An Engineered Empirical Bernstein Bound.

We derive a tightened empirical Bernstein bound (EBB) on the variation of the sample mean from the population mean, and show that it improves the performance of upper confidence bound (UCB) methods in multi-armed bandit problems. Like other EBBs, our EBB is a concentration inequality for the variation of the sample mean in terms of the sample variance. Its derivation uses a combination of probability unions and Chernoff bounds for the mean of samples and mean of sample squares. Analysis reveals that our approach can tighten the best existing EBBs by about a third, and thereby halves the distance to a bound constructed with perfect variance information. We illustrate the practical usefulness of our novel EBB by applying it to a multi-armed bandit problem as a component of a UCB method. Our method outperforms existing approaches by producing lower expected regret than variants of UCB employing several other bounds, including state-of-the-art EBBs.