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In this section we consider a generalization of the HALVING algorithm called the WEIGHTED-MAJORITY algorithm The WEIGHTED-MAJORITY algorithm makes predictions by taking a weighted vote among a pool of prediction algorithms and learns by altering the weight associated with each prediction algorithm These prediction algorithms can be taken to be the alternative hypotheses in H, or they can be taken to be alternative learning algorithms that themselves vary over time All that we require of a prediction algorithm is that it predict the value of the target concept, given an instance One interesting property of the WEIGHTED-MAJORITY algorithm is that it is able to accommodate inconsistent training data This is because it does not eliminate a hypothesis that is found to be inconsistent with some training example, but rather reduces its weight A second interesting property is that we can bound the number of mistakes made by WEIGHTED-MAJORITY in terms of the number of mistakes committed by the best of the pool of prediction algorithms The WEIGHTED-MAJORITY algorithm begins by assigning a weight of 1 to each prediction algorithm, then considers the training examples Whenever a prediction algorithm misclassifies a new training example its weight is decreased by multiplying it by some number B, where 0 5 B < 1 The exact definition of the WEIGHTED-MAJORITY algorithm is given in Table 71 Notice if f = 0 then WEIGHTED-MAJORITY is identical to the HALVING algorithm On the other hand, if we choose some other value for p, no prediction

7,600

Figure 3-4

DVD-Video Disc DVD-Video DVD-ROM2 LD Audio CD CD-ROM4 CD-R5 CD-RW5 CDV Video CD Photo CD CD-i Player yes no no yes no few6 yes part8 some9 no no DVD-ROM Drive depends1 yes no yes yes some6 yes part8 depends1 depends6,10 depends11 DVD/LD Player yes no yes yes no few6 yes usually3 some9 no no LD Player no no yes yes3 no yes3 no usually3 no no no CD Player no no no yes no yes no part8 no no no CD-ROM Drive no no no yes yes yes some7 part8 depends1 depends10 depends11 Video CD Player no no no yes no yes no part8 yes no no

a denotes the if* prediction algorithm in the pool A of algorithms wi denotes the weight associated i with ai

The term self refers to an object when sending a message, and it is also the receiver For instance, you might make a mental note to yourself to pick up milk on the way home from work (Listing 3-8) Listing 3-8 A method using the self keyword

1 2 3 4 5 6 7 8 9

For all i initialize wi c 1 For each training example (x, c(x)) c Initialize qo and ql to 0 am For each prediction algorithm ai c If ai(x)= O then qo t q0 +wi If ai(x)= 1 then ql c ql + wi If ql > qo then predict c(x) = 1 If qo > q1 then predict c(x) = 0 If ql = qo then predict 0 or 1 at random for c(x) For each prediction algorithm ai in A do If ai(x)# c(x) then wi +- Buri

- (void) goHome { [self pickupMilk]; } - (Milk*) pickupMilk { }

Computer requires hardware or software to decode and display audio/video DVD-ROM containing data other than standard DVD-Video files Most newer LD players can play audio from a CD and both audio and video from a CDV CD-ROM containing data other than standard CD digital audio CD-R/RW containing CD digital audio data DVD units require an additional laser tuned for CD-R readout wavelength Only MultiRead CD-ROM drives can read CD-RW discs CD digital audio part of disc only (no video) Not all DVD players can play Video CDs Computer requires software to read and display Photo CD graphic files Computer requires hardware or emulation software to run CD-i programs

Both methods are in the same object, and so the goHome method sends the message pickupMilk to itself, or self

algorithm will ever be eliminated completely If an algorithm misclassifies a training example, it will simply receive a smaller vote in the future We now show that the number of mistakes committed by the WEIGHTEDMAJORITY algorithm can be bounded in terms of the number of mistakes made by the best prediction algorithm in the voting pool

10 11

Like Java, you can nest Objective-C messages For instance, using Java, you might write the following:

DVD Unit DVD-ROM disc DVD-R(G) disc Reads Usually reads DVD-R(G) Unit Reads Reads, writes DVD-R(A) Unit Reads Reads, does not write Reads, writes DVD-RW Unit Reads Reads, often writes DVD-RAM Unit Reads Reads DVD RW Unit Reads Reads

Theorem 75 Relative mistake bound for WEIGHTED-MAJORITY D be any Let sequence of training examples, let A be any set of n prediction algorithms, and let k be the minimum number of mistakes made by any algorithm in A for the training sequence DThen the number of mistakes over D made by the WEIGHTED-MAJORITY algorithm using /3 = is at most

objMyObjectgetFoo(objMyFooIdentifiergetID());