  The contextual combinatorial semi-bandit problem with linear payoff functions
is a decision-making problem in which a learner chooses a set of arms with the
feature vectors in each round under given constraints so as to maximize the sum
of rewards of arms. Several existing algorithms have regret bounds that are
optimal with respect to the number of rounds $T$. However, there is a gap of
$\tilde{O}(\max(\sqrt{d}, \sqrt{k}))$ between the current best upper and lower
bounds, where $d$ is the dimension of the feature vectors, $k$ is the number of
the chosen arms in a round, and $\tilde{O}(\cdot)$ ignores the logarithmic
factors. The dependence of $k$ and $d$ is of practical importance because $k$
may be larger than $T$ in real-world applications such as recommender systems.
In this paper, we fill the gap by improving the upper and lower bounds. More
precisely, we show that the C${}^2$UCB algorithm proposed by Qin, Chen, and Zhu
(2014) has the optimal regret bound $\tilde{O}(d\sqrt{kT} + dk)$ for the
partition matroid constraints. For general constraints, we propose an algorithm
that modifies the reward estimates of arms in the C${}^2$UCB algorithm and
demonstrate that it enjoys the optimal regret bound for a more general problem
that can take into account other objectives simultaneously. We also show that
our technique would be applicable to related problems. Numerical experiments
support our theoretical results and considerations.
