Trading-Off Payments and Accuracy in Online Classification with Paid
  Stochastic Experts
Dirk van der Hoeven Ciara Pike-Burke Hao Qiu Nicolo Cesa-Bianchi
We investigate online classification with paid stochastic experts. Here,
before making their prediction, each expert must be paid. The amount that we
pay each expert directly influences the accuracy of their prediction through
some unknown Lipschitz "productivity" function. In each round, the learner must
decide how much to pay each expert and then make a prediction. They incur a
cost equal to a weighted sum of the prediction error and upfront payments for
all experts. We introduce an online learning algorithm whose total cost after
$T$ rounds exceeds that of a predictor which knows the productivity of all
experts in advance by at most $\mathcal{O}(K^2(\log T)\sqrt{T})$ where $K$ is
the number of experts. In order to achieve this result, we combine Lipschitz
bandits and online classification with surrogate losses. These tools allow us
to improve upon the bound of order $T^{2/3}$ one would obtain in the standard
Lipschitz bandit setting. Our algorithm is empirically evaluated on synthetic
data