Algorithms for Portfolio Management based on the Newton Method

Transcription

1 Algorihms for Porfolio Managemen based on he Newon Mehod Ami Agarwal Elad Hazan Sayen Kale Rober E. Schapire Princeon Universiy, Deparmen of Compuer Science, 35 Olden Sree, Princeon, NJ Absrac We experimenally sudy on-line invesmen algorihms firs proposed by Agarwal and Hazan and exended by Hazan e al. which achieve almos he same wealh as he bes consan-rebalanced porfolio deermined in hindsigh. These algorihms are he firs o combine opimal logarihmic regre bounds wih efficien deerminisic compuabiliy. They are based on he Newon mehod for offline opimizaion which, unlike previous approaches, explois second order informaion. Afer analyzing he algorihm using he poenial funcion inroduced by Agarwal and Hazan, we presen exensive experimens on acual financial daa. These experimens confirm he heoreical advanage of our algorihms, which yield higher reurns and run considerably faser han previous algorihms wih opimal regre. Addiionally, we perform financial analysis using mean-variance calculaions and he Sharpe raio. 1. Inroducion In he universal porfolio managemen problem, we seek online wealh invesmen sraegies which enable an invesor o maximize his wealh by disribuing i on a se of available financial insrumens wihou knowing he marke oucome in advance. The underlying model of he problem makes no saisical assumpions on he behavior of he marke (such as random walks or Brownian moion of sock prices (Luenberger, 1998)). In fac, he marke is even allowed o be adversarial. The simpliciy of he model permis he formulaion of he cenuries-old problem of wealh maximizaion as an online learning problem, Appearing in Proceedings of he 23 rd Inernaional Conference on Machine Learning, Pisburgh, PA, 06. Copyrigh 06 by he auhor(s)/owner(s). and he applicaion of machine learning algorihms. The sudy of such a model was sared in he 1950s by Kelly (1956) followed by Bell and Cover (1980; 1988), Algoe and Cover (1988). Absolue wealh maximizaion in an adversarial marke is of course a hopeless ask; we herefore aim o maximize our wealh relaive o ha achieved by a reasonably sophisicaed invesmen sraegy, he consan-rebalanced porfolio (Cover, 1991), abbreviaed CRP. A CRP sraegy rebalances he wealh each rading period o have a fixed proporion in every sock in he porfolio. We measure he performance of an online invesmen sraegy by is regre, which is he relaive difference beween he logarihmic growh raio i achieves over he enire rading period, and ha achieved by a prescien invesor one who knows all he marke oucomes in advance, bu who is consrained o use a CRP. An invesmen sraegy is said o be universal if i achieves sublinear regre. Of equal imporance is he compuaional efficiency of he online algorihm. So far, universal porfolio managemen algorihms were eiher opimal wih respec o regre, bu compuaionally inefficien (Cover, 1991), or efficien bu aained sub-opimal regre (Helmbold e al., 1998). In recen work, Agarwal and Hazan (05) inroduced a new analysis echnique which serves as he basis of algorihms ha are boh efficien and have opimal regre. These echniques were generalized by Hazan e al. (06) o yield even more efficien algorihms. These new algorihms are based on he well-sudied follow-he-leader mehod, a naural online sraegy which, simply saed, advocaes he use of he bes sraegy so far in he game for he nex ieraion. This mehod was firs proposed and analyzed by Hannan (1957) for he case of Lipschiz regre funcions, and laer simplified and exended by Kalai and Vempala (05) for linear regre funcions and also by Merhav and Feder (1992). Follow-he-leader based porfolio managemen schemes were also analyzed by Larson (1986), when he price relaive vecors are resriced o

