# Normal Bandits of Unknown Means and Variances: Asymptotic Optimality, Finite Horizon Regret Bounds, and a Solution to an Open Problem

Consider the problem of sampling sequentially from a finite number of N ≥ 2 populations, specified by random variables X^i_k, i = 1,... , N, and k = 1, 2, ...; where X^i_k denotes the outcome from population i the k^th time it is sampled. It is assumed that for each fixed i, { X^i_k }_k ≥ 1 is a sequence of i.i.d. normal random variables, with unknown mean μ_i and unknown variance σ_i^2. The objective is to have a policy π for deciding from which of the N populations to sample form at any time n=1,2,... so as to maximize the expected sum of outcomes of n samples or equivalently to minimize the regret due to lack on information of the parameters μ_i and σ_i^2. In this paper, we present a simple inflated sample mean (ISM) index policy that is asymptotically optimal in the sense of Theorem 4 below. This resolves a standing open problem from Burnetas and Katehakis (1996). Additionally, finite horizon regret bounds are given.

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