Contextual Bandits with Knapsacks for a Conversion Model

We consider contextual bandits with knapsacks, with an underlying structure between rewards generated and cost vectors suffered. We do so motivated by sales with commercial discounts. At each round, given the stochastic i.i.d.\ context \mathbf{x}_t and the arm picked a_t (corresponding, e.g., to a discount level), a customer conversion may be obtained, in which case a reward r(a,\mathbf{x}_t) is gained and vector costs \mathbf{c}(a_t,\mathbf{x}_t) are suffered (corresponding, e.g., to losses of earnings). Otherwise, in the absence of a conversion, the reward and costs are null. The reward and costs achieved are thus coupled through the binary variable measuring conversion or the absence thereof.