaed example of such adapive uning. The relevan algorihm, which we refer o as CBMS, is defined in (16) in Secion 4.2 of heir paper. To discuss is guaranees, we need he following noaion. Le l = min k l,k and l + = max k l,k denoe he smalles and larges loss in round, and le L = l l and L + = l l+ denoe he cumulaive minimum and maximum loss respecively. Furher le s = l + l denoe he loss range in rial and le S = max{s 1,..., s } denoe he larges loss range afer rials. Then, wihou prior knowledge of any propery of he daa, including T, S and L, he CBMS sraegy achieves regre bounded by 1 R CBMS 4 (L L )(L + ST L ) T ln K + lower order erms (1) (CesaBianchi e al., 2007, Corollary 3). Hence, in he wors case L = L + ST/2 and he bound is of order S T, bu when he loss of he bes exper L [L, L + ST ] is close o eiher boundary he guaranees are much sronger. The conribuions of his work are wofold: firs, in Secion 2, we develop AdaHedge, which is a refinemen of he CBMS sraegy. A (very) preliminary version of his sraegy was presened a NIPS (Van Erven e al., 2011). Like CMBS, AdaHedge is compleely parameerless and unes he learning rae in erms of a direc measure of pas performance. We derive an improved worscase bound of he following form. Again wihou any assumpions, we have R ah 2 S (L L )(L + L ) L + L ln K + lower order erms (2) (see Theorem 8). The parabola under he square roo is always smaller han or equal o is CMBS counerpar (since i is nondecreasing in L + and L + L +ST ); i expresses ha he regre is small if L [L, L + ] is close o eiher boundary. I is maximized in L a he midpoin beween L and L +, and in his case we recover he worscase bound of order S T. Like (1), he regre bound (2) is fundamenal, which means ha i is invarian under ranslaion of he losses and proporional o heir scale. Moreover, no only AdaHedge s regre bound is fundamenal: he weighs issued by he algorihm are hemselves invarian 1. As poined ou by a referee, i is widely known ha he leading consan of 4 can be improved o using echniques by Györfi and Oucsák (2007) ha are essenially equivalen o our Lemma 2 below; Gerchinoviz (2011, Remark 2.2) reduced i o approximaely AdaHedge allows a sligh furher reducion o
4 De Rooij, Van Erven, Grünwald and Koolen under ranslaion and scaling (see Secion 4). The CBMS algorihm and AdaHedge are insensiive o rials in which all expers suffer he same loss, a naural propery we call imelessness. An aracive feaure of he new bound (2) is ha i expresses his propery. A more deailed discussion appears below Theorem 8. Our second conribuion is o develop a second algorihm, called FlipFlop, ha reains he worscase bound (2) (up o a consan facor), bu has even beer guaranees for easy daa: is performance is never subsanially worse han ha of FollowheLeader. A firs glance, his may seem rivial o accomplish: simply ake boh FTL and AdaHedge, and combine he wo by using FTL or Hedge recursively. To see why such approaches do no work, suppose ha FTL achieves regre R fl, while AdaHedge achieves regre R ah. We would only be able o prove ha he regre of he combined sraegy compared o he bes original exper saisfies R c min{r fl, R ah } + G c, where G c is he worscase regre guaranee for he combinaion mehod, e.g. (1). In general, eiher R fl or R ah may be close o zero, while a he same ime he regre of he combinaion mehod, or a leas is bound G c, is proporional o T. Tha is, he overhead of he combinaion mehod will dominae he regre! The FlipFlop approach we describe in Secion 3 circumvens his by alernaing beween Following he Leader and using AdaHedge in a carefully specified way. For his sraegy we can guaranee R ff = O(min{R fl, G ah }), where G ah is he regre guaranee for AdaHedge; Theorem 15 provides a precise saemen. Thus, FlipFlop is he firs algorihm ha provably combines he benefis of Followhe Leader wih robus behaviour for anagonisic daa. A key concep in he design and analysis of our algorihms is wha we call he mixabiliy gap, inroduced in Secion 2.1. This quaniy also appears in earlier works, and seems o be of fundamenal imporance in boh he curren Hedge seing as well as in sochasic seings. We elaborae on his in Secion 6.2 where we provide he big picure underlying his research and we briefly indicae how i relaes o pracical work such as (Devaine e al., 2013). 1.2 Relaed Work As menioned, AdaHedge is a refinemen of he sraegy analysed by CesaBianchi e al. (2007), which is iself more sophisicaed han mos earlier approaches, wih wo noable excepions. Firs, Chaudhuri, Freund, and Hsu (2009) describe a sraegy called NormalHedge ha can efficienly compee wih he bes ɛquanile of expers; heir bound is incomparable wih he bounds for CBMS and for AdaHedge. Second, Hazan and Kale (2008) develop a sraegy called Variaion MW ha has especially low regre when he losses of he bes exper vary lile beween rounds. They show ha he regre of Variaion MW is of order VAR max T ln K, where VAR max T = max T s=1 ( ls,k 1 L,k ) 2 wih k he bes exper afer rounds. This bound dominaes our worscase resul (2) (up o a muliplicaive consan). As demonsraed by he experimens in Secion 5, heir mehod does no achieve he benefis of FTL, however. In Secion 5 we also discuss he performance of NormalHedge and Variaion MW compared o AdaHedge and FlipFlop. 1284
5 Follow he Leader If You Can, Hedge If You Mus Oher approaches o sequenial predicion include Defensive Forecasing (Vovk e al., 2005), and Following he Perurbed Leader (Kalai and Vempala, 2003). These radically differen approaches also allow compeing wih he bes ɛquanile, as shown by Chernov and Vovk (2010) and Huer and Poland (2005); he laer also consider nonuniform weighs on he expers. The safe MDL and safe Bayesian algorihms by Grünwald (2011, 2012) share he presen work s focus on he mixabiliy gap as a crucial par of he analysis, bu are concerned wih he sochasic seing where losses are no adversarial bu i.i.d. FlipFlop, safe MDL and safe Bayes can all be inerpreed as mehods ha aemp o choose a learning rae η ha keeps he mixabiliy gap small (or, equivalenly, ha keeps he Bayesian poserior or Hedge weighs concenraed ). 