Optimal Strategies from Random Walks


 Benedict McCarthy
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1 Optimal Strategies from Radom Walks Jacob Aberethy Divisio of Computer Sciece UC Berkeley Mafred K Warmuth Departmet of Computer Sciece UC Sata Cruz Joel Yelli Divisio of Physical ad Biological Scieces UC Sata Cruz Abstract We aalyze a sequetial game betwee a Gambler ad a Casio The Gambler allocates bets from a limited budget over a fixed meu of gamblig evets that are offered at equal time itervals, ad the Casio chooses a biary loss outcome for each of the evets We derive the optimal mimax strategies for both participats We the prove that the miimum cumulative loss of the Gambler, assumig optimal play by the Casio, is exactly a wellkow combiatorial quatity: the expected umber of draws eeded to complete a multiple set of cards i the geeralized Coupo Collector s Problem We show that this quatity ad the optimal strategy of the Gambler ca be efficietly estimated from a simple radom walk 1 Itroductio This paper aalyzes the problem of sequetial predictio ad decisio makig from the perspective of a two player game The game is played by a learer, called here the Gambler, who makes a sequece of bettig decisios The Gambler s oppoet is the Casio i which he plays Gambler vs Casio: 1 O each day, the Gambler arrives at the Casio with $1 The Casio presets evets ad each evet is played oce per day The Gambler chooses a distributio vector w [0, 1], where w i = 1, ad bets the portio w i of his $1 budget o evet i 2 O each day the Casio determies the outcome of each evet with the objective of wiig as much moey from the Gambler as possible I particular, after observig the distributio of the Gambler s bets the Casio decides betwee a loss or a o loss for all daily evets These choices are summarized by a loss vector l {1, 0} where l i = 1 implies that o evet i, the Gambler lost (For simplicity, we assume the oly relevat quatities are losses By shiftig our baselie we ca model wis as olosses) Supported by DARPA grat FA ad NSF grat DMS Supported by NSF grat CCR At the ed of each day, the Gambler leaves the Casio havig lost w l = i w il i ad the cumulative loss of the gambler is updated as L L + w l The Gambler also moitors the cumulative performace of each evet with a state vector s N, where s i is the curret total loss of evet i After icurrig loss l at the curret day, the state vector is updated to s s + l 4 The Gambler stops playig as soo as he observes that each eve has suffered more tha k losses, where k is some fixed positive iteger kow to both The Casio is aware of this decisio ad behaves accordigly Gamblig agaist a casio may seem a ulikely startig poit for a model of sequetial decisio makig we geerally cosider the typical eviromet for learig to be stochastic rather tha adversarial Yet are these two eviromets ecessarily icompatible? Amog the objectives of this paper is to address questios such as: What will be the Gambler s worstcase cumulative loss? ; ad What is the optimal bettig strategy? These questios, while clearly gametheoretic, are ultimately aswered here by cosiderig a radomized Casio rather tha a adversarial oe From this perspective, radomess may ideed be the Gambler s worst adversary Early work o sequetial decisio makig focused o the problem of predictig a biary outcome give advice from a set of experts I that settig, the goal of the predictor is to combie the predictios of the experts to make his ow predictio, with the objective of performig well, i hidsight, compared to the best expert The performace of both the learer ad the experts is measured by a loss fuctio that compares predictios to outcomes Oe of the early algorithms, the Weighted Majority algorithm [LW94], utilizes a distributio correspodig to the degree of trust i each expert It was observed by Freud ad Schapire [FS97] that the aalysis of the Weighted Majority algorithm ca be applied to the socalled hedge settig Rather tha predict a biary outcome, the learer ow plays some distributio over the experts o every roud, a loss value is assiged to each expert idepedetly, ad the learer suffers the expected loss accordig to his chose distributio I this case, the learer bears the exact burde of the Gambler  that of hedgig his bets so as to miimize his cumulative loss To emphasize that the Gambler/Casio game is useful for settigs other tha predictio, we use the term evet rather tha expert
