U.C. Berkeley CS270: Algorithms Lecture 4 Professor Vazirani and Professor Rao Jan 27,2011 Lecturer: Umesh Vazirani Last revised February 10, 2012
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1 U.C. Berkeley CS270: Algorthms Lecture 4 Professor Vazran and Professor Rao Jan 27,2011 Lecturer: Umesh Vazran Last revsed February 10, 2012 Lecture 4 1 The multplcatve weghts update method The multplcatve weghts method s a very useful framework applcable to a wde varety of learnng and optmzaton problems. The method was frst dscovered n the context of onlne learnng, and has been redscovered several tmes snce then. Algorthms n the multplcatve weghts framework mantan a probablty dstrbuton/ weghts over a certan set that s updated teratvely by a multplcatve rule. The analyss of these algorthms reles on quantfyng the change n an exponental potental functon. 1.1 An Infallble Expert The smplest example llustratng multplcatve weghts s the followng: There are n experts E 1,..., E n who predct the stock market every day. The predctons of the experts are bnary valued (up/down). It s known that at least one of the experts never makes a mstake. An onlne learnng algorthm sees the predctons of the experts every day and makes a predcton of ts own. The goal of the onlne algorthm s to mnmze the total number of mstakes made. The followng algorthm makes at most log n mstakes: 1. Intalze the set of experts who have not yet made a mstake to E = {E 1, E 2,..., E n }. 2. Predct accordng to the majorty of experts n the set E. 3. Update E by removng all the experts who predcted ncorrectly. Invarant: If the algorthm makes a mstake, at least half the experts n E are wrong. The sze of E gets reduced by at least 1/2 for every ncorrect predcton, so the total number of mstakes made s at most log n. The bound on the number of mstakes s ndependent of the number of days, the algorthm does not make more than log n mstakes. The presence of an nfallble expert ensures that E s non empty, so the algorthm can always make a predcton. Ths example s based on the unrealstc assumpton that some expert s always correct, but the followng features should be noted: () The algorthm mantans weghts on a set of experts whch are updated teratvely, the update rule beng multplcaton by 0 for all wrong experts. () If the algorthm makes a mstake, somethng drastc happens, the sze of the set E s reduced by 1/ Imperfect Experts and the Weghted Majorty Algorthm The prevous algorthm can not handle of mperfect experts as E would eventually be empty and no predcton can be made. Experts makng wrong predctons should not be dropped, but ther weghts can be reduced by a constant factor, say 1/2. The modfed algorthm s:
2 Notes for Lecture 4: Jan 27, Intalze w = 1, where w s the weght of the -th expert. 2. Predct accordng to the weghted majorty of experts. 3. Update weghts by settng w w /2 for all experts who predcted ncorrectly. Invarant: If the algorthm makes a mstake, the weght of the wrong experts s at least half the total weght. The weght of the wrong experts gets reduced by least 1/2, so the total weght s reduced by a factor of at least 3/4 for every mstake. After makng M mstakes the total weght s at most n. ( 3 M. 4) The potental functon for the analyss s W, the total weght of the experts. If the best expert makes m mstakes, the total weght W must be at least 1/2 m. Combnng the upper and lower bounds, ( ) 1 3 M 2 m W n 4 Takng logarthms we have a worst case bound on the total number of mstakes M made by the algorthm, m log n + M log(3/4) M m + log n log(4/3) 2.4(m + log n) (1) The weghted majorty algorthm does not make too many more mstakes compared to the best expert. The mstake bound can be mproved by usng a multplcatve factor of (1 ) n the experts algorthm. The followng standard approxmatons wll be used for the analyss, Proposton 1 For [0, 1/2] we have the approxmaton, 2 ln(1 ) < Proof: The Taylor seres expanson for ln(1 ) s gven by, ln(1 ) = N From the expanson we have ln(1 ) < as the dscarded terms are negatve. The other half of the nequalty s equvalent to the nequalty 1 e 2 for [0, 1/2]. By the convexty of the functon e x x2, the nequalty s true for all less than a threshold t. Substtutng = 1/2 we have 1/2 e 3/4 = 0.47 showng that the threshold t s more than 1/2. Proposton 2 For [0, 1] we have, (1 ) x < (1 x) f x [0, 1] (1 + ) x < (1 x) f x [ 1, 0] Proof: Note that (1 ) x s a convex functon so there exsts t such that (1 ) x (1 x) f and only f x [0, t]. Equalty holds for x = 1 so the threshold t s 1. The other nequalty s proved smlarly.