1.3 Ouline In he nex secion we presen and analyse AdaHedge and compare is worscase regre bound o exising resuls, in paricular he bound for CBMS. Then, in Secion 3, we build on AdaHedge o develop he FlipFlop sraegy. The analysis closely parallels ha of AdaHedge, bu wih exra complicaions a each of he seps. In Secion 4 we show ha boh algorihms have he propery ha heir behaviour does no change under ranslaion and scaling of he losses. We furher illusrae he relaionship beween he learning rae and he regre, and compare AdaHedge and FlipFlop o exising mehods, in experimens wih arificial daa in Secion 5. Finally, Secion 6 conains a discussion, wih ambiious suggesions for fuure work. 2. AdaHedge In his secion, we presen and analyse he AdaHedge sraegy. To inroduce our noaion and proof sraegy, we sar wih he simples possible analysis of vanilla Hedge, and hen move on o refine i for AdaHedge. 2.1 Basic Hedge Analysis for Consan Learning Rae Following Freund and Schapire (1997), we define he Hedge or exponenial weighs sraegy as he choice of weighs w,k = w 1,ke ηl 1,k Z, (3) where w 1 = (1/K,..., 1/K) is he uniform disribuion, Z = w 1 e ηl 1 is a normalizing consan, and η (0, ) is a parameer of he algorihm called he learning rae. If η = 1 and one imagines L 1,k o be he negaive loglikelihood of a sequence of observaions, hen w,k is he Bayesian poserior probabiliy of exper k and Z is he marginal likelihood of he observaions. Like in Bayesian inference, he weighs are updaed muliplicaively, i.e. w +1,k w,k e ηl,k. The loss incurred by Hedge in round is h = w l, he cumulaive Hedge loss is H = h h, and our goal is o obain a good bound on H T. To his end, i urns 1285
6 De Rooij, Van Erven, Grünwald and Koolen ou o be echnically convenien o approximae h by he mix loss m = 1 η ln(w e ηl ), (4) which accumulaes o M = m m. This approximaion is a sandard ool in he lieraure. For example, he mix loss m corresponds o he loss of Vovk s (1998; 2001) Aggregaing Pseudo Algorihm, and racking he evoluion of m is a crucial ingredien in he proof of Theorem 2.2 of CesaBianchi and Lugosi (2006). The definiions may be exended o η = by leing η end o. We hen find ha w becomes a uniform disribuion on he se of expers {k L 1,k = L 1 } ha have incurred smalles cumulaive loss before ime. Tha is, Hedge wih η = reduces o FollowheLeader, where in case of ies he weighs are disribued uniformly. The limiing value for he mix loss is m = L L 1. In our approximaion of he Hedge loss h by he mix loss m, we call he approximaion error δ = h m he mixabiliy gap. Bounding his quaniy is a sandard par of he analysis of Hedgeype algorihms (see, for example, Lemma 4 of CesaBianchi e al. 2007) and i also appears o be a fundamenal noion in sequenial predicion even when only socalled mixable losses are considered (Grünwald, 2011, 2012); see also Secion 6.2. We le = δ δ denoe he cumulaive mixabiliy gap, so ha he regre for Hedge may be decomposed as R = H L = M L +. (5) Here M L may be hough of as he regre under he mix loss and is he cumulaive approximaion error when approximaing he Hedge loss by he mix loss. Throughou he paper, our proof sraegy will be o analyse hese wo conribuions o he regre, M L and, separaely. The following lemma, which is proved in Appendix A, collecs a few basic properies of he mix loss: Lemma 1 (Mix Loss wih Consan Learning Rae) For any learning rae η (0, ] 1. l m h l +, so ha 0 δ s. 1 η (w 2. Cumulaive mix loss elescopes: M = ln 1 e ηl) for η <, L for η =. 3. Cumulaive mix loss approximaes he loss of he bes exper: L M L + ln K η. 4. The cumulaive mix loss M is nonincreasing in η. In order o obain a bound for Hedge, one can use he following wellknown bound on he mixabiliy gap, which is obained using Hoeffding s bound on he cumulan generaing funcion (CesaBianchi and Lugosi, 2006, Lemma A.1): δ η 8 s2, (6) 1286
7 Follow he Leader If You Can, Hedge If You Mus from which S 2 T η/8, where (as in he inroducion) S = max{s 1,..., s } is he maximum loss range in he firs rounds. Togeher wih he bound M L ln(k)/η from mix loss propery #3 his leads o R = (M L ) + ln K η + ηs2 T 8. (7) The bound is opimized for η = 8 ln(k)/(s 2 T ), which equalizes he wo erms. This leads o a bound on he regre of S T ln(k)/2, maching he lower bound on worscase regre from he exbook by CesaBianchi and Lugosi (2006, Secion 3.7). We can use his uned learning rae if he ime horizon T is known in advance. To deal wih he siuaion where T is unknown, eiher he doubling rick or a imevarying learning rae (see Lemma 2 below) can be used, a he cos of a worse consan facor in he leading erm of he regre bound. In he remainder of his secion, we inroduce a compleely parameerless algorihm called AdaHedge. We hen refine he seps of he analysis above o obain a beer regre bound. 2.2 AdaHedge Analysis In he previous secion, we spli he regre for Hedge ino wo pars: M L and, and we obained a bound for boh. The learning rae η was hen uned o equalise hese wo bounds. The main disincion beween AdaHedge and oher Hedge approaches is ha AdaHedge does no consider an upper bound on in order o obain his balance: insead i aims o equalize and ln(k)/η. As he cumulaive mixabiliy gap is nondecreasing in (by mix loss propery #1) and can be observed online, i is possible o adap he learning rae direcly based on. Perhaps he easies way o achieve his is by using he doubling rick: each subsequen block uses half he learning rae of he previous block, and a new block is sared as soon as he observed cumulaive mixabiliy gap exceeds he bound on he mix loss ln(k)/η, which ensures hese wo quaniies are equal a he end of each block. This is he approach aken in an earlier version of AdaHedge (Van Erven e al., 2011). However, we can achieve he same goal much more eleganly, by decreasing he learning rae wih ime according o η ah = ln K ah 1 (8) (where ah 0 = 0, so ha ηah 1 = ). Noe ha he AdaHedge learning rae does no involve he end ime T or any oher unobserved properies of he daa; all subsequen analysis is herefore valid for all T simulaneously. The definiions (3) and (4) of he weighs and he mix loss are modified o use his new learning rae: w,k ah = wah 1,k e ηah L 1,k w1 ah e ηah L 1 and m ah = 1 η ah ln(w ah e ηah l ), (9) wih w ah 1 = (1/K,..., 1/K) uniform. Noe ha he muliplicaive updae rule for he weighs no longer applies when he learning rae varies wih ; he las hree resuls of Lemma 1 are also no longer valid. Laer we will also consider oher algorihms o deermine 1287