2 ake values in a finie se and by Ordenlich and Cover (1996). The work in his paper and in Hazan e al. (06) bring ou he connecion of follow-he-leader o he Newon mehod for offline opimizaion. The previous algorihm of Helmbold e al. (1998) can be seen as a varian of gradien descen. On he oher hand, he new algorihm Online Newon Sep akes advanage of he second derivaive of he funcions. We noe ha his paper does no include comparisons o he recen algorihm of Borodin e al. (04), despie is excellen experimenal resuls. The reason is ha heir heurisic is no a universal algorihm since i does no heoreically guaranee low regre. To evaluae our algorihm, we reproduce he previous experimens of (Cover, 1991) and (Helmbold e al., 1998). Also, we es he algorihms on some more daases, and evaluae heir performance on addiional merics such as Annualized Percenage Yields (APYs), Sharpe raio and mean-variance opimaliy. The new algorihm ouperforms previous algorihms in nearly all hese experimens under all he performance merics we esed. 2. Noaion and preliminaries Le he number of socks in he porfolio be n. On every rading period, for = 1,..., T, he invesor observes a price relaive vecor r R n, such ha r (j) is he raio of he closing price of sock j on day o he closing price on day 1. A porfolio p is a disribuion on he n socks, so i is a poin in he n-dimensional simplex S n. If he invesor uses a porfolio p on day, his wealh changes by a facor of p r p r. Thus, afer T periods, he wealh achieved per dollar invesed is T (p r ). The logarihmic growh raio is T log(p r ). An invesor using a CRP p achieves he logarihmic growh raio T log(p r ). The bes CRP in hindsigh p is he one which maximizes his quaniy. The regre of an online algorihm, Alg, which produces porfolios p for = 1,..., T, is defined o be Regre(Alg) log(p r ) log(p r ). Since scaling r by a consan affecs he logarihmic growh raions of boh he bes CRP and Alg by he same addiive facor, he regre does no change. So we assume wihou loss of generaliy ha for all, r is scaled so ha max j r (j) = 1. We also make he assumpion ha afer his scaling, all he r (j) are bounded below by he marke variabiliy parameer α > 0. This has been called he no-junk-bond assumpion by Agarwal and Hazan, and can be inerpreed o mean ha no sock crashes o zero value over he rading period. Wih his seup, Cover (1991) gave he firs universal porfolio selecion algorihm which had he opimal regre O(log T ), wihou dependence on he marke variabiliy parameer α. His algorihm, however, needs Ω( n ) ime for compuing he porfolio p and is clearly impracical. Kalai and Vempala (03) gave a polynomial implemenaion of he algorihm using sampling of logconcave funcions from convex domains (Lovász & Vempala, 03b; Lovász & Vempala, 03a), and his resuls in a randomized polynomial ime algorihm, hough he polynomial is sill quie large. Helmbold e al. gave an algorihm which needs jus linear (in n) ime and space per period bu has O( T ) regre, under he no-junk-bonds assumpion. 3. Online Newon Sep Our algorihm, Online Newon Sep, is presened below. I akes parameers η, and β which are required for he heoreical analysis. I also akes a heurisic uning parameer, δ, which we se only for he purpose of experimenaion. (η, β, δ) On period 1, use he uniform porfolio p 1 = 1 n 1. On period > 1: Play sraegy p (1 η)p + η 1 n1, such ha: ( p = Π A 1 S n δa 1 1 b ) 1 where b 1 = (1 + 1 β ) 1 τ=1 [log τ (p τ r τ )], A 1 = 1 τ=1 2 [log(p τ r τ )] + I n, and Π A 1 S n is he projecion in he norm induced by A 1, viz., Π A 1 S n (q) = arg min (q p) A 1 (q p) p S n Figure 1. The Online Newon Sep algorihm. The Online Newon Sep algorihm, shown in Figure 1, has opimal regre and efficien compuabiliy. I is a Newon-based approach which uilizes he gradien (denoed ) and he Hessian (denoed 2 ) of he log funcion. I can be implemened very efficienly: all i needs o do, per ieraion, is compue an n n marix inverse, a marix-vecor produc, and a projecion ino he simplex. Aside from he projecion, all he

3 oher operaions can be implemened in O(n 2 ) ime and space using he marix inversion lemma (Brookes, 05). The projecion iself can be implemened very efficienly in pracice using projeced gradien descen mehods. We now proceed o analyze he algorihm heoreically, and show ha under he no-junk-bond assumpion, he algorihm has O(log T ) regre, whereas wihou he assumpion, he regre becomes O( T ). Theorem 1. The algorihm has he following performance guaranees: 1. Assume ha he marke has variabiliy parameer α. Then seing η = 0, β = α 8, and δ = 1, we n have [ ] Regre() 10n1.5 nt log α α Wih no assumpions on he marke variabiliy parameer, by seing η = n 1.25 T log(nt ), β = 1 8n 0.25, and δ = 1, we have T log(nt ) Regre() 22n 1.25 T log(nt ). Being a specializaion of he Online Newon Sep algorihm of Hazan e al., he analysis proceeds along he same lines. Firs, define = [log(p r )] = 1 p r r. Noe ha 2 [log(p r )] = 1 (p r ) r 2 r =, so A = τ=1 τ τ + I n. We will use his expression for A hroughou he analysis. Now define he funcions f : S n R as follows: f (p) log(p r ) + (p p ) β 2 [ (p p )] 2 where β = α 8 n. Noe ha f (p ) = log(p r ). Furhermore, by he Taylor expansion applied o he logarihm funcion (see also Lemma (2) in (Hazan e al., 06)), we ge ha, for all p S n : log(p r ) f (p). This implies ha max p log(p r ) log(p r ) max p so i suffices o bound he RHS of (1). Lemma 2. For all, we have p 1 = arg max f (p) β 2 p 2. p S n τ=1 f (p) f (p ), (1) Proof. For = 1, he uniform porfolio p 1 = 1 n 1 maximizes β 2 p 2. For > 1, expanding ou he expressions for f τ (p), muliplying by 2 β and geing rid of consans, he problem reduces o maximizing he following funcion over p S n : 1 [ ( p τ τ p + 2 p τ τ τ τ=1 = p A 1 p + 2b 1p. + 1 β τ ) ] p p p The soluion of his maximizaion is exacly he projecion Π A 1 K (A 1 1 b 1) as specified by Online Newon Sep. Proof. (Theorem 1) Par 1. We need o bound he RHS of (1), max p f (p) f (p ). A simple inducion (see (Hazan e al., 06)) shows ha for any p, β 2 p f (p +1 ) f (p) β 2 p 2. In Lemma 4 below, we bound f (p +1 ) f (p ). Since β 2 [ p 2 p 1 2 ] β 2, we can bound he regre as: Regre() 1 [ ] nt β n log α 2 + β 2. Now he saed regre bound follows by plugging in he specified choice of parameers. Par 2. The following lemma can be deduced from Theorem 2 in (Helmbold e al., 1998). The saed regre bound follows by using he lemma wih he specified choice of parameers wih he regre bound from par 1. Lemma 3. For an online algorihm Alg, le he derived algorihm SmoohAlg use he smoohened porfolio p = (1 η)p + η 1 n 1 where p is he porfolio compued by Alg on day. Then he regre can be bounded as: Regre(SmoohAlg) Regre(Alg) + 2ηT where Regre(Alg) is compued assuming he variabiliy parameer α is a leas η n. Lemma 4. [f (p +1 ) f (p )] 1 [ ] nt β n log α 2.

4 Lemma 4. For he sake of readabiliy, we inroduce some noaion. Define he funcion F 1 τ=1 f τ. Noe ha f (p ) = by he definiion of f. Finally, le be he forward difference operaor, for example, p = (p +1 p ) and F (p ) = ( F +1 (p +1 ) F (p )). We use he gradien bound, which follows from he concaviy of f : f (p +1 ) f (p ) f (p ) (p +1 p ) = p. (2) The gradien of F +1 can be wrien as: Therefore, F +1 (p) = [ τ β τ τ (p p τ )]. (3) τ=1 F +1 (p +1 ) F +1 (p ) = βa p. (4) The LHS of (4) is F +1 (p +1 ) F +1 (p ) = F (p ). (5) Puing (4) and (5) ogeher we ge βa p = F (p ). (6) Pre-muliplying by 1 β A 1, we ge an expression for he gradien bound (2): p = 1 β A 1 [ F (p ) ] = 1 β A 1 [ F (p )] + 1 β A 1. (7) Claim 1. The firs erm of (7) is bounded as follows: 1 β A 1 [ F (p )] 0. Proof. Since p τ maximizes F τ over S n, we have F τ (p τ ) (p p τ ) 0. (8) for any poin p S n. Using (8) for τ = and τ = +1, we ge 0 F +1 (p +1 ) (p p +1 ) + F (p ) (p +1 p ) = [ F (p )] p = 1 β [ F (p )] A 1 [ F (p ) ] (by solving for p in (6)) = 1 β [ F (p )] A 1 [ F (p )] 1 β [ F (p )] A 1 1 β [ F (p )] A 1. as required. (since A 1 0 p : p A 1 p 0) Now we bound he second erm of (7). Sum up from = 1 o T, and apply Lemma 5 (see Lemma 6 from (Hazan e al., 06)) below wih A 0 = I n and v =. 1 β A 1 1 [ ] β log AT A 0 1 [ ] nt β n log α 2. The second inequaliy follows since A T = T and n α, and so A T ( nt α ) n. 2 Lemma 5. For = 1, 2,..., T, le A = A 0 + τ=1 v τ vτ for a posiive definie marix A 0 and vecors v 1, v 2,..., v T. Then he following inequaliy holds: 3.1. Inernal regre [ ] v A 1 AT v log. A 0 Solz and Lugosi (05) exend he game-heoreic noion of inernal regre o he case of online porfolio selecion problems. The noion capures he following cause of regre o an online invesor: in hindsigh, how much more money could she have made, had she ransferred all he money she invesed in sock i, o sock j on all he rading days? Formally, for a porfolio p, define p i j as follows: p i j i = 0, p i j j = p i + p j, and p i j k = p k if k i, j. The inernal regre is defined o be max ij log(p i j r ) log(p r ). A sraighforward applicaion of he echnique of Solz and Lugosi (05) resuls in an algorihm, called IR-, ha achieves logarihmic inernal regre. In he full version of he paper, we prove: Theorem 6. Assume ha he marke has variabiliy parameer α. Then seing η = 0, β = α 8, and δ = 1, n we have [ ] InernalRegre(IR-) n3 nt α log α 2.