2 A cetral theme of much of the sequetial decisio makig literature is the use of socalled expoetial weights to determie the learer s distributio o each roud Use of the expoetial weightig scheme i the case of the Casio game results i the followig strategy for the Gambler: At a state s, bet w i = βsi j βsj o evet i, (1) where the factor β lies i [0, 1) From the aalysis of the Weighted Majority algorithm it follows that the cumulative loss of the Gambler usig the above strategy is bouded by l + k l 1 β 1 β Uder the assumptio that the loss of the best evet is at most k, the factor β ca be tued [FS97] so that the above boud becomes k + 2k l + l The expoetial weights framework, as well as other olie learig techiques, ca be motivated usig the method of relative etropy regularizatio [KW99] While the resultig algorithms are elegat ad i some cases ca be show to be asymptotically optimal [CBFHW96], they do ot optimally solve the uderlyig game Some improvemets have bee made usig, for example, biomial weights that lead to slightly better but still ooptimal solutios [CBFHW96] i a settig where the experts must produce a predictio While it is formally easy to defie the optimal algorithms usig miimax expressios, it has geerally bee assumed that actually computig a efficiet solutio is quite challegig [CBFH + 97] More recetly, however, a miimax result [ALW07] was obtaied for the specific game of predictio with absolute loss The resultig algorithm, Biig, is efficiet ad optimal i a slightly relaxed settig I this paper we show that the miimax solutio to the Gambler/Casio predictio game, which is idetical to the uderlyig game of the hedge settig with biary losses, ca be obtaied efficietly I additio, the game ca be fully aalyzed usig a simple Markov process: a radom walk o a dimesioal lattice The value of the game, that is the cumulative loss of a optimal Gambler, ca be iterpreted as the expected legth of such a radom walk The Gambler s optimal play, the portio of his budget he should bet o a give evet, ca similarly be iterpreted as maifestig a assessmet of the probability of a specific radom outcome of this walk The game s stoppig criterio, that is whe all evets have lost at least k + 1 times, may seem uusual at first yet fits quite aturally withi the experts framework Ideed, olie learig bouds are ofte tued with a explicit a priori kowledge of the cumulative loss of the best expert, which here would be k 1 While perhaps ot realistic i prac 1 Strictly speakig, i the expert settig it is assumed that at least oe expert has ot crossed the kmistake threshold, while here we stop the Gambler/Casio game whe the loss of the last expert/evet goes beyod this threshold It is easy to show that this slight modificatio, made for coveiece, icreases the worstcase loss of the Gambler by exactly 1 tice, k ca be estimated ad various techiques such as successive doublig ca be used to obtai earoptimal bouds [CBFH + 97] The paper is structured as follows I Sectio 2 we give a miimax defiitio of the optimal value of the game cosidered here I Sectio 21 we modify the game by restrictig the adversary s choices to uit loss vectors I Sectio 3, we the tur our attetio to a specific Markov process with a umber of relevat properties We apply this radomized approach to the Casio game i Sectio 4, where we prove our mai results I Sectio 5 we give recurreces ad exact formulas, based o sums over multiomials, for the value of the game ad for the optimal probabilities We set out a efficiet method to compute both the optimal strategy of the Gambler ad the value of the game I Sectio 6 we compare the optimal regret boud to previous results, ad i Sectio 7 we draw a coectio betwee our game ad a well studied versio of the coupo collector problem We also briefly summarize what is kow about the asymptotics of this problem We coclude with a discussio of our results ad list ope problems (Sectio 8) 2 The Value of the Game Assume that i each evet the Gambler has already suffered some losses specified by state vector s Defie V (s) to be the total moey lost by a optimal Gambler playig agaist a optimal Casio startig from the state s That is, V (s) is the amout of moey that the optimal Gambler will lose (agaist a optimal Casio) from ow util the ed of the game Roughly speakig, the value of the game is computed as: V (s)? = mi dist w max w l + V (s + l) l {0,1} The Gambler chooses w to miimize the loss while the Casio chooses l to maximize the loss, where the loss is computed as the loss w l o this roud plus the worst case loss V (s+l) o future rouds However, we have to be careful, as this recursive defiitio does t address the followig issues: Whe is the game over? What is the base case of V ( )? Is this recursio bouded? Do we eed to record the losses s i that go above k? We address these issues begiig with some simplificatios ad otatioal covetios First, we assume that the state vector s lies withi the set S = {0, 1,, k+1} Note that it is ot ecessary to record the losses of evets that have already crossed the k threshold We call such evets dead Sice the losses of dead evets are ot restricted, havig loss k + 3 is the same as loss k We therefore roud all states s ito the state space {0, 1,, k + 1} usig the otatio + which we defie below We use the otatio λ(s) to record the set of live evets; the statemet i / λ(s) is exactly the statemet s i = k + 1 Secod, as the game is defied recursively, we must guaratee that this recursio termiates If the Casio repeatedly chose l = 0,, 0, for example, the game would make o progress The same problem occurs if the Casio causes
