Contextual Pricing for Lipschitz Buyers

We investigate the problem of learning a Lipschitz function from binary feedback. In this problem, a learner is trying to learn a Lipschitz function $f:[0,1]^d \rightarrow [0,1]$ over the course of $T$ rounds. On round $t$, an adversary provides the learner with an input $x_t$, the learner submits a guess $y_t$ for $f(x_t)$, and learns whether $y_t > f(x_t)$ or $y_t \leq f(x_t)$. The learner's goal is to minimize their total loss $\sum_t\ell(f(x_t), y_t)$ (for some loss function $\ell$). The problem is motivated by \textit{contextual dynamic pricing}, where a firm must sell a stream of differentiated products to a collection of buyers with non-linear valuations for the items and observes only whether the item was sold or not at the posted price.