  In the classical contextual bandits problem, in each round $t$, a learner
observes some context $c$, chooses some action $i$ to perform, and receives
some reward $r_{i,t}(c)$. We consider the variant of this problem where in
addition to receiving the reward $r_{i,t}(c)$, the learner also learns the
values of $r_{i,t}(c')$ for some other contexts $c'$ in set $\mathcal{O}_i(c)$;
i.e., the rewards that would have been achieved by performing that action under
different contexts $c'\in \mathcal{O}_i(c)$. This variant arises in several
strategic settings, such as learning how to bid in non-truthful repeated
auctions, which has gained a lot of attention lately as many platforms have
switched to running first-price auctions. We call this problem the contextual
bandits problem with cross-learning. The best algorithms for the classical
contextual bandits problem achieve $\tilde{O}(\sqrt{CKT})$ regret against all
stationary policies, where $C$ is the number of contexts, $K$ the number of
actions, and $T$ the number of rounds. We design and analyze new algorithms for
the contextual bandits problem with cross-learning and show that their regret
has better dependence on the number of contexts. Under complete cross-learning
where the rewards for all contexts are learned when choosing an action, i.e.,
set $\mathcal{O}_i(c)$ contains all contexts, we show that our algorithms
achieve regret $\tilde{O}(\sqrt{KT})$, removing the dependence on $C$. For any
other cases, i.e., under partial cross-learning where $|\mathcal{O}_i(c)|< C$
for some context-action pair of $(i,c)$, the regret bounds depend on how the
sets $\mathcal O_i(c)$ impact the degree to which cross-learning between
contexts is possible. We simulate our algorithms on real auction data from an
ad exchange running first-price auctions and show that they outperform
traditional contextual bandit algorithms.
