  We propose a novel contextual bandit algorithm for generalized linear rewards
with an $\tilde{O}(\sqrt{\kappa^{-1} \phi T})$ regret over $T$ rounds where
$\phi$ is the minimum eigenvalue of the covariance of contexts and $\kappa$ is
a lower bound of the variance of rewards. In several practical cases where
$\phi=O(d)$, our result is the first regret bound for generalized linear model
(GLM) bandits with the order $\sqrt{d}$ without relying on the approach of Auer
[2002]. We achieve this bound using a novel estimator called double
doubly-robust (DDR) estimator, a subclass of doubly-robust (DR) estimator but
with a tighter error bound. The approach of Auer [2002] achieves independence
by discarding the observed rewards, whereas our algorithm achieves independence
considering all contexts using our DDR estimator. We also provide an
$O(\kappa^{-1} \phi \log (NT) \log T)$ regret bound for $N$ arms under a
probabilistic margin condition. Regret bounds under the margin condition are
given by Bastani and Bayati [2020] and Bastani et al. [2021] under the setting
that contexts are common to all arms but coefficients are arm-specific. When
contexts are different for all arms but coefficients are common, ours is the
first regret bound under the margin condition for linear models or GLMs. We
conduct empirical studies using synthetic data and real examples, demonstrating
the effectiveness of our algorithm.
