Improved Regret Bounds for Bandit Combinatorial Optimization

\textit{Bandit combinatorial optimization} is a bandit framework in which a player chooses an action within a given finite set $\mathcal{A} \subseteq \{ 0, 1 \}^d$ and incurs a loss that is the inner product of the chosen action and an unobservable loss vector in $\mathbb{R} ^ d$ in each round. In this paper, we aim to reveal the property, which makes the bandit combinatorial optimization hard. Recently, Cohen et al.~\citep{cohen2017tight} obtained a lower bound $\Omega(\sqrt{d k^3 T / \log T})$ of the regret, where $k$ is the maximum $\ell_1$-norm of action vectors, and $T$ is the number of rounds. This lower bound was achieved by considering a continuous strongly-correlated distribution of losses. Our main contribution is that we managed to improve this bound by $\Omega( \sqrt{d k ^3 T} )$ through applying a factor of $\sqrt{\log T}$, which can be done by means of strongly-correlated losses with \textit{binary} values. The bound derives better regret bounds for three specific examples of the bandit combinatorial optimization: the multitask bandit, the bandit ranking and the multiple-play bandit. In particular, the bound obtained for the bandit ranking in the present study addresses an open problem raised in \citep{cohen2017tight}. In addition, we demonstrate that the problem becomes easier without considering correlations among entries of loss vectors. In fact, if each entry of loss vectors is an independent random variable, then, one can achieve a regret of $\tilde{O}(\sqrt{d k^2 T})$, which is $\sqrt{k}$ times smaller than the lower bound shown above. The observed results indicated that correlation among losses is the reason for observing a large regret.