3 losses o oly dead evets We must therefore place additioal restrictios so that the dead state is reached evetually The simplest way to esure this is to forbid the Casio from iflictig loss o oly dead evets Yet this is ot sufficiet: with this restrictio aloe the Gambler would have a guarateed olosig strategy by bettig solely o dead evets We thus assume that either ca the Casio ca iflict losses o dead evets or ca the Gambler bet moey o them (keepig i mid that all such bets are i ay case ooptimal) We must eforce this explicitly i order to have a welldefied game We use two otatioal covetios to describe the above restrictios First, we write w λ(s) to describe the set {w w i = 0 i / λ(s)} where is the simplex We also abuse otatio slightly ad write l λ(s) to mea that l {0, 1} ad l i = 0 for all i / λ(s) We ow defie the value of the game precisely Defiitio 1 Defie the value V (s) of the game as follows At the dead state, V (d) := 0 For ay other s S, we defie V (s) recursively as V (s) := mi max w l + V (s + l) (2) 0 l λ(s) I our otatio, we commoly make use of several special states The state where the game begis is the iitial state, s = 0 Oce all evets have lost more tha k times the game is over ad we refer to this as the dead state d It will also be useful to cosider oelive states o i, where all evets except i are dead, ad the remaiig evet has exactly k losses By the game defiitio, it is easy to check that V (o i ) = 1, sice the Gambler must bet all of his moey o this evet, ad the Casio must iflict a correspodig loss, chargig the Gambler $1 ad edig the game Below, we iclude a list of otatios for referece: Notatio: S :={0,, k + 1} (the state space) 0 := 0, 0,, 0 = 0 (the iitial state) d := (k + 1) (the dead state) o i := d e i (ith oelive state) λ(s) := {i [] : s i k} (set of live evets) s +l := mi(s i + l i, k + 1) s := s i := {w R + : w = 1} 21 The Modified Game ( rouded additio) (elemetwise sum) (the simplex) We also cosider a modified game that we make easier for the Gambler I this ew game, we restrict the Casio to iflict loss o exactly oe evet i each roud, ie l must be a basis vector e 1,, e So for l = e i we have w l = w i We ca the precisely defie the value V ( ) of the modified game: Defiitio 2 Defie V (d) := V (d) = 0 Otherwise V (s) := mi max w i + V (s + e i ) (3) i λ(s) Oe of the cetral results of this paper is that the above game, while seemigly more restricted, is ultimately just as difficult for the Gambler as the origial game It is easy to show that V (s) V (s), sice the Casio has strictly more choices i the origial game We go further ad prove as our mai result i Theorem 12 that V (s) = V (s) Thus both games have the same worstcase outcome Both the aalysis of the modified game, as well as the proof of the above result, requires a differet formulatio of the Casio s actios 3 A Radomized Casio I Sectio 2 we preseted a gametheoretic aalysis of a wellkow sequetial predictio problem characterized as a game betwee a Gambler ad a Casio I the preset sectio, we cosider a differet framework, i which the Casio uses radom evets We will show that itroducig a radomized strategy of the Casio eables us to specify the optimal strategy of the Gambler 31 A Radom Walk o the State Graph Let us ow imagie that our Casio does ot fix outcomes determiistically, but istead chooses the outcome of each evet usig the followig radom process Assume we are at state s ad that, o each day, a evet i is chose uiformly at radom from {1,, } ad a loss is assiged to evet i I other words, the loss vector l is a uiformly sampled uit vector e i, ad after the loss the ew state is s +e i This process cotiues util we reach the dead state d We ca model this behavior as a Markov process o the state space as follows Cosider ay sequece of idices I 1, I 2, [], ad let S t := tm=1 e I m, where S 0 := 0 Assumig that we start at state s, this iduces a sequece of states s = s +S 0 s +S 1 s +S 2 s +S t Notice that this process has selfloops ; ie it is quite possible