Understanding Machine Learning: From Theory to Algorithms
20.7 Summary 281 In particular, δt=Jot(`t) =Jσ(Wtot)(`t+1)diag(σ′(Wtot))Wt =Jot+1(`t+1)diag(σ′(at+1))Wt =δt+1diag(σ′(at+1))Wt. I ...
282 Neural Networks Klivans & Sherstov (2006) have shown that for anyc >0, intersections ofnc halfspaces over{± 1 }nare n ...
20.9 Exercises 283 if we feed the network with the real number 0.x 1 x 2 ...xn, then the output of the network will bex. Hint:De ...
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Part III Additional Learning Models ...
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21 Online Learning In this chapter we describe a different model of learning, which is calledonline learning. Previously, we stu ...
288 Online Learning 21.1 Online Classification in the Realizable Case Online learning is performed in a sequence of consecutive ...
21.1 Online Classification in the Realizable Case 289 tional aspect of learning, and do not restrict the algorithms to be effici ...
290 Online Learning theorem21.3 LetHbe a finite hypothesis class. The Halvingalgorithm enjoys the mistake boundMHalving(H)≤log 2 ...
21.1 Online Classification in the Realizable Case 291 v 1 v 2 v 3 h 1 h 2 h 3 h 4 v 1 0 0 1 1 v 2 0 1 ∗ ∗ v 3 ∗ ∗ 0 1 Figure 21. ...
292 Online Learning x=j. Then, it is easy to show that Ldim(H) = 1 while|H|=dcan be arbitrarily large. Therefore, this example s ...
21.1 Online Classification in the Realizable Case 293 lemma21.7 SOAenjoys the mistake boundMSOA(H)≤Ldim(H). Proof It suffices to ...
294 Online Learning 21.2 Online Classification in the Unrealizable Case In the previous section we studied online learnability i ...
21.2 Online Classification in the Unrealizable Case 295 on roundtis P[ˆyt 6 =yt] =|pt−yt|. Put another way, instead of having th ...
296 Online Learning Weighted-Majority input:number of experts,d ; number of rounds,T parameter:η= √ 2 log(d)/T initialize:w ̃(1) ...
21.2 Online Classification in the Unrealizable Case 297 Summing this inequality overtwe get log(ZT+1)−log(Z 1 ) = ∑T t=1 log Zt+ ...
298 Online Learning corollary21.12 LetHbe a finite hypothesis class. There exists an algorithm for online classification, whose ...
21.2 Online Classification in the Unrealizable Case 299 Theorem 21.11 tells us that the expected number of mistakes ofWeighted-M ...
300 Online Learning 21.3 Online Convex Optimization In Chapter 12 we studied convex learning problems and showed learnability re ...
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