8 De Rooij, Van Erven, Grünwald and Koolen Single round quaniies for rial : l Loss vecor l = min k l,k, l + = max k l,k Min and max loss s = l + l Loss range w alg h alg = e ηalg L 1 / k e ηalg L 1,k Weighs played = w alg l Hedge loss m alg = 1 η alg δ alg v alg = h alg ( ln m alg = Var k w alg w alg e ηalg l ) Mix loss Mixabiliy gap [l,k ] Loss variance Aggregae quaniies afer rounds: (The final ime T is omied from he subscrip where possible, e.g. L = L T ) L, L, L+, Halg, M alg, alg, V alg τ=1 of l τ, l τ, l + τ, h alg τ, m alg τ, δτ alg, vτ alg S = max{s 1,..., s } Maximum loss range L = min k L,k Cumulaive loss of he bes exper R alg = H alg L Regre Algorihms (he alg in he superscrip above): (η) Hedge wih fixed learning rae η ah AdaHedge, defined by (8) fl FollowheLeader (η fl = ) ff FlipFlop, defined by (16) Table 1: Noaion variable learning raes; o avoid confusion he considered algorihm is always specified in he superscrip in our noaion. See Table 1 for reference. From now on, AdaHedge will be defined as he Hedge algorihm wih learning rae defined by (8). For concreeness, a malab implemenaion appears in Figure 1. Our learning rae is similar o ha of CesaBianchi e al. (2007), bu i is less pessimisic as i is based on he mixabiliy gap iself raher han is bound, and as such may exploi easy sequences of losses more aggressively. Moreover our uning of he learning rae simplifies he analysis, leading o igher resuls; he essenial new echnical ingrediens appear as Lemmas 3, 5 and 7 below. We analyse he regre for AdaHedge like we did for a fixed learning rae in he previous secion: we again consider M ah L and ah separaely. This ime, boh legs of he analysis become slighly more involved. Luckily, a good bound can sill be obained wih only a small amoun of work. Firs we show ha he mix loss is bounded by he mix loss we would have incurred if we would have used he final learning rae ηt ah all along: Lemma 2 Le dec be any sraegy for choosing he learning rae such ha η 1 η 2... Then he cumulaive mix loss for dec does no exceed he cumulaive mix loss for he sraegy ha uses he las learning rae η T from he sar: M dec M (η T ). 1288
9 Follow he Leader If You Can, Hedge If You Mus % Reurns he losses of AdaHedge. % l(,k) is he loss of exper k a ime funcion h = adahedge(l) [T, K] = size(l); h = nan(t,1); L = zeros(1,k); Dela = 0; end for = 1:T ea = log(k)/dela; [w, Mprev] = mix(ea, L); h() = w * l(,:) ; L = L + l(,:); [~, M] = mix(ea, L); dela = max(0, h()(mmprev)); % max clips numeric Jensen violaion Dela = Dela + dela; end % Reurns he poserior weighs and mix loss % for learning rae ea and cumulaive loss % vecor L, avoiding numerical insabiliy. funcion [w, M] = mix(ea, L) mn = min(l); if (ea == Inf) % Limi behaviour: FTL w = L==mn; else w = exp(ea.* (Lmn)); end s = sum(w); w = w / s; M = mn  log(s/lengh(l))/ea; end Figure 1: Numerically robus malab implemenaion of AdaHedge This lemma was firs proved in is curren form by Kalnishkan and Vyugin (2005, Lemma 3), and an essenially equivalen bound was inroduced by Györfi and Oucsák (2007) in he proof of heir Lemma 1. Relaed echniques for dealing wih imevarying learning raes go back o Auer e al. (2002). Proof Using mix loss propery #4, we have M dec T = T =1 which was o be shown. m dec = T =1 ( M (η) M (η) ) 1 T =1 ( M (η) M (η ) 1) 1 = M (η T ) T, We can now show ha he wo conribuions o he regre are sill balanced. Lemma 3 The AdaHedge regre is R ah = M ah L + ah 2 ah. Proof ah As δ ah 0 for all (by mix loss propery #1), he cumulaive mixabiliy gap is nondecreasing. Consequenly, he AdaHedge learning rae η ah as defined in (8) is nonincreasing in. Thus Lemma 2 applies o M ah ; ogeher wih mix loss propery #3 and (8) his yields M ah M (ηah T ) L + ln K η ah T = L + ah T 1 L + ah T. Subsiuion ino he rivial decomposiion R ah = M ah L + ah yields he resul. The remaining ask is o esablish a bound on ah. As before, we sar wih a bound on he mixabiliy gap in a single round, bu raher han (6), we use Bernsein s bound on he mixabiliy gap in a single round o obain a resul ha is expressed in erms of he variance of he losses, v ah = Var k w ah [l,k ] = k wah,k (l,k h ah )
10 De Rooij, Van Erven, Grünwald and Koolen Lemma 4 (Bernsein s Bound) Le η = η alg (0, ) denoe he finie learning rae chosen for round by any algorihm alg. The mixabiliy gap δ alg saisfies Furher, v alg δ alg g(s η ) s v alg (l + halg )(h alg l ) s2 /4., where g(x) = ex x 1. (10) x Proof This is Bernsein s bound (CesaBianchi and Lugosi, 2006, Lemma A.5) on he cumulan generaing funcion, applied o he random variable (l,k l )/s [0, 1] wih k disribued according o w alg. Bernsein s bound is more sophisicaed han Hoeffding s bound (6), because i expresses ha he mixabiliy gap δ is small no only when η is small, bu also when all expers have approximaely he same loss, or when he weighs w are concenraed on a single exper. The nex sep is o use Bernsein s inequaliy o obain a bound on he cumulaive mixabiliy gap ah. In he analysis of CesaBianchi e al. (2007) his is achieved by firs applying Bernsein s bound for each individual round, and hen using a elescoping argumen o obain a bound on he sum. Wih our learning rae (8) i is convenien o reverse hese seps: we firs elescope, which can now be done wih equaliy, and subsequenly apply Bernsein s inequaliy in a sricer way. Lemma 5 AdaHedge s cumulaive mixabiliy gap saisfies ( ah ) 2 V ah ln K + ( 2 3 ln K + 1)S ah. Proof In his proof we will omi he superscrip ah. Using he definiion of he learning rae (8) and δ s (from mix loss propery #1), we ge 2 = T =1 = ( ) = ( ) ln K 2δ + δ 2 η ( ) ( 1 + δ ) = ( ) ln K 2δ + s δ 2 ln K η ( ) 2δ 1 + δ 2 δ η + S. (11) The inequaliies in his equaion replace a δ erm by S, which is of no concern: he resuling erm S adds a mos 2S o he regre bound. We will now show δ η 1 2 v s δ. (12) This supersedes he bound δ /η (e 2)v for η s 1 used by CesaBianchi e al. (2007). Even hough a firs sigh circular, he form (12) has wo major advanages. Firs, inclusion of he overhead 1 3 s δ will only affec smaller order erms of he regre, bu admis a reducion of he leading consan o he opimal facor 1 2. This gain direcly percolaes o our regre bounds below. Second, (12) holds for unbounded η, which simplifies uning considerably. 1290