that s +S t = s +S t+1 This occurs whe (s +S t ) It+1 is already at k + 1 If we imagie the state space S as a dimesioal lattice, which we will call the state lattice, the the Markov process above ca be iterpreted as a radom walk o this lattice The walker starts at the iitial state 0, ad o every time iterval a positively directed sigle step is take alog a axis draw uiformly at radom If the walker has already reached the k + 1 boudary i this dimesio, he remais i place The walk stops oce the dead state d is reached We will show that the value V is 1/ times the expected total umber of radom draws that achieves this positio Thus V is the expected walk/path legth from s to d 32 Survival Probabilities We ow defie a survival probability at a state s We will show i the ext sectio that such probabilities are the basis for the Gambler s optimal strategy Defiitio 3 Assume we are at state s, ad let the radom state s +S t be the result of the above radom walk after t
4 steps Defie the ith survival probability p i (s) to be the probability that t : s +S t = o i Equivaletly, p i (s) = P r(λ(s +S t ) = {i} for some t) We call these survival probabilities sice p i (s) is the probability that, if the losses were assiged radomly to the evets i sequece, the ith evet would be the last odead evet Lemma 4 For ay s d, the vector p(s) := p i (s) i=1 defies a distributio o {1,, } Proof: The quatity i p i(s) is the probability that evetually there is exactly oe live evet This probability is exactly 1, give that the curret state is ot the dead state d We list some examples of survival probabilities: Whe s = 0 (or ay other symmetric state), we have p i (s) = 1, i bexause there is a uiform chace of survival Whe i is a dead evet, ie s i = k + 1, the p i (s) = 0 because o dead evet ca be the last remaiig live evet If there is oly oe remaiig live evet, ie λ(s) = {i}, the p i (s) = 1 Computig p i (s) for more geeral s requires a recursio, ad we leave this discussio for Sectio 5 33 Expected Path Legths Aother importat quatity we cosider is the legth of a radom path, ie the umber of steps i the radom walk o the state lattice required util the dead state d is reached Defiitio 5 For a sequece S 0, S 1,, let T (s) := mi{t 0 : s +S t = d} That is, T (s) is the legth of the radom path startig at s ad just eterig d Furthermore, let be the expected path legth τ(s) := E T (s) We ote that paths may be ifiitely log due to selfloops, yet such paths occur with probability 0 A key fact is that the expected path legth τ(s) ca be rewritte usig idicator variables: T (s) = 1[s +S t d], (4) t=0 ie T (s) is the umber of iitial segmets (icludig the empty segmet) of a radom path startig at s that has ot reached the dead state d We ow prove a relatioship betwee expected path legth τ(s) ad survival probabilities p i (s): Lemma 6 For ay state s ad evet i, p i (s) = 1 (τ(s) τ(s +e i )) Proof: Whe i / λ(s), the s = s +e i ad it is trivially true that p i (s) = 0 = 1 (τ(s) τ(s +e i )) The iterestig case is whe i λ(s) Ideed, Usig (4), we have τ(s) τ(s + e i ) = E T (s) E T (s + e i ) [ ] = E 1[s +S t d] 1[(s + e i ) +S t d] t=0 Sice the dead state d is a absorbig state we have that for ay path S, if s +S = d, the s + e i +S = d as well Equivaletly, if (s + e i ) +S d, the s +S d Thus i the differece betwee the expectatios, we oly eed be cocered with sequeces S t that are accouted for i the first expectatio but ot i the secod Therefore the above differece becomes [ ] = E 1[(s +S t d) ((s + e i ) +S t ) = d] t=0 We claim that ay sequece S t that satisfies the cojuctio must have the property that (S t ) i = k s i This is true because (s + e i ) +S t = d ad therefore (S t ) i k + 1 s i Also (S t ) j k + 1 s j, for j i This implies that s +S t = o i ad the above differece becomes E [ t=0 1[s +S t = o i ] ] The last term is exactly p i (s), the probability that s +S t evetually arrives at o i, times the expected umber of iteratios spet i state o i before arrivig at d To leave o i, the radom walk must make a step i the ith directio, ad thus the expected waitig time at o i is ca be computed as q q=1 (1 1 )q 1 } {{ } prob of q 1 loops 1 = }{{} prob of leavig The last lemma implies a importat fact about the state lattice Iterpret the state lattice as a directed graph with directed edges at all pairs (s, s + e i ) for each i λ(s) Also associate the edge (s, s + e i ) with the survival probability p i (s) Cosider startig at state s ad walkig through this directed graph: s s + e i1 s + e i1 + e i2 Corollary 7 Cosider ay two states s, s For ay path from s to s through the directed state graph, the sum of all edge weights p i ( ) alog this path is idepedet of the choice of path