5 4. Experimenal Resuls We implemened he algorihms presened in (Agarwal & Hazan, 05) and (Hazan e al., 06) as well as he algorihms of Cover (1991) 1, he Muliplicaive Weighs algorihm of Helmbold e al. (1998), and he uniform CRP. We also applied he echnique of Solz and Lugosi (05) o he algorihms of Helmbold e al. (1998) and his paper o ge varians which minimize inernal regre. We implemened Online Newon Sep wih parameers η = 0, β = 1, and δ = 1 8. Unless oherwise noed, we omi he resuls for IR- because i was inferior o. We performed ess on he hisorical sock marke daa from he New York Sock Exchange (NYSE) used by Cover and Helmbold e al. In addiion we randomly seleced porfolios of various sizes from a se of 50 randomly chosen S&P 500 socks 2 and performed experimens over he pas 4 years daa from 12 h December, 01 o 30 h November, 05 obained from Yahoo! Finance. Table 1. Abbreviaions used in he experimens. BCRP Bes CRP Uniform CRP Universal (Cover, 1991) (Helmbold e al., 1998) IR- Inernal regre varian of Online Newon Sep IR- Inernal regre varian of Performance Measures. The performance measures we used were Annualized Percenage Yields (APYs), Sharpe raio and mean-variance opimaliy Performance vs. Porfolio Size To measure he dependence of he performance of various algorihms on porfolio size we picked 50 ses of n random socks from he daa se, for values of n ranging from 5 o 40. All algorihms were run on he daa, rading once every wo weeks. The choice of rading period was o permi compleion of he Universal al- 1 Since we implemened Cover s algorihm by random sampling, here is a small degree of variabiliy in he measuremens recorded here. We used 1000 samples, which as suggesed by (Solz & Lugosi, 05), is sufficien o ge a good esimae of he behavior of ha algorihm. 2 The se of socks used was RTN, SLB, ABK, PEG, KMG, FITB, CL, PSA, DOV, NKE, AT, NEM, VMC, D, CPWR, NVDA, SRE, HPQ, CMX, LXK, GPC, ABI, PGL, QLGC, OMX, QCOM, KO, PMTC, SWK, CTXS, FSH, HON, COF, LH, KMG, BLL, WB, OMX, K, LUV, DIS, SFA, APOL, HUM, CVH, IR, SPG, WY, TYC, NKE. gorihm in reasonable ime. The rading period did no seem o affec he relaive performance of he algorihms. The resuls are shown in Figure 4. mean APY Universal IR Number of Socks Figure 2. Performance vs. Porfolio Size The improvemen in he performance of wih increasing number of socks is quie sark. The reason for his seems o be ha does an exremely good job of racking he bes sock in a given porfolio. Adding more socks causes some good sock o ge added, which proceeds o rack. Oher algorihms behave more like he uniform CRP and so average ou he increase in wealh due o he addiion of a good sock. Figure 3 shows how racks CMC, which ou-performs Kin-Ark for he es period, in a daase composed of Kin-Ark and CMC (also used by Cover) while oher algorihms have a nearly uniform disribuion on boh he socks. This is he reason ouperforms all oher algorihms on his daase, as can be seen in Figure 5. Fracion of CMC in porfolio BCRP IR Trading days Figure 3. How racks CMC.