11 Follow he Leader If You Can, Hedge If You Mus Firs noe ha (12) is clearly valid if η =. Assuming ha η is finie, we can obain his resul by rewriing Bernsein s bound (10) as follows: 1 2 v s δ 2g(s η ) = δ s f(s η )δ, where f(x) = ex 1 2 x2 x 1 η xe x x 2 x. Remains o show ha f(x) 1/3 for all x 0. Afer rearranging, we find his o be he case if (3 x)e x 1 2 x2 + 2x + 3. Taylor expansion of he lefhand side around zero reveals ha (3 x)e x = 1 2 x2 + 2x x3 ue u for some 0 u x, from which he resul follows. The proof is compleed by plugging (12) ino (11) and finally relaxing s S. Combinaion of hese resuls yields he following naural regre bound, analogous o Theorem 5 of CesaBianchi e al. (2007). Theorem 6 AdaHedge s regre is bounded by Proof Lemma 5 is of he form R ah 2 V ah ln K + S( 4 3 ln K + 2). wih a and b nonnegaive numbers. Solving for ah hen gives which by Lemma 3 implies ha ( ah ) 2 a + b ah, (13) ah 1 2 b b 2 + 4a 1 2 b ( b 2 + 4a) = a + b, R ah 2 a + 2b. Plugging in he values a = V ah ln K and b = S( 2 3 ln K + 1) from Lemma 5 complees he proof. This firs regre bound for AdaHedge is difficul o inerpre, because he cumulaive loss variance V ah depends on he acions of he AdaHedge sraegy iself (hrough he weighs w ah ). Below, we will derive a regre bound for AdaHedge ha depends only on he daa. However, AdaHedge has one imporan propery ha is capured by his firs resul ha is no longer expressed by he worscase bound we will derive below. Namely, if he daa are easy in he sense ha here is a clear bes exper, say k, hen he weighs played 1 as increases, hen he loss variance mus decrease: v ah 0. Thus, Theorem 6 suggess ha he AdaHedge regre may be bounded if he weighs concenrae on he bes exper sufficienly quickly. This indeed urns ou o be he case: we can prove ha he regre is bounded for he sochasic seing where he loss vecors l are independen, and E[L,k L,k ] = Ω( β ) for all k k and any β > 1/2. This is an imporan feaure of AdaHedge when i is used as a sandalone algorihm, and Van Erven e al. (2011) provide a proof for he previous version of he by AdaHedge will concenrae on ha exper. If w ah,k 1291
12 De Rooij, Van Erven, Grünwald and Koolen sraegy. See Secion 5.4 for an example of concenraion of he AdaHedge weighs. Here we will no pursue his furher, because he FollowheLeader sraegy also incurs bounded loss in ha case; we raher focus aenion on how o successfully compee wih FTL in Secion 3. We now proceed o derive a bound ha depends only on he daa, using an approach similar o he one aken by CesaBianchi e al. (2007). We firs bound he cumulaive loss variance as follows: Lemma 7 Assume L H. The cumulaive loss variance for AdaHedge saisfies V ah S (L+ L )(L L ) L + L + 2S. In he degenerae case L = L + he fracion reads 0/0, bu since we hen have V ah = 0, from here on we define he raio o be zero in ha case, which is also is limiing value. Proof We omi all ah superscrips. By Lemma 4 we have v (l + h )(h l ). Now T V = v (l + h )(h l ) S =1 1 T = ST (l + h )(h l ) s (l + h )(h l ) (l + h ) + (h l ) S (L+ H)(H L ), (14) L + L where he las inequaliy is an insance of Jensen s inequaliy applied o he funcion B defined on he domain x, y 0 by B(x, y) = xy x+y for xy > 0 and B(x, y) = 0 for xy = 0 o ensure coninuiy. To verify ha B is joinly concave, we will show ha he Hessian is negaive semidefinie on he inerior xy > 0. Concaviy on he whole domain hen follows from coninuiy. The Hessian, which urns ou o be he rank one marix 2 2 B(x, y) = (x + y) 3 ( ) ( ) y y, x x is negaive semidefinie since i is a negaive scaling of a posiive ouer produc. Subsequenly using H L (by assumpion) and H L + 2 (by Lemma 3) yields as desired. (L + H)(H L ) (L+ L )(L + 2 L ) (L+ L )(L L ) + 2 L + L L + L L + L This can be combined wih Lemmas 5 and 3 o obain our firs main resul: Theorem 8 (AdaHedge WorsCase Regre Bound) AdaHedge s regre is bounded by R ah 2 S (L+ L )(L L ) L + L ln K + S( 16 3 ln K + 2). (15) 1292
13 Follow he Leader If You Can, Hedge If You Mus Proof If H ah < L, hen R ah < 0 and he resul is clearly valid. Bu if H ah L, we can bound V ah using Lemma 7 and plug he resul ino Lemma 5 o ge an inequaliy of he form (13) wih a = S(L + L )(L L )/(L + L ) and b = S( 8 3 ln K + 1). Following he seps of he proof of Theorem 6 wih hese modified values for a and b we arrive a he desired resul. This bound has several useful properies: 1. I is always smaller han he CBMS bound (1), wih a leading consan ha has been reduced from he previously besknown value of 2.63 o 2. To see his, noe ha (15) increases o (1) if we replace L + by he upper bound L + ST. I can be subsanially sronger han (1) if he range of he losses s is highly variable. 2. The bound is fundamenal, a concep discussed in deail by CesaBianchi e al. (2007): i is invarian o ranslaions of he losses and proporional o heir scale. I is herefore valid for arbirary loss ranges, regardless of sign. In fac, no jus he bound, bu AdaHedge iself is fundamenal in his sense: see Secion 4 for a discussion and proof. 3. The regre is small when he bes exper eiher has a very low loss, or a very high loss. The laer is imporan if he algorihm is o be used for a scenario in which we are provided wih a sequence of gain vecors g raher han losses: we can ransform hese gains ino losses using l = g, and hen run AdaHedge. The bound hen implies ha we incur small regre if he bes exper has very small cumulaive gain relaive o he minimum gain. 4. The bound is no dependen on he number of rials bu only on he losses; i is a imeless bound as discussed below. 2.3 Wha are Timeless Bounds? All bounds presened for AdaHedge (and FlipFlop) are imeless. We call a regre bound imeless if i does no change under inserion of addiional rials where all expers are assigned he same loss. Inuiively, he predicion ask does no become more difficul if naure should inser sameloss rials. Since hese rials do nohing o differeniae beween he expers, hey can safely be ignored by he learner wihou affecing her regre; in fac, many Hedge sraegies, including Hedge wih a fixed learning rae, FTL, AdaHedge and CBMS already have he propery ha heir fuure behaviour does no change under such inserions: hey are robus agains such ime dilaion. If any sraegy does no have his propery by iself, i can easily be modified o ignore equalloss rials. I is easy o imagine pracical scenarios where his robusness propery would be imporan. For example, suppose you hire a number of expers who coninually monior he asses in your porfolio. Usually hey do no recommend any changes, bu occasionally, when hey see a rare opporuniy or receive suble warning signs, hey may urge you o rade, resuling in a poenially very large gain or loss. I seems only beneficial o poll he expers ofen, and here is no reason why he many resuling equalloss rials should complicae he learning ask. 1293