5 Proof: Assume the path s = s 1, s 2,, s T, s T +1 = s defied by a sequece of moves is i 1, i 2,, i T, where s t+1 = s t + e it By Lemma 6 the total weight sum is T p it (s t ) = t=1 T t=1 1 (τ(st ) τ(s t+1 )) = 1 (τ(s) τ(s )), which is idepedet of the choice of path Note that i the defiitio of the directed state graph above ad i the corollary we igore loops, which occur whe s = s +e i (or equivaletly i / λ(s)) Such loops out of state s are immaterial because they correspod to dead evets, ad i / λ(s) iff p i (s) = 0 4 The Optimal Strategy We ow have the all the tools to express V (s) i terms of the expected path legth τ(s), prove that V (s) = V (s), ad show that the optimal bettig strategy for the gambler is p(s) We prove two major theorems i this sectio We provide the mathematically precise argumet for each but, as formality ofte obscures the true ituitio, we also provide a Eglish Versio so that the reader sees a rough sketch Our mathematical proofs require iductio o the state space S, so we eed a measure of progress for state vectors s For ay s S, defie m(s) := (k + 1) s, the umber of steps required before reachig the dead state Clearly m(s) = 0 if ad oly if s = d Theorem 8 For all states s, V (s) = 1 τ(s) Now assume that m(s) > 0 The V (s) = mi (iduc) = mi (Lem 6) = max i max w i + V (s + e i ) i λ(s) max i λ(s) w i + 1 τ(s + e i) max p i(s) + 1 i λ(s) τ(s + e i) = 1 τ(s) 1 (τ(s) τ(s + e i)) + 1 τ(s + e i) We prove V (s) 1 τ(s) by a similar iductio Assume that the Gambler chooses the optimal distributio w which may ideed be differet from p(s) For ay i / λ(s), p i (s) is defied as zero For the optimal strategy wi = 0 as well because otherwise the Casio ca icurr ubouded loss by playig e i repeatedly Sice w ad p(s) are differet distributios o the live evets λ(s), there must exist some j λ(s) for which wj > p j(s) We ow have V (s) = max i λ(s) w i + V (s + e i ) (iduc) = max i λ(s) w i + 1 τ(s + e i) w j + 1 τ(s + e i) > p j (s) + 1 τ(s + e i) (Lem6) = 1 (τ(s) τ(s + e i)) + 1 τ(s + e i) = 1 τ(s) Proof: (Eglish Versio) Assume that the Gambler always plays accordig to the distributio vector p(s) The we may thik of the Casio s choices as a walk aroud the state graph ad, as we discussed at the ed of Sectio 3, a collectio of the weights p i ( ) alog the way, edig at d But as we proved i Corollary 7 for the weights p(), it does t matter what path is take: the Casio will always receive 1 (τ(s) τ(d)) = 1 τ(s) o ay path from s that just eded i d If the Gambler ever chooses a distributio w differet from p(s) at some state s, the the Casio ca simply let l = e j for ay j for which w j > p j (s), ad o this roud the casio will force loss greater tha p j (s) This meas that o some path startig from s, the Casio will accrue total weight/loss larger tha 1 τ(s), ad therefore that the distributio w at s was ooptimal for the Gambler We coclude that for the Gambler p() is the oly optimal assigmet of distributios to states Proof: (Formal Versio) We iduct o m(s) First we check the base case s = d I this case, the expected path legth is exactly 0 sice we have already reached the dead state Thus τ(s) = 0 = V (d) as desired Corollary 9 For ay s d, p(s) is the uique optimal probability vector for the learer for the game related to V Proof: See ed of last proof Corollary 10 For all s ad all i [], p i (s) = V (s) V (s +e i ) Proof: This follows from the previous theorem ad Lemma 6 We eed oe more lemma before we ca prove our mai result Lemma 11 For ay state s ad distict evets i, j λ(s), we have p i (s) < p i (s + e j ) This fact is ituitive: if losses are radomly assiged the the probability that the ith evet will survive last strictly icreases whe aother evet suffers a loss We prove this precisely below
6 Proof: To show that p i (s) p i (s + e j ) is straightforward Ay sequece S 0, S 1, S 2, that brigs s to the oelive state o i also brigs s + e j to o i Ideed, if s +S t = o i for some t the certaily (s + e j ) +S t = o i as well To show that this iequality is strict, we eed oly fid oe radom sequece for which s+e j is brought to o i but ot s Take ay sequece S 0, S 1, such that s +S t = d e i e j (where the oly evets remaiig are i ad j) ad where S t+1 = S t + e i The (s + e j ) +S t = o i but s +S t+1 = s +(S t + e i ) = o j Theorem 12 For all states s, V (s) = V (s) = 1 τ(s) Proof: (Eglish Versio) Imagie a gambler who plays the distributio p(s) at every state s We already kow that the Casio ca use its modified game strategy ad simply play uit vectors l = e i o each roud to force 1 τ(s) loss Yet sice l is urestricted, ca it obtai more? The aswer is No: cosider what happes if the Casio decides