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5 the next section. In practice, since most bandit policies can take prior information about the payoff probabilities as input, when restarting POL we can supply the previous payoff probability estimates as the prior (as done in our experiments. 4.2 Performance Bound for BMMP Policies Let S denote the sequence of bandit instances that are invoked, i.e., S = {S(1, S(2... S(N} where S(n denotes the index of the bandit instance invoked at the n th invocation. We compare the performance of BPOL with that of the optimal policy, denoted by OPT, where OPT has advance knowledge of S and the exact payoff probabilities of all bandit instances. We claim that bpol(n opt(n/2 O(f(N for any N, where bpol(n and opt(n denote the total expected reward obtained after N invocations by BPOL and OPT, respectively, and f(n denotes the expected number of mistakes made by POL after n invocations of the the regular multi-armed bandit problem (for UCB, f(n is O(ln n [3]. Our complete proof is rather involved. Here we give a high-level outline of the proof (the complete proof is given in [9]. For simplicity we focus on the C = 1 case; C 1 is a simple extension thereof. Since bandit arms generate rewards stochastically, it is not clear how we should compare BPOL and OPT. For example, even if BPOL and OPT behave in exactly the same way (activate the same arm on each bandit invocation, we cannot guarantee that both will have the same total reward in the end. To enable meaningful comparison, we define a payoff instance, denoted by I, such that I(i, n denotes the reward generated by arm i of bandit instance S(n for invocation n in payoff instance I. The outcome of running BPOL or OPT on a given payoff instance is deterministic because the rewards are fixed in the payoff instance. Hence, we can compare BPOL and OPT on per payoff instance basis. Since each payoff instance arises with a certain probability, denoted as P(I, by taking expectation over all possible payoff instances of execution we can compare the expected performance of BPOL and OPT. Let us consider invocation n in payoff instance I. Let B(I, n and O(I, n denote the arms of bandit instance S(n activated under BPOL and OPT respectively. Based on the different possibilities that can arise, we classify invocation n into one of three categories: Category 1: The arm activated by OPT, O(I, n, is of smaller or equal expected reward in comparison to the arm activated by BPOL, B(I, n. The expected reward of an arm is the product of its payoff probability and reward. Category 2: Arm O(I, n is of greater expected reward than B(I, n, but O(I, n is not available for BPOL to activate at invocation n due to budget restrictions. Category 3: Arm O(I, n is of greater expected reward than B(I, n and both arms O(I, n and B(I, n are available for BPOL to activate, but BPOL prefers to activate B(I, n over O(I, n. Let us denote the invocations of category k (1, 2 or 3 by N k (I for payoff instance I. Let bpol k (N and opt k (N denote the expected reward obtained during the invocations of category k (1, 2 or 3 by BPOL and OPT respectively. In [9] we show that bpol k (N = ( P(I I(B(I, n, n I I Similarly, opt k (N = I I ( P(I n N k (I n N k (I I(O(I, n, n Then for each k we bound opt k (N in terms of bpol(n. In [9] we provide proof of each of the following bounds: Lemma 1 opt 1 (N bpol 1 (N.

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