  A stochastic combinatorial semi-bandit is an online learning problem where at
each step a learning agent chooses a subset of ground items subject to
constraints, and then observes stochastic weights of these items and receives
their sum as a payoff. In this paper, we close the problem of computationally
and sample efficient learning in stochastic combinatorial semi-bandits. In
particular, we analyze a UCB-like algorithm for solving the problem, which is
known to be computationally efficient; and prove $O(K L (1 / \Delta) \log n)$
and $O(\sqrt{K L n \log n})$ upper bounds on its $n$-step regret, where $L$ is
the number of ground items, $K$ is the maximum number of chosen items, and
$\Delta$ is the gap between the expected returns of the optimal and best
suboptimal solutions. The gap-dependent bound is tight up to a constant factor
and the gap-free bound is tight up to a polylogarithmic factor.