to choose l larger tha a uit vector, eg let l = e i + e j for simplicity The o this roud it obtais p i (s) + p j (s), but it ca do better! We proved i Lemma 11 that survival probabilities strictly icrease ad therefore p i (s) < p i (s + e j ) Thus, a more patiet Casio could choose l = e j o this roud, obtai p j (s), ad the choose l = e i o the ext roud to obtai p i (s+e j ) As p j (s)+ p i (s+e j ) > p j (s)+ p i (s), the Casio oly does worse by playig ouit vectors Ideed, this suggests that the Gambler has a strategy by which the Casio ca iflict oly as much loss as i the modified game, ad thus the value V (s) is o differet from V (s) Proof: (Formal Versio) Certaily V (s) V (s), sice the Casio is give strictly fewer choices i the modified game Thus we are left to show that V (s) V (s) We proceed via iductio o m(s) By defiitio, V (s) = V (s) for the case s = d Now assume that, for all successive states s where m(s ) < m(s), V (s ) = V (s ) We proceed by directly aalyzig the recursive defiitio (2) Assume that the Gambler has chose the (possibly ooptimal) distributio w = p(s) to distribute his wealth o the live evets λ(s), ad let l {0, 1} be a optimal choice of the Casio (which ca deped o the Gambler s choice) By defiitio (1) of V (s), the chose loss vector ca t be 0 ad all evets with loss oe must be i λ(s) More precisely, V (s) = mi (id) = mi max w l + V (s + l) 0 l λ(s) max w l + V (s + l) 0 l λ(s) 0 l λ(s) p(s) max l + V (s + l) = p(s) l + V (s + l ) If l is ay uit vector e i, st i λ(s), the V (s) p(s) e i + V (s + e i ) = p i (s) + V (s + e i ) = V (s) ad i this case, V (s) = V (s) ad we are doe We ow prove by cotradictio that l ca have o more tha oe ozero coordiate Assume ideed that l > 1, ie it admits a decompositio l = e i + l for some i ad bit vector l 0 with li = 0 Applyig Lemma 11 repeatedly, we have that p i (s) < p i (s + l) ad therefore p(s) l + V (s + l ) = p i (s) + p(s) l + V (s + l ) (Lem 11) < p i (s + l) + p(s) l + V (s + l ) (Cor 10) = V (s + l) V (s + l ) + p(s) l + V (s + l ) = p(s) l + V (s + l) But the statemet p(s) l + V (s + l ) < p(s) l + V (s + l) implies l is a ooptimal choice for the Casio ad this cotradicts our assumptio that l was optimum Corollary 13 For ay s d, if the learer plays with the optimum probability vector p(s), the the oly optimal resposes of the adversary i the recurrece (2) for V is to choose a uit vector of a live evet Proof: Proved at the ed of the last theorem 5 Recurreces, Combiatorics ad Radomized Algorithms The quatities V (s), τ(s) ad p i (s) have a umber of iterestig properties that we lay out i this sectio 51 Some Recurreces The expected path legth, τ(s) satisfies a very atural recursio Whe s = d, the the path legth is determiistically 0 ad therefore τ(d) = 0 Otherwise, we see that the expected path legth is i=1 τ(s) = 1 + τ(s +e i ) (5) That is, the expected path legth is 1, for the curret step i the path, plus the expected path legth of the ext radom state Sice the ext state is chose radomly from the set {s +e i : i = 1,, }, the probability of ay give state is 1, hece the ormalizatio factor Of course, our origial quatity of iterest is V (s), ad as we showed i Theorem 12 V (s) = 1 τ(s) This immediately gives us a recursio for V : ( ) V (s) = τ(s +e i ) i=1 = 1 + i=1 V (s +e i ) This recurrece, while true for the fuctio V ( ), is ambiguous because V (s) ca occur o both sides of the equatio Ideed, wheever i / λ(s), V (s +e i ) = V (s) However, we ca rearrage all V (s) terms to obtai the followig welldefied recursio: V (s) = 1 + i λ(s) V (s + e i) (6) λ(s)
7 We ca fid a similar recurrece for p i ( ) For the oelive states o i we have p j (o i ) = 1 if i = j ad 0 otherwise If λ(s) > 1, the j=1 p i (s) = p i(s +e j ) As p i (s) is the probability of edig at state o i after executig the Markov chai, this formula is obtaied by coditioig o oe step of the Markov process That is, the probability of edig at state o i is P r(j chose)p r(radom process takes s +e j to o i ) j This recurrece suffers from the same problem as did our iitial recurrece for V ( ): p(s) ca occur o both sides of the equality We agai solve this problem by rearragig terms ad obtai p i (s) = 52 Combiatorial Sums j λ(s) p i(s + e j ) λ(s) A further aalysis gives us exact expressios for both p i (s) ad V (s) i terms of ifiite sums of multiomials Propositio 51 For ay state s S, p i (s) = ( ) ( ) r +1 r 1 r 1, r 2,, r r:s +r=o i Proof: By defiitio, p i (s) is the probability that s reaches the oelive state o i evetually To compute this probability, we cosider at what poit the Markov process exits the state o i ad ito d Recall the radom variable S t defied i Sectio 3 Take