The Multi-fidelity Multi-armed Bandit

We study a variant of the classical stochastic $K$-armed bandit where observing the outcome of each arm is expensive, but cheap approximations to this outcome are available. For example, in online advertising the performance of an ad can be approximated by displaying it for shorter time periods or to narrower audiences. We formalise this task as a \emph{multi-fidelity} bandit, where, at each time step, the forecaster may choose to play an arm at any one of $M$ fidelities. The highest fidelity (desired outcome) expends cost $\costM$. The $m$\ssth fidelity (an approximation) expends $\costm < \costM$ and returns a biased estimate of the highest fidelity. We develop \mfucb, a novel upper confidence bound procedure for this setting and prove that it naturally adapts to the sequence of available approximations and costs thus attaining better regret than naive strategies which ignore the approximations. For instance, in the above online advertising example, \mfucbs would use the lower fidelities to quickly eliminate suboptimal ads and reserve the larger expensive experiments on a small set of promising candidates. We complement this result with a lower bound and show that \mfucbs is nearly optimal under certain conditions.