  We study a sequential resource allocation problem between a fixed number of
arms. On each iteration the algorithm distributes a resource among the arms in
order to maximize the expected success rate. Allocating more of the resource to
a given arm increases the probability that it succeeds, yet with a cut-off. We
follow Lattimore et al. (2014) and assume that the probability increases
linearly until it equals one, after which allocating more of the resource is
wasteful. These cut-off values are fixed and unknown to the learner. We present
an algorithm for this problem and prove a regret upper bound of $O(\log n)$
improving over the best known bound of $O(\log^2 n)$. Lower bounds we prove
show that our upper bound is tight. Simulations demonstrate the superiority of
our algorithm.
