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

; Proceedings of Thirty Third Conference on Learning Theory, PMLR 125:3583-3634, 2020.

#### Abstract

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.