## A Parameter-Free Algorithm for Misspecified Linear Contextual Bandits

### Kei Takemura · Shinji Ito · Daisuke Hatano · Hanna Sumita · Takuro Fukunaga · Naonori Kakimura · Ken-ichi Kawarabayashi

Keywords: [ Applications ] [ Body Pose, Face, and Gesture Analysis ] [ Algorithms -> Regression; Probabilistic Methods; Probabilistic Methods ] [ Graphical Models ] [ Learning Theory and Statistics ] [ Decision Processes and Bandits ]

Abstract: We investigate the misspecified linear contextual bandit (MLCB) problem, which is a generalization of the linear contextual bandit (LCB) problem. The MLCB problem is a decision-making problem in which a learner observes $d$-dimensional feature vectors, called arms, chooses an arm from $K$ arms, and then obtains a reward from the chosen arm in each round. The learner aims to maximize the sum of the rewards over $T$ rounds. In contrast to the LCB problem, the rewards in the MLCB problem may not be represented by a linear function in feature vectors; instead, it is approximated by a linear function with additive approximation parameter $\varepsilon \geq 0$. In this paper, we propose an algorithm that achieves $\tilde{O}(\sqrt{dT\log(K)} + \eps\sqrt{d}T)$ regret, where $\tilde{O}(\cdot)$ ignores polylogarithmic factors in $d$ and $T$. This is the first algorithm that guarantees a high-probability regret bound for the MLCB problem without knowledge of the approximation parameter $\varepsilon$.