14 De Rooij, Van Erven, Grünwald and Koolen The oldes bounds for Hedge scale wih T or L, and are hus no imeless. From he resuls above we can obain fundamenal and imeless varians wih, for parameerless algorihms, he bes known leading consans (he firs iem below follows Corollary 1 of CesaBianchi e al. 2007): Corollary 9 The AdaHedge regre saisfies he following inequaliies: R ah T=1 s 2 ln K + S( 4 3 ln K + 2) (analogue of radiional T based bounds), R ah 2 S(L L ) ln K + S( 16 3 ln K + 2) (analogue of radiional L based bounds), R ah 2 S(L + L ) ln K + S( 16 3 ln K + 2) (symmeric bound, useful for gains). Proof We could ge a bound ha depends only on he loss ranges s by subsiuing he wors case L = (L + + L )/2 ino Theorem 8, bu a sharper resul is obained by plugging he inequaliy v s 2 /4 from Lemma 4 direcly ino Theorem 6. This yields he firs iem above. The oher wo inequaliies follow easily from Theorem 8. In he nex secion, we show how we can compee wih FTL while a he same ime mainaining all hese worscase guaranees up o a consan facor. 3. FlipFlop AdaHedge balances he cumulaive mixabiliy gap ah and he mix loss regre M ah L by reducing η ah as necessary. Bu, as we observed previously, if he daa are no hopelessly adversarial we migh no need o worry abou he mixabiliy gap: as Lemma 4 expresses, δ ah is also small if he variance v ah of he loss under he weighs w,k ah is small, which is he case if he weigh on he bes exper max k w,k ah becomes close o one. AdaHedge is able o exploi such a lucky scenario o an exen: as explained in he discussion ha follows Theorem 6, if he weigh of he bes exper goes o one quickly, AdaHedge will have a small cumulaive mixabiliy gap, and herefore, by Lemma 3, a small regre. This happens, for example, in he sochasic seing wih independen, idenically disribued losses, when a single exper has he smalles expeced loss. Similarly, in he experimen of Secion 5.4, he AdaHedge weighs concenrae sufficienly quickly for he regre o be bounded. There is he poenial for a nasy feedback loop, however. Suppose here are a small number of difficul early rials, during which he cumulaive mixabiliy gap increases relaively quickly. AdaHedge responds by reducing he learning rae (8), wih he effec ha he weighs on he expers become more uniform. As a consequence, he mixabiliy gap in fuure rials may be larger han wha i would have been if he learning rae had sayed high, leading o furher unnecessary reducions of he learning rae, and so on. The end resul may be ha AdaHedge behaves as if he daa are difficul and incurs subsanial regre, even in cases where he regre of Hedge wih a fixed high learning rae, or of FollowheLeader, is bounded! Precisely his phenomenon occurs in he experimen in Secion 5.2 below: AdaHedge s regre is close o he worscase bound, whereas FTL hardly incurs any regre a all. 1294
15 Follow he Leader If You Can, Hedge If You Mus I appears, hen, ha we mus eiher hope ha he daa are easy enough ha we can make he weighs concenrae quickly on a single exper, by no reducing he learning rae a all; or we fear he wors and reduce he learning rae as much as we need o be able o provide good guaranees. We canno really inerpolae beween hese wo exremes: an inermediae learning rae may no yield small regre in favourable cases and may a he same ime desroy any performance guaranees in he wors case. I is unclear a priori wheher we can ge away wih keeping he learning rae high, or ha i is wiser o play i safe using AdaHedge. The mos exreme case of keeping he learning rae high, is he limi as η ends o, for which Hedge reduces o FollowheLeader. In his secion we work ou a sraegy ha combines he advanages of FTL and AdaHedge: i reains AdaHedge s worscase guaranees up o a consan facor, bu is regre is also bounded by a consan imes he regre of FTL (Theorem 15). Perhaps surprisingly, his is no easy o achieve. To see why, imagine a scenario where he average loss of he bes exper is subsanial, whereas he regre of eiher FollowheLeader or AdaHedge, is small. Since our combinaion has o guaranee a similarly small regre, i has only a very limied margin for error. We canno, for example, simply combine he wo algorihms by recursively plugging hem ino Hedge wih a fixed learning rae, or ino AdaHedge: he performance guaranees we have for hose mehods of combinaion are oo weak. Even if boh FTL and AdaHedge yield small regre on he original problem, choosing he acions of FTL for some rounds and hose of AdaHedge for he oher rounds may fail if we do i naively, because he regre is no necessarily increasing, and we may end up picking each algorihm precisely in hose rounds where he oher one is beer. Luckily, alernaing beween he opimisic FTL sraegy and he worscaseproof Ada Hedge does urn ou o be possible if we do i in a careful way. In his secion we explain he appropriae sraegy, called FlipFlop (superscrip: ff ), and show ha i combines he desirable properies of boh FTL and AdaHedge. 3.1 Exploiing Easy Daa by Following he Leader We firs invesigae he poenial benefis of FTL over AdaHedge. Lemma 10 below idenifies he circumsances under which FTL will perform well, which is when he number of leader changes is small. I also shows ha he regre for FTL is equal o is cumulaive mixabiliy gap when FTL is inerpreed as a Hedge sraegy wih infinie learning rae. Lemma 10 Le c be an indicaor for a leader change a ime : define c = 1 if here exiss an exper k such ha L 1,k = L 1 while L,k L, and c = 0 oherwise. Le C = c c be he cumulaive number of leader changes. Then he FTL regre saisfies R fl = ( ) S C. Proof We have M ( ) = L by mix loss propery #3, and consequenly R fl = ( ) + M ( ) L = ( ). To bound ( ), noice ha, for any such ha c = 0, all leaders remained leaders and incurred idenical loss. I follows ha m ( ) = L L 1 = h( ) 1295 and hence δ ( ) = 0. By
