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Consider the problem of a controller sampling sequentially from a finitenumber 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 frompopulation $i$ the $k^{th}$ time it is sampled. It is assumed that for eachfixed $i$, $\{ X^i_k \}_{k \geq 1}$ is a sequence of i.i.d. uniform randomvariables over some interval $[a_i, b_i]$, with the support (i.e., $a_i, b_i$)unknown to the controller. The objective is to have a policy $\pi$ fordeciding, based on available data, from which of the $N$ populations to samplefrom at any time $n=1,2,\ldots$ so as to maximize the expected sum of outcomesof $n$ samples or equivalently to minimize the regret due to lack oninformation of the parameters $\{ a_i \}$ and $\{ b_i \}$. In this paper, wepresent a simple inflated sample mean (ISM) type policy that is asymptoticallyoptimal in the sense of its regret achieving the asymptotic lower bound ofBurnetas and Katehakis (1996). Additionally, finite horizon regret bounds aregiven.

An Asymptotically Optimal Policy for Uniform Bandits of Unknown Support