1 The Mistake Bound Model
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1 5-850: Advanced Algorthms CMU, Sprng 07 Lecture #: Onlne Learnng and Multplcatve Weghts February 7, 07 Lecturer: Anupam Gupta Scrbe: Bryan Lee,Albert Gu, Eugene Cho he Mstake Bound Model Suppose there are N experts who make predctons about a certan event every day for example, whether t rans today or not. At the begnnng of each tme step t, the experts gve ther predctons n a vector E t. We then also make a predcton about the outcome. Fnally, we see the actual outcome o t. he goal s to mnmze the number of tmes our predcton dffers from the outcome. Example.. Suppose E t = 0,, 0, 0, 0,,,, 0. o t =, so ths s a mstake. We predct 0, but the actual outcome s Fact.. If there s a perfect expert, then there s an algorthm that makes at most log N mstakes. Proof. he algorthm s gven by Lttlestone and Warmuth []. At each step, look at the all the experts who have made no mstakes so far. Predct what the majorty of them predct. Note that everytme we make a mstake, the number of experts who have not been wrong yet s cut n half. Snce there s at least one perfect expert, we can make at most log N mstakes. Fact.3. If the best expert makes m mstakes, there s an algorthm that makes at most mlog N+ + log N mstakes. Proof. hnk of tme as dvded nto epochs. In each epoch, we proceed as n the perfect expert scenaro, and keep track of all experts who have not made a mstake n that epoch. hs set halves wth every mstake the algorthm makes. When ths set becomes empty, we have made at most log N + mstakes, and every expert has made a mstake. hs epoch s over and we start the next epoch. Note that n each epoch, every expert makes a mstake. herefore the number of completed epochs s at most m, and the number of mstakes our algorthm makes n these epochs s at most mlog N +. In the last current epoch, the algorthm makes at most log N mstakes as n Fact.. he Weghted Majorty Algorthm Assgn a weght w to expert. Let w t weghts are w =. denote the weght of expert at tme t. Intally, all At each tme t, predct accordng to the weghted majorty of experts usng ther weghts. In other words, choose the outcome that maxmzes the sum of weghts of experts that predcted t. When we see the outcome, set w t+ = w t { f was correct f was ncorrect. heorem.. For all tmes t and experts, the number of mstakes the weghted majorty algorthm WM makes s at most.m + log N, where m s the number of mstakes expert makes.
2 Proof. he proof uses a potental functon argument. Let Φ t = [N] wt. Note that Φ = N Φ t+ Φ t for all t If WM makes a mstake at tme t, then the sum of weghts of the wrong experts s hgher than the sum of the weghts of the correct experts. hen Φ t+ = wrong = wrong = Φ t wrong 3 Φt w t+ + correct w t + correct w t w t+ w t herefore f expert makes m mstakes and WM makes M mstakes, then m = w t+ 3 M Φ t+ Φ = N 3 = m log log N + M log M m + log N.m + log N log 3 3 M Corollary.5. By changng the re-weghtng process to w t+ = w t { f was correct f was ncorrect the bound n heorem. s log N + m + O
3 Proof. Usng smlar analyss, we have m N N e M M Bernoull s Inequalty m log M + log N M log N m log log N + m log log N O + m + log N O + m + snce log + + for [0, ]. Remark.6. No determnstc algorthm A can do better than a factor of compared to the best expert. Proof. hen consder a scenaro wth experts A, B, the frst of whom always predcts for each tme step t, and the second of whom always predcts 0 for each tme step t. Snce A s determnstc, an adversary can fx all outcomes such that A s predctons are always wrong. hen at least of A and B wll have an error rate of 0.5, whle A s error rate s. 3 Randomzed Weghted Majorty he randomzed weghted majorty algorthm RMW proceeds as MW, except the predcton s probablstc based on the current weghts of the experts. As before, set w = for all. After tme t, predct After seeng o t, set w t+ 0 wth probablty otherwse w t w t :E t=0 w t { f was correct otherwse heorem.7. Gven any nput sequence of E and o, for any prefx of length, and for all, f RWM makes M mstakes and expert makes m mstakes then log N E[M] + m + O 3
4 Remark.8. he m + O log N gap from the best expert s called the regret. Remark.9. When descrbng randomzed algorthms, we must be careful n defnng what adversares can do. Oblvous Adversary Plans entre sequence up front: he nputs E, o, E, o, are pre-determned. Adaptve Adversary Sees our predcton before producng output: In order, t creates E sees predcton o t E Sem-Adaptve Adversary Lke the adaptve adversary, but our predcton happens n parallel wth the actual outcome; nether depends on the other. he adversares are equvalent on determnstc algorthms, because t always outputs the same predcton and the oblvous adversary could have calculated that n advance when creatng E t+. RWM works n the sem-adaptve model because predctons are not affected by the future. Proof of heorem.7. Defne the potental Φ t = wt as before. Let F t = ncorrect wt wt be the fracton of weght on ncorrect experts at tme t. Because we predct proportonally to the weghts of the experts, the probably that RWM makes a mstake at tme t s F t. herefore E[M] = t [ ] By our re-weghtng rules, Φ t+ = Φ t F t + F t = Φ t F t Boundng the sze of the potental after steps, F t m = w + Φ + = Φ = m ln ln N E[M] m ln + ln N = E[M] F t Ne F t = Ne where we used the nequalty +x e x n the frst lne. Fnally, note that ln for [0, ]. So we get the bound E[M] + + E[M] m + + ln N In the lterature, what we call the sem-adaptve adversary here s known as an adaptve adversary. here s no common name for what we call the adaptve adversary.