16 De Rooij, Van Erven, Grünwald and Koolen bounding δ ( ) as required. S for all oher we obain ( ) = T =1 δ ( ) = : c =1 δ ( ) : c =1 S = S C, We see ha he regre for FTL is bounded by he number of leader changes. This quaniy is boh fundamenal and imeless. I is a naural measure of he difficuly of he problem, because i remains small whenever a single exper makes he bes predicions on average, even in he scenario described above, in which AdaHedge ges caugh in a feedback loop. One example where FTL ouperforms AdaHedge is when he losses for wo expers are (1, 0) on he firs round, and keep alernaing according o (1, 0), (0, 1), (1, 0),... for he remainder of he rounds. Then he FTL regre is only 1/2, whereas AdaHedge s performance is close o he worscase bound (because is weighs w ah converge o (1/2, 1/2), for which he bound (6) on he mixabiliy gap is igh). This scenario is illusraed furher in he experimens, Secion FlipFlop FlipFlop is a Hedge sraegy in he sense ha i uses exponenial weighs defined by (9), bu he learning rae η ff now alernaes beween infiniy, such ha he algorihm behaves like FTL, and he AdaHedge value, which decreases as a funcion of he mixabiliy gap accumulaed over he rounds where AdaHedge is used. In Definiion 11 below, we will specify he flip regime R, which is he subse of imes {1,..., } where we follow he leader by using an infinie learning rae, and he flop regime R = {1,..., } \ R, which is he se of imes where he learning rae is deermined by AdaHedge (mnemonic: he posiion of he bar refers o he value of he learning rae). We accumulae he mixabiliy gap, he mix loss and he variance for hese wo regimes separaely: = δ ff τ ; M = m ff τ ; (flip) τ R τ R = δ ff τ ; M = m ff τ ; V = vτ ff. (flop) τ R τ R τ R We also change he learning rae from is definiion for AdaHedge in (8) o he following, which differeniaes beween he wo regimes of he sraegy: η ff = { η flip if R, η flop if R, where η flip = η fl = and η flop = ln K. (16) 1 Like for AdaHedge, η flop = as long as 1 = 0, which now happens for all such ha R 1 =. Noe ha while he learning raes are defined separaely for he wo regimes, he exponenial weighs (9) of he expers are sill always deermined using he cumulaive losses L,k over all rounds. We also poin ou ha, for rounds R, he learning rae η ff = η flop is no equal o η ah, because i uses 1 insead of ah 1. For his reason, he 1296
17 Follow he Leader If You Can, Hedge If You Mus % Reurns he losses of FlipFlop % l(,k) is he loss of exper k a ime ; phi > 1 and alpha > 0 are parameers funcion h = flipflop(l, alpha, phi) [T, K] = size(l); h = nan(t,1); L = zeros(1,k); Dela = [0 0]; scale = [phi/alpha alpha]; regime = 1; % 1=FTL, 2=AH end for = 1:T if regime==1, ea = Inf; else ea = log(k)/dela(2); end [w, Mprev] = mix(ea, L); h() = w * l(,:) ; L = L + l(,:); [~, M] = mix(ea, L); dela = max(0, h()(mmprev)); Dela(regime) = Dela(regime) + dela; if Dela(regime) > scale(regime) * Dela(3regime) regime = 3regime; end end Figure 2: FlipFlop, wih new ingrediens in boldface FlipFlop regre may be eiher beer or worse han he AdaHedge regre; our resuls below only preserve he regre bound up o a consan facor. In conras, we do compee wih he acual regre of FTL. I remains o define he flip regime R and he flop regime R, which we will do by specifying he imes a which o swich from one o he oher. FlipFlop sars opimisically, wih an epoch of he flip regime, which means i follows he leader, unil becomes oo large compared o. A ha poin i swiches o an epoch of he flop regime, and keeps using η flop unil becomes oo large compared o. Then he process repeas wih he nex epochs of he flip and flop regimes. The regimes are deermined as follows: Definiion 11 (FlipFlop s Regimes) Le ϕ > 1 and α > 0 be parameers of he algorihm (uned below in Corollary 16). Then FlipFlop sars in he flip regime. If is he earlies ime since he sar of a flip epoch where > (ϕ/α), hen he ransiion o he subsequen flop epoch occurs beween rounds and + 1. (Recall ha during flip epochs increases in whereas is consan.) Vice versa, if is he earlies ime since he sar of a flop epoch where > α, hen he ransiion o he subsequen flip epoch occurs beween rounds and + 1. This complees he definiion of he FlipFlop sraegy. See Figure 2 for a malab implemenaion. The analysis proceeds much like he analysis for AdaHedge. We firs show ha, analogously o Lemma 3, he FlipFlop regre can be bounded in erms of he cumulaive mixabiliy gap; in fac, we can use he smalles cumulaive mixabiliy gap ha we encounered 1297
18 De Rooij, Van Erven, Grünwald and Koolen in eiher of he wo regimes, a he cos of slighly increased consan facors. This is he fundamenal building block in our FlipFlop analysis. We hen proceed o develop analogues of Lemmas 5 and 7, whose proofs do no have o be changed much o apply o FlipFlop. Finally, all hese resuls are combined o bound he regre of FlipFlop in Theorem 15, which, afer Theorem 8, is he second main resul of his paper. Lemma 12 (FlipFlop version of Lemma 3) The following wo bounds hold simulaneously for he regre of he FlipFlop sraegy wih parameers ϕ > 1 and α > 0: ( ) ( ) ϕα ϕ R ff ϕ 1 + 2α S ϕ ; (17) ( ϕ R ff ϕ 1 + ϕ ) α S. (18) Proof The regre can be decomposed as R ff = H ff L = + + M + M L. (19) Our firs sep will be o bound he mix loss M + M in erms of he mix loss M flop of he auxiliary sraegy ha uses η flop for all. As η flop is nonincreasing, we can hen apply Lemma 2 and mix loss propery #3 o furher bound M flop M (ηflop T ) L + ln K η flop = L + T 1 L +. (20) Le 0 = u 1 < u 2 <... < u b < T denoe he imes jus before he epochs of he flip regime begin, i.e. round u i + 1 is he firs round in he ih flip epoch. Similarly le 0 < v 1 <... < v b T denoe he imes jus before he epochs of he flop regime begin, where we arificially define v b = T if he algorihm is in he flip regime afer T rounds. These definiions ensure ha we always have u b < v b T. For he mix loss in he flop regime we have M = (M flop u 2 Mv flop 1 ) + (Mu flop 3 Mv flop 2 ) (Mu flop b Mv flop b 1 ) + (M flop Mv flop b ). (21) Le us emporarily wrie η = η flop o avoid double superscrips. For he flip regime, he properies in Lemma 1, ogeher wih he observaion ha η flop does no change during he flip regime, give M = = b i=1 b ( ) M v ( ) i M u ( ) i = ( M (ηv i ) v i M (ηv i ) u i i=1 ( ) Mv flop 1 Mu flop 1 + b i=1 ( M ( ) v i L u i ) + ln K ) b = η vi i=1 ( ) Mv flop 2 Mu flop b i=1 ( M (ηv i ) v i L u i ) ( Mv flop i Mu flop i + ln K η ui +1 ) ( ) Mv flop b Mu flop b + b ui. (22) i=1 From he definiion of he regime changes (Definiion 11), we know he value of ui very accuraely a he ime u i of a change from a flop o a flip regime: ui > α ui = α vi 1 > ϕ vi 1 = ϕ ui
