Bandit Algorithms

30.7 Exercises 352 30.7 Construct an action set andi 6 =jandz,ξ∈Rdwithzj>0 such that a(z+ξ)i≥a(z− 2 zjej+ξ)i. ...

This material will be published by Cambridge University Press as Bandit Algorithms by Tor Lattimore and Csaba Szepesvari. This p ...

31.1 Adversarial bandits 354 regret by enlarging the competitor class as a way to ensure meaningful results. Let Γnm⊂[k]nbe the ...

31.1 Adversarial bandits 355 implementation of Exp4 is linear in the number of experts, which even for modestly sizedmis very la ...

31.2 Stochastic bandits 356 The next step is to apply Eq. (28.11) and the solution to Exercise 28.8 to bound the inner expectati ...

31.2 Stochastic bandits 357 If the locations of the change points were known then running a new copy of UCB on each interval wou ...

31.3 Notes 358 By Markov’s inequality, P   ∑ t∈Sj I{At= 2}≥ |Sj| 2  ≤^2 |Sj| E   ∑ t∈Sj I{At= 2}  ≤^2 E[T^2 (n)] L|Sj| ≤ ...

31.3 Notes 359 each arm once and subsequently At= argmaxi∈[k]  μˆγi(t−1) + √√ √√ α Tiγ(t−1) log (k ∑ i=1 Tiγ(t−1) ) . The id ...

31.4 Bibliographic remarks 360 whileVn≤ 2 cm^3 /^2 √ k/n. Tuningmso thatVn≤Vcompletes the proof. Note the algorithm achieving Eq ...

31.5 Exercises 361 on stochastic nonstationary bandits where the rewards are slowly drifting. The approach based on Brownian mot ...

This material will be published by Cambridge University Press as Bandit Algorithms by Tor Lattimore and Csaba Szepesvari. This p ...

32.1 Click models 363 A naive way to minimize the regret would be to create a finite-armed bandit where each arm corresponds to ...

32.1 Click models 364 k > m. This model is richer than the document-based model, which is recovered by choosingχ(k) =I{k≤m}. ...

32.1 Click models 365 a a′ 5 4 3 2 1 i j j i Figure 32.1Part (c) of Assumption 32.1 says that the probability of clicking in the ...

32.2 Policy 366 this model relative to the previous ones is that it offers more flexibility and yet it is not so flexible that l ...

32.3 Regret analysis 367 Step 3: Updating the relation For any pair of itemsi,j∈[`] defineStij= ∑t s=1UsijandNtij= ∑t s=1|Usij| ...

32.3 Regret analysis 368 bounded by Rn≤δnm`^2 + ∑` j=1 min{∑m,j− 1 } i=1  1 + 6(α(i) +α(j)) log ( c√n δ ) ∆ij  . Furthermore ...

32.3 Regret analysis 369 where in the second equality we added and subtractedPs− 1 (Csi= 1,Csj= 1). By the design of TopRank, th ...

32.3 Regret analysis 370 On the other hand, wheniandjare in the same block,Eu− 1 [Uuji|Uuji 6 = 0]≤ 0 almost surely by Lemma 32. ...

32.3 Regret analysis 371 Proof of Theorem 32.2 The first step in the proof is an upper bound on the expected number of clicks in ...