5 Extendng the game We can generalze the noton of experts who are rght or wrong to arbtrary gan/loss functons. At each tme t, suppose the algorthm produces a vector of probabltes p t = p t pt pt N N N s the probablty smplex so p 0 and pt =. In parallel, the adversary produces loss vector l t = l t lt N [, ]N. Our loss or cost s p t, l t. heorem.0. Consder a fxed. For all sequences of loss vectors, for all tmes, and for all ndces [N], there exsts a determnstc algorthm such that p t, l t = l t + + ln N Corollary.. For all log N, the average loss s bounded by p t, l t t Remark.. In the scenaro wth experts, the loss vector can be thought of as a vector of 0 and representng whether they are wrong. When the loss vector s non-negatve, we can use the multplcatve weghts algorthm to get a slghtly stronger bound. Proof of heorem.0. he Hedge algorthm [] proceeds as follows for a fxed. As usual, consder weghts w t p t = wt Φ t, so that pt =. After each round, set w t+ Note that Φ = N, and w t t l t + ntalzed to w =, and set Φ t = wt. Choose probabltes e lt. Φ t+ = w t+ = w t t l e w t lt + lt usng the nequalty e x + x + x x [, ] w t + w t l t = Φ t + Φ t p t, = Φ t + p t, Φ t e p t, l t t l l t usng + x e x because l t So lt e = w t+ Φ + Φ e p t, l t = lt ln N + t p t, l t = t p t, t l lt + + ln N 5
6 ln N Note that f we choose ɛ =, then ɛ + ln N ɛ = ln N, so that the regret term s sublnear ln N n tme. hs ndcates that the average regret of Hedgeɛ converges towards the best expert, so that Hedgeɛ s n some sense learnng. For future reference, we state the analogous result for gans g t nstead of losses l t,.e., g t = l t. heorem.3. For every 0 < ɛ, there exsts an algorthm Hedge g ɛ such that for all tmes > 0, for every sequence of gan vectors g,..., g, and for every {,..., n}, at every tme t, Hedge g ɛ produces p t N such that g t, p t g t, e ɛ ln N ɛ where e s the th vector n the standard bass of R N. Note that the frst term on the rght hand sde represents the gan of the th expert, and the last two terms represents the regret of not havng always chosen the th expert. We also state a corollary of.3 that we wll use n a future lecture regardng zero-sum games. Corollary.. Let ρ. For every 0 < ɛ, for all tmes ρ ln N, for all sequences of ɛ gan vectors g,..., g wth each g t [ ρ, ρ] N, and for all {,..., N}, at every tme t, Hedge g ɛ produces p t N such that, g t, p t g t, e ɛ. 5 Bandts A further extenson to ths game occurs when one consders the scenaro where the player, havng made a predcton t randomly based on p t, s only gven access to l t nstead of the entre loss vector l t. An example of a realzaton of ths problem comes from the analyss of slot machnes, or as they are also affectonately known, one-armed bandts. Consder the followng algorthm: We proceed as n the RWM algorthm n defnng p t. Upon seeng l t for choce t, we construct l t as follows: lt j = 0 f j t l t j p t j f j = t It turns out that ths approxmate algorthm acheves a farly good estmate of the underlyng loss vector. It can be shown that the followng result holds: Remark.5. E[ l t ] = l t. 6
7 References [] Nck Lttlestone, Manfred K. Warmuth he Weghted Majorty Algorthm. Informaton and Computaton 08:-6, manfred/pubs/j.pdf [] Yoav Freund, Robert E. Schapre A Decson-heoretc Generalzaton of On-lne Learnng and an Applcaton to Boostng Journal of Computer and System Scences 55:9-39,
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