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The purpose of this paper is to provide further understanding into thestructure of the sequential allocation ("stochastic multi-armed bandit", orMAB) problem by establishing probability one finite horizon bounds andconvergence rates for the sample (or "pseudo") regret associated with twosimple classes of allocation policies $\pi$.  For any slowly increasing function $g$, subject to mild regularityconstraints, we construct two policies (the $g$-Forcing, and the $g$-InflatedSample Mean) that achieve a measure of regret of order $ O(g(n))$ almost surelyas $n \to \infty$, bound from above and below. Additionally, almost sure upperand lower bounds on the remainder term are established. In the constructionsherein, the function $g$ effectively controls the "exploration" of theclassical "exploration/exploitation" tradeoff.

Asymptotic Behavior of Minimal-Exploration Allocation Policies: Almost Sure, Arbitrarily Slow Growing Regret