19 Follow he Leader If You Can, Hedge If You Mus By unrolling from low o high i, we see ha b b ui ϕ 1 i ub ϕ 1 i ub = i=1 i=1 i=1 ϕ ϕ 1 u b. Adding up (21) and (22), we herefore find ha he oal mix loss is bounded by b M + M M flop + ui M flop + ϕ ( ) ϕ ϕ 1 u b L + ϕ 1 + 1, i=1 where he las inequaliy uses (20). Combinaion wih (19) yields R ff ( ϕ ϕ ) +. (23) Our nex goal is o relae and : by consrucion of he regimes, hey are always wihin a consan facor of each oher. Firs, suppose ha afer T rials we are in he bh epoch of he flip regime, ha is, we will behave like FTL in round T + 1. In his sae, we know from Definiion 11 ha is suck a he value ub ha promped he sar of he curren epoch. As he regime change happened afer u b, we have ub S α ub, so ha S α. A he same ime, we know ha is no large enough o rigger he nex regime change. From his we can deduce he following bounds: 1 α ( S) ϕ α. On he oher hand, if afer T rounds we are in he bh epoch of he flop regime, hen a similar reasoning yields In boh cases, i follows ha α ( S) α. ϕ < α + S; < ϕ α + S. The wo bounds of he lemma are obained by plugging firs one, hen he oher of hese bounds ino (23). The flop cumulaive mixabiliy gap is relaed, as before, o he variance of he losses. Lemma 13 (FlipFlop version of Lemma 5) The cumulaive mixabiliy gap for he flop regime is bounded by he cumulaive variance of he losses for he flop regime: 2 V ln K + ( 2 3 ln K + 1)S. (24) 1299
20 De Rooij, Van Erven, Grünwald and Koolen Proof The proof is analogous o he proof of Lemma 5, wih insead of ah, V insead of V ah, and using η = η flop = ln(k)/ 1 insead of η = η ah = ln(k)/ ah 1. Furhermore, we only need o sum over he rounds R in he flop regime, because does no change during he flip regime. As i is sraighforward o prove an analogue of Theorem 6 for FlipFlop by solving he quadraic inequaliy in (24), we proceed direcly owards esablishing an analogue of Theorem 8. The following lemma provides he equivalen of Lemma 7 for FlipFlop. I can probably be srenghened o improve he lower order erms; we provide he version ha is easies o prove. Lemma 14 (FlipFlop version of Lemma 7) Suppose H ff L. variance for FlipFlop wih parameers ϕ > 1 and α > 0 saisfies V S (L+ L )(L ( L ) ϕ + L + L ϕ 1 + ϕ ) α + 2 S + S 2. Proof The sum of variances saisfies V = R v ff T =1 v ff S (L+ H ff )(H ff L ) L + L, The cumulaive loss where he firs inequaliy simply includes he variances for FTL rounds (which are ofen all zero), and he second follows from he same reasoning as employed in (14). Subsequenly using L H ff (by assumpion) and, from Lemma 12, H ff L + γ, where γ denoes he righhand side of he bound (18), we find which was o be shown. V S (L+ L )(L + γ L ) S (L+ L )(L L ) + Sγ, L + L L + L Combining Lemmas 12, 13 and 14, we obain our second main resul: Theorem 15 (FlipFlop Regre Bound) The regre for FlipFlop wih doubling parameers ϕ > 1 and α > 0 simulaneously saisfies he wo bounds R ff where c 1 = R ff c 1 ( ϕα ϕ 1 + 2α + 1 ) R fl + S S (L+ L )(L L ) L + L ϕ ϕ 1 + ϕ α + 2. ( ϕ ϕ ), ( ln K + c 1 S (c ) ln K + ) ln K S, This shows ha, up o a muliplicaive facor in he regre, FlipFlop is always as good as he bes of FollowheLeader and AdaHedge s bound from Theorem 8. Of course, if 1300
On 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 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 informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
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 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 informationState Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University
Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween
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 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 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 informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
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 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 information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
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 informationTask is a schedulable entity, i.e., a thread
RealTime Scheduling Sysem Model Task is a schedulable eniy, i.e., a hread Time consrains of periodic ask T:  s: saring poin  e: processing ime of T  d: deadline of T  p: period of T Periodic ask T
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationPATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
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 information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
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 informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
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 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 informationUnderstanding Sequential Circuit Timing
ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor
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 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 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 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 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 informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
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 informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