ay r for which s +r = o i ad coditio o S t = r The p i (s) = P r(s t = r)p r(s t+1 = r + e i S t = r) r:s +r=o i The first probability is exactly ( ) r r 1,r 2,,r r ad the secod probability is exactly 1/ Sice V (s) ca be writte as a expected path legth, we ca obtai a similar expressio as a sum of multiomials for V (s): Propositio 52 ( ) ( ) r +1 r 1 V (s) = ( r + 1) r i=1 1, r 2,, r r:r +s=o i 53 Radomized Approximatios Computig the exact value V (s) for large but oasymptotic values of the state vector is difficult because we have o polyomial time algorithm O the other had, fidig a radomized approximatio to V (s) ca be doe very efficietly Ideed, as we ow have a represetatio of V (s) i terms of the legth of a radom walk, we ca simply ru the radom walk S 1, S 2, several times, ote the legth T (s), ad retur the mea Such radom approximatios require that the distributio o T (s) has lowvariace, yet this certaily holds i the case at had While the radom walk requires at least (k + 1) iteratios to fiish, a simple argumet shows that with probability 1 δ the radom walk completes i less tha k log(k/δ) rouds Algorithm 1 Radom Approximatio to V (s) Iput: state s t 0 for i = 1,, NUMITER do z s repeat Sample i {1,, } uar z z +e i t t + 1 util z = d ed for t Retur NUMITER If R(s) is the radom variable retured by the above algorithm, the clealy ER(s) = V (s) By icreasig NUMITER, the variace of this estimate ca be reduced quickly A radomized approximatio for p(s) ca be obtaied similarly Agai the above algorithm approximately com Algorithm 2 Radom Approximatio to p(s) Iput: state s d p 0 for i = 1,, NUMITER do z s repeat Sample i {1,, } uar z z +e i util z = o j for some j p p + e j ed for p Retur NUMITER putes p(s) i the followig sese: If R(s) is the radom variable retured by the above algorithm, the clearly ER(s) = p(s) Agai icreasig NUMITER, reduces the variace of the estimate 54 A Simple Strategy i a Radomized Settig I the particular case of bettig agaist the Casio, it may be ecessary for the Gambler to compute p i (s) i order to place his bets optimally I a alterative settig, however, a radomized algorithm may be sufficiet Let us cosider the case i which the Gambler chooses to bet accordig to the outcome of several coi tosses Further assume that the Casio ca observe his strategy but caot see the outcome of the coi tosses or his fial bets I this sceario, the Gambler ca eve bet all of his moey o a radom evet I {1,, } draw accordig to some distributio as log
8 22 Radomly Computig V compariso of the regret for = 2, 10, 100 ad k = 1,, The Optimal Boud vs the Hedge Boud The value V (0) k Samples 1 Sample Exact Value of V Regret Optimal, =2 Hedge, =2 Optimal, =10 Hedge, =10 Optimal, =100 Hedge, = The parameter k Figure 1: We illustrate the accuracy of the radomized approximatio to V (0) stated i Algorithm 1 The plot compares the exact value of V (0) to that obtaied by usig either 1 or 10 samples of the radom walk Here = 100 as E1[I = i] = p i (s) for all i, ad ideed his expected loss would be p(s) l For this sceario, radomly approximatig p is ot ecessary: oly oe sample is eeded! To be precise, the Gambler ca take the state s, ru the radom walk util the state reaches o i for some i, ad the bet his full dollar o evet i This bet will be correct i expectatio, ie he will pick evet i with probability p i (s), ad thus his expected loss will be exactly p l The key here is that samplig from the distributio p(s) may be quite easy eve whe computig it exactly may take more time Note that the above method based o oe sample is similar to the way the Radomized Weighted Majority algorithm approximates the Weighted Majority algorithm (more precisely the WMC algorithm of [LW94]) More precisely NUMITER=1 of Algorithm 53 correspods to WMR, ad NUMITER correspods to WMC 6 Compariso to Previous Bouds As metioed i the itroductio, the boud obtaiable base o expoetial weights [FS97] is k + 2k log + log (7) ad ca be show to be asymptotically optimal [Vov98] 2 Havig computed the miimax solutio to the same game, we ca compute the gametheoretically optimal boud of V (0) usig Algorithm 1 For small values of ad k, these bouds do differ quite substatially We preset i Figure 2 a 2 A slightly better but more complicated boud tha (7) was give i [Vov98] I the full paper we compare the optimal boud to this oe as well T he parameter k Figure 2: We compare the optimal regret boud we obtai from V (0) to that foud i [FS97], which we refer to as the hedge boud While asymptotically optimal, we observed that the hedge boud of k + 2k log + log is ot tight for small values of ad k 7 Coectios to classic problems of probabilistic eumerative combiatorics