We study the generalized linear contextual bandit problem within the constraints of limited adaptivity. In this paper, we present two algorithms, $\texttt{ B-GLinCB}$ and $\texttt{ RS-GLinCB}$, that address, respectively, two prevalent limited adaptivity settings. Given a budget $M$ on the number of policy updates, in the first setting, the algorithm needs to decide upfront $M$ rounds at which it will update its policy, while in the second setting it can adaptively perform $M$ policy updates during its course. For the first setting, we design an algorithm $\texttt{ B-GLinCB}$, that incurs $\tilde{ O}(\sqrt{ T})$ regret when $M = \Omega( \log{ \log T} )$ and the arm feature vectors are generated stochastically. For the second setting, we design an algorithm $\texttt{ RS-GLinCB}$ that updates its policy $\tilde{ O}(\log^2 T)$ times and achieves a regret of $\tilde{ O}(\sqrt{ T})$ even when the arm feature vectors are adversarially generated. Notably, in these bounds, we manage to eliminate the dependence on a key instance dependent parameter $\kappa$, that captures non-linearity of the underlying reward model. Our novel approach for removing this dependence for generalized linear contextual bandits might be of independent interest.
Categories: cs.LG stat.ML