Lecture 4. Instructor: Haipeng Luo


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1 Lecture 4 Instructor: Hapeng Luo In the followng lectures, we focus on the expert problem and study more adaptve algorthms. Although Hedge s proven to be worstcase optmal, one may wonder how well t would actually perform when dealng wth a practcal problem that s probably not the worst case or even relatvely easy. Indeed, the regret bound we proved for Hedge only says that for all problem nstances, Hedge s regret s unformly bounded by O T ln. However, deally we want to have an algorthm that enjoys a much smaller regret n many easy stuatons, but n the worst case stll guarantees the mnmax regret O T ln. Dervng adaptve algorthms and adaptve regret bounds s exactly one way to acheve ths goal. Smallloss Bounds We start wth the arguably smplest adaptve bound, sometmes called smallloss bound or frst order bound. Recall that we proved the followng ntermedate bound for Hedge: R T = L T L T ln = p t l t, where L T s the cumulatve loss vector, s the best expert and we defne L T = T p t, l t to be the cumulatve loss of the algorthm. By boundedness of losses the last term above can be bounded by L T. If <, then rearrangng gves R T ln L T. Therefore, f { for a moment } we assume we knew the quantty L T ahead of tme and was able to set = mn, ln L T, then we arrve at { } R T max ln, ln ln /LT ln L T L T = O LT ln ln. The fnal bound above s the socalled smallloss bound, whch essentally replaces the dependence on T n the mnmax bound T ln by the loss of the best expert L T. ote that L T s bounded by T, therefore the smallloss bound s not worse than the mnmax bound. More mportantly, t can be much smaller than T when the best expert ndeed suffers very small loss. In partcular, f the best expert makes no mstakes at all and have L T = 0, then the smallloss bound s only Oln, ndependent of T. Ths s one typcal example of adaptve bounds that we are amng for. Of course, one obvous ssue n the above dervaton s that the learnng rate has to be set n terms of the unknown quantty L T. In fact, ths becomes an even more severe problem n a nonoblvous envronment snce L T can depend on the algorthm s actons and thus, makng the defnton of crcular. Fortunately, there are many dfferent ways to address ths ssue, and we explore one of them here. The dea s to use a more adaptve and tmevaryng learnng rate schedule. Specfcally, the algo
2 rthm predcts p t exp t L t where t = ln L t. ote that L t = t τ= p τ, l τ s the cumulatve loss of the algorthm up to round t and s thus avalable at the begnnng of round t. Ths s sometmes called a selfconfdent learnng rate snce the algorthm s confdent that ts loss s close to the loss of the best expert and thus uses t as a proxy for the loss of the best expert to tune the learnng rate. We next prove that ths algorthm ndeed provdes a smallloss bound. Theorem. Hedge wth adaptve learnng rate schedule ensures R T 3 L T ln 9 ln. Proof. Let Φ t = ln = exp L t. In Lecture we already proved Summng over t and rearrangng gve L T Φ 0 Φ T T Φ t t Φ t t p t, l t t t = ln T T ln exp T L T = L T ln L T p t l t To bound the term T t p t, l t, note that p t, l t = L t = L t L t L t L t L t L t L t L t L t LT L 0 L T, and thus T t p t, l t L t L t L t dx x t = = p t l t. Φ t t Φ t t p t l t Φ t t Φ t t t p t, l t Φ t t Φ t t. L t L t L t L t L t L t L t L t L T ln. To bound Φ t t Φ t t, we prove that Φ t n ncreasng n and thus Φ t t Φ t t. It suffces to prove that the dervatve s nonnegatve. Indeed, drect calculaton shows that wth