Theorem 12 shows that a optimal strategy for the Casio requires uit vector plays This leads to alterative iterpretatios of the game i terms of well studied radom processes For example, oe ca easily cofirm that our game also describes the radom process uderlyig a geeralized form of the Coupo Collector s Problem [? ] i which the collector buys cereal boxes oe by oe i order to obtai K = k+1 complete sets of baseball cards, assumig oe card is radomly placed withi each cereal box The value of our game, V (0, 0), is i fact the expected umber of cereal boxes, per baseball card, eeded to obtai the desired K complete sets Specifically, the probability geeratig fuctio for the geeralized Coupo Collector s Problem is [MW03] G,K (z) = e t/z t K 1 1 t j dt (K 1)! 0 j! j k Takig the derivative at z = 1 ad dividig by, we derive the expected umber of steps to obtai K sets, which is also the value of our game, viz V (0 ) = t K e t 1 t j dt (8) (K 1)! 0 j! j k Equatio (8) gives us a elegat closed form for the twocard case ( = 2): V ( 0, 0 ) = K + K ( ) 2K 2 2K K
9 From (8) we also obtai the well kow asymptotic expressio for the value, for large ad fixed K, V (0 ) log + (K 1) log log [1 + o(1)] The same asymptotic form appears i the aalysis of a evolvig radom graph [ER60] The radom walk o the state lattice provides yet aother iterpretatio of the same dyamics For K >> >> 1, the law of large umbers gives [NS60] V ( 0 ) = K + O(K 1/2 ) 8 Coclusio We showed i Corollary 13 that agaist the optimal learig algorithm the optimal strategy of the adversary is to choose oe of the uit loss vectors as his respose Curiously eough it ca be show that this is also true of the Weighted Majority algorithm (1) That is, ay trial i which q > 1 experts icurred a uit of loss ca be split ito q trials i which a sigle expert has a uit of loss, ad doig this always icreases the loss of the algorithm for all update factor β [0, 1) This observatio about the Weighted Majority algorithm might actually lead to improved loss bouds for this algorithm, perhaps i the way the parameter β is tued There remais also a deep questio regardig the techiques itroduced i this paper: how geeral is this method of computig the value of a game based o a radom path? Ca it hadle slightly more ivolved problems? Examples we have cosidered iclude competig agaist msized sets of experts, discussed i [WK06], i which the loss of the algorithm is compared to the loss of the best msubset Aother example is the problem of competig agaist permutatios of objects [HW07], where the loss of a permutatio is liearly assiged Our prelimiary ivestigatio suggests that similar techiques ca be adapted to also hadle such more complex problems I the full paper we hope to delieate the scope of our ew method of optimal algorithm desig Refereces [ALW07] J Aberethy, J Lagford, ad M K Warmuth Cotiuous experts ad the Biig algorithm I Proceedigs of the 19th Aual Coferece o Learig Theory (COLT06), pages Spriger, Jue 2007 [CBFH + 97] Nicolò CesaBiachi, Yaov Freud, David Haussler, David P Helmbold, Robert E Schapire, ad Mafred K Warmuth How to use expert advice J ACM, 44(3): , 1997 [CBFHW96] N CesaBiachi, Y Freud, D P Helmbold, ad M K Warmuth Olie predictio ad coversio strategies Machie Learig, 25:71 110, 1996 [ER60] P Erdos ad A Reyi O the evolutio of radom graphs Publ Math Ist Hug Acad Sci, 5A:17 61, 1960 [FS97] Yoav Freud ad Robert E Schapire A decisiotheoretic geeralizatio of olie learig ad a applicatio to Boostig J Comput Syst Sci, 55(1): , 1997 Special Issue for EuroCOLT 95 [HW07] D Helmbold ad M K Warmuth Learig permutatios with expoetial weights I Proceedigs of the 20th Aual Coferece o Learig Theory (COLT07) Spriger, 2007 [KW99] Jyrki Kivie ad Mafred K Warmuth Averagig expert predictios I Computatioal Learig Theory, 4th Europea Coferece, EuroCOLT 99, Nordkirche, Germay, March 2931, 1999, Proceedigs, volume 1572 of Lecture Notes i Artificial Itelligece, pages Spriger, 1999 [LW94] N Littlestoe ad M K Warmuth The Weighted Majority algorithm Iform Comput, 108(2): , 1994 Prelimiary versio i i FOCS 89 [MW03] A Myers ad H S Wilf Some ew aspects of the CoupoCollector s problem SIAM J Disc Math, 17:1 17, 2003 [NS60] D Newma ad L Shepp The Double Dixie Cup problem Amer Math Mothly, 67: , 1960 [Vov98] V Vovk A game of predictio with expert advice J of Comput Syst Sci, 56(2): , 1998 Special Issue: Eighth Aual Coferece o Computatioal Learig Theory [WK06] M K Warmuth ad D Kuzmi Radomized PCA algorithms with regret bouds that are logarithmic i the dimesio I Advaces i Neural Iformatio Processig Systems 19 (NIPS 06) MIT Press, December 2006
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