6 4.2. Random socks from S&P 500 We esed he average APYs (over 50 rials of 10 random socks from he S&P 500 lis menioned before) of he algorihms, for differen frequencies of rebalancing, namely daily, weekly, fornighly and monhly. As can be seen in figure 4 he performance of he algorihm is superior o all oher algorihms in all he 4 cases. As is expeced he performance of all algorihms degrades as rading frequency decreases, bu no very significanly. The simple sraegy of mainaining a uniform consan-rebalanced porfolio seems o ouperform all previous algorihms. This raher surprising fac has been observed by Borodin e al. (04) also Universal IR daily weekly fornighly monhly Trading frequency Figure 4. Performance vs. Trading Period Cover s Experimens We replicaed he experimens of Cover and Helmbold e al. on Iroquios Brands Ld. and Kin Ark Corp., Commercial Meals (CMC) and Kin Ark, CMC and Meicco Corp., IBM and Coca Cola for he same 22 year period from 3 rd July, 1962 o 31 s December, As can be seen from Figure 5, ouperforms all oher algorihms excep on he Iroquios Brands Ld. and Kin Ark Corp. daase. Figure 6 shows how he oal wealh (per dollar invesed) varies over he enire period using he differen algorihms for a porfolio of IBM and Coke. The algorihm, and is inernal regre varian IR-, ouperform even he bes consan-rebalanced porfolio Sock volailiy We ook he 50 sock daa se used in previous experimens which had a hisory for 1000 days raded fornighly and sored hem according o volailiy and creaed wo ses: he 10 socks wih larges and small- APY Universal IR 0 Iroq.&Kin Ark CMC&Kin Ark CMC&MEI IBM&KO Figure 5. Four pairs of socks esed by Cover (1991) and Helmbold e al. (1998). Wealh achieved per dollar BCRP IR Trading days Figure 6. Wealh achieved by various algorihms on a porfolio consising of IBM and Coke. es price variance. Then we applied he differen algorihms on he wo differen ses. Figure 7 shows ha he performance of increases wih marke volailiy whereas he performance of oher algorihms decreases Margins loans In line wih Cover (1991) and Helmbold e al. (1998), we also esed he case where he porfolio can buy socks on margin. The daa se we esed on was he 22 year IBM and Coca Cola daa menioned earlier. Resuls for his case are given in Table 4. The margin purchases we incorporae are 50% down and 50% loan. The algorihm in fac enhances is performance edge over oher algorihms if margin loans are allowed.

7 Table 2. Sharpe raios for various algorihms on differen daases. Universal IR- Iro.&Kin-Ark CMC & Kin-Ark CMC & Meicco IBM & Coke Table 3. Minimum variance CRPs for various algorihms on differen daases. The number o he lef of he slash is he volailiy of he minimum variance CRP and he number o he righ is he volailiy of he algorihm on he daase. Universal IR- Iro. & Kin-Ark / / / / / CMC & Meicco / / / / / mean APY low volailiy Universal IR high Figure 7. Performance of algorihms on high and low volailiy daases. Table 4. Incorporaing margin loans. Algorihm APY, no margin APY wih margin Universal IR Sharpe Raio and Mean-Variance Opimal CRPs I is a well-known fac ha one can achieve higher reurns by invesing in riskier asses (Luenberger, 1998). So i is imporan o rule ou he possibiliy of he algorihm achieving higher reurns compared o oher algorihms by rading more riskily. Parameers like he Sharpe raio and he opimal mean-variance porfolio are used o measure his risk versus reward radeoff. Sharpe raio is defined as Rp R f σ p where R p is he average yearly reurn of he algorihm, which indicaes reward, R f is he risk-free rae (ypically he average rae of reurn of Treasury bills), and σ p is he sandard deviaion of he reurns of he algorihm, which indicaes is volailiy risk. Higher he Sharpe Raio he beer is he algorihm a balancing high rewards wih low risk. The mean-variance opimal CRP for an algorihm is he CRP which achieves he same reurn as he algorihm bu has minimum variance. This is he leas risky CRP one could have used in hindsigh o produce he same reurns. The closer he volailiy of he CRP o ha of he algorihm, he beer he algorihm is avoiding risk. Table 2 shows ha has eiher he bes or slighly smaller Sharpe raio among all algorihms. In Table 3, i can be seen ha has comparable volailiy o he minimum variance CRP, implying ha does no ake excessive risk in is porfolio selecion. In he case of IBM & Coke and Kin-Ark & CMC, beas he Bes CRP in hindsigh. Hence he concep of he opimal mean-variance CRP does no apply and he resuls for his case are omied Running imes As expeced, runs slighly slower han, bu boh are much faser han Universal. We measured he running ime (in seconds) of hese algorihms on he 22 year daa ses menioned earlier. The machine used was a dual Inel 933MHz PIII processor wih 1GB operaed wih Linux Fedora Core 3 operaing sysem. The average running ime, on he four daa ses we considered, was 4882 seconds for he Universal algorihm 3, whereas and ook 3.7 and 26.7 seconds, respecively. This clearly shows he significan advanage of over Universal and ha i is comparable wih in erms of compuaional efficiency. 3 Wih 1000 samples.