3 p t exp L t, Φ t = ln exp L t = L t exp L t = = exp L t = ln p t ln exp L t j L t = ln = ln = = = j= p t ln j= exp L tj exp L t p t ln p 0, t where the last step s by the fact that entropy s maxmzed by the unform dstrbuton. To sum up, we have proven that R T = L T L T 3 L T ln. Solvng for L T leads to L T 3 ln L T 9 ln. 4 Fnally squarng both sdes and usng a b a b gve whch completes the proof. L T 9 ln L T 3 L T ln, Besdes enjoyng a better theoretcal regret bound, ths algorthm s also ntutvely more reasonable snce t tunes the learnng rate adaptvely based on observed data. In general, learnng rate tunng s an mportant topc n machne learnng and could be of great practcal mportance. Quantle Bounds Smallloss bounds mprove the dependence on T n the mnmax regret bound to L T. Is t possble to mprove the other term ln n the mnmax bound to somethng better? To answer ths queston, consder agan Hedge wth a fxed learnng rate for smplcty, and note that we proved n Lecture, L T ln ln exp L T T. = Wthout loss of generalty, assume L T L T so that expert s the th best expert. Prevously we obtaned the fnal regret bound by lower boundng = exp L T by max exp L T = exp L T. In general, however, for each we have exp L T j j= exp L T j exp L T, j= and we therefore have the followng regret bound aganst the th best expert: L T L T ln T. Wth optmally tuned to ln /T, the bound becomes T ln. Ths s called the quantle bound and t states that the algorthm suffers at most ths amount of regret for all but / fracton of 3
4 the experts. Of course, at the end of the day what we care about s actually the loss of the algorthm. So assumng we had the knowledge of L T for a moment, then we could pck the optmal to acheve L T mn L T T ln [], 3 whch s a strctly better bound compared to L T T ln. To understand the mprovement, consder the case when s huge but there are many smlar experts so that for example the top % of them all have the same cumulatve loss. Then bound 3 s at most whch s ndependent of. L T % T ln % = L T T ln00, Just as n the prevous dscusson, one obvous ssue n the dervaton of bound 3 above s agan that the learnng rate needs to be tuned based on unknown knowledge. To address the ssue, here we explore a qute dfferent approach. The dea s to have dfferent nstances of Hedge runnng wth dfferent learnng rates, and have a master Hedge to combne the predctons of these metaexperts. To ths end, we use Hedge to denote an nstance of Hedge runnng wth learnng rate. The algorthm s shown below. Algorthm : Hedge wth Quantle Bounds Input: master learnng rate > 0, base learnng rates,..., M Intalze: M Hedge algorthms Hedge,..., Hedge M, C 0 j = 0 for all j [M] for t =,..., T do let p j t be the predcton of Hedge j on round t compute p t = M j= q tjp j t where q t j exp C t j play p t and observe loss vector l t [0, ] update C t j = C t j pass l t to Hedge,..., Hedge M. p j t, l t for all j [M] By Eq., we have for each Hedge j and each expert p j t, l t L T ln T j. j On the other hand, for the master Hedge, we have for each metaexpert j, M q t j p j t, l t C T j ln M T. j= ote that by constructon, we have M j= q tj p j t, l t = p t, l t and C T j = T p j t, l t. Therefore summng up the above two nequaltes lead to p t, l t L T ln T j ln M T = ln T j T ln M, j j where the last step s by pckng the optmal = ln M/T. ote that the above holds for all j and all. Therefore, suppose we have a for each, there s an j such that j ln T j = O T ln, and b M s much smaller than, then we obtan bound 3. Settng M = and j = ln j /T would clearly satsfy a, but not b. Fortunately, t turns out that one only needs to create M ln metaexperts and stll satsfy a. Specfcally, let j = ln ln j and M = log T ln. 4
5 ow clearly for each, there exst a j such that j ln /T j and therefore p t, l t L T ln T j T ln M j ln ln T ln /T T ln M /T = 3 T ln T ln M. It remans to show that M s small enough. Indeed, snce ln x x/, x, we have ln = ln, and therefore M = Oln ln. So as least for the case when / s larger than Oln ln, the term T ln M s domnated by T ln n the regret bound. We summarze the result n the followng theorem. ln Theorem. Algorthm wth = T, j = ln j T and M = log ln ln ensures L T mn L T 3 T ln O T lnln ln. Ths dea of combnng algorthms usng Hedge s useful for many other problems. It s usually a quck and easy way to verfy whether some regret bound s possble or not n theory. However, the resultng algorthm mght not be so elegant and practcal. In the next lecture, we wll study a dfferent algorthm that not only guarantees a quantle bound n fact even better than the one proven here, but also enjoys several more useful propertes. 5
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