Normal Bandits of Unknown Means and Variances

Wesley Cowan, Junya Honda, Michael N. Katehakis.

Year: 2018, Volume: 18, Issue: 154, Pages: 1−28


Consider the problem of sampling sequentially from a finite number of $N \geq 2$ populations, specified by random variables $X^i_k$, $ i = 1,\ldots , N,$ and $k = 1, 2, \ldots$; where $X^i_k$ denotes the outcome from population $i$ the $k^{th}$ time it is sampled. It is assumed that for each fixed $i$, $\{ X^i_k \}_{k \geq 1}$ is a sequence of i.i.d. normal random variables, with unknown mean $\mu_i$ and unknown variance $\sigma_i^2$. The objective is to have a policy $\pi$ for deciding from which of the $N$ populations to sample from at any time $t=1,2,\ldots$ so as to maximize the expected sum of outcomes of $n$ total samples or equivalently to minimize the regret due to lack on information of the parameters $\mu_i$ and $\sigma_i^2$. 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 \cite{bkmab96}. Additionally, finite horizon regret bounds are given.