3 Notes for Lecture 4: Jan 27, Analyss wth multplcatve factor 1 Clam 3 The number of mstakes M made by the experts algorthm wth multplcatve factor of (1 ) s bounded by, M 2(1 + )m + 2 ln n (2) Proof: In ths case, we have that the best expert has weght at least (1 ) m. Moreover, n each step where a mstake s made, at least half the weght s reduced by a factor of (1 ), whch mples that after M mstakes the weght s at most (1 /2) M n. Thus, we have the followng bound on the total weght, Takng logs we have, (1 ) m W (1 /2) M n m ln(1 ) ln n + M ln(1 /2) The approxmaton from proposton 1 s used on both sdes of the nequalty to replace the ln(1 )s by expressons dependng on, m( + 2 ) ln n M/2 Rearrangng and dvdng by /2 we have the mstake bound, M 2m(1 + ) + 2 ln n The constant n the bound (2) s better than that n (1) but the followng example shows that t s not possble to acheve a constant better than 2 wth the weghted majorty strategy. Suppose there are two experts A and B where A s rght on odd numbered days whle B s rght on even numbered days. For all, the weghted majorty algorthm makes ncorrect predctons after round 1 as the ncorrect expert gets assgned more than 1/2 the weght. The algorthm always predcts ncorrectly, whle the best experts s wrong half the tme showng that the factor of 2 s tght. A probablstc strategy that chooses experts wth probabltes proportonal to ther weghts performs better than the determnstc weghted majorty rule. The expected number of mstakes made by the probablstc strategy s, M (1 + )m + ln n (3) 1.4 Probablstc experts algorthm We generalze the settng by allowng the losses suffered by the experts to be real numbers n [0, 1] nstead of bnary values. The loss suffered by the -th expert n round t s denoted by l (t) [0, 1]. The probablstc algorthm s the followng:
4 Notes for Lecture 4: Jan 27, Intalze w = 1, where w s the weght of the -th expert. 2. Predct accordng to an expert chosen wth probablty proportonal to w, the probablty of choosng the -th expert s w() W where W s the total weght. 3. Update weghts by settng w w (1 ) l(t) for all experts. Clam 4 If L s the expected loss of the probablstc experts algorthm and L s the loss of the best expert then, L ln n + (1 + )L (4) Proof: The potental functon W (t) := w s the sum of the weght of the experts for round t. The expected loss suffered by the algorthm durng round t s gven by, L t = w l (t) W (t) W (t + 1) can be calculated drectly as the weght w gets updated to w (1 ) l(t) whch s less than w (1 l (t) ) from proposton 2 as all the losses are n [0, 1]. [If the loss belongs to [0, ρ], the update rule s w (1 ) l/ρ, the analyss yelds L ρ ln n + (1 + )L ]. W (t + 1) (1 l (t) )w = w = w (1 = W (t)(1 L t ) w l (t) ) w l (t) w The ntal value of the potental functon s W (0) = n, and so the fnal value can be bounded by, W (T ) n (1 L t ) (5) If the best expert ncurs a total loss of L, then W (T ) (1 ) L as the total weght s at least the weght of the best expert. Combnng the upper and lower bounds and takng logarthms we have, (1 ) L n (1 L t ) L ln(1 ) ln n + ln(1 L t ) (6) Usng the approxmaton from proposton 1 to replace the ln(1 )s by expressons dependng on,
5 Notes for Lecture 4: Jan 27, L ( + 2 ) ln n t T L t (7) Rearrangng and usng the fact that the expected loss L = t T L t we have, L ln n + L (1 + ) (8) Exercse: If we run the multplcatve weghts algorthm wth gans g [0, 1] updatng the weght of an expert usng the rule w.(1 + ) g, a smlar analyss yelds, G (1 )G ln n (9) 2 Wrap-Up The multplcatve weghts method s very smple way to acheve provably good bounds n the sense of dong as well as the best expert n retrospect. In future lectures we wll see the the multplcatve weghts method appled to problems lke fndng optmal strateges for zero sum games and boostng.
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