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2 De Rooij, Van Erven, Grünwald and Koolen he regre of he more inuiive bu less robus Follow-he-Leader (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 K-dimensional 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 per-rial 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 Follow-he-Leader (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 Follow-he-Leader 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 wors-case 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 so-called doubling rick: seing a budge on he saisic, and resaring he algorihm wih a double budge when he budge is depleed (Cesa-Bianchi and Lugosi, 2006; Cesa-Bianchi 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. (Cesa-Bianchi and Lugosi, 2006; Auer e al., 2002). The Hedge sraegy presened by Cesa-Bianchi, 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) (Cesa-Bianchi 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 wors-case 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 wors-case 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

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 Cesa-Bianchi 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 Follow-he-Leader, 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 Hedge-ype algorihms (see, for example, Lemma 4 of Cesa-Bianchi e al. 2007) and i also appears o be a fundamenal noion in sequenial predicion even when only so-called 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 well-known bound on he mixabiliy gap, which is obained using Hoeffding s bound on he cumulan generaing funcion (Cesa-Bianchi and Lugosi, 2006, Lemma A.1): δ η 8 s2, (6) 1286

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 Follow-he-Leader (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 Cesa-Bianchi 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()-(m-mprev)); % 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.* (L-mn)); 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 ime-varying 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 (Cesa-Bianchi 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 Cesa-Bianchi 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 Cesa-Bianchi 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 lef-hand 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 Cesa-Bianchi 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 wors-case 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 sand-alone 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 Follow-he-Leader 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 Cesa-Bianchi 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 semi-definie 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 semi-definie 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 Wors-Case 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

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()-(m-mprev)); Dela(regime) = Dela(regime) + dela; if Dela(regime) > scale(regime) * Dela(3-regime) regime = 3-regime; 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 i-h 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 sraigh-forward 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 righ-hand 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 Follow-he-Leader and AdaHedge s bound from Theorem 8. Of course, if 1300

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