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 informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationEconomics 140A Hypothesis Testing in Regression Models
Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationRealtime Particle Filters
Realime Paricle Filers Cody Kwok Dieer Fox Marina Meilă Dep. of Compuer Science & Engineering, Dep. of Saisics Universiy of Washingon Seale, WA 9895 ckwok,fox @cs.washingon.edu, mmp@sa.washingon.edu Absrac
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
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 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 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 informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationThe Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas
The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he
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 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 informationInventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds
OPERATIONS RESEARCH Vol. 54, No. 6, November December 2006, pp. 1079 1097 issn 0030364X eissn 15265463 06 5406 1079 informs doi 10.1287/opre.1060.0338 2006 INFORMS Invenory Planning wih Forecas Updaes:
More informationChapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )
Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More information2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity
.6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationINTRODUCTION TO FORECASTING
INTRODUCTION TO FORECASTING INTRODUCTION: Wha is a forecas? Why do managers need o forecas? A forecas is an esimae of uncerain fuure evens (lierally, o "cas forward" by exrapolaing from pas and curren
More information11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.
11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationPrice elasticity of demand for crude oil: estimates for 23 countries
Price elasiciy of demand for crude oil: esimaes for 23 counries John C.B. Cooper Absrac This paper uses a muliple regression model derived from an adapaion of Nerlove s parial adjusmen model o esimae boh
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationCan Individual Investors Use Technical Trading Rules to Beat the Asian Markets?
Can Individual Invesors Use Technical Trading Rules o Bea he Asian Markes? INTRODUCTION In radiional ess of he weakform of he Efficien Markes Hypohesis, price reurn differences are found o be insufficien
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationMACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR
MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR The firs experimenal publicaion, which summarised pas and expeced fuure developmen of basic economic indicaors, was published by he Minisry
More informationSupplementary Appendix for Depression Babies: Do Macroeconomic Experiences Affect RiskTaking?
Supplemenary Appendix for Depression Babies: Do Macroeconomic Experiences Affec RiskTaking? Ulrike Malmendier UC Berkeley and NBER Sefan Nagel Sanford Universiy and NBER Sepember 2009 A. Deails on SCF
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationMachine Learning in Pairs Trading Strategies
Machine Learning in Pairs Trading Sraegies Yuxing Chen (Joseph) Deparmen of Saisics Sanford Universiy Email: osephc5@sanford.edu Weiluo Ren (David) Deparmen of Mahemaics Sanford Universiy Email: weiluo@sanford.edu
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationRelationships between Stock Prices and Accounting Information: A Review of the Residual Income and Ohlson Models. Scott Pirie* and Malcolm Smith**
Relaionships beween Sock Prices and Accouning Informaion: A Review of he Residual Income and Ohlson Models Sco Pirie* and Malcolm Smih** * Inernaional Graduae School of Managemen, Universiy of Souh Ausralia
More informationA TwoAccount Life Insurance Model for ScenarioBased Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul
universiy of copenhagen Universiy of Copenhagen A TwoAccoun Life Insurance Model for ScenarioBased Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:
More informationThe naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1
Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces imeseries smoohing forecasing mehods. Various models are discussed,
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 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 informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
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 informationAppendix D Flexibility Factor/Margin of Choice Desktop Research
Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\22348900\4
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
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 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 information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationAn empirical analysis about forecasting Tmall airconditioning sales using time series model Yan Xia
An empirical analysis abou forecasing Tmall aircondiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationCooperation with Network Monitoring
Cooperaion wih Nework Monioring Alexander Wolizky Microsof Research and Sanford Universiy November 2011 Absrac This paper sudies he maximum level of cooperaion ha can be susained in perfec Bayesian equilibrium
More information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationChapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationDoes Option Trading Have a Pervasive Impact on Underlying Stock Prices? *
Does Opion Trading Have a Pervasive Impac on Underlying Sock Prices? * Neil D. Pearson Universiy of Illinois a UrbanaChampaign Allen M. Poeshman Universiy of Illinois a UrbanaChampaign Joshua Whie Universiy
More informationImpact of scripless trading on business practices of Subbrokers.
Impac of scripless rading on business pracices of Subbrokers. For furher deails, please conac: Mr. T. Koshy Vice Presiden Naional Securiies Deposiory Ld. Tradeworld, 5 h Floor, Kamala Mills Compound,
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 informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More information