8 5. Conclusions We experimenally esed he recenly proposed algorihms of (Agarwal & Hazan, 05; Hazan e al., 06) for he universal porfolio selecion problem. The Online Newon Sep algorihm is exremely fas in pracice as expeced from he heoreical guaranees. Moreover, i seems o be beer han previous algorihms a racking he bes sock. I would be ineresing o combine he ani-correlaed heurisic of Borodin e al. (04) wih he bes sock racking abiliy of our algorihm. Anoher open problem is o incorporae ransacion coss ino he algorihm, as done by Blum and Kalai (1999) for Cover s algorihm. Acknowledgemens We would like o hank Sanjeev Arora and Moses Charikar for helpful suggesions. Elad Hazan and Sayen Kale were suppored by Sanjeev Arora s NSF grans MSPA-MCS , CCF , ITR We would also like o hank Gilles Solz for providing us wih he daa ses for experimens and helpful suggesions. References Agarwal, A., & Hazan, E. (05). New algorihms for repeaed play and universal porfolio managemen. Princeon Universiy Technical Repor TR Algoe, P., & Cover, T. (1988). Asympoic opimaliy and asympoic equipariion properies of logopimum invesmen. Annals of Probabiliy, 2, Bell, R., & Cover, T. (1980). Compeiive opimaliy of logarihmic invesmen. Mahemaics of Operaions Research, 2, Bell, R., & Cover, T. (1988). Game-heoreic opimal porfolios. Managemen Science, 6, Blum, A., & Kalai, A. (1999). Universal porfolios wih and wihou ransacion coss. Machine Learning, 35, Borodin, A., El-Yaniv, R., & Gogan, V. (04). Can we learn o bea he bes sock. Journal of Arificial Inelligence Research, 21, Brookes, M. (05). The marix reference manual. [online] Cover, T. (1991). Universal porfolios. Mahemaical Finance, 1, Hannan, J. (1957). Approximaion o bayes risk in repeaed play. In M. Dresher, A. W. Tucker and P. Wolfe, ediors, Conribuions o he Theory of Games, III, Hazan, E., Kalai, A., Kale, S., & Agarwal, A. (06). Logarihmic regre algorihms for online convex opimizaion. To appear in he 19h Annual Conference on Learning Theory (COLT). Helmbold, D., Schapire, R., Singer, Y., & Warmuh., M. (1998). On-line porfolio selecion using muliplicaive updaes. Mahemaical Finance, 8, Kalai, A., & Vempala, S. (03). Efficien algorihms for universal porfolios. Journal Machine Learning Research, 3, Kalai, A., & Vempala, S. (05). Efficien algorihms for on-line opimizaion. Journal of Compuer and Sysem Sciences, 71(3), Kelly, J. (1956). A new inerpreaion of informaion rae. Bell Sysems Technical Journal, Larson, D. C. (1986). Growh opimal rading sraegies. Ph.D. disseraion, Sanford Univ., Sanford, CA. Lovász, L., & Vempala, S. (03a). The geomery of logconcave funcions and an O (n 3 ) sampling algorihm (Technical Repor MSR-TR-03-04). Microsof Research. Lovász, L., & Vempala, S. (03b). Simulaed annealing in convex bodies and an O (n 4 ) volume algorihm. Proceedings of he 44h Symposium on Foundaions of Compuer Science (FOCS) (pp ). Luenberger, D. G. (1998). Invesmen science. Oxford: Oxford Universiy Press. Merhav, N., & Feder, M. (1992). Universal sequenial learning and decision from individual daa sequences. 5h COLT (pp ). Pisburgh, Pennsylvania, Unied Saes. Ordenlich, E., & Cover, T. M. (1996). On-line porfolio selecion. 9h COLT (pp ). Desenzano del Garda, Ialy. Solz, G., & Lugosi, G. (05). Inernal regre in on-line porfolio selecion. Machine Learning, 59,