Taking a hint: How to leverage loss predictors in contextual bandits?

We initiate the study of learning in contextual bandits with the help of loss predictors. The main question we address is whether one can improve over the minimax regret $\mathcal{O}(\sqrt{T})$ for learning over $T$ rounds, when the total error of the predicted losses relative to the realized losses, denoted as $\mathcal{E} \leq T$, is relatively small. We provide a complete answer to this question, with upper and lower bounds for various settings: adversarial and stochastic environments, known and unknown $\mathcal{E}$, and single and multiple predictors. We show several surprising results, such as 1) the optimal regret is $\mathcal{O}(\min\{\sqrt{T}, \sqrt{\mathcal{E}}T^\frac{1}{4}\})$ when $\mathcal{E}$ is known, in contrast to the standard and better bound $\mathcal{O}(\sqrt{\mathcal{E}})$ for non-contextual problems (such as multi-armed bandits); 2) the same bound cannot be achieved if $\mathcal{E}$ is unknown, but as a remedy, $\mathcal{O}(\sqrt{\mathcal{E}}T^\frac{1}{3})$ is achievable; 3) with $M$ predictors, a linear dependence on $M$ is necessary, even though logarithmic dependence is possible for non-contextual problems. We also develop several novel algorithmic techniques to achieve matching upper bounds, including 1) a key \emph{action remapping} technique for optimal regret with known $\mathcal{E}$, 2) computationally efficient implementation of Catoni’s robust mean estimator via an ERM oracle in the stochastic setting with optimal regret, 3) an underestimator for $\mathcal{E}$ via estimating the histogram with bins of exponentially increasing size for the stochastic setting with unknown $\mathcal{E}$, and 4) a self-referential scheme for learning with multiple predictors, all of which might be of independent interest.