  We consider the adversarial linear contextual bandit setting, which allows
for the loss functions associated with each of $K$ arms to change over time
without restriction. Assuming the $d$-dimensional contexts are drawn from a
fixed known distribution, the worst-case expected regret over the course of $T$
rounds is known to scale as $\tilde O(\sqrt{Kd T})$. Under the additional
assumption that the density of the contexts is log-concave, we obtain a
second-order bound of order $\tilde O(K\sqrt{d V_T})$ in terms of the
cumulative second moment of the learner's losses $V_T$, and a closely related
first-order bound of order $\tilde O(K\sqrt{d L_T^*})$ in terms of the
cumulative loss of the best policy $L_T^*$. Since $V_T$ or $L_T^*$ may be
significantly smaller than $T$, these improve over the worst-case regret
whenever the environment is relatively benign. Our results are obtained using a
truncated version of the continuous exponential weights algorithm over the
probability simplex, which we analyse by exploiting a novel connection to the
linear bandit setting without contexts.
