Bandit Algorithms

37.9 Bibliographical remarks 472 setI(p)⊆Pd− 1 by I(p) = { q∈Pd− 1 : ∑d i=1 (pi−qi)I{Φai=f}= 0 for alla∈[k] andf∈[F] } This is t ...

37.10 Exercises 473 due to Foster and Rakhlin [2012]. The policy presented here is a simplification of that algorithm [Lattimore ...

37.10 Exercises 474 apples cannot be sold. Good apples yield a unit reward when sold, while the sale of a bad apple costs the co ...

37.10 Exercises 475 37.13(Lower bound depending on the number of feedbacks) Consider G= (L,Φ) given by L= ( 1 0 1 0 ··· 1 0 0 1 ...

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

38.1 Problem setup 477 then samplesSt+1from the probability vectorPAt(St) and the next round begins (Fig. 38.1). Observe stateSt ...

38.1 Problem setup 478 A memoryless policy that does not randomize is called amemoryless deterministic policy. To reduce clutter ...

38.1 Problem setup 479 course the measurePdepends on the policy, Markov decision process and the initial distribution. For most ...

38.2 Optimal policies and the Bellman optimality equation 480 38.2 Optimal policies and the Bellman optimality equation We now d ...

38.2 Optimal policies and the Bellman optimality equation 481 the existence of which you will prove in Exercise 38.7. For eachk∈ ...

38.2 Optimal policies and the Bellman optimality equation 482 Theorem38.2.The following hold: (a) There exists a pair(ρ,v)that s ...

38.3 Finding an optimal policy ( ) 483 38.3 Finding an optimal policy ( ) There are many ways to find an optimal policy, includi ...

38.3 Finding an optimal policy ( ) 484 Theorem38.5.There exists a state ̃s∈Ssuch that the solutionvof Eq.(38.7) satisfies the Be ...

38.3 Finding an optimal policy ( ) 485 innonly, provided that the constraints satisfy certain structural properties. Let K⊂Rnbe ...

38.4 Learning in Markov decision processes 486 bought by adding this slack is that now the linear program in Eq. (38.9) satisfie ...

38.5 Upper confidence bounds for reinforcement learning 487 transition matrix is unknown while the reward function is given. Thi ...

38.5 Upper confidence bounds for reinforcement learning 488 in sequential context. The algorithm that establishes Theorem 38.6 i ...

38.5 Upper confidence bounds for reinforcement learning 489 1:Input S,A,r,δ∈(0,1) 2:t= 0 3:fork= 1, 2 ,...do 4: τk=t+ 1 5: Findπ ...

38.6 Proof of upper bound 490 equation forM ̃kyields a value functionvkand constant gainρk∈Rthat satisfy Eq. (38.16). A minor de ...

38.6 Proof of upper bound 491 Step 1: Failure events and decomposition LetKbe the (random) number of phases and fork∈[K] letEk={ ...