Bilinear Bandits with Low-rank Structure.

We introduce the bilinear bandit problem with low-rank structure where an action is a pair of arms from two different entity types, and the reward is a bilinear function of the known feature vectors of the arms. The problem is motivated by numerous applications in which the learner must recommend two different entity types as one action, such as a male / female pair in an online dating service. The unknown in the problem is a $d_1$ by $d_2$ matrix $\mathbf{\Theta}^*$ with rank $r \ll \min\{d_1,d_2\}$ governing the reward generation. Determination of $\mathbf{\Theta}^*$ with low-rank structure poses a significant challenge in finding the right exploration-exploitation tradeoff. In this work, we propose a new two-stage algorithm called "Explore-Subspace-Then-Refine" (ESTR). The first stage is an explicit subspace exploration, while the second stage is a linear bandit algorithm called "almost-low-dimensional OFUL" (LowOFUL) that exploits and further refines the estimated subspace via a regularization technique. We show that the regret of ESTR is $\tilde{O}((d_1+d_2)^{3/2} \sqrt{r T})$ (where $\tilde{O}$ hides logarithmic factors), which improves upon the regret of $\tilde{O}(d_1d_2\sqrt{T})$ of a naive linear bandit reduction. We conjecture that the regret bound of ESTR is unimprovable up to polylogarithmic factors.